Basic concepts in Portfolio Analysis Wolfgang Marty (3. Version 18.07.2011)

Table of contents 1. Introduction 2. Return analysis 2.1 Interest rate, return and compounding 2.2. Return contribution and attribution 2.3. Time weighted rate of return (TWR) 2.4. The money weighted rate of return (MWR) 2.5. TWR attribution versus MWR attribution 3. Risk measurement 3.1. Absolute risk 3.2. Value at Risk 3.3. Relative risk 3.4. Risk decomposition on asset level 3.5. Risk decomposition with factor analysis 4. Performance measurement 4.1. Investment Process and Portfolio Construction 4.2. Portfolio optimisation 4.3. Absolute Risk adjusted measures 4.4. Capital asset pricing model 4.5. Relative Risk adjusted measures 5. Investment controlling 5.1. Investment controlling and the investment process 5.2. Building blocs and set up of investment controlling 5.3 The benefits of GIPS reporting 5.4. Some critical issues in performance attribution Index References

Preface While getting acquainted with a series of topics relating to the banking industry I realized that scientific methods and terminologies describe many aspects of financial markets. Many notions that are used in finance have a scientific background. In the daily activity of the finance industry, it is often not even realized that mathematical concepts are applied. The use of mathematics ranges from basic arithmetic manipulations such adding positions on an account or calculating interest payments, to sophisticated problems such that investing times series or portfolio optimization. In my view the connection between science and financial mathematics is particularly apparent in portfolio analysis. The illustration of this fact is one of the scopes of this book. In traditional asset management industry I was confronted with statements like the following: ‘I am a practitioner and I do not need the abstraction of a theory.’ Nothing could be further from the truth. Using portfolio analysis, I intend to explain why concepts that are based on mathematics can enhance the decision making process of the participants in the financial community. Furthermore the approach we describe here alleviates the unification of the terminology and provides a concise use on the notions. Financial markets are inundated by data. In other words they are produced literally continuously. In portfolio analysis, the amalgamation and reconciliation of data is important. We try to answer the following question: What information is relevant in portfolio analysis? We intend to explain the figures that asses a specific portfolio. In this book the basic notions in portfolio analysis are introduced by a mathematical notation. For more advanced concepts and ideas we limit our exposition to intuitive descriptions. The book does not only contain traditional material but also current topics in portfolio analysis. We also refer to the relevant literature.

Acknowledgments This book is based on several courses and seminars held at CS business school. The transformation of a series of presentations in a lecture book is a not to be underestimated challenge. Many discussions throughout my daily activities consisted substantially to the here presented material. I am particular grateful to Dominik Büsser and Petra Illic for revising my first daft of this book. I would like to give my thanks to my employer Credit Suisse and in particular to the following colleagues in alphabetical order: Rolf Bertschi, Ralph Häfliger, Dr. Stefan Illmer, Thomas Kellersberger, Kurt Oberhänsli, Elias Mulky and Thomas Widmer. Their thoughts and ideas added great value to various parts of this book. The continuing discussions with Professor Janos Mayer from the University of Zurich helped me concerning mathematical aspects of this book. In addition, I could establish an excellent platform with Yuri Shestopaloff for exchanging thoughts on different issues in performance measurement.

Conventions This book consists of 5 chapters. The chapters are divided into sections. (1.2.3) denotes formula (3) in Section 2 of Chapter 1. If we refer to formula (2) in Section 2 of Chapter 1 we only write (2) otherwise we use the full reference (1.2.3). Within the chapters, definitions, assumptions, theorems and examples are numerated continually, e.g. Theorem 2.1 refers to Theorem 1 in Chapter 2. They are numerated continually. The end of a theorem, lemma or example is marked with ◊ . Square brackets [ ] contain references. The details of the references are given at the end of the book.

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1. INTRODUCTION

A portfolio is a set of investments. The return of a portfolio is probably the most important information for the investor. In most cases the investor has an intuitive perception of the return. If the value of the portfolio rises the return is positive and if the value of the portfolio declines the return is negative. However, there is no unique method for calculating the return. The discussion of different approaches for return calculation is subject of this book.

Generally speaking prices are provided by market or by theoretical considerations. In the following we assume that the prices of the different investments are available and as a consequence it is possible to calculate the return. Returns can be calculated from past prices in the market place (ex post), from actual prices or from projected prices (ex ante) (see Figure 1.1). The return depends of the input data and the more frequent the data is available the more meticulously the return of the investments and the portfolio can be monitored.

Now - 1 year now now + 1 year

P0P-1-1

Time axis

ex post ex ante

futurepast

P1

Now - 1 year now now + 1 year

P0P-1-1

Time axis

ex post ex ante

futurepast

P1

The setting for portfolio analysis Figure 1.1 There are two main different concepts for measuring the return. The first concept starts with the evaluation of the portfolio, parts of a portfolio or an investment at different points

2

of time. The return is then calculated while time elapses, i.e. it always refers to a time span. A value for a return is always accompanied by a time span. For instance, a return over multiple years is usually annualized. This concept neutralizes external cash flows in the portfolios. Thus, it is appropriate for comparing portfolio managers, i.e. it is the base for peer analysis. The unit of the return is percentage. This concept is subject of Section 2.3 entitled Time weighted rate of return (TWR). Examples are e.g. the total return of an investment, profit and loss considerations. The second concept is based on an arbitrage relationship, i.e. a condition which avoids a situation with a profit without risk. We refer here to the saying: ‘There is no free lunch in the financial market.’ The corresponding return measurement is based on the value of portfolio, parts of a portfolio or an investment at one point in time. This concept includes cash flow and it is thus the return from the client’s perspective. Profit and Loss are also considered. This concept is subject of the Section 2.4 entitled Money weighted rate of return (MWR). The Internal rate of return (IRR) equation is such an example (see e.g. Section 2.5 and Section 2.8) of the MWR approach. TWR and MWR are applied on an absolute and a relative basis. We discuss in section 2.5 the relationship between TWR and MWR. With the development of portfolio analysis risk considerations have become more influential. Here we assess the risk of a portfolio by the elementary measure standard deviation used throughout in statistics in other words we quantify the risk by a value for the fluctuation of the return. Intuitively this means that if the return is constant there is no risk and the bigger the risk value is the bigger the uncertainty of the returns is. As will be showed in Chapters 2 and 3 the return and the risk can be considered on an absolute and on a relative basis. The tracking error originally stems from Control theory and is a relative risk measure. We will see that the concept of returns

3

is easier to understand than to the concept of risk. Returns are additive, i.e. when referring the weights of the investments then the sum of the return of the investment adds up to the returns of the portfolio. Adding risks of investments obeys a generalization of the theorem of Pythagoras which states that under the assumption that a and b are rectangular the sum of a2 and b2 is equal to c2 (Figure 1.2).

ba

c

ba

c A geometrical interpretation: Adding risk Figure 1.2 The following statement explains the important notion of performance:

Performance encompasses risk and return.

Remark 1.1: In physics, power is introduced through work by unit of time and the transition from finance to physics is reflected by performance to power, by return to work and by risk to time. Performance deals with the past, the present and the future as seen in Figure 1.1. The time axis and the Price Pi (i = −1, 0, 1) of an asset in the past on the left side and potential outcome in the future on the right side. In the past we deal with statistics whereas in the present and the present we are confined to applying models providing forecasts and projections. The calculations have to reflect cash flows such as dividends from equities or coupons from bonds and liabilities

4

that are typically important for pension funds. Many assets do not have a price history. Estimations to their future behaviour can solely depend on future scenarios and a statistical price analysis is not possible. Such examples are Initial Public Offering (IPO), recently built house or bonds just launched in the market. The expost return analysis is a time series analysis and an exante return is the same as a return forecast. The expost risk analysis of a portfolio is a time series analysis of the returns of the portfolio, i.e. all past changes in the portfolio are reflected in the risk figure. An exante risk analysis, however, is based on the actual portfolio and the following mathematical question: • What are the main drivers of the risk in the portfolio? A set of factors intends to explain the risk of a portfolio. The choice of the factors requires intense research. They are modelled by time series and do not depend of the portfolios under investigation. They are estimated by the underlying market universe of the portfolio. The factors are objective estimations of the considered markets and the portfolio is subjective. Thus a risk model can also be used for risk management. Updating the risk model is a slow process, i.e. updating once a month is sufficient. Up to our knowledge the data is mostly based on monthly frequency and in some cases on daily frequency. We proceed by summarizing the main topics addressed here:

1. We introduce some basic notions for describing and assessing properties of portfolios.

2. The decomposition of the portfolio into its constituents is discussed.

3. The portfolio is investigated on an absolute basis and relative to a reference portfolio.

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The following definition brings the notion of return and risk together. Definition 1.1: Portfolio Analytics is concerned with quantifying the sources of the return and assessing the risk of a portfolio. It does not only measure the evolution of the wealth over a certain time period but also provides a comprehensive discussion of the performance of specific portfolios. We proceed with the following remarks: Remark 1.2: In portfolio analytics we consider mostly variance or covariance as risk measures. They are introduced in Chapter 3. Remark 1.3: We distinguish between performance measurement on a portfolio level and performance measurements on segment or on constituent level and on a multi period level. We intend to discuss the complete portfolio analysis problem. Performance attribution is an important part of Portfolio Analytics. The performance attribution is essentially the decomposition of a real number, i.e. 5 = 1 + 4 or 5 = 2 + 3. In the arithmetic just considered the operation usually goes from right to left. In performance attribution in the operation goes from left to right. The return and the risk number of a portfolio is decomposed. However, what is paramount is that this decomposition reflects the requirement of the client, portfolio managers and the adviser. In this book we focus on the return attribution. On the risk side we will only give an introduction in decomposing to the risk. We present a performance attribution to the TWR and MWR. IRR with multi solutions we illustrate by easy examples. Performance Attribution for TWR is common practise, but the approach presented here for the MWR is new. Chapter 4 introduces a holistic approach from a portfolio’s view, i.e. return and risk considerations are simultaneously taken in account. We start with the part portfolio construction in

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the investment process. We describe the key ingredient for the portfolio construction. The material presented here is basic and traditional but it can be considered as summary of voluminous text books. We show by an example how constraints can change optimal portfolios. In Chapter 5 we focus on the part investment controlling in the investment process. We compare many portfolios and give an introduction in Global Investment Performance Standard (GIPS). We illustrate a performance review.

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2. RETURN ANALYSIS 2.1. Interest rate, return and compounding Retail accounts are facilities that give the clients of a bank the possibility to deposit their money. The client lends money to the bank whereas the bank borrows money from the client. Money on accounts is also called a retail deposit. The bank will recompense their clients by paying interest rate. Such accounts bear no or only little interest rate. Low interest rates can be justified by the fact that the costs of the banks for the infrastructure, staff and paperwork are substantial. In the following we present the basic interest calculation. The Beginning value BV and the End value EV of a money account are related to the interest rate r: EV = BV (1+ r). (2.1.1) BV and EV are amounts in US Dollar ($). However, our considerations in the following are applicable to any reference currency. In relationship (1) the time span between BV and EV is not specified. Assuming that an investor puts a 100$ on an account and gets 200$ after 10 years and another investor earned starting from 100$ to 200$ in a year. Interest rates are usually quoted on a yearly or annual basis. However, other time spans like days, months or 6 months (semi-annual) can also be appropriate in specific situations. It is seen that an interest rate is always referred to a time spam. It is not related to a point on the time line. Sometimes interest rate is always called a growth rate. In most cases the relationship (1) is applied in 3 versions. First if we know BV and r, EV can be calculated. BV is the present value und EV is the future value. The future value is on the right side of the present value on the time axis (Figure 1.1).

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Example 2.1: If BV = 100$ and r = 5% then EV = 100$ . (1 + 0.05) = 105$. ◊ Secondly if EV and BV are known and if we assume than the investor starts from an initial lump sum BV >0 then the solution r of (1) is called the return expressed by

BV

BVEVr

−= = 1−

BVEV . (2.1.2)

We divide the profit or loss (EV − BV) by the invested capital BV. EV cannot be negative as a complete loss translates in EV = 0, i.e. that is we have EV ≥ 0, i.e. by (1) r ≥ −1. Example 2.2: BV = 100$ and EV = 200$ then

BV

BVEVr

−= = ,1

$100$100$200=

−

i.e. in percentage r = 100%. ◊

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Thirdly if EV and r with r > −1 is given we have

BV = r1

EV+

.

By using the discount factor d defined by

d = r1

1+

(2.1.3)

we find BV = d EV. Broadly speaking interest rates with the accompanying discount factors translate money amounts though time. Definition 2.1: Compounding is the reinvestment of the income to earn more income in the subsequent periods. If the income and the gains are retained within the investment vehicle or reinvested, they will accumulate and contribute to the starting balance for each subsequent period’s income calculation. In the following we look at an arbitrary but fixed time base period. We proceed by assuming that the interest is distributed twice and is reinvested by the same interest rate at the middle of the period. We consider

EV2 = ⎥⎦⎤

⎢⎣⎡

⎟⎠⎞

⎜⎝⎛ +⋅

2r1BV . ⎟

⎠⎞

⎜⎝⎛ +

2r1

If interest is distributed n-times per period we find

10

EVn = BV n

nr1 ⎟⎠⎞

⎜⎝⎛ + . (2.1.4)

We define e by the limes

n

n n11lim ⎟⎠⎞

⎜⎝⎛ +

∞→= e.

The number e appears in many contexts in science and is called after its founder Leonard Euler. The 14 digit numerical value is e = 2.71828’18284’5905. The faculty n! is defined by n! = n (n − 1) (n − 2)….. For calculating e numerically it is favourable to use the series

e = 1+ !1

1 +!2

1 +!3

1 +!4

1 +!5

1 +….

and find

n

n nr1lim ⎟⎠⎞

⎜⎝⎛ +

∞→ = er

which yields

n

n nr1limBVEV ⎟⎠⎞

⎜⎝⎛ +=

∞→∞ = BV er. (2.1.5)

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We see that the end values EV1, EV2,…., ∞EV are increasing with the compounding frequency. ∞EV is the Ending value using continuously compounding as the money is reinvested momentarily. The above formulae are assumed that the beginning value BV, the interesting rate r and the compounding assumption are given and the ending values EV1, EV2,…., EVn,…, ∞EV are computed. Example 2.3: For r = 0.05 (= 5%) annually we get in decimals and BV = 100$ EV2 = 105.06250 (Semi annual), EV4 = 105.09453 (Quarterly), ∞EV = 105.127110 (Continuous). ◊ If the beginning value BV, the ending value EV and the number of compounding annually are given the underlying interest is based on (4)

r(n) = n1BVEVn ⎟⎟

⎠

⎞⎜⎜⎝

⎛− .

Consistency with (2) yields r = r(1). If the beginning value BV and the ending value EV are given the underlying return is found based on (5)

∞r = ln BVEV ,

where ln is the natural logarithm, i.e. the logarithm for the basis e.

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In the following we look at N base periods with n = 1 in the above formulae. We distinguish different ways of compounding. Simple or arithmetic simple return means that the interests are adding EV = BV . (1 + N . r). (2.1.6) We use simple interest calculation if the earned income is withdrawn from the account. Geometric compounding is mostly used in practice EV = BV . (1 + r)N. (2.1.7) For N = 1 (6) is the same as (7). Note that the time unit is year. Then the continuous versions, i.e. the formulae for arbitrary t are EV1(t) = BV . (1 + t . r), resp. (2.1.8a) ∞EV (t) = BV . (1 + r)t. (2.1.8b) We illustrate (8) with r = 0.01 (= 10%)

r (8a) (8b) (8b)-(8a)0 1.000 1.000 0.0000000

0.1 1.010 1.010 0.00042340.2 1.020 1.019 0.00075510.3 1.030 1.029 0.00099420.4 1.040 1.039 0.00113990.5 1.050 1.049 0.00119120.6 1.060 1.059 0.00114710.7 1.070 1.069 0.00100700.8 1.080 1.079 0.00076970.9 1.090 1.090 0.00043431 1.100 1.100 0.0000000

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Remark 2.1: (8a) and (8b) are the same for t = 1. We consider (8a) and (8b) for t ∈ [0, 1]. For no compounding we have EV(t) = BV(1 + tr), ∀t ≥ 0 and for continuous compounding we have EV(t) = BV t1re . (2.1.9) For t = 1 there follows from (8a) and (8b) r1 = ln(1+ r). thus r1 t = ln(1 + r) t, ∀t ≥ 0 and we conclude EV(t) = BV t)r1ln(e + = BV . (1 + r)t,∀t ≥ 0. In (8a) there is no compounding and in (8b) we have continuously compounding. A similar formula can be derived for arbitrary compounded period. Remark 2.2: Considering semi-annual compounding by

2)2r1( + (2.1.10)

is faulty as the time is scaled by 2. In other words a semi-annular interest stream of 1$ at the end of the year is

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2r1)r1)(

2r1( 5.0

1 +−++

where r1 is a reinvestment assumption of the first cash flow. This formula assumes that the proceeds are paid out after the first half year and the interest is r1 for the second half of the year.

EV = BV2r)r1)(

2r1( 5.0

1 ⎟⎠⎞

⎜⎝⎛ +++ .

Example 2.4: We consider a semi-annual Bond with Coupon C = 8%, yield to maturity r = 4% and time to maturity t = 0.5 years. Referring to discount factor (3) the coupon stream in the first year based on (8b) and (10) after half a year is

( ) 5.0r1

2C

+ = 3.922323,

2r1

2C

+ = 3.921569

and after one year is

r1

2C

+ = 3.846154, 2

2r1

2C

⎟⎠⎞

⎜⎝⎛ +

= 3.698225

We note that they are not the same. The size of the difference depends of the value for C and r. ◊

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Furthermore continuous compounding EV = BV . erN is mostly used in the academic literature because it allows to formulate theoretical contexts in an elegant and efficient way. The continuous version for arbitrary time t is EV = BV . ert.

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2.2 Return Contribution and Attribution In the following we start by a set of investments like equities or bonds. Definition 2.2: A portfolio is a set of investments. Building blocs of the analysis of a portfolio are investments. On the time axis we consider a beginning portfolio value PB and an end portfolio value PE. Definition 2.3: The simple or arithmetic rate of return rp of a portfolio P is

PB

PBPErP−

= = 1−PBPE , (2.2.1)

i.e. it is measured as the change of the portfolio value relative to its beginning value over a prespecified time span in the past. Remark 2.2: We denote by R1 the set of real numbers. By multiplying PB and PB by the scalar λ ∈ R1 we have

PB

PBPErP−

=PB

PBPE⋅λ

⋅λ−⋅λ= .

Thus we conclude that rP is invariant by λ ∈ R1. Using

PB1

=λ

we conclude that we can assume without loss of generality PB = 1. Thus rP does not depend on the size of the portfolio.

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Remark 2.3: (2) and (9) express the same. However, (1) is applied much more often than (2) and (9) in prominent in portfolio analysis as PE and PB are measured in the market place followed by calculating the return. Example 2.4: A portfolio moved from 80 to 100. What is its return in decimals and in %?

25.080

80100r =−

= which is in percentage equal to 25%.

In this formula we assume that there is no cash flow between PB and PE. Furthermore we did not specify how long it takes to achieve this return. Is this the return over a day, a month or a year? ◊ Definition 2.4: The investment universe is given by the securities the portfolio manager is allowed to invest in. In the following we consider a portfolio of n stocks with Price PBj, 1 ≤ j ≤ n, PEj, 1 ≤ j ≤ n and number of units Nj. The net asset value of the portfolio is

j

n

1jj PBNPB ⋅= ∑

=, j

n

1jj PENPE ⋅= ∑

=, resp. (2.2.2)

which yields by (1)

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=⋅

⋅−⋅

=

∑

∑∑

=

==

j

n

1jj

j

n

1ijj

n

1jj

PPBN

PBNPENr

(2.2.3a)

j

n

1jj

j

jjj

n

1jj

j

n

1jj

jj

n

1jj

PBN

PB)PBPE(

PBN

PBN

)PBPE(N

⋅

−⋅⋅

=⋅

−⋅

∑

∑

∑

∑

=

=

=

= .

By defining for j = 1, 2,….,n the weights

i

n

1ii

jjj

PBN

PBNw

∑=

⋅

⋅= , j = 1, 2,….,n. (2.2.3b)

and with the return rj of an asset j

,PB

PBPEr

j

jjj

−= j = 1, 2,….,n. (2.2.3c)

By inserting (3b) and (3c) in (3a) we find that the (absolute) return rP of a portfolio P is the weighted average of the return of the return:

rP = ∑ ==

n

1jjjrw w1 r1 + w2 r2 + ….. + wn rn. (2.2.4)

The terms wj rj, 1 ≤ j ≤ n are called the return contribution. They are the contribution of the return of the asset j to the return of the portfolio whereas the return rj allows the comparison of the returns

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between the assets j. They are nonweighted or unweighted returns. Remark 2.4: With (3b) we introduce the weights at the beginning of the period. From (3b) there follows

∑=

n

1jjw = ∑

∑=

=⋅

⋅n

1ji

n

1ii

jj

PBN

PBN = 1. (2.3.5a)

In portfolio theory this condition is called the budget constraint. If there is no short selling we have the nonnegativity constraint wj ≥ 0, j = 1, 2,….,n. (2.3.5b) Form (5) it follows 0 ≤ wj ≤ 1, j = 1, 2,….,n. Example 2.5: We consider the following price movement of 3 stocks A, B and C:

Stock Beginning End Return A 120 160 33.3% B 100 120 20.0% C 30 50 66.6%

Table 2.1 The returns of the stocks follow from a beginning value and an ending value. Further we assume the weights

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w1 = 15%, w2 = 25%, w3 = 60%. The return contribution shows

Stock Weights Return Absolute

Contribution A 15% 33.3% 5% B 25% 20.0% 5% C 60% 66.6% 40%

Portfolio return 50% Table 2.2

◊

We see that the elements or the basis of the return analysis are the returns of the individual securities in the portfolio. In an absolute return contribution analysis the return is decomposed according to a breakdown of the investment universe. Often the return is broken down into the return of the different countries or the different industries in the investment universe. Definition 2.5: A segment is a set of investments of the investment universe. We illustrate (4) by considering a breakdown of the universe into two segments. One part consists of the first m securities with returns r1,….,rm and in the second part we have the n − m securities with returns rm+1,….,rn. With the abbreviations

W1 = ∑=

m

1jjw , W2 = ∑

+=

n

1mjjw ,

21

R1 = ∑∑∑===

+++ m

1jj

mmm

1jj

22m

1jj

11

w

rw......w

rw

w

rw ,

R2 = ∑∑∑+=+=

++

+=

++ +++ n

1mjj

nnn

1mjj

2m2mn

1mjj

1m1m

w

rw........w

rw

w

rw

(4) is the same as .RWRWr 2211P ⋅+⋅= (2.2.6) It is seen that (6) is the special case n = 2 in (3). R1 and R2 are the return of the 2 segments. If, for instance, R1 > R2 the return of segment 1 is higher than the return of segment 2. The overall return of the portfolio is then the weighted sum of the returns of the two segments. We summarize as follows

The return is linear, i.e. the return of a portfolio is equal to the sum of the weighted returns of its investments.

In return contribution it is assumed that we only measure the return of a single portfolio manager or in other words it is a one-dimensional breakdown of the return. It is given for instance by the theme of a fund or by the contract between the client and an asset management firm. If the portfolio manager’s goal is to maximize the absolute return, the return analysis has to account for this. Here we have an important principle in return analysis or more general in performance measurement. The analysis of the performance has to account for the investment strategy of the portfolio manager. Given appropriate risk consideration in an absolute return strategy the portfolio manager tries to maximizes the absolute return of his or her

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portfolio. We see that the elements of the return analysis are the returns of the individual securities in the portfolio. In an absolute return contribution analysis the return is decomposed according to a breakdown of the investment universe in segments. Often the return is broken down into the return of the different countries or the different industries in the Investment Universe. For evaluation the return of an absolute contribution we need: Definition 2.6: A risk-less asset is defined as an investment that has no capital loss over a predetermined period and the risk-free return rf is the return that we can earn on such an investment. Typical examples of risk-less assets are Teasury-Bills or Certificates of deposits. They are deemed to be risk-less as the issuers of these money market instruments are guaranteed to repay the money the investor has invested and to pay the interest rate due. Definition 2.7: The excess return is the difference between the return of a portfolio rP and the risk-free return rf rP − rf. If we consider a relative return analysis the return is measured against a reference portfolio. Definition 2.8: A reference portfolio is called a benchmark portfolio or shortly a benchmark. In the finance industry there are two kinds of Benchmarkes. For equity or fixed income portfolios most investors consider industry standards benchmarks. In the equity world the most important index providers are MSCI and FTSE and in the fixed

23

income area names like J.P. Morgan, Citigroup, Barclays and Merrill Lynch are market leaders in producing indices. For balanced portfolios, i.e. Portfolios that consist of different asset classes, investors mostly use tailor-made benchmarks. In most cases the benchmark is a mix of equity, bond and money market indices. The proportion of the different asset classes is subject to discussions between the client and the portfolio manager. It is important to realize that the performance of the portfolio manager is measured against the Benchmark. Thus a relative return contribution has to be considered. Relative return measures the contributions of the securities in the portfolio relative to the contributions of securities in the Benchmark. We proceed by adopting the following notation for a Benchmark portfolio. We consider n investments with Beginning Price BBi, 1 ≤ i ≤ n, and end prices BEi, 1 ≤ i ≤ n with Mi units. The asset value of the portfolio is

i

n

1ii BBMBB ⋅= ∑

=, i

n

1ii BEMBE ⋅= ∑

=, resp.. (2.2.7a)

Similar to (3b), with

i

n

1ii

jjj

BBM

BBMb

∑=

⋅

⋅= , 1 ≤ j ≤ n (2.2.7b)

we have the weights of the benchmark securities in benchmark. The return of the Benchmark (Portfolio) is

jn

1jjB rbr ∑=

=. (2.2.7c)

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Definition 2.9: With (4) and (7) the arithmetical relative return ARR is the difference ARR = .rr BP − With (7) we have for the relative return

.r)bw(ARR jj

n

1jj −= ∑

= (2.2.8)

This decomposition is called Brinson-Hood-Beebower (BHB). From

)b(w0 jn

1jj −∑=

=

we have

)r()b(w0 Bjn

1jj −−∑=

=

and by adding (8) there follows

ARR = ).rr()bw( Bjj

n

1jj −−∑

= (2.2.9)

This decomposition is called Brinson-Fachler (BF). From (7) and (9) it is seen that return analysis is not a unique process and in fact in many practical situations (9) is used instead of (7). In the decomposition if term of the sum in (7) is positive the portfolio manager contributed positively to the relative return otherwise

25

negatively. Thus (9) allows a meaningful discussion of the relative return. Summarizing, in return contribution we analyze the portfolio return in view of different investment universes. It is a one dimensional process. If the portfolio manager is allowed to invest in assets outside the benchmark we have bi = 0 for these assets and (4), (8) and (9) can also be used for the return analysis. However, assets outside the benchmark contribute with the absolute return to the relative return of the portfolio. From the Portfolio managers view, investments outside the benchmark are thus considered to be risky. Example 2.6: We consider the same 3 stocks as in Example 2.5 and the universe of the portfolio and the benchmarks are the same with w1 = 15%, w2 = 25%, w3 = 60% in (3a) and b1 = 25%, b2 = 25%, b3 = 50% in (7) (see Table 3). Inputs are in bold. Table 2.3 shows two decompositions of the relative return 1.00% of the portfolio versus the benchmark. In the last line is the return of the portfolio −1.50% and the benchmark 2.50%, i.e. we find that the portfolio outperforms the benchmark by 1.00%. The value added indicates that stock A and C are underperforming the benchmark and B is outperforming the benchmark. However, Stock B has a neutral position and does not contribute to the relative return. As A is an underweighting position versus the benchmark, the investing decision is favourable, that is the contribution is positive, namely 2.00% on an absolute basis (BHB) and 1.75% on a relative basis

26

(BF). As C is an overweighting position versus the benchmark, the investing decision is unfavourable, that is the contribution is negative, namely -1.00% on an absolute basis (BHB) and -0.75% on a relative basis (BF).The following table shows the BHB and BF decomposition:

Portfolio BenchmarkValue added rj − rB

Under/ Over

weight

Contri- Bution BHB

Contri- Bution

BF

Re- turn weight weights

A -20% 15% 25% -17.5% -10% 2.00% 1.75%

B 30% 25% 25% 32.5% 0 0.00% 0.00%

C -10% 60% 50% -7.5% 10% -1.00% -0.75%

Re- turn -1.50% -2.50% 1.00% 1.00%

Table 2.3 If we consider a capitalization weighted benchmark, the benchmark weights are the capitalization weights of the securities in the benchmark portfolio. However, from (7) it is seen that they can be chosen differently from capitalization weights. It is seen that in nature the benchmark portfolio is the same as the investment portfolio and the elements of the return analysis are the securities in the investment universe. ◊ We proceed by explaining the analysis of the Money managers’ performance. We attribute the return to the appropriate decision makers and to the different steps of the investment

27

process. We analyse different factors of the investment process. As in the previous section we distinguish between an absolute and relative attribution: Definition 2.10: Return Attribution is the decision oriented decomposition of the return. Here we only confine to an explanatory illustration for absolute attribution. The absolute attribution is no used widely. However, as the following example shows the formulae already get intricate. Base for the attribution is:

• Segmentation of the investment universe and (or) the

benchmark universe

• Multi level investment process Example 2.7: We consider the Example 2.6 exposed in Table 2.3 and we assume the Universes of the portfolio and the benchmark are the same and consists of stock A with weight w1 = 15%, stock B with weight w2 = 25% and stock C with weight w3 = 60%,. We introduce a two step investment process and 3 Portfolio Managers (PM).

• PM1 decides on the Country allocation (W1, W2),

• PM2 decides on the portfolio in Country X (w1, w2),

• PM3 decides on portfolio in Country Y (w3). We ask what the return of the individual managers is. We have a two level decision. First the PM1 (country manager) decides on W1 = w1 + w2 and W2 = w3. Referring to the notation in (7) the country benchmarks rX, rY are

28

3Y221

21

21

1X rr,r

bbbr

bbbr =

++

+= .

Using the data from Example 2.6 we find

21

1bb

b+

= 0.5, 21

2bb

b+

= 0.5, rX = 25% and rY = 10%.

PM1’s return is measured by

( ) 33221

21

21

121PM1 rwr

bbbr

bbbwwr +⎟⎟

⎠

⎞⎜⎜⎝

⎛+

++

+= = −4%.

rPM1 is the return arising from investing in country X and Y. PM1 has taken in Country X overweight position and this was a wrong decision. It is called the asset allocation effect. The PM2 (Country X) decides on w1 and w2 and his return is

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+−+= 221

21

21

1212211PM2 r

bbbr

bbbwwrwrwr =

= 4.5% − 0.4 . 5% = 2.5%. rPM1 is the return from deciding on the securities in X. The PM2 has taken the right decision as he has taken an underweight in Stock A and an overweight in Stock B. It is called the stock picking effect. If the PM3 (Country Y) is not allowed to buy stocks outside the benchmark there is no decision to be taken for PM3. We see the definition of the universe is crucial in return attribution. If PM3 had more choices it could influence the return. Moreover, it is not clear that the allocation is an asset allocation effect or a sector

29

effect (chapter issues in performance). PM1 and PM3 decisions coincide. In the following section we will see that decision makers can have common influence on part of the return. ◊ In the return attribution it is assumed that the investment process consists of different steps and different decision makers. It is a multi step process. Again absolute and relative return attribution can be considered. In the following we focus on relative return attribution that consists of two layers. As in the return contribution a breakdown of the Investment Universe (e.g. Countries or Industries) is considered. In most cases Investors are interested in a country and (or) industry breakdowns. However, also classifications according to the security capitalization in the equity market or duration buckets in the fixed income area are possible. In the following we distinguish between arithmetic attribution (addition) and geometric attribution (product). a. The arithmetic attribution In the following Wj, 1 ≤ j ≤ n, are the weights of the portfolio part with return Rj, 1 ≤ j ≤ n, and Vj, 1 ≤ j ≤ n, is the weight which is in most cases the market capitalization of the corresponding benchmark segment with return Bj,1 ≤ j ≤ n. Following Definition 2.9 the arithmetic relative return is

ARR = ( )∑∑∑===

⋅−⋅=⋅−⋅n

1ijjjj

n

1jjj

n

1jjj BVRWBVRW (2.2.10)

and consider the identity

30

Wj . Rj − Vj . Bj = (2.2.11) (Wj − Vj ) . Bj + (Rj − Bj ) . Vj + (Wj − Vj ) . (Rj − Bj ) (see Figure 2.1). This identity allows the discussion of the relative return. We proceed by the first part of the sum in (10) we refer to Brinson-Hood-Beebhower (see (7)) and introduce by BHB

iA in the following the asset allocation effect BHB

jA in segment j based on BHB

jA = (Wj − Vj ) . Bj. By referring to Brinson-Fachler (see (8)) we denote by BF

iA in the following the asset allocation effect BF

jA in segment j based on BF

jA = (Wj − Vj ) . (Bj − rB). We note that we can BHB

jA consider as the special case when the benchmark return vanishes in BF

jA . The asset allocation effect is the overweight or underweight of the segment of the breakdown of the benchmark. However, the weight of the constituents within the segments stays the same. BHB

jA > 0 ( BHBjA < 0) says

whether the asset manager decisions to overweight or underweight the segment adds positively (negatively) to the return of the portfolio. BF

jA > 0 ( BFjA < 0) says whether the asset

manager decisions to overweight or underweight the segment adds positively (negatively) to the relative return of the portfolio

31

versus the benchmark. As in (8) on portfolio level BHBjA and BF

jA add up to the same:

Atot = )rB()VW(A Bj

n

1jjj

n

1j

BHj −−= ∑∑

== = ∑

=

n

1j

BHBjA . (2.2.12a)

It measures the return of the portfolio manager that takes the asset allocation decisions relative to the considered breakdown of the investment universe into account. There are two possibilities for a positive asset allocation effect namely an overweight of that segment followed by an outperformance or an underweight followed by an underperformance of the segment relative to the benchmark. For a negative asset allocation effect an analogous statement holds. Stock selection Sj effect in segment j defined by Sj = (Rj − Bj ) . Vj .

Vi

return

weights

benchmark

portfolio

Bi

Wi

Ri

Vi

return

weights

benchmark

portfolio

Bi

Wi

Ri

Brinson-Hood-Beebower Figure 2.1

32

measures the effect of the security selection within the universe. We can measure the overall stock selection in the portfolio or the stock selection within a specific segment. In the relative performance attribution there is a third component of the return called interaction Ij effect in segment j defined by Ij = (Wj − Vj ) . (Rj − Bj ). This interaction effect is a part of the performance that cannot be uniquely attributed to a particular decision maker. For instance if the asset allocation effect and the stock select effect is positive then the overall return is not barely the sum of the two components but returns cumulate and as a consequence the return is bigger. It is important to calculate and to account for the interaction effect but it is a very subjective decision which decision maker is the owner of the interaction effect. The asset allocation effect, the Stock selection effect and the interaction effect are called the management effect. With Wj . Rj − Vj . Bj = BHB

jA + Sj + Ij and

∑∑==

==n

1ijtot

n

1jjtot II,SS (2.2.12b)

we find by (10), (11)

33

ARR = =⋅−⋅ ∑∑==

n

1jjj

n

1jjj BVRW )ISA( jj

n

1jj ++∑

= =

∑=

n

1i

BHBjA + ∑

=

n

1jjS + ∑

=

n

1jjI = ∑

=

n

1i

BFjA + ∑

=

n

1jjS + ∑

=

n

1jjI =

Atot + Stot + Itot. We note that in general: Wj . Rj − Vj . Bj ≠ BF

jA + Sj + Ij and summarize

We conclude the discussion on the BF and the BHB decomposition of the arithmetic relative return. The return attribution is a reverse problem, i.e. the return is known and observable and the decomposition of return is not unique. In the following examples we illustrate different decomposition based on different requests of a user of a return attribution system. Example 2.8: We expand the portfolio in Example 2.6 by stock D. The input in Table 2.4 in bold:

The BF decomposition does not hold on segment level but on portfolio level the relative return is equal to the three management effects. The BHB decomposition holds on segment level and on portfolio level.

34

Portfolio BenchmarkValue added rj − rB

Under/ Over

weight

Contri- Bution BHB

Re- turn weight weights

A -20% 15% 25% -19.5% −10% 2.00% 1.95%

B 30% 25% 25% 30.5% 0 0.00% 0.00%

C -60% 10% 20% -59.5% −10% 6.00% 5.95%

D 30% 50% 30% 30.5 20% 6.00% 6.10%

Re- turn 13.5% −0.5% 14% 14%

Table 2.4 We consider the 2 segments. A segment consists of Stock A and B and another segment consists of Stock C and D. The following Table 2.5 shows the absolute return of the benchmark and the portfolio:

Segment j Wj Rj Vj Bj

1 40% 11.25% 50.00% 5.00%

2 60% 15.00% 50.00% −6.00%

Total 13.50% −0.50%

Table 2.5

35

Table 2.6 illustrates (10) - (12):

Relative Return BHB BF S IA

2.000% −0.500% −0.550% 3.125% −0.625%

12.500% −0.600% −0.550% 10.500% 2.100%

Total 14.000% −1.100% −1.100% 13.625% 1.475%

Table 2.6 This example shows that all investment decisions exempt the in investment in Stock B are favourable and the portfolio outperformances the benchmark by 14.00%. ◊ Example 2.9 (two investment processes): We assume rP − rB = 2.0% over a month. We decompose the relative return of a balanced or multi asset class portfolio in two different ways. The first decomposition assumes a three step investment process which starts with the benchmark definition, asset allocation and stock picking. Doing such an analysis assumes also that the asset allocation as well as the stock picking each is done by one decision maker. Following (11) the report of an attribution system might find over a month for the Asset allocation effect with return 2.0%, the Stock picking effect with return −0.7% and for the interaction effect the return −0.3%. Then the decomposition is 2.0% = 3.0% − 0.7% − 0.3%.

36

In the second decomposition of the relative return we consider an investment process that is a more complex decision making process. We assume that the investment process consists of six steps and that we have the following returns: benchmark definition, the internal benchmark selection with a return 0.3%, an asset allocation effect with a return 1.0%, fixed income asset allocation with a return −0.5%, an equity asset allocation with a return 0.8% and portfolio implementation or stock picking with a return 0.7%. The interaction has a return −0.3%. Then we have 2.0% = 0.3% + 1.0% − 0.5% + 0.8% + 0.7% − 0.3%. Looking at the figures, in the first decomposition one would conclude that the balanced account manager is a poor stock picker but if the second investment process would be correct, on the contrary the balanced account manager would be a good stock picker. Neglecting the real investment process and not reflecting it in the performance attribution potentially causes wrong interpretation and can lead to wrong conclusions. ◊ Example 2.10 (two different benchmarks): We consider a European equity mutual fund which a formal external benchmark is the MSCI Europe. We assume that over a period the relative return 2.0% of the portfolio versus MSCI Europe is decomposed in the asset allocation effect with a assumed return 3.0%, a the stock picking effect with return -0.7% and the interaction effect -0.3%, i.e. 2.0% = 3.0% − 0.7% − 0.3%.

37

In order to measure this mutual fund against the index MSCI Europe and to decompose its excess return is not appropriate if one wants to measure the quality of the asset manager. The product management of the mutual fund company positioned this mutual fund internally as a growth product with the internal benchmark of a European growth index. To get the right picture the set up of the performance attribution has to be changed in a way that the excess return versus the external benchmark is split in four effects: a) benchmark selection with a return 1.8%, b) asset allocation effect with a return −0.5% c) a stock picking effect with a return 1.0% and d) an interaction with a return −0.3%. We have the following decomposition 2.0% = 1.8% − 0.5% + 1.0% − 0.3%. Such a return decomposition ensures that the contribution of the positioning of the product and the contribution of the asset manager can be isolated and assessed. In our case the signs of the asset allocation and stock picking effects changed just due the fact that we changed the relevant benchmark to the European growth index. ◊ Example 2.11 (two different segmentations): Performance attribution software is normally quite flexible in setting up the analysis, so that the performance analysts can choose between decomposing the return by using different segments such as countries or sectors. Depending on the chosen segment to decompose the return the performance attribution may come up with different management effects. We assume an asset manager of a European equity account and consider the following decomposition of the relation return 2.0% into asset allocation

38

effect, stock selection and interaction by different segmentations of the benchmark MSCI Europe. With a sector approach 2.0% = 3.0% − 0.7% − 0.3% and with a country approach 2.0% = −0.5% + 2.7% − 0.2% we come up with different management effects. Setting up the performance attribution in a wrong way may lead to a wrong interpretation and to wrong conclusions. This again shows that setting up the performance attribution according to the investment process is essential to get a meaningful analysis and appropriate feedback into the investment process. ◊ In (2.2.2) is assumed that o the portfolio is invested in a single currency. An illustration is a

fixed portfolio with bonds that comprises only of one currency or o the return arising of different currencies is not of interest.

Equity of internationally active companies does not relate to a single currency and a currency attribution is not requested.

b. The geometrical attribution Definition 2.11: The geometrical relative return is

GRR = 1r1r1

B

P −++ . (2.2.13)

39

Remark 2.5 (absolute return of the portfolio): If rB = 0 then GRR = ARR = rP. Remark 2.6 (Interpretation of GRR): Based (12) we find

1r1r1GRRB

P −++

= = B

BPr1rr

+− =

BBBBBE1

BBBBBE

PBPBPE

−+

−−

−

.

According to Remark 2.2 we scale PB or BB such that we can assume PB = BB and it follows:

GRR =

PBBEPB

BEPE −

= BE

BEPE − = BEPE − 1.

We conclude that

o GRR can be expressed by the end value of the portfolio and the benchmark more specifically the GRR is the rate of return of the portfolio ending value and the benchmark ending value.

Example 2.12: We suppose that difference the portfolio and the benchmark is

40

ARR = 5%. Firstly this can be found with rp = 10%, rB = 15%. This yield GRR = 4.35%. Secondly assuming rp = 40%, rB = 45% we find GRR = 3.45%. The arithmetic relative return is the same for both cases. The user of the geometrical relative return can argued that with a higher benchmark return a higher portfolio return is easier to achieve that a lower portfolio return with a lower benchmark return. The geometrical relative return is relative to benchmark return. ◊ Remark 2.7: From Example 2.11 we see that GRR measures the outperformance of the performance relative to the return of the benchmark.

Time

Segment

T

n

Decomposition

Geometric Linking

Geometric Linking

Time

Segment

T

n

Time

Segment

T

n

Decomposition

Geometric Linking

Geometric Linking

Figure 2.2

41

We proceed with the decomposition on segment level. The asset allocation effect for segment j, j = 1,....,n is defined by

Aj = (Wj − Vj) . ⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

+1

r1B1

B

j , (2.2.14a)

the stock selection effect is defined by

Sj = Wj . ⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

+1

B1R1

j

j . S

j

r1B1

+

+ (2.2.14b)

and the notional portfolio with returns rS is defined by

rS = ∑=

N

1jjj BW (2.2.14c)

and the geometrical relative return for segment j 1 + GRRj = (1 + Aj) (1+ Sj).

We proceed by

∑=

=n

1jjtot AA = ∑

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

+−

n

1j B

jjj 1

r1B1

)VW( =

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−

n

1j B

jBjj r1

Br)VW( =

)rB)(VW(r1

1Bj

n

1jjj

B−−

+∑=

=

42

B

BSr1rr

+− =

B

Sr1r1

+− − 1

and

∑=

=n

1jitot SS = ∑

= ⎟⎟⎠

⎞⎜⎜⎝

⎛

+

+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

+⋅

n

1j S

j

j

jj r1

B1 1

B1R1

W =

∑= ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛

+

−⋅

n

1j S

j

j

jjj r1

B1

B1BR

W =

S

SPr1rr

+−

= S

Pr1r1

++

− 1.

From (13) we have

1 + GRR = B

Pr1r1

++

= S

Pr1r1

++

B

Sr1r1

++

.

We define the asset allocation effect of for the portfolio

Atot = B

Sr1r1

++ − 1

and the selection effect of for the portfolio

Stot = S

Pr1r1

++

− 1

and have

GRR + 1 = (1 + Atot) (1 + Stot)

43

Note: We do not propose any relationship between GRR and GRRj, j = 1,....,n. c. The currency attribution The return of the portfolio depends of the currency and the return of the portfolio can be expressed in different currency. In the following we discuss the relation between the different returns. Definition 2.12: The base currency relates to the investor’s accounting currency. Definition 2.13: The local currency relates to the currency in which the investment is made. We start with an investment value PB in the base currency. We assume an investment in a market or an investment in the market whose return rL is not measured in the base currency, but in the local currency. Definition 2.14: The currency exchange rate that translate at time t from local currency, base currency, resp. to the base currency, local currency, resp. is denoted by ec,t, t,ce , resp. We have

C,t

C,t e1e = .

The return rC from the currency exchange rate is

44

rC =

local currency

base currencyPB PE

eB,c eE,c

local currency

base currencyPB PE

eB,c eE,c

Evaluation in different currencies Figure 2.3

B,C

B,CE,C

eee −

=C,E

C,B

ee

− 1 (2.2.15a)

rc =

B,C

B,CE,C

e1

e1

e1

− =

E,C

B,C

ee

− 1. (2.2.15b)

The end value of the investment PE is

PE = PB ,e

)r1(e

E,C

LB,C +.

By (15b) the total return rT in the base currency is

45

rT = PB

PBPE − = PB

PBe

)r1(e PB

E,C

LB,C −+

= 1e

)r1(e

E,C

LB,C −+

= (1 + rL) (1 + rC) − 1,

i.e. rT = rL + rC + rC rL. The expression (15) is independent on PB. We see that we have geometric linking of the market and the currency return. Example 2.13: We consider a CHR investor that buy a bond denominated in Danish kroner and Polish Zloty. We want to calculate the return in CHF. The bonds have the following prices for at the beginning PB, DKK = 111.110, PB, POL= 111.160 and the end PE,DKK = 111.946, PE,POL = 111.049. In Table 2.7 and Table 2.8 bold figures. In Table 2.8, the exchange in Polish Zloty and Danish Kroner are given against US Dollar. With the value of Table 2.7 against the Swiss Franc we find for example

5.2470 = 1.155356.06270 .

t1 t2 CHF/US 1.15535 1.07835 Table 2.7

46

US US CHF CHF t1 t2 t1 t2 DKK 6.06270 6.08235 5.24750 5.64149 POL 3.32075 3.37715 2.87424 3.13177 Table 2.8 1. For the Danish Bond we have by (12b)

rDKK = 15.641495.24750

− = −0.06984

i.e. the CHF has a negative return because the Danish Crone is inflating. (13) yields

rT = 111.110111.946

5.641495.24750 − 1= −0.07121.

2. For the Polish Bond we have

rPOL = 3.131772.87424 − 1 = −0.08223,

i.e.

rT = 111.160111.049

3.131772.87424 − 1 = −0.08315.

◊ We proceed by considering the currency attribution of a segment j with a unique currency rT,j = rC,j + rM,j + rC,j rM,j,1 ≤ j ≤ n

47

rT,P = ∑=

=n

1jj,Tjrw ( )( )∑

=++

n

1jj,Mj,Cj,Cj,Mj rrrrw =

∑=

+n

1jj,Mjrw ∑

=+

n

1jj,Cjrw ∑

=

n

1jj,Cj,Mj rrw .

If the investments in the base currency are in the segment 1 we have r1,C = 0, i.e.

∑=

+n

1jj,Mjrw ∑

=+

n

2jj,Cjrw ∑

=

n

2jj,Cj,Mj rrw .

The total return of the portfolio consists of three parts. The first and the second part are the sum of the market return and the currency return of the segment. The third part cannot be assigned to the currency return and the market return.

48

2.3. The time weighted rate of return a. Absolute return measurement We consider t0 = 0,....,tN = T (2.3.1) as time points and corresponding cash flows C1,....,CN−1 on the time axis and k−1rP,k (2.3.2a) denotes the return of the portfolio between tk−1 and tk, k = 1,....,N (see Figure 2.4).

t0 = 0 tN = T

t

C1

tN-1

CN−1Ck

t1 t2t0 = 0 tN = T

t

C1

tN-1

CN−1Ck

t1 t2

The time axis and cash flows Figure 2.4 Based on the beginning portfolio values PBk and the ending portfolio values PEk+1 we calculate the return k−1rP,k by

k−1rP,k = 1k

1kkPB

PBPE

−

−− , k = 1,...,N. (2.3.2b)

In order to measure the return between time 0 and time k we link the returns multiplicatively by

49

0rP,k = (1 + 0rP,1) . (1 + 1rP,2) ...... (1 + k−1rP,k) − 1, k = 1,...,N (2.3.2c) where 0rP,k is the return of the portfolio P from 0 to k. Furthermore it is assumed in (2c) that the returns are compounded at the end of each sub period. As we multiply we allude to an area as geometrical object and the formula is called geometrically linked returns. Definition 2.15: For equidistant knots tk = k, k = 0,..., N = T, we consider the geometric mean return Pr ( )N T,P0P r1r += − 1. (2.3.3a) It is seen when compounding with pr over the time period we get

pr by (3a), i.e. ( )N pP r1r += − 1. (2.3.3b) Definition 2.16: For annual data over N years the annualized return Pra is defined by (3b). For monthly date over N months the annualized return Pra ( )12N pP r1ra += . Remark 2.8: We see that we have here by (3) two different attributions along the time axis. The first attribution is according to (2c) and the second is described by (3) where the returns over 0rP,K is uniformly distributed or attributed with pr defined by (3b). We note that the returns in (2) are commutative, i.e. the returns stay unchanged by considering for example r1 in period 2

50

and r2 in period 1. The commutative law of the returns and the independence of the size of the portfolio by geometrically linking are illustrated by the following example. Example 2.14: a. We assume N = 2 with t0 = 0, t1 = 1 and t2 = 2 in (1) and PB0 = 1$, PB1 = 100$, 0rP,1 = 5%, 1rP,2 = 7% in (3). Then we have with (3b) PE1 = PB0 (1 + 0rP,1) = 1$ (1 + 0.05) = 1.05$ and PE2 = PB1 (1 + 0rP,2) = 100$ (1 + 0.07) = 107$. (2c) yields 0rP,2 = (1 + 0rP,1) . (1 + 1rP,2) − 1 = 0.1235. There is an inflow C1 = 100 − 0.07 = 98.93 at t1 =1. It is seen that C1 that nor infer the calculation of 0rP,2. b. We assume N = 2 with t0 = 0, t1 = 1 and t2 = 2 in (1) and PB0 = 100$, PB1 = 1$, 0rP,1 = 5%, 1rP,2 = 7%. Then we have with (3b) PE1 = PB0 (1 + 0rP,1) = 100$ (1 + 0.05) = 1.05$ and PE2 = PB1 (1 + 0rP,2) = 1$ (1 + 0.07) = 107$. (2c) yields 0rP,2 = (1 + 0rP,1) . (1 + 1rP,2) − 1 = 0.1235.

51

There is an outflow C1 = 1 – 107 = –106 at t1 =1. It is seen that C1 that nor infer the calculation of 0rP,2. We see that there is a cash inflow in a. and a cash outflow in b. at time 1, although the result is the same in both cases. We see that the cash flows are reinvested by 1rP,2. In a and b we find by (3a) for the annualized return ( )N T,P0P r1r += − 1 = ( )2 2,P0r1+ − 1 = ( )2 1235.01+ -1 = 0.059953. c. We assume N = 2 with t0 = 0, t1 = 0.5 and t2 = 1 in (1) and PB0 = 1$, PB1 = 100$, 0rP,1 = 5%, 1rP,2 = 7% in (3). Then we have with (3b). Following case a above we find 0rP,1 = (1 + 0rP,1) . (1 + 1rP,2) − 1 = 0.1235. d. We assume N = 2 with t0 = 0, t1 = 0.5 and t2 = 2 in (1) and PB0 = 100$, PB1 = 1$, 0rP,1 = 5%, 1rP,2 = 7%. Then we have with (3b). Following case b above we find 0rP,1 = (1 + 0rP,1) . (1 + 1rP,2) − 1 = 0.1235. We see that in both cases c and d we have ( )1 1,P0P r1r += − 1 = 0.1235. We conclude that the return does not dependent the cash flow but is dependent from the underlying time span. ◊

52

The mean defined in the following definition does not respect compounding. It is used extensively in Chapter 4 on risk. Definition 2.17: For equidistant knots tk = k, k = 0,..., N = T, we consider the arithmetic mean return Pr of a portfolio P . It is calculated by the sum of the returns in (2a) divided by the number N of returns in (2a):

.rN1r

N

1kk,P1kP ∑

=−= (2.3.4)

By expanding (2.2.2) we denote by Nj,k the units and Pj,k the prices of asset j at the time tk, 0 = 1,….,N − 1. With

∑=

=n

1jkj,kj,k PNPB , ∑

=+=

n

1j1kj,kj,k PNPE

there follows that (2a) is equivalent to

1 + k−1rP,k = ∑

∑

=

=+

n

1jkj,kj,

n

1j1kj,kj,

PN

PN.

With (2b) we have by chain linking for k = p,....,q, 0 ≤ p < q ≤ T

1 + prP,q = ∏∑

∑

=

=

=+1-q

kpn

1jkj,kj,

n

1j1kj,kj,

PN

PN

and for p = 0 and q = N we find

53

0rP,T = ∏∑

∑

=

=

=+1N-

0kn

1jkj,kj,

n

1j1kj,kj,

PN

PN − 1. (2.3.5)

At the end of each period, we divide by the portfolio value and observe the price movement of the securities in the portfolio over the next period. In other words we neutralize the portfolio value and the number N of shares outstanding remains unchanged over one time period. Definition 2.18: (5) is called the time weighted rates of return calculation (TWR). As potential changes in the money invested in the portfolio does not affect the return. We simply look at relative changes over multiple time periods. Time weighted rates of returns are used by the index providers. Most market indices are calculated on a daily basis, i.e. the time unit is days. The value of the market portfolio is calculated at the end of each business day and formula (5) is evaluated. The returns are published by data providers. It is well known that indices can be rebased at each point in time. This again reflects the fact that (5) is independent of the portfolio size, i.e. if we consider an index with base currency $ and rebase the portfolio to 100 we observe the development of an initial portfolio of 100$. Let us assume N i,k = λk . Ni,k−1 , λk ∈ R1 for k = 1,….,N − 1 (2.3.6) and consider (5) over two periods [λk−1, λk] and [λk, λk+1], 0 ≤ p < k < q ≤ T and apply (6) twice

54

(1 + k−1rP,k)(1 + krP,k+1) = ∑ ∑

∑ ∑

= =

= =+

⋅

⋅

n

1j

n

1jkj,kj,1-kj,1-kj,

n

1j

n

1j1kj,kj,kj,1-kj,

PNPN

PNPN =

(2.3.7)

∑

∑

=−

=+

λ⋅n

1jk1kj,1-kj,

n

1j1kj,kj,

PN

PN =

∑

∑

=−

=+

n

1i1kj,1-kj,

n

1j1kj,1-kj,

PN

PN = (1 + k−1rP,k+1).

By applying (6) to the denominator and using (7) repeatedly we have

0rT = 1

100

1 −

∑

∑

=

=n

i,i,i

n

iT,iT,i

PN

PN

and consequently

0rT = 0

0PV

PVPVT − ,

i.e. the return reduces to the return based on the change of the portfolio value at time 0 and T. The portfolio values at the time t1 to tK-1 do not affect the return as intermediate Portfolio value are dropping. We summarize:

55

Theorem 2.1: If (6) holds then the return of a portfolio over multiple periods is equal to the geometrically linked returns of the portfolio over the sub periods. Proof: The assertion follows by applying (7) to the interval [p, q], 0 ≤ p < q + 1 ≤ T consecutively. ◊ We distinguish two cases: Case 1:

∑∑==

− =n

1jk,jk,j

n

1jk,j1k,j PNPN , k = 1,….,N − 1 (2.3.8)

This case includes the case λk = 1, k = 1,….,N − 1 in (6) in other words as the number of securities are held constant the buy and hold strategy is reflected. However, case 1 leaves open the possibility of a rebalancing within the portfolio and the portfolio is restructured (see Figure 2.5). By evaluating (7) for the buy and hold strategy and the rebalanced portfolio the value added from restructuring the portfolio can be measured. In both case the return is only determined by the initial value and the terminal value of the portfolio. Furthermore this case is important in understanding the computation of Total Return Indices. If an index reinvests overnight or instantaneously it is assumed that a cash flow such as dividends or coupons stay in the index, however it is distributed between the securities of the index, i.e. the portfolio value is unchanged, but the cash flow is distributed such that the ratio of the market capitalization is unchanged.

56

Theorem 2.2: If (8) holds then the return of a portfolio over multiple periods is equal to the geometrically linked returns of the portfolio over the sub periods. Proof: The assertion follows by considering the interval [p, q], 0 ≤ p < q + 1≤ T and hence we have by considering (7) over two periods [λk−1, λk] and [λk, λk+1]

(1 + k−1rP,k)(1 + krP,k+1) = ∑ ∑

∑ ∑

= =

= =+

⋅

⋅

n

1j

n

1jkj,kj,1-kj,1-kj,

n

1j

n

1j1kj,kj,kj,1-kj,

PNPN

PNPN =

∑

∑

=

=+

n

1j1-kj,1-kj,

n

1j1kj,kj,

PN

PN =

1k

1kPBPE

−

+ = (1 + k−1rP,k+1).

◊ Case 2: There exists a k, k = 1,…., N − 1 with

∑≠∑==

−n

1ik,ik,i

n

1ik,i1k,i PNPN . (2.3.9)

C h a n g e s o f P r i c e s

N i , k P i , k

N i , k - 1 P i , k - 1

Changes of U

nits

N i , k - 1 P i , k

N i , k P i , k + 1

Chain linking at a seam Figure 2.5

57

This case reflects a change in the portfolio value due to an external cash flow as we divide in (5) by the portfolio value. However, the cash flow does not affect the return calculation as the return depends only the movement between the knots tk, k = 0,…., N. Taking the argument further chain linking return measurement does not take into account the timing of external cash flows. This property underpins the importance of this methodology in the index industry. The finance industry is only interested in objective market movements. For the difference of the return from a client’s or a portfolio manager’s perspective we refer to the Section 2.5. Furthermore this case is important in understanding the computation of Total Return Indices. If an index reinvests overnight or instantaneously it is assumed that a cash flow such as dividends or coupons stay in the index, however it is distributed between the securities of the index, i.e. the portfolio value is unchanged, but the cash flow is distributed such that the ratio of the market capitalization is unchanged. Theorem 2.3: If (9) is holds then following is implied: If the external cash flow in tk C = ∑

=−

n

1jk,j1k,j PN − ∑

=

n

1jk,jk,j PN

is distributed with (6) the return over [λk-1, λk+1] is independent of the portfolio value in tk otherwise the return over [λk-1, λk+1] is dependent on the value ∑

=−

n

1jk,j1k,j PN

and the allocation of the external cash flow C in Nj,k,1 ≤ j ≤ n in tk.

58

Proof: The assertion follows from Theorem 2.1 or from (1 + k−1rP,k)(1 + krP,k+1) =

∑ ∑

∑ ∑

= =

= =+

⋅

⋅

n

1j

n

1jkj,kj,1-kj,1-kj,

n

1j

n

1j1kj,kj,kj,1-kj,

PNPN

PNPN.

◊ b. Relative return measurement Let tp, tk and tq, p < k < q denote times on the time axis (Figure 2.4). The following formulae are valid on a portfolio level. Based on (2c) we consider the geometrically linked return between the time from tp to tk and from tk to tq of the portfolio. prP,q = (1 + prP,k)(1 + krP,q) − 1 = prP,k + krP,q +prP,k

. krP,q (2.3.10a) and the benchmark prB,q = (1 + prB,k)(1 + krB,q) − 1 = prB,k + krB,q +prB,k

. krB,q. (2.3.11b) Definition 2.19: The term q,kk,p rr •• ⋅ with • either a portfolio or a benchmark is called cross term. Definition 2.20: The arithmetical relative returns pARRq, pARRk, kARRq, p < k< q, p = 0,....,N, k = 0,....,N, q = 0,....,N is defined by pARRq = prP,q − prB,q, pARRk = prP,k − prB,k, kARRq = krP,q − krB,q.

59

For pARRq and p < k < q we find pARRq = prP,k + krP,q + prP,k

. krP,q − prB,k − krB,q −prB,k . krB,q =

prP,k − prB,k + krP,q − krB,q + prP,k

. krP,q −prB,k . krB,q = (2.3.12)

pARRk − kARRq − prP,k

. krP,q − prB,k . krB,q.

We proceed by introducing two different decomposition of the relative arithmetical return for two periods. We refer to [1, Bonafede] and [7, Feibel]. Based on (12) the first decomposition of the cross terms is prP,k . krP,q − prB,k . krB,q = prP,k . krP,q − prP,k . krB,q + prP,k

. krB,q − prB,k . krB,q = prP,k (krP,q − krB,q) + krB,q (prP,k − prB,k), thus by (12) again we have pARRq = (1 + krB,q) . (prP,k − prB,k) + (1 + prP,k) . (krP,q − krB,q), i.e. pARRq = (1 + krB,q) . pARRk + (1 + prP,k) . kARRq. (2.3.13a) We see in (13a) that in addition to the sum of the arithmetical relative return the relative return in the first period is compounded with the total benchmark return in the second period and the relative return of the second period in compounded with the total portfolio return in the first period. We refer to [9, Feibel, page 278].

60

Based on (12) the second decomposition of the cross terms is prP,k . krP,q − prB,k . krB,q = prP,k . krP,q − prB,k . krP,q + prB,k . krP,q − prB,k . krB,q = krP,q (prP,k − prB,k) + prB,k (krP,q − krB,q), thus by (12) pARRq = (1 + krP,q) . (prP,k − prB,k) + ( 1 + prB,k) . (krP,q − krB,q), i.e. pARRq = (1 + krP,q) . pARRk + (1 + prB,k) . kARRq. (2.3.13b) We see in (13b) that in addition to the sum of the arithmetical relative return the relative return in the first period is compounded with the total portfolio return in the second period and the relative return of the second period in compounded with the total benchmark return in the first period. We proceed by an example that illustrates (13): Example 2.15: We consider two periods with t0 = 0, t1 =1 and t2 = 2 in (1) with returns P,10r = 2%, B,10r = 10%,

P,21r = 10%, and B,21r = 2%. We find 0ARR2 = 0. (13a) yields the decomposition 0 = − (1 + 0.1) . 8% + (1 + 0.1) . 8% and (13b) yields the decomposition

61

0 = − (1 + 0.02) . 8% + (1 + 0.02) . 8%. This example shows two decompositions of the return over time. ◊

The attribution of the return over time is not unique and depends of reinvestment assumptions. The attribution of the return over time is not unique and depends of reinvestment assumptions.

pARRq is in general critically dependent on the compounding knots between p and q (see Theorem 2.3). This property is encountered by the absolute return (see (2.1.6) and (2.1.7)) and the relative return. As exposed so far the value for pARRq is not necessary unique. In the following we assume that compounding pattern is given and we respect all compounding times tk, k = 0,....,N. We use the iterative character by (13) by q = k + 1. In addition we respect all compounding time tk. pARRk+1 = (1 + krB,k+1) . pARRk + (1 + prP,k) . kARRk+1 (2.3.14a) resp. pARRk+1 = (1 + krP,k+1) . pARRk + (1 + prB,k) . kARRk+1. (2.3.14b) We continue by considering the segmentation of the portfolio over time (Figure 2.5). The weights change in time but are fixed at each time knots. By adapting the notation from (2.2.10) for the segment level we consider k = 0,....,N − 1

kj,

n

0jkj,1kP,k R W r ∑

=+ = , kj,

n

0jkj,1kB,k B V r ∑

=+ = . (2.3.15)

62

Definition 2.21: The arithmetical relative return kARRj,k+1, k = 0,....,N − 1 in segment j is defined by 1k,jk ARR + = W j,k Rj,k − V j,k Bj,k. (2.3.16) Based on the first decomposition (14a) we consider the iteration for 1

k,jp ARR with respect to k 1

1k,jp ARR + = (2.3.17a) (1 + krB,k+1) . 1

k,jp ARR + (1 + prP,k) . kARR j,k+1 with p = 0, 1,....,T − 2, k = p + 1, p + 2 ....., T − 1 and kARRj,k+1 is given by (15). The initial values are

1

1p,jp ARR + = pARRj,p+1 = Wj,p . Rj,p − Vj,p . Bj,p

(2.3.17b) p+1ARRj,p+2 = Wj,p+1 . Rj, p+1 − Vj, p+1 . Bj, p+1. Reinvestment assumption: The return on p to k is compounded with the relative return of segment j in the second period form k to k + 1 and the segment return in the second period is compounded with the relative return of the first period. Based on the second decomposition (14b) we consider the iteration for 2

k,jp ARR with respect to k 2

1k,jp ARR + = (2.3.18a) (1 + krP,k+1) . 2

k,jp ARR + (1 + prB,k) . kARR j,k+1

63

with p = 0, 1,....,T − 2, k = p + 1, p + 2, ....., T − 1 and kARRj,k+1is given by (15). The initial values are

2

1p,jp ARR + = pARRj,p+1 = Wj,p . Rj,p − Vj,p . Bj,p

(2.3.18b) p+1ARRj,p+2 = Wj,p+1 . Rj, p+1 − Vj, p+1 . Bj, p+1. Reinvestment assumption: The segment return from k to k + 1 is compounded with the relative return in the second period form p to k and the segment relative return in the second period is compounded with the total return of the benchmark in the first period. The iterations (17) and (18) introduce i

1k,jp ARR + , k > p, i = 1, 2. They calculates the arithmetic relative return if a new data points kARRj,k+1 is available. The following Lemma ensures that they are a decomposition of pARRk+1. Lemma 2.1: We claim that

∑=

+

n

1j

i1k,jp ARR = pARRk+1, i = 1,2 (2.3.19)

where is calculated by (17), (18), respectively. Proof: We start by the right side from (19) and combining with (17a), (18a) we have

∑=

+

n

1j

11k,jp ARR =

(1 + krB,k+1) . ∑=

n

1j

1k,jp ARR + (1 + prP,k) . ∑

=+

n

1j1k,jk ARR ,

64

∑=

+

n

1j

21k,jp ARR =

(1 + krB,k+1) . ∑=

n

1j

2k,jp ARR + (1 + prP,k) .∑

=+

n

1j1k,jk ARR , resp.

We adopt a proof by induction and start with k = p + 1. The assertion follows from (13) and (14). For general k the assertion follows from the induction assumption, (13) and (14). ◊ Example 2.16 (continued): We consider again two periods with t0 = 0, t1 =1; t2 = 2 with returns P,10r = 2%, B,10r = 10%,

P,10r = 10% and B,21r = 2% and assume in addition the portfolio and the benchmark consists of 2 segments with the following data: W1,0 = W1,1 = W2,1 = W2,2 = V1,0 = V1,0 = V1,0 = V1,0 = 0.5; R1,0 = R1,0 = R1,0 = R1,0 = 2%; B1,0 = B1,0 = B1,0 = B1,0 = 10%. (13a) yields the decomposition 1

2,j0 ARR = 0, j = 1, 2, i.e. 0 = (1 + 0.1) . 0.5 (2% − 10%) + (1 + 0.1) . 0.5 (10% − 2%) and (13b) yields the decomposition

65

22,j0 ARR = 0, j = 1, 2,

i.e. 0 = − (1 + 0.02) . 8% + (1 + 0.02) . 8%. This example shows two decompositions of the return over time. ◊ We proceed by investing the management effects over multiple periods by adopting the approach of Brinson-Hood-Beehower. We have a set of management effect in each segment and at each time point. Definition 2.22: The asset allocation effect kAj,k+1, k = 0,....,N − 1 in segment j is defined by

Management effect

Time

Segment

T

n

Management effect

Time

Segment

T

n

The full portfolio return problem (management effect) Figure 2.6 kAj,k+1 = (W j,k − V j,k ) . Bj,k. (2.3.20a) The stock selection effect kSj,k+1, k = 0,....,N − 1 in segment j is defined by

66

kSj,k+1 = (Rj,k − Bj,k ) . V j,k (2.3.20b) The interaction effect kIj,k+1, k = 0,....,N − 1 in segment j is defined by kIj,k+1 = (W j,k − V j,k ) . (Rj,k − Bj,k). (2.3.20c) Based on the decomposition in (27) and (31) we consider the iteration for the management effect 1

1k,jp A + = (1 + krB,k+1) . 1k,jp A + (1 + prP,k) . kA j,k+1 (2.3.21a)

1

1k,jpS + = (1 + krB,k+1) . 1k,jpS + (1 + prP,k) . kS j,k+1 (2.3.21b)

1

1k,jpI + = (1 + krB,k+1) . 1k,jpI + (1 + prP,k) . kI j,k+1 (2.3.21c)

2

1k,jp A + = (1 + krB,k+1) . 2k,jp A + (1 + prP,k) . kA j,k+1 (2.3.21d)

2

1k,jpS + = (1 + krB,k+1) . 2k,jpS + (1 + prP,k) . kS j,k+1 (2.3.21e)

2

1k,jpI + = (1 + krB,k+1) . 2k,jpI + (1 + prP,k) . kI j,k+1 (2.3.21f)

with p = 0, 1,....,T − 2, k = p + 1, p + 2 and the initial values are given by (20). Lemma 2.2: Based on (21) we claim that

pARR k+1 = ∑=

+

n

1j

i1k,jp A +∑

=+

n

1j

i1k,jpS +∑

=+

n

1j

i1k,jpI , i = 1,2.

67

Proof: We adopt a proof by induction and start k = p + 1. We consider for i = 1,2

pARRp+2 = ∑=

+

n

1j

i2p,jp A +∑

=+

n

1j

i2p,jpS +∑

=+

n

1j

i2p,jpI =

)ISA( i2p,jp

i2p,jp

n

1j

i2p,jp ++

=+ ++∑ .

The assertion follows from (13), (20) and (21). For general k the assertion follows from induction assumption (13), (20) and (21). ◊ Definition 2.23: For the geometrical relative return pGRRq pGRRk, kGRRq, p < k< q, p = 0,....,N, k = 0,....,N, q = 0,....,N we define

pGRRq = r1

r1

qB,p

qP,p

+

+ − 1,

pGRRk = r1

r1

kP,p

kP,p

+

+−1,

kGRRq = r1

r1

qB,k

qP,k

+

+−1.

As a consequence of the definition we have

1 + pGRRq = r1

r1

qB,p

qP,p

+

+ =

)r)(1r(1)r)(1r(1

qB,kkB,p

qP,kkP,p

++

++ =

68

(1 + pGRRk)(1 + kGRRq). We see that the geometrical relative return has no cross term. Example 2.17: We consider two periods with returns

P,10r = 2%, P,21r = 10%, B,10r = 10% and B,21r = 4%. For the geometrical linked return we have for the portfolio 1.02 * 1.10 − 1 = 1.122 − 1 = 0.22 or 22% and for the benchmark 1.10 * 1.04 − 1 = 1.144 − 1 = 0.144 or 14.4%. The geometrical relative return is

144.1

1.122 − 1 = −0.01923 or –1.923%.

◊

69

2.4. The money weighted rate of return In this section we concentrate on methods for calculating rates of return that take cash flows into account. We consider

∑−

= ++

+=

1N

1kT

Tkt

k0 )r1(

PV)r1(

CPV (2.4.1a)

where PV0, PVT, respectively is as in the previous section initial, terminal value of the portfolio, respectively and Ck are the cash flows at time tk, k = 0, 1, 2,...., N with t0 = 0, tN = T. The solutions of equation (1a) are called internal rates of returns (IRR). We denote the solutions of (1a) with IR. The fundamental properties are: • (1a) is based on the arbitrage condition that the value today is

equal to the discounted cash flow in the future. • The investment assumption is that the cash flows are

reinvested by the internal return. (1a) has in general not an explicit solution, however, there are several numerical methods like for instance the Secant, Newton-Raphson or modified Newton-Raphson methods for computing a solution of the equation for r. Starting with an initial approximation for the value of IR, more accurate approximations for IR are then computed. In [22, Shestopaloff] different methods for computing IR are compared. In numerical analysis it is well known for instance that the Secant method has the advantage that we do not need any derivatives although the convergence speed is slower than by applying the Newton-Raphson method.

70

Remark 2.5 (Invariance of the solution IR): The

multiplication by TPV

1 or 0PV

1 of (1a) leaves the solution of (1a)

invariant. Thus without loss of generalization we can assume

∑+

=+

+−

=

1N

1kTkt

k0 )r1(

1)r1(

CPV , 0 < t1 <....< tN = T (2.4.1b)

or

∑−

= +=

++

1

1 111

N

kT

Tkt

k

)r(

PV

)r(

C , 0 < t1 <....< tN = T. (2.4.1c)

More generally the solutions of (1) is invariant under multiplication by λ ∈ R1 (see Remark 2.3). Remark 2.6 (Normalization of the solution IR to [0,1]):

By considering the transformation Tt

t~ kk = , k = 0, 1,...., N in (1a)

we have

∑−

= +=

++

1

10 11

N

k

Tkt

~k

rPV

)r(

CPV , 0 < 1t

~ <....< Nt~ = 1. (2.4.1d)

With d = IR+1

1 in (1a) and with the solution IR1 of (1d) we

consider d1 = 11

1IR+

. Then we find with

d1 = dT that d1 satisfy (1d)

71

∑+

+−

=

1N

1k kt~

k0

)r1(

CPV = ∑+

−

=

−1N

1k

kt~

1k0 dCPV = ∑+−

=

−1N

1k

Tkt

1k0 dCPV =

∑+−

=

−1N

1kkt

k0 dCPV = dT = d1.

Furthermore we have IR1 = (1 + IR)-T − 1. (2.4.1e) The basic idea of equation (1a) is to discount all cash flows at a point in time. In (1a) we discount to t = 0. It is seen that the IR is independent on the time and does only depend on the cash flows. Definition 2.24: The notion money weighted rate of return (MWR) refers to a method for assessing the return reflecting the timing and size of cash flows. Thus IRR method is a type of a MWR. It is based on a time arbitrage condition. Furthermore if there are no cash flows, the solution of (1a) is equal to 0r1 in (2.3.5), i.e. the time weighted rate of return is equal to the money weighted rate of return. The difference of TWR and MWR is called the timing effect as the timing of the cash flows explains the difference between the two concepts (see Section 2.5b). As solving (1a) is mathematically challenging we concentrate on the analytics of (1a). It is seen that (1a) is a nonlinear equation for the unknown r. By considering the discount factor (1.1.3) we consider the function P defined

P(d) = k1N

1kk0

TT dCPVdPV ⋅∑−−

−

=. (2.4.2)

72

We proceed by illustrating (2) Example 2.18 (Different signs of the cash flow): We consider one cash flow and equidistant knots, i.e. t0 = 0, t1 = 1, t2 = 2. From (1a) we have PV0 = C1 d + PVT d2. Hence 0 = PVT d2 + C1 d − PV0. (2.4.3a) The solutions d1/2 are

( )

T

T02

112/1 PV2

PVPV4CC-d

+±= , (2.4.3b)

i.e.

.1d

1RI2/1

2/1 −= (2.4.3c)

We note that in this special case the equation (2a) can be solved explicitly and has two solutions. We consider a) PV0 = 2$, C1 = 4/3$, PV2 = 1$ in (1b) or PV0 = 1$, C1 = 2/3$, PV2 = 0.5$ in (1c).

73

The case represents a bond, i.e. there is an investment at the beginning and two out flows.

0 1 2

t[years]C1

PV 0

PV2

0 1 2

t[years]C1

PV 0

PV2

Cash flow positive Figure 2.6 Then (3b) yields d1 = 0.8968 and d2 = −2.2301 and by (3c) the solutions for the internal rate return are IR1 = 0.11506, IR2 = −1.4484. Figure 2.7 depicts the function (2) with the choices of the parameter considered here

-6-5-4-3-2-10123

-3 -2 -1 0 1 2 PVT=1

PVO=1

Minimum negative-cash negative (Bond) Figure 2.7

74

b) PV0 = 2$, C1 = −4/3$, PV2 = 1$ in (1b) or PV0 = 1$, C1 = −2/3$, PV2 = 0.5$ in (1c).

0 1 2

t[years]PV

C1PV0

2

0 1 2

t[years]PV

C1PV0

2

Cash flow negative Figure 2.8 Then (3b) yields d1 = 2.230139 and d2 = −0.89681 and by (3c) the solutions for the internal rate return are IR1 = −0.55160, IR2 = −2.11507. In a) (b)) the minimum is positive (negative). In both case the y axis is negative and there is a unique positive solution. We see that the minimum of the Figure 2.8 to 2.9 change from negative to positive. This example can be solved explicitly because the knots are equidistant and we have only a cash flow.

75

-6

-5

-4

-3

-2

-1

0

1

2

3

-2 -1 0 1 2 3 POT=1.0

P0T=0.5

Minimum positive - Cash positive Figure 2.9 ◊ In the following we assume T = 1, 2,.... and t1 = 1,..., t2 = 2,..., tN−1 = N − 1. Then (1) is a polynomial. Follow the fundamental theorem of algebra the equation P(d) = 0 has in general T zeros or solutions that can be complex. We denote by C the set of complex numbers and i is the unit of the imaginary numbers. We consider the special case where there are no cash flows during the considered time span from 0 to T. For Ck = 0, k = 1,...., N − 1, the condition P(d) = 0 is then equal to

0

TT

PVPVd = ,

Hence

T

0T

PVPVd = 1

76

and consequently

T0

T1,....,T PV

PVd = exp(2πi/k) ∈ C, k = 1,....,T,

where T0

TPVPV is the positive root. By (1.1.3) we find for the

internal rate of rate

T

T,....,1 d1RI = − 1 ∈ C.

In T is odd we have a real positive zero d and T − 1 complex zeros d whereas if T is even we have a positive and a negative real zero d. In addition we have T − 2 complex zeros d. If there is no cash flow MWR is equal TWR. TWR is not a special case of MWR when the cash flow is zero. As illustrated in Section 2.5b TWR measure the return in view of the portfolio (see also 2.13). MWR measure the return in view of the client. Example 2.19: We assume T = 2 and no cash flow at T = 1 and PV0 = 99$, PV2 = 120$. in (1). Then

99

1202 =d

with

99

1201 =d = 1.1010,

99120

2 −=d = −1.1010.

77

We have IR1 = 0.1010, IR2 = −2.1010. From a business aspect IR1 = 0.1010 has to be considered as IRR. ◊ Negative real zeros can also be found in cases with 2 periods and one cash flow by solving the quadratic equation explicitly. To sum it up it is seen that the solution of (1) is not unique. As we are assuming that the interest rates are positive it is seen from Figure 2.10 that we are interested in varying r from −1 to infinity and as a consequence d is supposed to be in the interval [ ∞− ,−1] and [1,∞ ]. In Figure 2.10 we illustrate the discount factor z and z3. In the following we assume T in R1. As interest rates are small real numbers we develop the discount factors in a neighborhood of 0. More precisely, in the theory of complex analytic the discounts factors are called meromorphic functions. They have a singularity in r = −1. Furthermore for r < 1 the Taylor expansion

.....rk!a....r

2!ar

1!a1

r)(11 kk221

t +++++=+

, t ∈ R1 (2.4.4)

with

( )∏ −+−==

k

1j

kk )1jt(1a

holds. For t = 1 we find¨

.....r)1(....rr1r1

1 kk2 +⋅−+++−=+

78

-4

-3

-2

-1

0

1

2

3

4

-5 -3 -1 1 3 5

r values

d va

lues

Discount factors Figure 2.10 and for k = 1 we have with (4) ta1 −= . With T = 1 and 0 < tk < 1, k=1,....,N − 1 in (1) we find by

multiplying with r1

1+

∑ ++

=⋅+−

=−

1N

1k11kt

k0 PV

r)(1CPVr)(1 .

We consider the first order approximation and have by (4) with t = tk − 1

79

∑ +−−=⋅+−

=

1N

1k1kk0 PV1))r(t(1CPVr)(1 .

Such an approximation can be good or bad in nature. The accuracy of the approximation depends on the magnitude of the increments r and Ck. The smaller the interest rates, the time period and the cash flows are the better the series converges or in other words the better the first order approximation is. In addition, the approximation is better the more terms in the Taylor Series (4) of (1) are reflected. We solve

.)t(1CPV

PVCPVr 1N

1kkk0

1N

1k0k1

∑

∑−

=

−

=

−+

−−= (2.4.5a)

r is not IR because our derivation contains the approximation by the Taylor Series (4). By replacing Ck by −Ck, k = 1,....,N − 1 we have: Definition 2.25: The Modified Dietz return MD is defined by

.)t(1CPV

PVCPVDM 1N

1kkk0

1N

1k0k1

∑

∑−

=

−

=

−+

−−= (2.4.5b)

In the denominator we have the sum of the value of the initial portfolio and the time weighted cash flows. In the following Original Dietz return OD the cash flow are centered in the middle of the interval

80

Definition 2.26: The Original Dietz return OD is defined by:

.D

∑

∑−

=

−

=

⋅+

−−= 1N

1kk0

1N

1k0k1

C0.5PV

PVCPVO (2.4.6)

Example 2.20 (Compare Example 2.14): a. We assume N = 2 with t0 = 0, t1 = 0.5 and t2 = 1 in (1) and PV0 = 1$, PV1 = 107$, C1 = 98.95$. Then we have with (3b)

MD = OD = 10

011C5.0PV

PVCPV⋅+

−− =95.985.01

195.98071⋅+

−− = 0.1397.

b. We assume N = 2 with t0 = 0, t1 = 0.5 and t2 = 1 in (1) and PV0 = 100$, PV1 = 1.07$, C1 = –104. Then we have with (3b)

MD = OD = 10

011C5.0PV

PVCPV⋅+

−− =1045.0100100104071.

⋅−−+

= 0.1056.

◊ Definition 2.27: The Profit and Loss PL is defined by

PL = ∑ −−−

=

1N

1k0k1 PVCPV

and the average investment capital AIC of a portfolio is defined as follows

81

⎢⎢⎢⎢

⎣

⎡

∑ =−+

≠=

−

=

1N

1kkk0

IR

0IR),t(1CPV

0IR,IRPL

AIC , (2.4.7a)

AICMD = ∑−

=−+

1N

1kkk0 )t(1CPV , (2.4.7b)

AICOD = ∑−

=⋅+

1N

1kk0 C0.5PV . (2.4.7c)

Lemma 2.1 (AIC is continuous): With (7a) we have

∑−

=→−+=

1N

1kkk0

0)t(1CPVIR

IRAIClim .

Proof: We consider an Portfolio with beginning Value PV0, ending value PVT and cash flows Ck ≠ 0, k = 1,...., N − 1, N ≥ 2 over the time unit interval. The IR is the solution of

PV0 (1+ IR) + ∑=

+1N-

1k

kt-1k )I(1C R = PVT.

We subtract the total cash flow

PV0 (1+ IRi) + ∑=

+1N-

1k

kt-1k )I(1C R − ∑

=

1N-

1kkC = PVT − ∑

=

1N-

1kkC .

By

PL = PVT − PV0 − ∑=

1N-

1kkC

82

we have

PV0 . IR + ∑=

+1N-

1k

kt-1k )I(1C R − ∑

=

1N-

1kkC = PL

and the average invested capital we find

IR

IRAIClim

0→

( )IR

RlimPV

IR

∑=

→

−++=

1N-

1k

kt-1k

00

1)I(1C.

The above relationship shows that the average invested capital is equal to the initial value corrected by difference of the cash flows and the discounted cash flow to the invested start point divided by the internal rate of return. In the absence of cash flow the average invested capital is equal to the beginning value of the portfolio. For IR → 0 we have by the rules of hospital

MDAIC )t(1C k

1N-

1kk0 −+= ∑

=PV .

◊ If we neglect second and higher terms in the Taylor series (4) we get for the difference of the discount factor and the first order approximation t

.r)t(1r)(1

1t ⋅−−

+

83

This difference is called the second order effect. In Figure 2.11 it is illustrated that the error term grows with the power t reflecting the time and the interest rate r. Again considering equation (1d) in Figure 2.12 we evaluate (1) between r = 0 and r = 1.6 for a bond with 4 coupon 0.2 and a last coupon including the face value 1.2. Then for r = 0 the sum of the Face Value and the Coupon amounts to 2.0. It is seen if a Price PV0 lies between 0 and 2, a numerical procedure will compute the corresponding internal rate of return. Also for r = 0.2 the price of the bond PV0 is equal to 1. Furthermore it is seen that the discount factor are monotonic decreasing functions and so is the sum of discount factors and as a consequence there is a unique solution r for a given price between 0 and 2. Cases where the cash flows are not positive throughout are subject to further research. Example 2.21 (Sign change of interest, Interval [0,2]): We consider one cash flow and equidistant knots, i.e. t0 = 0, t1 = 1, t2 = 2. We assume PV0 = 60$, C1 = 40$, PV2 = 30$. By (2) the solutions for the quadratic polynomial are d1 = 0.8968 and d2 = −2.23014 and the solutions for the internal rate return are IR1 = 0.1151, IR2 = −1.4484.

84

0.00

0.01

0.02

0.030.04

0.05

0.06

0.07

0.08

0.000.01

0.020.04

0.050.06

0.070.08

0.100.11

0.120.13

0.140.16

interest rate

seco

nd o

rder

err

or

t=1t=2t=3t=4t=6

Second order effect Figure 2.11

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.00 0.50 1.00 1.50

r value

PV0

Convexity of a Bond (C = 2%) Figure 2.12

85

We assume PV0 = 60$, C1 = 30$, PV2 = 30$. Then we have by (2) IR1 = 0.00, IR2 = −1.50. We assume PV0 = 60$, C1 = 20$, PV2 = 30$. By (2) the solutions for the quadratic polynomial are d1 = 1.2573 and d2 = −1.5907 and the solutions for the internal rate return are IR1 = −0.2057, IR2 = −1.6287. In all three cases the solution that can be identified as an interest rate is unique although the quadratic equation has two distinct solutions. ◊ From the example above it can be seen that from PV0 < C1 + PV2 resp. PV0 > C1 + PV2 it follows IR1 > 0 resp. IR1 < 0. Example 2.22 (Sign change of interest, Interval [0,1]): As in the previous example we consider one cash flow PV0 + C1 dα = PV1 d, 0 < α < 1

86

α PV0 C1 PV1 IRR AIC(IRR) MD AIC(MD) OD AIC(OD)0.25 30 20 60 21.51% 46.50 22.22% 45.00 25.00% 40.00

30 30 60 0.00% 52.50 0.00% 52.50 0.00% 45.00 30 40 60 -15.63% 63.99 -16.67% 60.00 -20.00% 50.00

0.5 30 20 60 25.36% 39.44 25.00% 40.00 25.00% 40.00 30 30 60 0.00% 45.00 0.00% 45.00 0.00% 45.00 30 40 60 -19.57% 51.09 -20.00% 50.00 -20.00% 50.00

0.75 30 20 60 27.66% 36.15 28.57% 35.00 25.00% 40.00 30 30 60 0.00% 37.50 0.00% 37.50 0.00% 45.00 30 40 60 -22.46% 44.52 -25.00% 40.00 -20.00% 50.00

Table 2.7 but opposite to Example 2.9 the time intervals are non equidistant, and do not have length 1. We calculate the figures in bold. The values for the IRR are computed by the Excel Routine XIRR. However, for the value 25.36% (α = 0.5) we found by using (1e) and (7a) 0.2536 = (1 + 0.1196) . (1 + 0.1196) − 1,

AICIR = 36.25

032006 −− = 39.44.

By (5)

MD = 1k0

011C)t(1PV

PVCPV−+−− =

4510

= 22.22%,

by (7b) AICMD = 30 + (1 − 0.25) . 20 = 45. By (5)

OD = 2010

50304060

C5.0PVPVCPV

10

011 −=−−

=+

−− = −20%,

87

by (7c)

AICOD = 30 + 21

. 40 = 50.

Example 2.23 (Transition from 2 real solutions to no real solution by continuing changing the cash flow): We assume PV0 = −30$, C1 = 80$, PV2 = −30$. Then the quadratic equation for the internal rate return yields the solution IR1 = −0.54858, IR2 = −1.21525. We assume PC0 = −30$, C1 = 60$, C2 = −30$. Then we have IR1 = 0.00, IR2 = 0.00. We assume C0 = −30$, C1 = 40$, C2 = −30$. Then there is no real solution. We see that the solution is behaving different in all tree cases. ◊

88

2.5. TWR attribution versus MWR attribution a. Absolute Decomposition In this section we consider a portfolio with n constituents at discrete points in time tk, k = 0,...., K with t0 = 0 and tK = T. For the value B 0PV of the portfolio and the value BPVj,0, j = 1,...., n of its segments at the beginning t0 = 0 of the time period considered we have ∑

==

n

1jj,00 PVBPVB (2.5.1a)

and for the value PV TE of the portfolio and the value PVEj,T, j = 1,...., n of its segments at the end tK = T of the time period we have ∑

==

n

1jj,TT PVEPVE . (2.5.1b)

Remark 2.5: Contrary to (2.3.5), (2.3.7), resp., (1) specifies the time the portfolio is considered. In (1) the products in (2.3.5), (2.3.7), resp. are summarized to values of segments, i.e. therefore this notation allows the investigating of not only investments like equity or bonds but also cash and cash flows. Remark 2.6: The units of the equation are currency. For our exposition here we choose the US$. The cash flows PCj,k, j = 1,...., n, k = 1,...., N − 1 with the segment j of the portfolio P is composed of internal cash flow PICj,k and external cash flow PECj,k with PCj,k = PICj,k + PECj,k (2.5.1c)

89

The full portfolio return problem (Cash Management) Figure 2.12 and )PEC(PICPCPC k,j

n

1jkj,

n

1jkj,k +== ∑∑

==.

We start with an initial investment PVB0 and consider a series of in and out flows PCk, k = 1,…, N − 1 in the portfolio followed by a terminal value PVET IRtot = IR(PVB0, PC1,…., PCN-1, PVET) (2.5.2) is the solution of an equation that equals the initial investment and the present value of the future cash flows including the terminal value. If there are no cash flows IR is equal to TWR. In the case of cash flows the difference of TWR and IR is reflected in the Timing effect and the management effect as described in the following Section 2.5b.

PCji,k

Time

S egment

T

n

PCji,k

Time

S egment

T

n

90

Example 2.24 (Difference between TWR and MWR [14, Illmer]): We assume N = 2, t1 = 1, t2 = 2, PVB0 = 200$ and 0rP,1 = 0.25, 1rP,2 = −0.2 in (2.3.2). Then PVE1 = 200$ (1 + 0.25) = 250$. We distinguish 3 cases: a) PIC1 = 0, PEC1 = 0, i.e. by (1c) PC1 = 0. By (2.3.5) we have for the TWR

10

2120 PVPV

PVPVr⋅⋅

= − 1 = 250200200250

⋅⋅ − 1 =

(1 + 0.25)(1 − 0.2) − 1= 0. We have PL = 0 and the solution of (2.4.1) is d1/2 = 1,

91

i.e. for MWR we find IR1/2 = 0. b) PIC1 = 0, PEC1 = 50, i.e. by (1c) PC1 = 50. By (2.3.5) we have for the TWR

10

2120 PVPV

PVPVr⋅⋅

= − 1= 300200240250

⋅⋅ − 1 = 0.

We have PL = −10$. and the solution of (2.4.1) d1 = 1.144675, d2 = −0.72801 i.e. we find for MWR IR = −0.12639 c) PIC1 = 0, PEC1 = −50,

92

i.e. by (1c) PC1 = −50. By (2.3.5) we have for the TWR

10

2120 PVPV

PVPVr⋅⋅

= − 1= 200200160250

⋅⋅ − 1 = 0

and we have PL = 10$ and the solution of (1) is d1 = 0.848386, d2 = −1.47339, i.e. for MWR we find IR = 0.178709. In all tree cases the TWR vanishes as TWR is insensitive to cash flows after the first period and the MWR, however, reflects the PL, i.e. selling in a bear market is a good decision and buying in a bear market is a bad decision. In addition MWR equals TWR if there is no cash flow. The discussion between MWR and TWR is in the next section. ◊ With PVBj,k we denote the Portfolio Value of segment j, j = 1,…., n, at the begining of period k, k = 0,…., N − 1, and with PVEj,k the Portfolio Value of segment j, j = 1,…., n at the end of

93

period k = 1,…, N. Then the weights Wj,k of segment j at the beginning of period k are

∑=

= n

1jk,j

k,jk,j

PVB

PVBW , k = 0,…., N − 1, j = 1,…., n,

and for the return Rj,k over the period [k, k + 1], k = 0,…., N − 1 of segment i we have

k,j

k,j1k,jk,j PVB

PVBPVER

−= + , k = 0,…., N − 1, j = 1,…., n.

As a consequence we have for the market value of the portfolio PVEj,k+1 at time k + 1 in segment j: PVEj,k+1 = (1 + R j,k) PVBj,k. (2.5.3) Furthermore PCj,k denotes the portfolio cash flow in or out of segment j at time k given by PCj,k = PVEj,k − PVBj,k, k = 1,…., N − 1. (2.5.4) The IRj in segment j is the given by IRj = IR(PVBj,0, PCj,k, k = 1,…., N − 1, PVEj,T) = (2.5.5) IR(PVBj,0, Rj,k, Wj,k, k = 0,….,N − 1). The total Profit and Loss PLtot over the investment period [O, T] is equal to the sum of the PLj, j = 1,…., n of the individual segments: ∑

==

n

1jjtot PLPL . (2.5.6)

94

Definition 2.25: Assuming that there exists a solution IRj ≠ 0, IRtot ≠ 0, resp. the average invested capital for segments

IRjICA , IR

totICA , resp. is

j

jIRj IR

PLICA = , (2.5.7a)

tot

totIRtot IR

PLIC =A , resp.. (2.5.7b)

Using the average invested capital for the internal rate of return we have IR

totAIC totIR = j

n

1j

IRj IRAIC∑

=,

thus we find a decomposition of IR IRtot = j

n

1jIRtot

IRj IR

AIC

AIC∑=

. (2.5.8) Definition 2.26: By defining the return contribution RCj by jIR

tot

IRj

j IR*AIC

AICRC = (2.5.9)

we have .RCIR

n

1jjtot ∑

== (2.5.10)

We proceed by discussing three special cases:

The IR of the portfolio is equal to the return contribution defined

by (10).

95

Case 1 (No cash flows): If PCj,k = 0, j = 1,...., n, k = 1,...., N − 1, K ≥ 2 and T = 1

j

0,jT,j

j

jIRj IR

PVBPVEIRPL

ICA−

== = PVBj,0 and

tot

T,tot0,tot

tot

totIRtot IR

PVBPVEIRPLICA

−== = PVB0.

Thus by (1a) =IR

totAIC ∑=

n

i

IRiAIC

1.

Case 2 (IR = 0): We consider an segment j with beginning Value PVj,0, ending value PVj,T and cash flows PCj,k ≠ 0, k = 1,...., N − 1, N ≥ 2 over the time unit interval. The IRj is the solution of PVj,0 (1+ IRj) + ∑

=+

1N-

1k

ktjkj, )RI(1PC = PVj,T

We subtract the total cash flow PVj,0 (1+ IRj) + ∑

=+

1N-

1k

ktjk,j )RI(1PC − ∑

=

1N-

1kkj,PC = PVj,T − ∑

=

1N-

1kkj,PC .

By PLj = PVj,T − PV j,0 − ∑

=

1N-

1kkj,PC

we have PVj,0 . IRj + ∑

=+

1N-

1k

ktjkj, )RI(1PC − ∑

=

1N-

1kkj,PC = PLj

and assuming IRj ≠ 0 for the average invested capital we find

96

IR

jAIC = ( )

j

1-N

1k

ktjkj,

0,jj

j

IR

1)RI(1PCPV

IR

PL ∑=

−++= .

The above relationship shows that the average invested capital is equal to the initial value corrected by difference of the cash flows and the discounted cash flow to the invested start point divided by the internal rate of return. In the absence of cash flow the average invested capital is equal to the beginning value of the portfolio. For IRj → 0 we have by the rules of hospital MD

jAIC ∑=

−=1N-

1kkkj,0,j tPCPV .

We conclude that if IRj = 0, j = 1,...,n there follows IRtot = 0 and by (1) =IR

totAIC ∑=

n

1j

IRjAIC .

Case 3 (No compounding, Modified Dietz) We are concerned with the modified Dietz formula on portfolio level

∑ ∑

∑ ∑−

= =

−

= =

+

−−

= 1N

1k

n

1jkk,j1

1N

1k

n

1j0k,jT

tottCPV

PVCPVDM

and on a segment level

∑

∑−

=

−

=

+

−−= 1N

1kkk,jj,T

1N

1k0,jk,jj,T

jtCPV

PVCPVDM , j = 0,1, 2, ....,n

we find

97

.MD

CtPV

tCPVMD j

n

1j1-N

1k

n

1jkj,k0

1N

1kkkj,0

tot ∑∑ ∑

∑

=

= =

−

= ⋅+

−=

By considering the first term of the product in the above sum is the ratio of average invested capital of the individual assets and the average invested capital of the whole portfolio and the second term is the Modified Dietz of the individual assets. By using the abbreviation

∑ ∑

∑

= =

−

=

+

−= 1N-

1k

n

1jkj,k0

1N

1kkk0,i

MDj

CtPV

tCPVw , j = 1,2,…., n.

It seen that the Modified Dietz of the whole portfolio is decomposed into the weighted average of the Modified Dietz of the individual assets:

∑=

=n

1jj

MDjtot .DMwDM

By using the definition of the average invested capital for the modified Dietz we find

=MDtotAIC ∑

=

n

1j

MDjAIC .

We note that this decomposition has a weighting scheme that does not add up to one as we have in general there is no relationsship between

IRtotAIC and ∑

=

n

1j

totjAIC .

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Summarizing by (6), (7) and (8), the IR is decomposed such that the decomposition is consistent with the profit and loss of the portfolio. In the following we use only the AIC for the internal rate of return and suppress the superscript in AIC throughout. Example 2.25: We consider a portfolio with initial value PV0 = 100$ over two years. In the first year the value of the portfolio goes to 130$ and in the second year the value goes to 218$. The portfolio consists of two segements the first one has a initial value of 1,0PV = 60$ and the second one has an initial value of 0 2,PV = 40$. In the first period the return of the first, second segment resp. is 01,R = 25%, 02,R = 37.25%, (2.5.11a) resp. and in the second period the returns are R1,1 = 100%, R2,1 = 40%, resp.. (2.5.11b) There is no external cash flow, i.e. PEC1 = 0, PEC2 = 0 (see (1c)) and there is an internal cash flow of PC1,1 = PIC1,1 = −15$, PC2,1 = PIC2,1 = 15$ (2.5.11c) in the middle of the 2 year period. We proceed by the following calculations. Based on (3) and (11) we have

99

PVE1,1 = (1 + 0.25)*60$ = 75$ PVE2,1 = (1 + 0.375)*40$ = 55$. Furthermore by (4) PVB1,1 = 75$ − 15$ = 60$, PVB2,1 = 55$ + 15$ = 70$. Again with (3) we have PVE1,2 = (1 + 1.0) * 60$ = 120$, PVE2,2 = (1 + 0.4) * 70$ = 98$, PVE2 = PVE1,2 + PVE2,2 = 218$. With (6), (7) and (8) we have PL1 = 45, PL2 = 73, PLtot = 118, IR1 = 54.5, IR2 = 38.9, IRtot = 47.6. 1AIC =

5.5475 = 1.3761,

2AIC =

9.3843 = 1.1053,

totICA =

6.47118 = 2.4789,

The return contributions (9) are

100

RC1 = totAIC

IC1A IR1 = 118

6.47*75 = 30.2542, RC2 =

totAICIC2A IR2 =

1186.47*43 = 17.3457,

RC1 + RC2 = 47.6. ◊ b. Relative Decomposition In this section we discuss the impact of the client on the overall performance of an account or in other words we ask the question: what is the performance of an asset manager and what part of the overall performance originates from the client’s decisions? We want to highlight some of the differences between the two return calculation methodologies: • The time weighted rate of return (TWR) is the return of an

account that measures the return in such a way that the return figures do not depend on changes in the invested capital o TWR measures the return from the asset manager’s

perspective (assuming that he has no control over external cash flows)

o TWR allows a comparison with a benchmark as well as with peers and the competitors

o The calculation, the decomposition and the reporting of TWR is common practice in the asset management industry

o The presentation of TWR is one of the main principles of the GIPS standards

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• The money weighted rate of return (MWR) measures the account return in such a way that the return figures depend on changes in the invested capital o MWR measures the return form the client’s perspective

(assuming that the client has control over external cash flows)

o MWR allows no comparison with peers and the competitors, however, with an appropriate adjusted benchmark

o The calculation, the decomposition and reporting of MWR is not common practice in the asset management industry

o MWR is not addressed by the GIPS standards in detail. The last mentioned point leads very often to the fact that the performance of an asset management account is very often reported to the existing client only from the asset manager’s perspective. The asset manager argues always that they have to report their performance and not the one from the client – neglecting the impact of the (external) cash flows. In the following we discuss in addition the return attribution from a client’s point of view and illustrate the decomposition of the money weighted rate of return (MWR) and its relationship to the return attribution based on the time weighted rate of return concept. As mentioned above today it is common practice in the asset management industry to calculate and to report the time weighted rate of return (TWR) on a total portfolio level to existing as well as to prospective clients. The TWR is insensitive to changes in the money invested in the account and therefore allows a comparison of the account return across peer groups and against a benchmark or an index. This property might be the main reason why it is also common practice to analyze and to decompose the TWR and not the MWR of the account. However, there is also a need for calculating and decomposing the MWR because it is the MWR which covers the timing effect of cash flows into or out of the

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account. The MWR reflects the timing of cash flows and is consistent with the absolute profit and loss of the account. Due to these characteristics the MWR is the true return from a client’s point of view if it is the client’s decision to invest money into or to withdraw money from the account.

As it is not common practice to run a performance or return attribution from a client’s perspective which covers the timing effect of (external) cash flows, in the following we illustrate an intuitive procedure to decompose the MWR of an account on the total portfolio level. The basic idea of this approach is to decompose the MWR of an account such that the most important investment decisions from a client’s perspective are reflected:

• The “benchmark effect”, which is the return contribution due to

the decision to invest the initial money into a specific benchmark strategy and which is equal to the benchmark return over the investment period,

• The “management effect”, which is the return contribution due to the decision to change the asset allocation and stock selection of the account relative to the benchmark over the investment period, and

• The “timing effect”, which is the return contribution due to the decision to change the money invested in the benchmark strategy and in the active asset allocation of the account over the investment period.

Figure 2.13 illustrates the so called decision-oriented return decomposition approach and shows the relationship between the MWR and the TWR of an account where the TWR of an account is the sum of the benchmark effect and the management effect. In addition, for investment periods with no (external or non-discretionary) cash flows the MWR is equal to the TWR and there is no timing effect. The different return contributions can be further decomposed according to the specific investment decisions. For

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example, the management effect is often split up into the asset allocation, stock selection and currency management effect and the timing effect can be decomposed in the effect from changing the money invested in the benchmark strategy and in the effect from changing the money invested in the active asset allocation.

Account MWR

Benchmark effect + Management effect + Timing effect

Account TWR

Account MWR

Benchmark effect + Management effect + Timing effect

Account TWR Overview of the decomposing approach Figure 2.13 The return contributions are calculated using the benchmark return and the MWR as well as the TWR of the relevant account. The TWR of the account is calculated assuming no cash flows but considering the active asset allocations over the investment period. In measuring the benchmark return it is also implicitly assumed that no money is invested into or is withdrawn from the benchmark portfolio. On the opposite, the MWR of the account reflects not only the active asset allocations but also - and often more importantly - the timing effect of the (external) cash flow decisions. After calculating the overall returns of the benchmark and the account, the benchmark effect equals the benchmark return, the management effect is the difference of the TWR of the account and the benchmark return and the timing effect is the difference between the MWR and the TWR of the account. In order to isolate the timing effect completely it is necessary to calculate not only the “true” TWR but also the “true” MWR. The

104

“true” TWR is not affected by any (external) cash flow. It is best practice to calculate the “true” TWR on a daily basis and then to link the daily returns geometrically over the investment period. On the opposite the “true” MWR covers the total timing effect of all (external) cash flows and is calculated using the internal rate of return methodology over the whole investment period. Using instead an approximation method often results in a residual return component relative to the fictitious “true” return (TWR or MWR) whose missed evidence may lead to misleading feedback into the investment process. This section illustrated why neither the MWR calculation nor the MWR decomposition should be neglected but rather incorporated into the performance reporting and evaluation process or investment controlling. Not considering the MWR concept and with this ignoring the timing effect of the (external or non-discretionary) cash flows bears the risk of misinterpretation and of wrong feedback into the investment process. The MWR concept adds value and is by no means outdated. Moreover, all participants are encouraged to remember the basics in order to get back the lost insights of the MWR concept and should reintroduce the MWR concept to the area of performance measurement as well as to the area of performance attribution. In (2.2.9) and (2.2.10) we decompose the relative return of a portfolio. Essentially the formula is valid for a single period, i.e. the weights are given at the beginning of the investment period and it is assumed that there is no cash flow. In the following section we introduce and discuss the case of the IR Attribution. We show that the concept of the MWR is more general than the TWR approach. In a Portfolio the weights are given at the begining of a period, however, at the end of a period they are different due to market changes. Similar to the previous section we consider a benchmark with n constituents at discrete points in time tk, k = 1,...., K with

105

t0 = 0 and tK = T. For the value of the benchmark value BBV0 and its segments j,0BBV , j = 1,...., n at t0 = 0 we have ∑

==

n

1jj,00 BBVBBV

and for the value of the portfolio value EBVT and its segments EBVi,T, j = 1,...., n at tK = T we have ∑

==

n

1jj,TT EBVEBV .

The cash flow is composed of internal cash flow IBC and external cash flow EBC )BEC(BICBCBC k,j

n

1jkj,

n

1jki,k +== ∑∑

==.

The weights of the benchmarks are given by

∑=

= n

1ik,i

k,ik,i

BVB

BVBV

and for the return in a period we have

k,j

k,j1k,jk,j BVB

BVBBVEB

−= +

. As a consequence we have for the market value of the benchmark at time k + 1 in segment j: BVEj,k+1 = (1+Bj,k) BVBj,k (2.5.12) Furthermore k

jBC denotes the derived portfolio cash flow, Benchmark cash flow, respectively in or out of segment i at time k. There is

106

PCj,k = PVEj,k − PVBj,k and BCj,k = BVEj,k − BVBj,k (2.5.13) Definition 2.27: A Notional (or reference) Portfolio is a sythetic portfolio against a portfolio is measured with a passive Strategy. Definition 2.28: Starting with the portfolio or the benchmark EVj,k ),( •• , EVtot,k ),( •• , k = 1,2,….,K, resp. (2.5.14a) denotes the ending value using for the first argument W or B and for the second argument R or M. We note ∑

=••

n

1jT,j ),(EV = EVtor,T ),( •• , k = 1,2,….,K. (2.5.14b)

We introduce the following four versions of IR and the corresponding Profit and Loss: 1. (Portfolio) IRj(R, W) = IR(Rj,k, Wj,k, k = 0,….,K − 1) = (2.5.15a) IR(PVBj,0, PCj,k, k = 1,…., N − 1, EVj,T(R,W)) is the IRi using the returns and the weight of the portfolio. (5) and (15a) are the same. In addition we have PLj(R, W) = PVBj,0 − ∑

−

=

1K

1kk,jPC − EVj,T(R,W) (2.5.15b)

where PLj in (6) is the same as the left side of (14b).

107

2. (Notional Portfolio) IRj(R, M) = IR(Rj,k, Mj,k, k = 0,…., N − 1) = (2.5.15c) IR(PVBj,0, BCj,k, k = 1,…., N − 1, EVj,T(R,M)) is the IR using the returns of the portfolio and the weight of the benchmark and PLj(R, M) = PVBj,0 − ∑

−

=

1N

1kk,jBC − EVj,T(R,M), (2.5.15d)

3. (Notional Portfolio) IRj(B, W) = IR(Bj,k, Wj,k, k = 0,…., N − 1) = (2.5.15e) IR(BVBj,0, PCj,k, k = 0,…., N − 1, EVj,T(B,W)) is the IR using the returns of the benchmark and the weight of the portfolio and PLj(B, W) = PVBi,0 − ∑

−

=

1N

1kk,jPC − EVj,T(R,W) (2.5.15f)

4. (Benchmark) IRj(B, M) = IR(Bj,k, Mj,k, k = 0,…., N − 1) = (2.5.15g) IR(BVBj,0, BCj,k, k = 0,…., N − 1, EVj,T(B, M)) is the IR using to the returns of the benchmark and the weight of the portfolio and PLj(B, M) = PVBj,0 − ∑

−

=

1N

1kk,jPC − EVj,T(B,M). (2.5.15h)

108

By (2), (4) and (15a) we have IRtot = IRtot(R, W) and by the left side in (6) we have PLtot = PLtot(R, W). Similary to (15c) - (15h) we introduce the notation IRtot(R, M), PLtot(R, M), IRtot(B, W), PLtot(B, W), IRtot(B, M) and PLtot(B, M). Definition 2.29: Referring to Definition 2.9, we define the average invested capital ),(ICA j •• , resp. for the first argument R or B and for the second argument W or M by

),(IR),(PL

),(ICAj

jj ••

••=•• , (2.5.16a)

),(IR),(PL),(ICA

tot

tottot ••

••=•• , resp.. (2.5.16b)

We proceed by the identity X − Y= (2.5.17) Z − Y + V − Y + X – Z – V + Y, ∀X∈ R1, ∀Y∈ R1, ∀Z∈ R1, ∀V ∈ R1. We apply (17) to (compare (2.2.10)) AICj(R, W) − AICj(B, V) = AICj(B, W) − AICj(B, V) + AICj(R, V) − AICj(B, V) + AICj(R, W) − AICj(B, W) − AICj(R, V) + AICj(B, V), AICtot(R, W) − AICtot (B, V) =

109

AICtot (B, W) − AICtot (B, V) + AICtot (R, V) − AICtot (B, V) + AICtot (R, W) − AICtot (B, W) − AICtot (R, V) + AICtot (B, V). and by using the notation for IR in (15) )W,R(IR

)W,R(AIC W)(R,AIC

jtot

j − V)(B,IR)M,B(AIC M) (B,AIC

jtot

j = W)(B,IR

)W,B(AIC W)(B,AIC

jtot

j − V)(B,IR)M,B(AIC M)(B,AIC

jtot

j + V)(R,IR

)M,R(AIC M)(R,AIC

jtot

j − V)(B,IR)M,B(AIC M)(B,AIC

jtot

j +(2.5.18) W)(R,IR

)W,R(AIC W)(R,AIC

jtot

j − )W,B(IR)W,B(AIC W)(B,AIC

jtot

j − V)(R,RI

)M,R(AIC M)(R,AIC

jtot

j + V)(B,IR)M,B(AIC M)(B,AIC

jtot

j . We proceed by PLtot ),( •• = ∑

=••

n

1jj ),(PL . (2.5.19)

Hence PLtot (R,W) − PLtot(B,W) = ( )∑

=−

n

1jjj )W,B(PL)W,R(PL =

(∑

=

n

1jj W)(B,PL − V)(B,PL j + V)(R,PL j − V)(B,PL j + (2.5.20)

W)(R,PLi − )W,B(PLi − V)(R,PLi + )V)(B,PLi .

110

With (19)

),(AIC1

tot •• Ptot ),( •• = ∑

=••

••

n

1ji

tot),(PL

),(AIC1 .

Hence with (14) we find IRtot(R, W) − IRtot(B, V) = )

)V,B(AIC)V,B(PL

)W,R(AIC)W,R(PL

(tot

jn

1j tot

j −∑=

= ∑

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

n

1jj

tot

jj

tot

j )V,B(IR)V,B(AIC

)V,B(AIC)W,R(IR

)W,R(AIC)W,R(AIC

= ∑

=⎜⎜⎝

⎛n

1jj

tot

j W)(B,IR)W,B(AIC W)(B,AIC

− V)(B,IR)V,B(AIC V)(B,AIC

jtot

j + V)(R,IR

)M,R(AIC M)(R,AIC

jtot

j − V)(B,IR)M,B(AIC M)(B,AIC

jtot

j + (2.5.21) W)(R,IR

)W,R(AIC W)(R,AIC

jtot

j − )W,B(IR)W,B(AIC W)(B,AIC

jtot

j − V)(R,RI

)M,R(AIC M)(R,AIC

jtot

j + ⎟⎟⎠

⎞V)(B,IR

)M,B(AIC M)(B,AIC

jtot

j.

Definition 2.30: The return contribution ),(RCi •• ).,(IR

),(AIC

),(AIC),(RC jIR

tot

IRj

j ••∗••

••=•• (2.5.22)

We decompose the relativ return by IRtot(R,W) − IRtot(B,V) = Atot + Stot + Itot. (2.5.23a)

111

The Asset Allocation Atot effect is given by Atot =∑

=

n

1jjA [ ]∑

=−=

n

1jjj )V,B(RC)W,B(RC . (2.5.23b)

The Stock Picking effect Stot is given by Stot ∑

==

n

1jjS [ ]∑

=−=

n

1jij )V,B(RC)V,R(RC . (2.5.23c)

The Interaction Itot is given by ∑=

=

n

1iitot II =

(2.5.23d) [ ] [ ]∑

=−−−

n

1jijij )M,B(RC)W,B(RC)M,R(RC)W,R(RC .

1. External cash flow This cash flow can be injected by the client. If we have external cash flows the value of the IR is not equal to the TWR and the return of the client perspective is not equal to the return achieved by the portfolio manager. 2. Internal cash flow This cash flow might be tipically caused by a change of a different asset allocation pursued by the portfolio manager or due to the rebalancing of the benchmark. We pursue different strategies: 1) If we have no external cash flows in the portfolio, benchmark, resp. 0IPC

n

0i

ki =∑

=, 0IBC

n

0i

ki =∑

=, resp.,k = 1,…,N − 1

112

2) Buy and hold in portfolio, benchmark, resp. k = 1,…,N − 1 0IPCk

i = , 0IBCki = , resp..

We note that 1) follows 2). Example 2.25 (continued): We consider a Benchmark with initial value BBV0 = 100 $ over two years. In the first year the value of the portfolio goes to 170$ and in the second period the value goes to 284$. The benchmark consists of two segment. The first one has a value of BVB1,0 = 20$ and the second one of BVB2,0 = 80$. In the first period the return of the first, second segment is B1,0 = 150%, B2,0 = 50%, (2.2.24a) resp. and in the second period the return is B1,1 = 20%, B2,1 = 120%, resp.. (2.2.24b) There is no external cash flow, i.e. BEC1 = 0, BEC2 = 0 and there is an internal cash flow of BIC1 = 40$, BIC2 = −40$ (2.2.24c) in the middle of the 2 year period. We discuss the relative return portfolio versus benchmark and proceed first with the calculation for the benchmark. Based on (12) and (24) we have EV1,1(B,M) = BVE1,1 = (1 + 1.50)*20$ = 50$,

113

EV2,1(B,M) = BVE2,1 = (1 + 0.50)*80$ = 120$, EV1(B,M) = BVE1 = BVE1,1 + BVE2,1 = 170$. Furthermore by (13) BVB1,1 = 50$ + 40$ = 90$, BVB2,1 = 120$ − 40$ = 80$. Again with (12) and (22) we have EV1,2(B, M) = BVE1,2 = (1 + 0.2) * 90$ = 108$, EV2,2(B, M) = BVE2,2 = (1 + 1.2) * 80$ = 176$, EV2(B, M) = BVE2 = BVE1,2 + BVE2,2 = 284. Adapting the notation (14) and (20) IRIC1A (B, B) =

9.5248 = 0.7940,

IR

2ICA (B, B) = 975

136.

= 1.7918, IR

totICA (B, B) = 5.68

184 = 2.6861. RC1(B, B) = IR

tot

IR1

AICICA IR1(B, B) =

1845.68*48

= 17.8695, RC2(B, B) = IR

tot

IR2

AICICA IR2(B, B) =

184568136 .* = 50.6304,

114

RC1(B, B) + RC2(B, B) = 68.5. The argument ),( •• yields the following numerical results: (R, W) in (13a) and (13b)

i Cash flow PL(R, W) IR(R, W) 1 − 60 −15 120 75 54.5 2 − 40 15 98 43 38.9 tot −100 0 218 118 47.6

Table 2.4 (R, M) in (13c) and (13d)

i Cash flow PL(R, M) IR(R, M) 1 −60 -15 162 117 77.3 2 −40 15 165 110 85.2 tot −100 0 327 227 80.8

Table 2.5 (R, W) in (13e) and (13f)

i Cash flow PL(R, W) IR(R, W) 1 −20 40 130 70 73.8 2 −80 -40 98 58 38.4 tot −100 0 228 128 50.9

Table 2.6 (B, M) in (13g) and (13h)

i Cash flow PL(B,M) IR(B,M) 1 −20 40 108 48 52.9 2 −80 -40 176 136 75.4 tot −100 0 284 184 68.5

Table 2.7

115

We note that the IRR in the Tables 4 to 7 are non weighted. The relative consideration portfolios versus benchmarks yields the following results. For the PL we find according (46) 75 − 48 = 27, −43 − 136 = −93, 118 − 184 = 66. Asset

Allocation Stock Picking

Interaction Effect

Total

1 22 69 -64 27 2 -78 -26 11 -93 tot -56 43 -53 -66

Table 2.8 By the Tables 4 to 7 we find for the difference of the portfolio and benchmarks −0.2087 = 0.476 − 0.685. For the effects we find according to (21) – (23) Asset

Allocation Stock Picking

Interaction Effect

Total

1 0.1001 0.2379 -0.2139 0.1240 2 -0.2753 -0.2247 0.0573 -0.3328 tot -0.1752 0.1230 -0.1565 -0.208

Table 2.9 ◊

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3. RISK MEASUREMENT It is the task of every performance measurement department to calculate return figures. This is without doubt important information for the investor because the return reports of the achievement with a specific portfolio. However, a return figure is just one realization of the portfolio in the past. A further question is how this return is achieved. Here risk considerations are requested. It can well be that two portfolios have the same returns but the risks are significantly different. Return calculations are deterministic and risk relates to randomness. Risk calculation uses methods from statistics and from the theory of probability. Generally returns are the result of precise calculations and risk figures are rather estimations that are based on a model. We see that different risks can assume different characteristics and occurrences. In this chapter we focus on risk and in the following chapters we discuss return and risk together, i.e. the performance is discussed. The term risk was originally already used in the 16th century by Italian merchants for danger or hazard. The expression refers to potential losses and damages in context of a company or an enterprise. Risks are pitfalls that have to be avoided. The notion risk is used in many areas and contexts. A general definition can be as follows: Risk is the potential for an occurrence of an undesired negative consequence of an event. In the American Hermitage Dictionary risk is defined as the possibility of suffering losses or losses due to an event that will quite probably occur. In financial markets, different risks are distinguished. We proceed with the description of some types of risks. With market risk we refer to risks that are common to an entire class of assets or liabilities. The value of investments may decline over a given time period simply because of economic

117

changes or other events that impact large portions of the market. Asset allocation and diversification can protect against market risk because different portions of the market tend to underperform at different times. Market risk is also called systematic risk. Credit risk is usually an important issue in bond investments. It reflects the possibility that a bond issuer can default by failing to repay principal and - or interest in a timely manner. Bonds issued by a government or an authority, for the most part, are immune from default since Governments can print more money. Bonds issued by corporations are more likely to be defaulted on, since companies can go bankrupt. Credit risk is also called default risk. Liquidity risk refers to the fact that an investor can have the difficulty of selling an asset. Unfortunately, an insufficient secondary market may prevent the liquidation or limit the funds that can be generated from an asset. Some assets are highly liquid and have low liquidity risk (such as a stock of a publicly traded company) while other assets are highly illiquid and are highly risky (such as a house). The call risk is reflected by the cash flow resulting from the possibility that a callable bond will be redeemed before maturity. Callable bonds can be called by the company that issued them, meaning that bonds have to be redeemed to the bondholder usually so that the issuer can issue new bonds at a lower interest rate. This forces the investor to reinvest the principal sooner than expected, usually a lower interest rate. Political risk is the risk of changes in a country’s political structure or policies caused by tax laws, tariffs, expropriation of an asset or restriction in repatriation of profits for example, a company may suffer from loss in the case of expropriation or tightened foreign exchange repatriation rules or from increased credit risk if the government changes policies to make it difficult to pay creditors.

118

Legal risk is a description of the potential for loss arising from the uncertainty of legal proceeding, such as bankruptcy and potential legal proceedings. The currency risk refers to the fact that business operations and the value of investment are affected by changes of currency rates. The risk usually influences business but it can also affect individual investors who make international investments. It is also called exchange rate risk. Inflation risk assesses the fact that the purchasing power of the currency shrinks the value of assets. Inflation causes money to decrease in value at some rate and does so whether the money is invested or not. The financial risk is the aggregation of the risks described above. In financial markets a security whose return is likely to change much is said to be risky and a security whose return does not change much is said to carry little risk. Generally we can say that equity has higher risk than a money market instrument. In the following we discuss the risk of a portfolio in view of some of the financial risks discussed above.

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3.1. Absolute Risk We introduce some notions from descriptive statistics. The overall idea is to merge a series of data into one value. We start by a return series of the portfolio P rP,k, k = 1, 2,…., N (3.1.1) of periodic returns, i.e. rP,k is the return between tk-1 = k − 1, tk = k, k = 1,..., N = T. Remark 3.1 (about collecting representative statistical samples): The statisticians’ "camps" argue about the problem for a long time. Ideally, we need a uniform and representative general assembly of samples. Assuming that the statistical model we chose is right and adequate to considered process, then, the method of collecting samples is defined by the chosen statistical model. Note that we always make some assumptions, explicit or implicit, about the nature of our process when we choose methods of collecting samples. Some people may not realize this, when they collect samples, but it is impossible to pick up the method for collecting samples without such assumptions. A typical assumption is that the sample is independent and identically distributed. The correlation or comovement time is assumed inferior to the periodic time of the sample. It dictates the periodicity of the sample (Figure 3.1). Further equidistant knot underpins the independency of the sample. So, the method of collecting samples is defined by several factors: the nature of the process or phenomenon we study, our model, our measurement tools and our purposes. Average is a generic term for statistics describing the middle of the data set. The geometric mean return defined in Definition 3.1 and the arithmetic mean return defined in Definition 3.2 are examples of averages.

120

Equidistant knots Figure 3.1 Definition 3.1: A population is defined as all members of a specified group. Definition 3.2: The variance var(P) of a portfolio P is the sum of the squared deviations from the arithmetic mean Pr defined by (2.3.4c) divided by the number of returns in (1):

var(P) = ∑=

−N

1k

2PkP, )r(r

N1 (population variance). (3.1.2)

Definition 3.3: The standard deviation std(P) of a portfolio P is defined by std(P) = )Pvar( . (3.1.3) Often the term absolute risk is used similarly with standard deviation or particularly in finance with volatility. Remark 3.2 (intuition): The standard deviation is a measure of how widely the actual returns are dispersed from the arithmetic mean return. r does not mean a lot if the variance is big. On the other hand if var(P) = 0 the mean is equally to the return series

Correlation function

Samples

Time

121

(1). In [23, Taleb] these two cases (small or big var(P)) are called Mediocristan or Extremistan. Remark 3.3 (unit): The unit of the standard deviation and the return is percentage or decimals. The standard deviation has the same unit as the return series. Example 3.1: We consider N = 1, 2, 3, …. rP,k = 5%, k = 1, 3,…., 2N − 1, rP,k = 15%, k = 2, 4,…., 2N. Then we have Pr = 10% and std(P) = 5%. ◊ Remark 3.4 (annualizing): The variance is additive, i.e. if we have f, f = 0,1,2,…. variances per year that are independent and identically distributed we can multiply by f but the standard deviation, risk (volatility) is not additive. The variance is annualized by f whereas the risk and volatility is multiplied by f . As in science, the first attempt for analysing data in finance like the return series (1) is the normal distribution. It is the benchmark in the theory of distributions. It assumes that the N in (1) and (2) is big and tends or converges to the mean μ and the standard deviation σ. The normal distribution has the probability density function

122

f(x) = ,exp2

1 22

)x(

2σ

μ−−

πσ ∀x ∈ R1. (3.1.4)

It can be characterised by returns that are equally distant from the mean return and have the same relative frequency of observations. Most of the returns are close to the average mean return and there are relatively few extremely high and low returns. Sometimes, the normal distribution is called by the nickname bell shape distribution. Statistically, the following statements hold:

• About 68.3% of the observed returns will be within the range of one standard deviation above and below the mean return.

• About 90% of the returns will be within ± 1.65 standard

deviations.

• About 95.4% of the returns will be within ± 2.0 standard deviations.

• About 99.7%, i.e. almost all of the returns will be within

± 3.0 standard deviations. If returns are normally distributed and linked to asset prices by the natural logarithm, then the asset price follows a log normal distribution with probability density function

f(x) = ,exp2x

1 22

))x(ln(

2σ

μ−−

πσ ∀x > 0.

This means that continuously compounded, normally distributed returns and log normally distributed assets prices are consistent.

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Definition 3.4: A sample is a subset of a population. We estimate statistically figures like variance, etc. by a sample of a population, i.e. a specific aspect of a population is estimated. The estimation of the historical mean is unbiased, i.e. the expected value of the estimation is the average. If the average is unknown, i.e. is estimated by (2), it can be shown that the estimator for the variance (3) is biased, meaning there is a systematic error in the estimation. The unbiased estimator with N samples for the variance is

Evar(P) = ∑=

−N

1k

2PkP, )r(r

1-N1 (empirical or sample variance).(3.1.5a)

Asymptotically, i.e. N ∞→ , the two estimators are the same. The corresponding standard variation is then Estd(P)= )Pvar(E (3.1.5b) For N ∞→ we have the transition from the statistics to the theory of probabilities. Referring to (4), the normal distribution is important in the theory of probabilities.

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3.2. The Value-at-Risk (VaR) In 1993 the G30 (an influential international body consisting of senior representatives of the private and public sectors and academia) published a seminal report for the first so-called off balanced sheet products, like derivatives, in a systematic way. Value-at-Risk (VaR) as a market risk measure was born and RiskMetrics set an industry-wide standard. In highly dynamic world with round-the-clock market activity, the need for instant market valuation of trading position (known as marking-to-market) became a necessity. Moreover, in markets where so many positions were written on the same underlying managing risk based on simple aggregation of normal positions became satisfactory. Banks pushed to be allowed to consider netting effects, i.e. the compensation of long versus short positions on the same underlying. Today value at risk is probably the most widely used risk measure in institutions and has made its way into Basel II capital-adequacy. It is deterministic to asses the maximum loss. Value at Risk is a straightforward extension form maximum loss. The idea is simply to replace maximum loss by maximum loss which is not exceeded with a high probability the so called confidence level α. Definition 3.3: Quantile is a generic term for grouping when sampling in descriptive statistics. Examples are quartiles = 4, quintiles = 5, deciles = 10 or percentiles = 100. In probabilistic terms, VaR refers to thus simply a quantile of the loss distribution. When we are communicating the risk associated with a portfolio or an investment it is useful to convert the standard deviation in percentage into dollar terms. Definition 3.4: Value at risk VaRα(P) of a portfolio P at t = 0 with value PV is the estimate of maximizing dollar loss we

125

could expect to experience, over the time horizon, with a stated level of confidence α. Thus input parameter for calculating value of risk figures are

• the level of confidence α • an set of returns (1) with a frequency typically days or

months

Typical value for are α = 0.95 or α = 0.99 in market risk management the time horizon is usually 1 or 10 days, in credit management and operational risk management in usually one year. Remark 3.5: The level of confidence is also used in conjunction with the test of a hypothesis. When a hypothesis is statistically tested, two types of errors can be made. The first one is that we reject the null hypothesis while it is actually true, and is referred to as a type I error. The second one, a type II error, is that the null hypothesis is not rejected while the alternative is true. The probability of a type I error is directly controlled by researchers through their choice of the significance level α. When a test is performed at the 5% level, the probability of rejecting the null hypothesis while it is true is 5%. A confidence interval α means

• a probability 1 − α for an error of the first kind. • a probability α for an error of the second kind.

So we see that there is a trade off between the two errors. Remark 3.6: VaR has been fundamentally criticized as risk measure on the grounds that is has poor aggregation properties.

126

We proceed by describing three types of methodology for calculating value at risk figures: a. Historical VaR We start with a frequency distribution or a histogram and select the α% worst returns. (100 − α)% of the returns does not fall under the value of the maximum of these worst returns. Example 3.2: We assume that we have in (1) N = 100, i.e. we consider 100 returns of a portfolio P with portfolio value PV an r1,P < r2,P < r3,P < ..... < r100,P. Then we have )P(VaRH

%5 = r5,P PV and )P(VaRH

%1 = r1,P PV. ◊ We see that the general case shows some minor complications like equal sign or the fact that the number of return points is not divisible by the confidence level. Furthermore the number of data points has to exceed a minimal number of points for calculating )P(VaRH

α . The methodology is solely empirically and does not use a model. It is thus non-parametric.

127

b. Parametric VaR We start by assuming that the loss function in normal distributed (see Figure 3.2). The short fall probability under the normality assumption is

.dxexp2

1)z(z 22

)x(

2 ∫πσ

=Φ∞−

σ

μ−−

Then )P(VaRP

%z is )P(VaRP

%z = − (μ + z σ) . PV For z1 = −1.64448 we have Φ(z1) = 0.05 and )P(VaRP

%5 = − (μ + z1σ) . PV and for z2 = −2.3263 we have Φ(z1) = 0.01 and )P(VaRP

%1 = − (μ + z1σ) . PV. μ, σ, resp. is estimated by (1), (5), resp.. c. Monte Carlo simulation Starting point is the transformation of market conditions into the change to portfolios. This method is calculation intensive and depends on the distribution of the randomly applied markets conditions. It can be preferably used for portfolios that have strongly non-linear payoffs.

128

ProbabilityProbability

The normal distribution Figure 3.2

129

3.3. Relative Risk Referring to (3.1.1), we proceed by considering a second periodic return series ,r k,P k = 1, 2, ….,N (3.3.1)

of a portfolio P . Definition 3.5: The covariance cov(P, P ) of the portfolios P and P is the sum of the products of the return (3.1.2) of P and the return (1) of P divided by the number N of returns:

cov(P, P ) = )r(r)r(rN1

Pk,P

N

1kPkP, −∑ −

=. (3.3.2a)

Covariance is a statistical measure of the tendency of two data series to move together. It measures the direction and the degree of the association of the returns of the portfolios P and P . It is difficult to interpret as anything other than the average product of the difference between the deviations of the portfolio returns from the arithmetic mean in (2). Furthermore, it is also difficult to use for portfolio comparison because it is impacted by the absolute size of the returns. Referring to the empirical Variance in (3.1.5) the same is valid for the covariance Ecov(P,P ) = (3.3.2b)

)rr()r(r1-N

1PkP,

N

1kPkP, −−∑

= (empirical or sample covariance).

130

Definition 3.6: The correlation corr(P, P ) is defined by

corr(P, P ) =)Pvar(E)Pvar(E

)P,Pcov(E . (3.3.3)

Remark 3.7: We note that the covariance and variances used in (3) is not influenced from the choice between sample or population in (3.1.2), (3.1.5) resp. and (2). Remark 3.8: By the Cauchy Schwarz in equality [8, Kreysig] it can be shown that −1 ≤ corr(P, P ) ≤ 1. This mathematical property is valid for any series (3.1.1) and (1). Remark 3.9: The numerations of the entries in the formulae do not influence the numerical values, i.e. any permutation of the measurement leaves the numerical value unchanged. There are no dependencies between the measurements. Correlation normalizes the covariance to the Interval [-1,1] and be used for direct comparison of different portfolios. It has no units and is a scalar. A correlation close to 1 (-1) indicates that the two time series move together (in the opposite direction). A correlation close to 0, however, indicates that they are out of synchrony. A correlation equal to 1 does not mean that the returns series in (1) and (6) are the same. Definition 3.7: The coefficient of determination R(P, P ) - squared or R2(P, P ) is defined by

131

R2(P, P ) = ( )2 )P Corr(P, . R2(P, P ) is the proportion of variability in the returns of the portfolio P return that relates to the variability of the returns of the Portfolio P . It measures the degree of association of the portfolios P and P . A high R2(P, P ) indicates that the portfolios P and P are probably exposed to similar risk exposure that are driving return. We proceed by the tracking error. The notion was first introduced in Control Theory. The tracking error is a measurement of the relative risk of a portfolio versus the benchmark. The benchmark portfolio is denoted by B with the periodic return series ,r k,B k = 1, 2,…., N. Definition 3.8 (model independent): The tracking error TE(1) is defined as follows

TE(1) = ∑=

N

1k

2kd

N1 (3.3.4a)

where dk = BkB,PkP, rrrr +−− , k = 1, 2,…., N. (3.3.4b) Lemma 3.1: If the tracking error TE(1) is zero, there follows rP, k = Pr + ck, k = 1,2,….,N, rB, k = Br + ck ,k = 1,2,….,N with

132

∑=

N

1kkc = 0.

Proof: By assumption of the Lemma

∑=

N

1k

2kd

N1 = 0,

hence rP,k – rB,k = Pr − Br , k = 1,2,….,N. We assume that there exists a ck ∈ R1 such that rP,k – rB,k = Pr + ck − ck − Br , k = 1, 2,….,N. The right side can be broken in different ways. The only way which leads not evidently to the assertion of the Lemma is rP,k = crP + k, k = 1,2,….,N. By summing up we have

∑=

N

1kk,Pr = N )cr(

N

1kkP ∑

=+

and by (2.3.4), the Lemma is shown. ◊ We proceed by illustrating the computation of the tracking error

133

Table 3.1 The returns in (4) can be calculated differently. We have 4 versions. We consider 1a)

rP,k = 1k

1kkP

PP

−

−− , rB,k = 1k

1kkB

BB

−

−−, k = 1,….,N, (3.3.5a)

1b)

rP,k = 1k

kPPln

−, rB,k = ,

BBln

1k

k

− k = 1,….,N (3.3.5b)

for using (4). We proceed by

k

kk B

PQ = , k = 1,….,N, (3.3.5c)

and consider 2a)

rP,k − rB,k = 1k

1kkQ

QQ

−

−−, k = 1,….,N, (3.3.5d)

Time Portfolio Value Benchmark Value t0 P0 B0 t1 P1 B1 t2 P2 B2 t3 P3 B3 . . . tN PN BN

134

2b)

rP,k − rB,k = 1k

kQQln

−, k = 1,….,N, (3.3.5e)

for applying (4). By calculating the difference rP,k − rB,k with (5b) and compare to the difference in (5d) we see than that they are algebraically different. We note, however, that cases (5b) and (5e) are the same since

1k

kQQln

− = ln Qk − ln Qk-1 = ln

k

kBP − ln

1k

1kBP

−

− =

ln Pk − ln Bk − ln Pk-1 + ln Bk-1 = ln 1k

kPP

− − ln

1k

kBB

−.

We proceed with a second definition of the tracking error: Definition 3.9 (Regression portfolio versus benchmark): The tracking error TE(2) is defined by TE(2) = std(P) )B,P(1 2ρ− . (3.3.6) Theorem 3.1: The tracking error TE(2) minimizes the distance from the line that regresses the portfolio returns against the benchmark returns. Proof: The linear regression assumes that the values of one the variable are at least partial dependent of the other. If the there is a linear relationship between return of the portfolio and the benchmark then the Tracking error is zero. We proceed with the Ansatz

135

rP,k = α + β rB,k + εκ, k = 1,....,N, N ≥ 2. In this regression rP,k is the dependent variable and rB,k is the independent variable. The method of the ordinary least squares (OLS) yields α = Pr − β Br with

β = )Bvar()B,Pcov( . (3.3.7)

We consider the error εκ = rP, k − α − β rB, k = rP, k − Pr − β (rB, k − Br ) and find

∑=

εN

1k

2k = ( )∑ −β−−

=

n

1k

2Bk,BPk,P )rr(rr =

( ) ( )( ) ( )( )∑=

−β+−−β−−N

1k

2Bk,B

2Pk,PBk,B

2Pk,P rrrrrr2rr .

With

( ) ( )∑=

−−βN

1kPk,PBk,B rrrr2 = ( )∑

=−β

N

1k

2Bk,B

2 rr2 .

We find (6) by using (7)

136

TE(2) = std(P) )Pvar()Bvar(1 2β− = std(P) )B,P(1 2ρ− .

◊ Remark 3.10: Theorem 3.1 is a time series analysis and in 3.4 we consider a cross sectional analysis, we regress over a universe at a specific point of time. Lemma 3.2: If the portfolio returns are a linear combination of the benchmark return, then TE(2) = 0. Proof: That follows from the fact that εk = 0. ◊ Lemma 3.3: TE(1) and TE(2) satisfy the inequality: TE(2) ≤ TE(1). Proof: Based on Theorem 3.1 we see TE(2) minimizes all linear combination and following Lemma 3.1 TE(1) is such a specific linear combination. ◊ The notion tracking error stems from control theory and is defined as measure of the difference between the desired output from the measured output. Originally the aim was to achieve a tracking error equal to zero. In passive asset allocation the investor wants a vanishing tracking error. This endeavour is in line with the original definition of the tracking error. As the expression says in finance the error between the return of the portfolio and the benchmark is measured. Today, however, the tracking error is also

137

used in active §asset allocation. A tracking error significantly differently from zero means that the portfolio manager has a considerably deviation of his portfolio from the benchmark.

Portfolio Returns

Benchmark Returns

Regression Line

Observation 5

Observation 6

Residual ɛ5

Residual ɛ6

Portfolio Returns

Benchmark Returns

Regression Line

Observation 5

Observation 6

Residual ɛ5

Residual ɛ6

The regression Figure 3.3 The ex post risk of a portfolio P is based on its historical returns (1). There are fundamental questions when using time series:

• What is an optimal length of a particular time series to be used for a performance analysis of a given portfolio?

• What data frequency like daily, monthly should be

used? In order to assess the quality of an asset manager is the averaging of the asset allocation and the stocking the effect over the total observation period. Normally the performance attribution calculates a total figure for the whole reporting period for the different management effects and does not show the management effects over time for example on a monthly basis. In Figure 3.4 we compared the management effects for the whole period with the

138

ones on a monthly basis. The total figures indicate that the asset manager was quiet a bad stock picker and a good asset allocater during the reporting period. But this is the wrong conclusion because the positive asset allocation effect was mainly generated in the first two months of the reporting period and the monthly asset allocation effect over the last seven months were constantly negative. With respect to the stock picking effect there is a similar situation but vice versa. The monthly stock picking effect was negative over the first 8 months but constantly positive over the last seven months. It is now up to further analysis to get more insight into the figures and to come up with the “right” conclusions.

Do the management effects vary over time? Figure 3.4

Asset allocation effectStock picking effect

139

3.4. Risk decomposition on asset level The last section is based of the historical return series of a portfolio. For instance a shift from equity to bond a year ago is reflected in the analysis. The exposition in the last section is thus called ex post risk analysis. In this section we decompose the risk of a portfolio into segments (Definition 2.5) or individual investments. The weights of the portfolio are as of today that why the approach in this section is called ex ante risk analysis. However, the returns of the invested investments or segments are historical. The time series of the returns can provide estimations for the variance and covariance. We need for the following some concepts from a special discipline in mathematics called linear algebra. A real matrix A is defined by

A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

n,N1,N

n,11,1

a..a..

a..a

where ai,j, i = 1,….,N, j = 1,….,n are real numbers, i.e. for short A is an element in RNxn. A real vector v is defined by

v =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

v.

.v

where vj, j = 1,….,n are real numbers, i.e. for short v is an element in Rnx1 = Rn. The Null vector v is defined by

140

v =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

0.

.0

.

The transpose AT of A is defined by

AT =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

N,n1,n

N,11,1

aa

aa

,

for short AT is an element in RnxN. The transpose vT of v is defined by vT = [ ]n1 vv , for short vT is an element in R1xn = Rn.

Matrices are used for assessing the variance and the standard

deviation of a portfolio. The components of a vector are holdings

of the portfolio, benchmark, resp. or the returns of an investment

or a segment of the portfolio, benchmark, resp..

141

We proceed by introducing the Matrix multiplication, i.e. the multiplication of two Matrices

A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

n,m1,N

n,11,1

a..a..

a..a

and

B = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

M,n1,n

M,11,1

b..b..

b..b.

We consider C = A . B where the element of C is defined

C = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

M,n1,N

M,11,1

c..c..

c..c

by

∑=

=n

1kj,kk,ij,i bac , 1 ≤ i ≤ N, 1 ≤ j ≤ M.

Example 3.3: We consider the matrices

A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100210321

, B = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

123012001

and calculate the product C = A . B

142

C = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

1232583814

.

◊ Definition 3.11: A ∈ Rnxn is called positive definite if

∈∀>∑∑= =

j

n

1j

n

1iijj,i v,0vva R1, ∈∀ iv R1 (3.4.1a)

for v ≠ 0 is satisfied and A ∈ Rnxn is positive semi-definite if

∈∀≥∑ ∑= =

jn

1j

n

1iijj,i v,0vva R1, ∈∀ iv R1. (3.4.1b)

is satisfied. On one side an investor could argue that there is no expost risk because there are no unprecedented events in the past on the other side another investor could have the stance that there is no risk in the future as no event has materialized. We conclude that risk to do with probability. The idea of the exante risk model is based on the actual weighting on the portfolio and the covariance matrix S of the constituents, however, is based on historical data. In other words the weights of the portfolio are subjective and thus dependent of the investor and the historical behaviour of the market are considered to be objective. In the following we consider a portfolio P and a benchmark P with n constituents Cj with returns k,Pr , k,Br , k,jr , 1 ≤ j ≤ n, 1 ≤ k ≤ N (3.4.2)

143

at time tk. Here the frequency of the time point is arbitrary. In most performance system monthly or daily data are used. Similarly to (3.1.3) and (3.3.2) we introduce

var(Ci) = ∑ −=

N

1k

2iki, )r(r

N1 , i = 1,….,n (3.4.3a)

and

cov(Ci, Cj) = )r(r)r(rN1

jkj,N

1kiki, −∑ −

=,1 ≤ i, j ≤ n, i ≠ j (3.4.3b)

The calculation of exante risk figures are based on these two inputs. Exante Risk considerations can be perceived as risk management by applying different weight schemes. With

,rN1r

N

1kk,jj ∑=

=1 ≤ j ≤ N

the input data is given

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

=

nN,n11,n

nN,111,1

rr...rr..........

rr...rr

R (3.4.4a)

whereas

144

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

=

nN,nnN,1

11,n11,1

T

rr......rr......

rr......rr

R . (3.4.4b)

Lemma 3.3 (Variance Covariance matrix based on return data): Considering (2) for the matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

n,n1,n

n,11,1

s..s....

s..s

S

defined by

S = N1

RTR (3.4.5)

we have a) sjj = var(Cj, Cj), 1 ≤ j ≤ n, sij = cov(Ci, Cj), ji ≠ ,1 ≤ i, j ≤ n. b) S is positive semi-definite, i.e. S satisfies (1b). Proof: a) The assertion follows from (1) and (3). b) We multiply S by wT ∈ Rn and w ∈ Rn and find wT A w ∈ R1, ∀w ∈ Rn.

145

With (5) and v ∈ Rn v = w R we have

wT S w = N1

wT RT R w = N1 vT v, ∀w ∈ Rn.

By components

[ ]Tn1 v..v

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

n

1

v..v

the right side is 0v.......v 2

n21 ≥++ .

Thus the assertion (1b) wT S w ≥ 0, ∀w ∈ Rn is shown. ◊ Remark 3.11: S is called the Risk matrix and is the starting point for the portfolio optimisation theory (Chapter 4). Example 3.2: We consider N = 3 data points, and n =1 segment and choose the vector

146

w = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

321

and calculate wT w

[ ]T321

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

321

= 14.

Then

N1

wT w is the variance of an investment. We consider N = 1

data points and n = 3 segments and choose the vector

wT = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

321

and calculate wT w.

T

321

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ [ ]321 =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

963642321

.

Evaluating the vector

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−01

2

147

we see that the matrix is only semi-positive definite. Then wT w is the variance covariance matrix of a portfolio with 3 segments. ◊ From the Cholesky decomposition [10, Neumaier] there follows that the inverse of Lemma 3.1 is somehow also true: every positive definite matrix can be derived from a not unique time series. From the analysis of a real value function of a single variable the element of S can be perturbed if S satisfies the strict equality (1a). However, there is no statement about the magnitude of perturbation. If S is semi definite the elements of S can in general not be perturbed. Theorem 3.2: We consider a portfolio P and benchmark B with n constituents Cj, 1 ≤ j ≤ n with return history (2) and the vector w for the weights wj for the portfolio and the vector b for the weights bj for the benchmark, 1 ≤ j ≤ n. Then the following holds: a) For the variance of the portfolio P we have Var(P) = SwwT =

=∑ ∑= =

n

1j

n

1iijj,i wws

∑ ∑=

≠=

n

1j

n

ji1i

ijij )C,Ccov(ww + ( )∑=

n

1jj

2j )Cvar(w =

148

[ ]Tn1 w..w

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

n,n1,n

n,11,1

s..s....

s..s

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

n

1

w..w

and the corresponding risk is

std(P) = ∑∑= =

=n

1j

n

1iijj,i

T wwsSww . (3.4.6)

b) For the tracking error TE(1) defined in (3.3.4) we have

TE(1) = ∑ ∑ −−=−−= =

n

1j

n

1iiijjj,i

T )bw)(bw(s)bw(S)bw( =

[ ]Tnn11 bw..bw −−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

n,n1,n

n,11,1

s..s....

s..s

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

nn

11

bw..

bw

.

Proof: From (2.2.4) and (3.1.2) we have

( ) ∑ =⎟⎟⎠

⎞⎜⎜⎝

⎛∑ −∑ =−=

= ==

N

1k

2n

1jjk,jj

N

1k

2Pk,P )rr(w

N1rr

N1)Pvar(

∑ ∑ ∑= = =

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

N

1k

n

1jik,i

n

1ijk,jij )rr)(rr(ww

N1

149

∑ ∑ ∑= = =

−−n

1j

n

1iik,i

N

1kjk,jij )rr()rr(

N1ww .

By Lemma 3.1 a) we have

∑ ∑=

≠=

=n

1j

n

ji1i

ijij )C,Ccov(ww)Pvar( + ∑=

n

1jj

2j )Cvar(w .

The proof of b) is the same as a). ◊ Furthermore the correlation matrix is defined by

ρ =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ρ

ρ

1......

..1

1,n

n,1

(3.4.7)

where

ρi,j = j,ji,i

j,i

sss

Lemma 3.4: The correlation matrix (7) is positive semi-definitive. Proof: As S is positive semi-definite we have

∈∀≥∑∑= =

j

n

1j

n

1iijj,i v,0vvs R1, ∈∀ iv R1

150

we consider

i,i

ii s

vv = ,

hence

∈∀≥∑=

j

n

1jij

ji

j,i v,0vvss

s R1, ∈∀ iv R1

and conclude that the correlation matrix is also semi-positive definite. ◊ In the following we investigate the structure of the Risk matrix and the correlation matrix. Example 3.3: We consider a portfolio P with investment or segments C1 und C2 and risk

S = ⎟⎟⎠

⎞⎜⎜⎝

⎛)Cvar()C,Ccov(

)C,Ccov()Cvar(

221

211 .

With matrices the risk of the portfolio is

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

221

21121 w

w)Cvar()C,Ccov(

)C,Ccov()Cvar(ww)Pvar( ,

i.e. var(P) = ( )21w var(C1) + 2w1 w2 cov(C1, C2) +( )22w var(C2).

151

We consider 3 special cases: 1. cov(C1, C2) = var(C1) * var(C2) (ρ = 1) var(P) = var(C1 + C2) 2. cov(C1, C2) = 0 (ρ = 0) var(P) = var(C1) + var(C2) 3. cov(C1, C2) = var(C1) * var(C2) (ρ = −1) var(P) = var(C1 − C2) The example illustrates the Theorem of Pythagoras in Figure 1.2. ◊ An eigenvalue λ ∈ R1 and an eigenvector v ∈ Rn of a quadric matrix A ∈ Rnxn is defined by A v = λ v. For calculating the eigenvalue and eigenvalue we refer to a text in linear algebra, see for e.g. [10, Neumaier]. Lemma 3.5: The matrix A ∈ Rnxn is positive definite only and only if the eigenvalues of matrix are positive. Proof: [10, Neumaier]. ◊

152

The following two theorems yield a further criterion for checking that a matrix is positive definite. Lemma 3.6 (Gerschgorin, weak version): Let A ∈ Rnxn be an arbitrary quadric matrix with

∑=α

≠=

n

1j1j

j,ij a

then there is an eigenvalue λ of A and an index i such that ii,ia α≤−λ Proof: [16, Varga]. ◊ Lemma 3.7 (Gerschgorin, strong version): Let A ∈ Rnxn be an irreducible matrix

∑≠=

=αn

1j1j

ijj a .

Is λ an eigenvalue of A so is either iα=−λ iia for at least one k or iα<−λ iia for all k.

153

Proof: [16, Varga]. ◊ Example 3.4: Let

S =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

141

41

411

41

41

411

.

Based of Gerschgorin the matrix is positive definitive, thus A represents a correlation matrix. This matrix can be perturbed without loosing the positive definiteness. ◊ Example 3.5: We consider the matrix

S = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

111111111

.

For the following we refer to [10, Neumaier]. We illustrated the calculation of the Eigenvalues. With

I = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100010001

154

we calculate the characteristic polynomial:

P(λ) = det (A − λ ID) = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

λ−−−λ−−

−λ−

111111

111 =

(1 − λ) det ⎥⎦

⎤⎢⎣

⎡λ−−

−λ−11

11 + det ⎥

⎦

⎤⎢⎣

⎡λ−−

−11

11

+ det ⎥⎦

⎤⎢⎣

⎡−λ−

−11

11 = 0,

where det stands for the determinant defined by

det ⎥⎦

⎤⎢⎣

⎡dcba

= ad − cd.

Hence P(λ) = λ2(λ − 3) = 0, thus S is positive semi-definite and as a consequence has the property of a correlation matrix. For λ = 3 we have

v1 = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−11

1

and for λ = 0 we have

155

v2 = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

101

, v3 = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

110

for the eigenvectors. ◊ Example 3.6: With

S = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

111111111

the vector

v = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

111

yields vT S v = −3, i.e. the matrix is not positive semi-definite. It is seen that 3 assets cannot have mutually correlations −1. ◊

156

3.5. Risk decomposition with factor analysis A. Introduction Factor models are a well-accepted way of reducing the number of variable analyzing the return and risk of financial investments. If we have a portfolio of n assets the number of covariances, correlations, resp. is

2

)1n(n − ,

i.e. there is quadratic growth with n. We start by illustrating the concept of the factor analysis by supposing a set of given data and we attempt to find a factor that allows the reproduction data 1 and data 2. In Figure 3.5 Data 1 (Data 2) is approximately 1.5 (2.0) times the factor.

Basic idea of factor analysis Figure 3.5 In factor analysis it is the aim to asses the main drivers of the risk and to discuss the risk of a portfolio in term of sensitivity. If we invest e.g. in Royal Dutch a potential factor analysis will include the oil price as a factor and asses the sensitivity of the portfolio with

data 2 data 1

factor

157

Royal Dutch respect to oil price. We proceed by introducing the following notations for 1 ≤ i ≤ N and 1 ≤ p ≤ M: ri,t is the observed return of the security i at time t

(independent variable); βi,p,t is the factor return of the security i at time t (dependent

variable); fp,t are the factors to be estimated (slopes);

b: is a vector with returns that are known a priori (deterministic) like currencies or the risk free rate.

There are different types of factor like

• Macroeconomic Factor • Fundamental Factor • Statistically significant independent factor

Here we do not describe the process of identification of a set of factor and proceed by considering two types of factor analysis. B. Cross-Sectional Model (Regression over securities) The following basic linear relationship is only over a time period k and has no relationship with the prior time periods. It decomposes the return ri,k, 1 ≤ i ≤ N of securities of a prespecified Universe and it is assumed. The characteristics of the securities are assumed to be assigned linearly to the different factors: ri,k = βi,1,k f1,k + βi,2,k f2,k +….+ βi,M,k fM,k + b = (3.5.1)

158

∑ ⋅β=

M

1pt,pt,p,i f + b

Obviously the relationship for fp,t, 1 ≤ p ≤ M cannot be satisfied exactly as M ≤ N in realistic situations, however, there are techniques for minimizing the error ε

ri,t = ∑ ⋅β=

M

1pt,pt,p,i f + b + ε

in a certain sense. It explains security returns via measured sensitivities to systematic factors and estimated returns associated with those factors. Changes in security characteristics are captured and relates well to portfolio management. We consider

k,ir~ = ∑ ⋅β

=

M

1pk,mk,p,i f + b

as approximation of ri,k. For the return k,Pr

~ of the portfolio with n securities we have

k,Pr~ = ∑ ⋅

=

n

1ik,ik,i r~w + b.

We consider

159

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

=

NN,M11,M

NN,111,1

ff...ff..........

ff...ff

F (3.5.2)

and consider the covariance matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

M,M1,M

M,11,1

t..t....

t..t

T

defined by

T = N1

FTF.

Following Lemma 3.1 T is positive semi-definite. Furthermore we denote

βi =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

β

β

iM

1i

160

The exposure of security i to the factor βi,p, 1 ≤ p ≤ M. Following Theorem 3.2 the risk of a asset is the

std(Ci) = ∑ ∑ ββ=ββ= =

M

1p

M

1iqq,ip,iq,pi

Ti tT

and covariance of

cov(Ci, Cj) = ∑ ∑ ββ=ββ= =

M

1p

M

1qq,ip,jq,pi

Tj tT

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ββ

ββ

=ββ=β

M,N1,N

M,11,1

N1

.............

...

...

Theorem 3.3: We consider a portfolio P with a return series (3.1.1) and n constituents Cj, 1 ≤ j ≤ n with return history (3.4.4) and weights wj, 1 ≤ j ≤ n. Then the following holds: a) for the variance of the portfolio P we have var(P) = wTw TT ββ =

161

=∑ ∑ ∑ ∑ ββ= = = =

n

1j

n

1i

M

1p

M

1qq,jq,iq,pij tww

∑ ∑=

≠=

n

1j

n

ji1i

ijij )C,Ccov(ww + ∑=

n

1jjj )Cvar(w =

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ββ

ββ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ββ

ββ

nw..1w

n,M..1,M

..n,1..1,1

M,Mt.M,1t..1,Mt.1,1t

T

M,n.1,n

..

..M,1.1,1

Tnw..1w (3.5.3a)

and the corresponding risk is

std(P) = ∑ ∑ ∑ ∑ ββ=ββ= = = =

n

1j

n

1i

M

1p

M

1qq,jq,iq,pij

TT twwwTw .

b) For the tracking error TE(1) we have

TE(1) = ∑ ∑ −−=−ββ−= =

n

1j

n

1iiijjj,i

TT )bw)(bw(s)bw(T)bw( =

[ ] .

bw

bw

tt

ttbwbw

nn

11

n,M1,M

n,11,1

M,MM,1

1,M1,1

T

M,n1,n

M,11,1

T1n11

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ββ

ββ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ββ

ββ

−− (3.5.3b)

◊ The risk measures in (3.4.6) are called exante because there only depend on the weights wj, 1 ≤ j ≤ n of the portfolio and the benchmarks bj, 1 ≤ j ≤ n as of today are not dependents These weights are subjective and the variance and the covariance in (3.4.6) are objective, i.e. they are only dependent of the markets.

162

One way is to estimate them from statistical techniques, but they also can be derived from sheer economical considerations. C. Times series (Regression over time) Different to (1) we assume βi,p = βi,p,t, 1 ≤ p ≤ M i.e. the factors do not depend on t, thus we consider

ri,k = βi,1 fi,k + βi,2 f2,k + + βi,n fn,k + b = ∑ ⋅β=

M

1pk,pp,i f + b,

i.e. we vary over time whereas in B. we vary on the universe. Here we explain security return over time via measured return associated with systemic factors and estimated sensitivities to those factors. The classic example is the CAPM model. It is a one factor model. The factor is also estimated by least regression method. We will discuss it in the Chapter 4. D. The principal component An analysis aimed at finding a small number factor that describes most of the variation in a large number of correlated variables. We proceed by illustrating the factor analysis to two typical investment processes in portfolio management. Key is that performance measurement mirrors the investment process. In Factor analysis we assume an investment process where all investment decisions are taken simultaneously and independently. Example 3.6: We illustrate a typical equity investment process (see Figure 3.4). We invest first in regions then in sectors and followed by stock selection.

163

Portfolio Return or Risk

Region A

Sector 1

Stock 1

Sector 2

Stock 2 Stock 1 Stock 2

Region B

Sector 3

Stock 1

Sector 4

Stock 2 Stock 1 Stock 2

Portfolio Return or Risk

Region A

Sector 1

Stock 1

Sector 2

Stock 2 Stock 1 Stock 2

Region B

Sector 3

Stock 1

Sector 4

Stock 2 Stock 1 Stock 2

The equity investment process Figure 3.4 The equation (3.5.1) assumes the following form

lock,ir − k,fr = ∑ ⋅

=

M

1pt,ik,p,i fe + ∑ ⋅δ

sec/indsec/indsec/ind f +

∑ ⋅δregion

regionregion f + ∑ ⋅β⋅δcountry

t,countryt,icountry f + εi,t

where loc

k,ir is the local return of the ith security k,fr is the risk-free rate, ei,j,t is the exposure of the ith security to the jth factor, fj,t is the factor return, βj,t is the exposure of the ith security to its country factor and εj,t is the residual of the ith security with expected value zero. Exposure to region/country/industry group assumes either 0 or 1 depending whether or not the security is found in the region, country or industry group. Regression over the equity universe yields a line in the matrix F in (2). The following table shows a specification of the factors

164

Equity factors (Source Wilshire) Figure 3.5 ◊ Example 3.7: We illustrate a typical Fixed Income process (see Figure 3.6). We first invest in currencies followed by durations and then specific issues. At t = 0 the price Pi,0 of the ith bond in local currency with time to maturity ti,N is given by the total sum of the discounted cash flow:

[ ] kt)kt(s)kt(rN

kk,i,i eCFP +−

=∑=

00

Here r is the spot rate and s is a spread to the spot rate at time tk, 0 ≤ k ≤ N. To calculate a return on this bond, it necessary to express the price of the bond at some later small or instantaneous time t

165

Portfolio Return or Risk

Currency A

Currency B

Duration 1

Duration 2

Issue 1

Issue 2

Issue 3

Issue 4

Issue 5

Issue 6

Portfolio Return or Risk

Currency A

Currency B

Duration 1

Duration 2

Issue 1

Issue 2

Issue 3

Issue 4

Issue 5

Issue 6

A fixed Income investment process Figure 3.6 [ ] .eCFP )tkt()t(ds)t(s()t(dr)t(rN

0kk,it,i

−+++−

=∑=

Following the work of Nelson Siegel for representing the shape of the yield curve and the fact the dynamic of the yield curve can describe by 3 basic movements it is proposed that dr(t) is seen by an expansion of 3 factors around dr(t) = 0:

dr(t) = x1 + x2 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎠⎞

⎜⎝⎛−

7t

e1 + x3 7t ⎟

⎠⎞

⎜⎝⎛ −

7t1

e (3.5.4)

166

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35 40

f1

f2

f3

Yield curves factor structure Figure 3.7 where x1, x2, x3 are coefficients for the exponentials in the above expansion. We identify the factors

f1(t) = 1, f2(t) = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎠⎞

⎜⎝⎛−

7t

e1 , f3(t) = 7t ⎟

⎠⎞

⎜⎝⎛ −

7t1

e

(see Figure 3.7). We see that f2 and f3 are zero at t = 0. At large t f3(t) approaches 0 and, relative to f1, f2 dominates the yield whilst at intermediate t (t = 7) f3 dominates f2 relative to f1. Consequently f1 dominates the short part of the curves f2 affects the long end and f3 impacts the middles part of the yield curve. We decompose the change of the spread ds(t) linearly

dsi = jj

j,i λχ∑

is the spread factor and χi,j is the identity function which takes the value 0 or 1 depending on whether the security is found in bucket.

167

With (4) we find Pi,t =

∑=

ε+λλ∂

∂χ++

⎟⎠⎞

⎜⎝⎛ −

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛−

−++− ∑N

0k

kt)ijdj

P

j

ij)kt(s7

tk1

e7t

3x7kt

e12x1x)kt(r(

k,i eCF .

Over time P is a function of P = P(x1, x2, x3, λj, t). The differential is

.dPP1dx

xP

P1dx

xP

P1dx

xP

P1dt

tP

P1dP

P1

ij

jj

i

i3

3

i

i2

2

i

i1

1

i

i

i

ii

iε+∑ λ

λ∂∂

+∂∂

+∂∂

+∂∂

+∂∂

=

We introduce the local return loc

k,ir of bond i at time k and the yield lock,ir by holding the bond over time t

loc

k,ir − yieldk,ir = i

i

ii

idt

tP

P1dP

P1

∂∂

− .

The exposures are denoted by D1i,k, D2i,k, D2i,k and the factor returns are fk,i, k = 1, 2, 3.

lock,ir − f

tr = D1i,k f k,1 + D2i,k fk,2 + D3 i,k f k,3 + ∑ λ λj

jjf + εi

=∂∂

==λ= 011x1

i

it,i x

PP11D ,eCFt

P1 N

1kkt))kt(s)kt(r(

kt,iki

∑=

+−

=∂∂

==λ= 0222

12x

i

it,i x

PP

D

168

iP

1,eCFe1t

N

1kkt))kt(s)kt(r(

kt,i7kt

k∑⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

=

+−⎟⎠⎞

⎜⎝⎛−

=∂∂

==λ= 0333

13x

i

it,i x

PP

D iP

1.eCFe

7tt

N

1kkt))kt(s)kt(r(

kt,i7kt1

kk∑

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

+−⎟⎠⎞

⎜⎝⎛ −

Similarly for the spread factor

=λ∂

∂=

=λ= 0j1xj

i

ij,k,i

PP11SD ∑χ

=

+−N

1kkt))kt(s)kt(r(

kt,ikj,i eCFt

Fixed Income factors (Source Wilshire) Figure 3.8

169

The factor returns of the model are estimated with the regressing of factor exposure against the return of a universe of bonds. They are classified according Figure 3.8 and the regression function is loc

k,ir − k,fr = Euroc,i ∉χ ( )+++∑ χ

∈c,k,3k,ic,k,2k,ic,k,1k,i

countriescc,i f3Df2Df1D

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∑+++χ∈

∈

CountriesEuroc

Euro,k,ck,iEuro,k,3k,iEuro,k,2k,iEuro,k,1k,iEuroc,i f1Df3Df2Df1D

+ ∑ χ

∈∈

CurrencycSectors

c,k,sk,ic,s,i f1SD + +∑ χ

∈∈

Currencycqualitiesq

c,k,qk,ic,q,i f1SD

∑ χ

∈∈

Currencycqualitiese

c,k,ek,ic,e,i f1SD + i

USAc=χ +∑∈ factorsothera

k,ak,i,a fA εi,k.

Regression over the bond universe yields a line in the matrix F in (2). ◊ We proceed by some conclude remark of the chapter. 1. A comparison or evaluation of expost versus expost Risk figure

is left to further research. It is seen from Examples 3.6 and 3.7 that exante absolute risk and exante tracking error is based the choice of the factors in the factor analysis.

170

2. It is seen that this chapter is based on the Definition 3.1 for the variance and the Definition 3.5 for covariance. This definition is basic and therefore often criticised in literature because e.g.

• The centre around which standard deviation is calculated is

the arithmetical average historical return. However, prevailing market condition can be substantially different and are also thus also differentially forecasted from historical indications.

• Standard deviation treats positive and negative return

deviations from the mean with equal weight. As Standard is perceived as risk, only negative deviations should be counted as solely they are averse market movements for the investor.

Downside risk is an asymmetrical type of risk. A possible starting point is the decomposition of the variance

∑ −+∑ −=∑ −

>=

≤==

n

rir1i

2i

n

rir1i

2i

n

1i

2i )r(r

n1)r(r

n1)r(r

n1 .

Further exposition of down risk is beyond scope of this text and we refer to the literature [9, Feibel]. However up to our knowledge there are not studies on the superior performance of downside risk against symmetrical risk measures.

171

4. PERFORMANCE MEASUREMENT 4.1. Investment Process and Portfolio Construction We start with considering the investment process depicted in Figure 4.1. The discussion of return and risk in the previous Chapters 2 and 3 is relevant to all part of the investment process. We proceed with focusing on the different steps of the investment process. The topic of this chapter is the asset allocation, portfolio construction and rebalancing. The evaluation of the investment strategy is in Chapter 5 (Investment controlling).

Portfolio construction

Portfolio analytics /Performance analysis

Reb

alan

cing

Ass

et a

lloca

tion

proc

ess

Investment target, risk profile and benchmark

Actual market data andforecasts

Portfolio construction

Portfolio analytics /Performance analysis

Reb

alan

cing

Ass

et a

lloca

tion

proc

ess

Investment target, risk profile and benchmark

Actual market data andforecasts

Constraint to be respected

The investment process Figure 4.1 Definition 4.1: An asset class is a group of securities that exhibit similar characteristics, behave similar in the marketplace and are subject to the same laws and regulations. The three main asset classes are equities (stocks), fixed-income (bonds) and cash equivalents (money market instruments) (compare www. investopedia. com).

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The asset allocation process is the decision oriented process of investing in the different asset classes. A typical investment strategy is reflected by the asset allocation process followed by the portfolio construction. In Example 2.7 the asset allocation process is the decision marking process between equities markets in different countries and the portfolio construction identifies individual equities. In most situations the investment process is much more complex. In this section we describe some approaches for the decision making process of the pursued investment strategy which is based on actual market data and forecasts. The definition of the investment universe (Definition 2.4) and applicable constraints on the portfolio and its constituents are: • Liquidity needs • Expected cash flow • Investable funds (i.e. asset and liability) • Time horizon • Tax considered • Regulatory and legal circumstances • Investor preferences, prohibition, circumstances and unique

needs • Proxy voting responsibility and guidelines Running performance attribution implicitly assumes that the asset manager has no guideline restriction for example with respect to the minimum or maximum weight of a specific security or asset class within the portfolio. If this assumption is not true and there are restrictions in a way that the asset manager is not able to invest according to the passive investment alternative – the benchmark – the calculated management effects may be misleading in a way that limiting an investment may have a positive or negative impact on the excess return. For example, there are equity indices which are dominated by a specific security – assuming for instance 60% of the index consists of this security – where a maximum weight

173

limit of 10% is an implicit bet on this specific security by predefining the minimum underweight of this security of 50%. The question is who is responsible for the return contribution due to this investment restriction. The same is true if there are minimum limits for specific securities. Portfolio construction is either

a) Creating a portfolio if there is new money to invest or b) Rebalancing a portfolio if there exists already a portfolio.

a. Active investment strategy We proceed with the following definition: Definition 4.2: The objective of an active (passive) investment strategy is to outperform (mimic) the benchmark. Portfolios that try to outperform (mimic) the benchmark are said to be actively (passively) managed. An active investment process requires research of financial markets, currencies, individual securities, yield curves etc. and is thus much more costly than a passive investment process. An active investment strategy attempts to take advantage of changes in the risk premium and consequentially to respond tactically to changing market conditions. We proceed by describing approaches for analyzing and forecasting different investments like bonds, equities, commodities or particular financial markets. In the following we give a tour d’horizon on the different discipline for assessing financial markets. For more extensive and in-depth information we recommend [16, Malkiel]. Referring to [16] the Firm-Foundation Theory and the Castle-in-the Air Theory are the starting point of the following two sections on fundamental research and technical research.

174

Portfolio ConstructionAsset Allocation

FundamentalResearch

TechnicalResearch

EconomicResearch

PoliticalResearch

Portfolio ConstructionAsset Allocation

FundamentalResearch

TechnicalResearch

EconomicResearch

PoliticalResearch

Some different areas of financial market research Figure 4.2 In the following we give a tour d’horizon on the different discipline for assessing financial markets. For more extensive and in-depth information we recommend [16, Malkiel]. Referring to [16] the Firm-Foundation Theory and the Castle-in-the Air Theory are the starting point of the following two sections on fundamental research and technical research. Fundamental research Introduction and Terminology For the investor in a stock of a company is vital to know the ‘right value’ of the common stock is. Fundamental research strives to be relatively immune to the optimism and pessimism of the crowed and makes a sharp distinction between a stock’s market price and a theoretical price. There might be many such theoretical prices and in literature they are called inner or intrinsic value of a common stock. The Fundamental research is based on the idea that the market price adjusts itself to the long term to its intrinsic or true value, which depends on the company and economic data. In estimating the firm-foundation value of a security, the fundamental’s job is to estimate e.g. the firm’s future stream of earning, dividends, sales level, operating costs, corporate tax

175

rate, depreciation policies and the costs of its capital requirements. Current deviations from the intrinsic value can be used profitably, as the market will correct the valuation. Market participants trade rationally and research is worthwhile. Overall fundamental research is mostly company research, already interconnected is the research of the industry. Evaluation a stock: some first approaches The most basic and easy understand model for evaluating a common stock is the discounted cash flow model. It is based on the arbitrage condition that the value of a common stock today is the same as the present value of the future dividends C1 and the value of the stock PV1 discounted to today, hence we have

PV0 = r1

C1+

+ r1

PV1+

.

By further assuming that the dividends Cj, j = 1,....,N are known, iteration to N periods yields

PV0 = ∑= +

N

1jj

i

)r1(C + PVN

where PVN is the value of the stock after N periods. We see that this is a version of the IRR equation considered in (2.4.1). By assuming that the dividends are known for all future periods we have

PV0 = ∑∞

= +1jj

i

)r1(C . (4.1.1)

176

A first question is whether instead of the dividends, the earnings or earnings plus non-cash earnings should be discounted in (1) [16]. As (2.1.1) in used in various versions, (1) is solved in different ways. Firstly, PV0 is computed by using estimates on the dividends and an appropriate discount rate. Secondly, the idea is to calculate r out of the input parameter PV0, Cj, j = 1,....,n,..... An appropriate r can be calculated by the security market line and the risk premium of the stock. Thirdly, the equation is divided by the earning the left hand of the equation would be the normal price earning at which the stock should sell. Here are some realization of the growth of Cj, j = 1,...., n,... in (1): 1. Constant growth over an infinite amount of time 2. Growth for a finite number of years at a constant rate, then

growth at the same rate as a typical firm in the economy from that point on.

3. Growth for a finite number of years at a constant rate, followed by a period during which growth declines to a steady-state level over a second period of years. Growth is then assumed to continue at the steady-state level into the indefinite future.

Technical research Introduction and Terminology Technical research is based on the graphical representation of financial data. The value of technical analysis can be expressed by the old Chinese proverb ‘A picture is worth ten thousand words’. Technical research provides an abstract access to financial market as every investment or a particular financial market can be studied by graphical representations. Examples for such investments are gold, currencies markets or equities markets. For instance a graph can contain price, return or other market data measured vertically in the graph and

177

quantified by the unities on the vertical axis. The vertical axis contains usually the time scale of the graph. A chartist or a technical analyst seeks to identify price patterns and trends in financial markets and attempt to exploit these patterns. Although the study of price charts is primary, technicians use also various further methods and approaches for investing financial market. They are called technical indicators. A typical example are moving averages MA(N) defined by

MA(N) = .rN1 N

1kk,P∑

=

Referring to Figure 2.3 we assume here tN+1 = N + 1 and MA(N) is a forecast at time N + 1. Furthermore for N = 1 we have the simplest forecast model tomorrow is equal to today. The idea is that the forecast of the future development depends exclusively on the historical direction of prices and the volume of trading. Most chartists believe that the market movement is only 10 percent logical and 90 percent psychological. Essentially it is assumed that financial markets have a memory and it is possible to conclude their future behavior from the past behavior. Some properties of technical analysis are

• Based on strict rules and guidelines

• Provides objectivity in an investment process

• The theoretical framework can be applied immediately

• Applicable for any time horizon and market

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Description of some important concepts

TrendMoving average

Momentum

Rate of change

PatternElliot wave

TrendMoving average

Momentum

Rate of change

PatternElliot wave

Base concepts and their schematic representation Figure 4.3 We proceed with some basic technical indicators: A trend is a line that connects at least two points. In an up trend low points are connected. In a down trend high points are connected. A trend, however, is more significant the more points can be connected. ‘The trend is my friend until it breaks’ is a saying from technical analysis because future movements of the financial markets are anticipated and invested accordingly. In Physics the momentum is an important notion. It is the increase and decreased of the speed of an object. A momentum is a rate of the change. Momentum indicators give a first signal before a trend is changing. It shows whether a trend is accelerating or decelerating. The Elliot wave principle is a form of technical analysis that investor use to forecast trends in the financial market by identifying extremes in investor psychology. Ralph Nelson Elliott (1871-1948) developed the concept in the 1930. He proposed that market prices unfold in specific patterns which practitioners called Elliot waves or simply waves.

179

The wave principle posits that collective investor psychology moves from optimism to pessimism and back again in natural sequence. We proceed with the buy and sell behavior of a investor. The experience shows that the investor is also too late when he buys or sells. He buys when an up trend already is on his way and he sells when a down trend is already partially realized in the market place. An optimist is convinced to buy and a pessimist mood is to sell.

confidence

Conviction = in the mood to buy

caution

scorn Capitulation = in the mood to sell

complacency

concern

confidence

Conviction = in the mood to buy

caution

scorn Capitulation = in the mood to sell

complacency

concern

Selling and Buying behavior Figure 4.4

The investor must learn when to buy when he is pessimistic and

is anxious and when to sell when he is optimistic and euphoric.

The most important goal of the technical analysis is detecting buy and sell signals, i.e. to indicate when there is a turning point. Economic Research We distinguish between Micro Economic and Marco Economic Research. Fundamental research is an area that can be perceived as Micro Economics whereas Marco economic

180

deals with the economics of a country or a region. Economics projections have an important impact on the performance of a portfolio. A holder of a balance portfolio might switch from an asset class to the other depending of the changing of the economic situation. In a booming phase an investor should be invested in equity. In declining markets, bonds are preferred, and in a recession or a economic crisis commodity and cash or money market like instruments are generally recommended (see Figure 4.5).

The investment cycle Figure 4.5 Here are some definitions of macro economic variables (see www.investorwords.com). Inflation is the rate at which the general level of prices for goods and services is rising, and, subsequently, purchasing power is falling. Central banks attempt to stop severe inflation, along with severe deflation, in an attempt to keep the excessive growth of prices to a minimum. Stagflation occurs when the economy is not growing but prices are increasing. This is not a good situation for a country. This happened to a great extent during the 1970s, when world oil prices rose dramatically, fuelling sharp inflation in developed countries. For these countries, including the U.S., stagnation increased the inflationary effects. A general decline in prices is called deflation, often caused by a reduction in the supply of money or credit. Deflation can be caused also by a decrease in government, personal or investment spending. The opposite of inflation, deflation has

Equities Bonds Commodity Cash Equities Bonds Commodity Cash

181

the side effect of increased unemployment since there is a lower level of demand in the economy, which can lead to an economic depression. Central banks attempt to stop severe deflation, along with severe inflation, in an attempt to keep the excessive drop in prices to a minimum. Extremely rapid or out of control inflation is called hyperinflation. There is no precise numerical definition to hyperinflation. Hyperinflation is a situation where the price increases are so out of control that the concept of inflation is meaningless. Interest rate is the amount charged, expressed as a percentage of principal, by a lender to a borrower for the use of assets. Interest rates are typically noted on an annual basis, known as the annual percentage rate. The assets borrowed could include, cash, consumer goods, large assets, such as a vehicle or building. Interest is essentially a rental, or leasing charge to the borrower, for the asset's use. In the case of a large asset, like a vehicle or building, the interest rate is sometimes known as the lease rate. The Gross Domestic Product (GDP) is the monetary value of all the finished goods and services produced within a country's borders in a specific time period, though GDP is usually calculated on an annual basis. It includes all of private and public consumption, government outlays, investments and exports less imports that occur within a defined territory that we have GDP = C + G + I + NX where: C is equal to all private consumption, or consumer spending, in a nation's economy. G is the sum of government spending. I is the sum of all the country's businesses spending on capital. NX is the nation's total net exports, calculated as total exports minus total imports (NX = Exports – Imports). The Gross National Product (GNP) is an economic statistic that includes GDP, plus any income earned by residents

182

from overseas investments, minus income earned within the domestic economy by overseas residents. The Consumer price Index (CPI) is a measure that examines the weighted average of prices of a basket of consumer goods and services, such as transportation, food and medical care. The CPI is calculated by taking price changes for each item in the predetermined basket of goods and averaging them; the goods are weighted according to their importance. Changes in CPI are used to assess price changes associated with the cost of living. Sometimes it is referred to as headline inflation. The economic variables can be measured in the past, but economists also publishing forecasts for them on a regular basis (see Figure 1.1). Every major financial institution provides forecasts and formulates a house view called economic outlook. The forecasts are debated in media, but also applied to the portfolios of the clients. A forecast is tied to a time horizon. We distinguish between short term und long term forecasts. Usually the maximal time horizon is one year. The accuracy of the forecast is often not tested. The delivery of periodic forecasts is one of the important tasks of economic research. Example 4.1 (impact of a macro economic variable): An inflation link bond protects the bond investor against the risk of raising inflation. ◊ A further area of economic research is the examination of dependency between economic variables. There is a substantial interest for business cycles (Figure 4.4. and 4.5) in economics [10, Gabisch, 13, Lorenz]. With t as time variable, S as savings function, I as investment function, Y as economic activity or real income of companies, K as capital stock, α > 0 and δ > 0, the Kaldor Model [10, 13] is given by ordinary differential equation

183

dtdY = α (I(Y, K) − S(Y, K))

(4.1.2)

dtdK = I(Y, K) − δ(Y, K)

and is a prototype of a dynamical system which generates business cycles. In the first equation economic of (2) economic activity is seen to tend towards a level where savings and investments are equal. A discrepancy between investments and savings induces a change in the level of economic activity which proceeds until the discrepancy is eliminated. If savings and investments are linear functions of the level of activity, the economic system would lead to hyperinflation with full employment, or a state of complete collapse with zero employment [10]. As economic systems in general do not behave in this manner, it is concluded that savings and investment cannot be realistically characterised in terms of linear functions. They have to be assumed nonlinear. The second equation of (2) represents the accumulation of capital stock. If there is no investment it is seen that the capital stock is depreciating by a factor δ. By using the Poincare Bendixson theorem it is shown in [13] that (2) exhibits business cycles. The numerical computation of business cycles is left to future research. Political research Investment decision depends also on political considerations. Often an investor avoids a country because a government is not stable enough. The buyer of government bond can face the danger that the bond default or change its conditions (Haircut). A caution portfolio manager decides only to invest in countries with political stability. Similarly before an election in a country a portfolio manager might choose a portfolio close to the benchmark or a caution portfolio manager decides only to invest in countries with political stability. A left oriented government will augment its

184

dents regardless of economic aspects whereas a right oriented government purports more free financial markets. Another investor might lend its money only to governments that favor sustainable asset. These might be governments that seek to reduce pollution and the accompanying emissions. An investor might choose a country with a low tax bracket and low regulations of financial markets. Quantitative Research Generally Data provide important information for the investor. The extraction of relevant data is the task of Quantitative Research applying the full apparatus of mathematically techniques especially from statistics and the theory of probability. Qualitative Research is accompanying Quantitative Research. Qualitative research describes crucial facts from financial markets. Often a conjecture from Qualitative Research is confirmed. Quantitative Research encompasses the development of Models. Modern Portfolio Theory belongs in particular to quantitative research. It quantifies financial markets. Modern Portfolio Theory needs both return and risk as input. Asset Allocation and portfolio construction are forward looking processes because they need forecasts. With fundamental and technical research forecasts can be generated. There are participants that beliefs that the research above is useless as all inferences are always in the prices. The random walk or efficient market theory refers to the fact that security prices fully reflect all available information. This is a very strong hypothesis. The efficient market hypothesis has historically been broken into three categories each one dealing with a different type of information: With week information efficiency of a market is expressed the fact that future stock prices cannot be predicted on the basis of past stock prices. With semi-strong information efficiency of a market is indicated that even when using published available information future prices are not predicable.

185

The strong information efficiency of a market referred to the fact that no information– not even unpublished development – can be of use for predicting future prices, i.e. the current market price reflects also the relevant non-public (insider) information. The stronger the information efficiency in the capital market is the less interesting and less informative it is putting effort in financial market research. b. Passive investment strategy A passive manager uses a range of different approaches – full replication, stratified sampling, optimization techniques. Under full replication we understand the investment in all securities with the corresponding weight of the benchmark. In stratified sampling the investor considers the constituents of the benchmarks that influence the benchmark most. More precisely, it allows the portfolio to match the basic characteristics of the index without requiring full replication. Characteristics of the benchmark like for instance the overall duration, the convexity and maturity buckets of a fixed income Benchmark are sequentially matched. Risk considerations such as investigating correlation structures do not enter in the portfolio construction. Optimization techniques intend to find a tracking portfolio with fewer securities than the benchmark but that has similar or even same characteristics as the benchmark. In many cases the number of securities is fixed and a portfolio with low tracking error and other desirable proprieties is selected. A portfolio tracking the benchmark has a tracking error equals or almost equal to zero. The passive investment style needs no forecasts and is therefore objective.

186

4.2. Portfolio optimization Portfolio optimisation is an approach for constructing portfolios. The risk as introduced in Chapter 3 does not interfered in the previous Section 4.1. Furthermore opposite to for instance technical research and fundamental research portfolio optimisation reflects the interaction between different investments. Harry Markowitz published his seminal work in portfolio theory in the 1950s. Together with William F. Sharpe he conducted a scientific study of portfolios invested in US equity market in 1952. Harry Markowitz, William F. Sharpe and Merton H. Miller received the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 1990. In financial markets many greedy investor wanted to earn more money with less risk. Abnormality of a particular strategy can be measured by Modern Portfolio Theory (MPT). Portfolio optimisation is a part of MPT. In Section 4.3 we continue with MPT. The basic ideas of MPT are still regularly discussed in media see e.g. [17, Marty] and in literature e.g. [4, Copland] and [23, Taleb]. For a comprehensive introduction in MPT we refer to the text books [8, Elton] and [21, Sharpe]. We proceed by introducing the assets considered in this section. Remark 4.1: We distinguish between risk-less assets and risky assets. Risk-less assets are introduced in Definition 2.6. As described in Section 3 an asset can bear different risks. In the following we think of risk as degree of fluctuation of the return expressed in volatility (see Definition 3.2). Risky assets can assume significant volatility, i.e. the investor can experience substantial gain and loss and they have not a volatility ‘close’ to zero. Typical risky assets are equities and MPT originates from the investigation of equities.

187

The problem of portfolio optimization under appropriate condition is solved theoretically and optimal portfolios can by numerically computed. They are many companies like Ibottson, Barra and Wilshire that offer products with portfolio optimisation software. As input they need the forecasts of the investors and the risk is estimated by factor analysis (See Section 3.4). However, its practical benefit is highly dependent on the accuracy of the associated forecasts. Here lies the key, since even forecast realizations that differ only slightly from expectations can have a major impact. Definition 4.3 (compare to [4, Copland page 169]): The set of efficient or optimal portfolios is the set of mean-variance choices from the investment opportunity set where for a given variance (or standard deviation) no other investment opportunity offers a higher mean return or equivalently for a given mean return no other investment opportunity offers less variance (or standard deviation). Remark 4.2: The notion of a benchmark is introduced in Definition 2.8. We see in the following that we can optimize with respect to a money market rate (absolute optimization) or with respect to a benchmark (relative optimization). A mathematical formulation for the optimal portfolio problem is as follows. We denote (see (2.2.3a)) by w the vector of the weights for the portfolio

w =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

w...

w

, (4.2.1)

by b the vector of the weights for the benchmark

188

b =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

b...

b

(4.2.2)

where

1bn

1ii =∑

=, bi ≥ 0, (4.2.3)

by p the vector of the weights for the positions

p =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

p...

p

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

w...

w

−

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

b...

b

, (4.2.4)

and by r the vector of returns (see (2.2.3b))

r =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

n

1

r...r

. (4.2.5)

Furthermore by referring to (2.3.4c) with the matrix S defined by (3.4.5) the optimal portfolios w ∈ R1 are defined as those a) that yield a maximum return μ ∈ R1 for a given risk σ ∈ R1, i.e.

189

μ =

wmax wTr

under the condition ABS1) absolute optimization σ = wT S w REL1) relative optimization σ = pT S p b) that minimize the risk σ ∈ R1 for a given return μ ∈ R1, i.e. ABS2) absolute optimization σ =

wminwT S w

REL2) relative optimization σ =

pminpT S p

under the condition wTr = μ The portfolio optimal portfolio problem together with the constraints

11

=∑=

n

iiw (4.2.6a)

and wi ≥ 0 (4.2.6b)

190

are called the (Markowitz) Standard Model. If only the budget constraint (6a) is assumed and no short selling constraint is considered the optimisation model is referred as Black Model [2, Bruce, page 159]. Optimal portfolios are hence to be found on the efficient frontier (Figure 4.6 and 4.7).

Absolute optimization Figure 4.6

relative optimization Figure 4.7

Tracking Error Benchmark

Efficient Portfolio x

Return

x

Asset A

Asset B

Risk

Return

Efficient Frontier

Inefficient Portfolios

191

Remark 4.2: From (4) and (6a) follows for the positions

0pn

1ii =∑

=.

We see that the positions can be positive or negative where with (6b) we have only positive weights. Remark 4.3: The relative optimization assumes the benchmark portfolio is considered as the portfolio with risk zero and therefore the reference portfolio. The absolute optimization assumes only vanishing risk for specified correlation structures. Remark 4.4: Positions are discussed in Table 2.3. We consider three positions: An overweight, neutral and underweight of securities versus the benchmark. Remark 4.5: The problem ‘maximum risk’ is not tackled in the framework discussed here. Remark 4.6: The (Markowitz) Standard Model for two assets can be solved explicitly and needs not a numerical algorithm. We refer to the following Examples 4.2 and 4.4. Example 4.2 (absolute Optimization with 2 assets): We consider a Portfolio which consists of 2 assets A and B. The covariance matrix is

S = ⎥⎦

⎤⎢⎣

⎡−

−25.1

5.13 (4.2.7)

and the return is (5)

192

r = ⎥⎦

⎤⎢⎣

⎡2.07.0

. (4.2.8)

The return of the portfolio is μ = 0.7 w1 + 0.2 w2. (4.2.9) By (3.4.5) and (3.4.6) in Theorem 3.2 the variance of the portfolio is with (7) Var(P) = w1

2 var(A) + 2 w1 w2 cov(A, B) + w22 var(B) =

(4.2.10) 3 w1

2 – 3 w1 w2 + 2 w22.

In the following we assume the condition (6), i.e. w2 = 1 − w1, w1 ∈ [0, 1] and with yields by (9) 2μ = w1 + 0.4, μ ∈ [0.2, 0.7] hence the parameterization of the weight is w1 = 2μ − 0.4, (4.2.11) w2 = 1.4 − 2μ. By vector

⎥⎦

⎤⎢⎣

⎡

2

1ww

= ⎥⎦

⎤⎢⎣

⎡−4.14.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.2, 0.7]. (4.2.12)

These weights also comprise portfolios that are not efficient. They can are also be on the inefficient branch of the efficient

193

frontier. In the risk σ return μrepresentation we find by (10) and (11) σ(μ) = 3 (2μ − 0.4)2 – 3 (2μ − 0.4)(1.4 − 2μ) + 2 (1.4 − 2μ)2 = 3 (−4μ2 + 3.6μ − 5.6) + 2 (1.96 − 5.6 μ + 4μ2) = −14μ2 + 10.8μ − 17.8 + 3.92 − 11.2μ + 8μ2 = −4μ2 − 0.4μ − 12.88. In view of the absolute optimisation problem (see (ABS1) and (ABS2)) we have uniqueness of the risk return relationship. The maximum return portfolio is (ABS1) is

⎥⎦

⎤⎢⎣

⎡

2

1ww

= ⎥⎦

⎤⎢⎣

⎡−4.14.0

+ 1.4 ⎥⎦

⎤⎢⎣

⎡−11

= ⎥⎦

⎤⎢⎣

⎡01

and μ = 0.7.

We proceed by the minimum risk portfolio (ABS2). By (6) and (10) we have Var(P) = 8w1

2 + 7w1 + 2. The first derivative with respect to w1

1w

Var(P)∂

∂ = 16 w1 – 7.

By requesting

1w

Var(P)∂

∂ = 0

we find

194

wg =

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

169

167

= ⎥⎦

⎤⎢⎣

⎡0.56250.4375

and by (12) μg = 0.41875. The Standard Model, i.e. (6) is respected by

⎥⎦

⎤⎢⎣

⎡

2

1ww

= ⎥⎦

⎤⎢⎣

⎡−4.14.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.41875, 0.7]

and the Black Model, i.e. only (6a) and not (6b) is respected by

⎥⎦

⎤⎢⎣

⎡

2

1ww

= ⎥⎦

⎤⎢⎣

⎡−4.14.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.41875, ∞ ].

The short selling condition (6a) does affect the minimum risk solution in this example. ◊ If a matrix S ∈ Rnxn has a unique matrix S-1 ∈ Rnxn with S-1S = SS-1 = I then S-1 is called the inverse of S. Lemma 4.1: If S is positive definite then there exists an inverse Matrix S-1. Proof: Following [10, Neumaier] we have

195

SST = I. The assertion follows from ST = S-1. ◊ The minimum variance portfolio is discussed in detail in the relevant literature [11, Kleeberg]. It offers a variety of interesting characteristics. For example, it does not depend on forecasts. Lemma 4.2 (global minimum risk portfolio): It is assumed that S is positive definite. Furthermore the inverse of S is denoted with S-1. With

ID =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

1...1

the solution wg of σ =

wminwT S w

for the Black model is

wg = IDSID

IDS1

1

−

−.

Proof: The Lagrange function is L = wT S w + λ ID. The derivative is

196

iw

L∂∂ = Swi + λ, i = 1,....,n

and from the optimality condition we have w = λS-1 ID and from (6a)

λ = IDSID

11−

which yields the results. The minimal condition has to be checked by the second derivative. ◊ In the following example S is only semi positive definite and not positive definite. Example 4.3 (Minimum risk portfolio): We consider the correlation matrix

ρ =⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

111111111

.

As shown in Example 3.5 this matrix has an Eigenvalue 0 with multiplicity 2 and Eigenvalue 3. There is a two dimensional Eigenspace to the Eigenvalue 0 with the basis

v1 = λ1 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

011

, λ1 ∈ R1, v2 = λ2 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−101

, λ2 ∈ R1 (4.2.15)

and a one dimensional Eigenspace with Eigenvalue 3

197

v3 = λ3 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−111

, λ3 ∈ R1.

For minimal risk portfolio we consider the Eigenspace of the Eigenspace 0. In the Standard Markowitz model we have the conditions w1 ≥ 0, (4.2.16a) w2 ≥ 0, (4.2.16b) w3 ≥ 0 (4.2.16c) and w1 + w2 + w3 = 1. (4.2.17) Let v = v1 + v2 where

v = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

www

there follows from (15) λ2 = 0, hence (16c) w3 = 0 with (17) w1 + w2 = 2λ1,

198

thus (17) λ1 = 0.5 here is only one minimal risk portfolio with weight by (15)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

www

= ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

05.05.0

.

In the Black model we have the condition w1 + w2 + w3 = 1 there is only one minimal risk portfolio with weights

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

www

= ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−111

.

◊ Example 4.4 (relative Optimization with 2 assets): We again assume the return (9) and the risk (10). We consider the benchmark weights (4) with n = 2

⎥⎦

⎤⎢⎣

⎡

2

1bb

= ⎥⎦

⎤⎢⎣

⎡5.05.0

.

The return is the same as in (11) and the risk is Var(P) = p1

2 var(A) + 2 p1 p2 cov(A, B) + p22 var(B) =

(4.2.13) 3 p1

2 – 3.0 p1 p2 + 2 p22.

199

The REL1 is solved by

⎥⎦

⎤⎢⎣

⎡

2

1

ww

= ⎥⎦

⎤⎢⎣

⎡0.00.1

, ⎥⎦

⎤⎢⎣

⎡

2

1

pp

= ⎥⎦

⎤⎢⎣

⎡− 5.0

5.0. (4.2.14)

The return is μ = 0.7 and (13) with (14) yields Var(P) = 2.0. The REL2 is solved by

⎥⎦

⎤⎢⎣

⎡

2

1

ww

= ⎥⎦

⎤⎢⎣

⎡

2

1

bb

= ⎥⎦

⎤⎢⎣

⎡5.05.0

, ⎥⎦

⎤⎢⎣

⎡

2

1

pp

= ⎥⎦

⎤⎢⎣

⎡0.00.0

.

Thus from (8) we have μ = 0.45 and Var(P) = 0. The Standard Model, i.e. (6) is respected

⎥⎦

⎤⎢⎣

⎡

2

1ww

= ⎥⎦

⎤⎢⎣

⎡−4.14.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.45, 0.7]

and the positions are

⎥⎦

⎤⎢⎣

⎡

2

1

pp

= ⎥⎦

⎤⎢⎣

⎡−9.09.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.45, 0.7]

and the Black Model, i.e. only (6a) and not (6b) is respected is

⎥⎦

⎤⎢⎣

⎡

2

1ww

= ⎥⎦

⎤⎢⎣

⎡−4.14.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.45, ∞ ]

and the positions are

⎥⎦

⎤⎢⎣

⎡

2

1

pp

= ⎥⎦

⎤⎢⎣

⎡−9.09.0

+ 2μ ⎥⎦

⎤⎢⎣

⎡−11

, μ ∈ [0.45, ∞ ].

In the risk σ return μ space we have

200

σ(μ) = 3 (2μ − 0.9)2 – 3 (2μ − 0.9)(0.9 − 2μ) + 2 (0.9 − 2μ)2 = 4 (2μ − 0.9)2 = 16μ2 − 14.4 μ + 3.24. ◊ Asset A and Asset B represent risk and return of two risky securities such as equities. Figure 4.6 portrays the risk and return on various portfolios comprising securities A and B. In actual practice, however, portfolios comprising many risky equities are optimized. In such cases, the efficient frontier looks the same or at least similar, depending on the boundary conditions and the input parameters. Alongside risk and return, an additional important parameter for portfolio optimization is correlation. This is a measure for the movement of securities relative to each other. In mathematical terms, the correlation must lie between the two extremes of –1 and +1. A and B are linked in three different ways in Figure 4.6, corresponding to three different correlation values. For a correlation of +1, the risk-return curve is a straight line linking both securities. For a correlation of –1, the efficient portfolios are at least partially linear, and one of the portfolios is risk-free. This is on the return axis in Figure 4.6. And for a correlation value between –1 and +1, A and B are linked by a curve; optimization programs derive portfolios that are on the efficient frontier. What's key is that risk declines in line with correlation. Figure 4.6 shows that only the portfolios above the minimum variance portfolio are optimal. The part of the efficient frontier below the minimum variance portfolio is described as the "inefficient branch" in the relevant jargon. What's clear is that optimization is one approach to quantitative modeling available to the financial industry, allowing forecasts to be expressed in the form of a concrete portfolio

201

recommendation. Critics say that optimization is very easily skewed by forecasting errors. If there is a benchmark, optimization can be conducted in as stable a manner as required; for, if the investor then buys that benchmark, he is actually immune to portfolio optimization and the forecasts made. The well-known approach of Black und Littermann is essentially based on this concept. Figure 4.6 and 4.7 are extreme simplifications. They merely reflect the risk and return for different portfolios and assume that the risk and return of the individual investments can be defined. But it is precisely the forecasts and the estimates of risk that are critical. It is also important that the optimal portfolios move in line with changing forecasts and changing market data. Hence, the optimal portfolio is constantly in a state of flux and has to be optimized continuously. "Stability is the enemy of optimality" is therefore often heard among specialists. As mentioned before mathematics has solved the portfolio optimization problem. However, the practical benefit of the optimized portfolio is highly dependent on the accuracy of the financial market forecasts. And it is precisely their verification that forms the subject of many empirical studies, and offers the basis for further development of Markowitz's original portfolio optimization theory [6, Diderich, Marty], [19, Rustem, Marty].

The basic findings of MPT can be expressed as follows:• Higher risk is always accompanied by higher average return.• The interaction of the different investments in the portfolio

is reflected.

In the literature it was pretended that the efficient frontier is continuously differentiable but in [25, Varös] it is shown that the efficient frontier is in general only continuous by the following example.

202

Example 4.5 (non differentiability of the efficient frontier): We consider in (3.4.5)

S = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

7523123113

133

and (5)

r = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

531

.

An algorithm for computing optimal portfolios yields the following results. a) We assume conditions (6). For the global minimum variance portfolio we find μmin = μG = 1.2, 2

Gσ = 2.8, wG = (0.95, 0, 0.05). The maximal value is μmax = 5. The efficient portfolios consist of the following pieces. Stability intervals are μ ∈ [1.2, 1.5]: σ2(μ) = 5μ2 − 12μ + 10; μ ∈ [1.5, 2.0]: σ2(μ) = μ2 + 1; μ ∈ [2.0, 3.0]: σ2(μ) = 2μ2 − 4μ + 5; μ ∈ [3.0, 5.0]: σ2(μ) = 10 μ2 − 48 μ + 65. Except for μ = 3 is σ2(μ) differentiable on [1.2, 5.0]. σ2(μ) has a kink at μ = 3. In Figure 4.8 we see the standard Markowitz model.

203

Efficient portfolio for the Standard Model Figure 4.8 b) We assume only condition (6a). For the global minimum variance portfolio we get μmin = μG = 0, 2

Gσ = 1.0, wG = (2.0, −1.5, 0.5) and μmax = ∞ . We have μ ∈ [0, ∞ ]: σ2(μ) = μ2 + 1. The efficient portfolios are along the straight line

w(μ) = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

5.05.1

2 + μ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

25.000.175.0

= ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

μ−μ+−μ−

25.05.05.1

75.02

for μ ≥ 0.

204

Figure 4.8 shows the numerical realisation of the Standard Model and the Black Model.

Efficient portfolios for the Standard and the Black Model Figure 4.9 With this example we have shown that the Standard Model can behave significantly different from the Black model. ◊

205

4.3. Absolute risk-adjusted measures One aspect of portfolio analysis is the examination of the portfolio along the time axis. The comparison of a specific portfolio to similar portfolio is another aspect of portfolio analysis, i.e. peer analysis as of now is the contents of the section. Certainly the investor looks first at returns and then at the risk taken. We proceed by considering measures that consists of return and risk that is we introduce risk-adjusted return measures. Definition 4.6: We assume that a return series (3.1.1) is given. The coefficient of variation CVA is the ratio of the standard variation (3.1.2) to the arithmetic mean return (2.3.4c)

CVA(P) = Pr

)P(std .

We note that the denominator and the numerator in this ratio have the same unit. The coefficient of variation reflects the basic finding that for higher return we expect higher risk, i.e. if Portfolio Managers 1 has achieved double as much return and risk and a Portfolio Managers 2, they are having the same CVA and their skills are the same. If the return Pr of the portfolio is positive that it can be said qualitatively that a low CVA is desirable. Example 4.6 (higher risk but lower coefficient of variation): We consider a return series (3.1.1) with arithmetic mean Pr . We define for σ ∈ R1 and N = 1,…. rP,k = Pr − σ, k = 1, 3,…., 2N − 1, rP,k = Pr − σ, k = 2, 4,…., 2N.

206

Then we have std(P) = σ and

CVA(P) = Prσ .

In Example 3.1 we have Pr = 10% and σ = 5% thus

CVA(P) = 21 .

With Pr = 10% and σ = 25% we find

CVA(P) = 52 .

We see that the relation between the risks and the CVAs is inverse. ◊ We continue with an illustration from physics. We consider two drivers who drove from A to B, C to D, resp.. We assume that two drivers drove with constant speed. The question is: Which driver is faster? If the distance, the time, resp. is same the question can be answers by the time, the distance, resp. the two drivers drove. If the distances and the times are different we need to normalize or gauge either by the distance, by the time, resp.. The speed defined by the distance divided by time has to be used instead

207

of distance (time), resp.. Here is the comparison to portfolio analysis. If we want to compare two portfolios with both different returns and risks. Definition 4.7: The risk adjusted return RAR is the inverse of CVA, i.e.

RAR = )P(std

rP .

Following Definition 2.6 and similarly to (3.1.1) we denote with rf,k, k = 1, 2,…., N the risk-free return in period k, k = 1,….,N and following Definition 3.2 the arithmetic mean return fr the considered series of risk-free return is

∑=

=N

0kk,ff r

N1r .

As risky investments, risk-free investments enter also as part in modern portfolio theory. The following risk-adjusted return measure is named after Dr. William Sharpe. It reflects the fact that the investor cannot earn a return above the risk-free return without risk. Definition 4.8: The Sharpe Ratio SR is the difference between the annual arithmetic mean return and annual arithmetic mean risk-free return divided by the annualized deviation of the fund, i.e.

SR = )P(std

rr fP −.

208

Remark 4.3: CVA, RAR and SR are used for classifying a set of portfolios. They map return and risk figures in the set of the real numbers. If a portfolio manager is judged by SR, he tried to maximize his SR because he attempts to achieve a high return with little as possible risk. Even the return an equal the SR can be different. If this return is positive the Portfolio manager with less risk is superior to that with more risk. For negative return this judgment is opposite which make no sense. Strictly speaking the domain of applicability for SR is limited to positive return. The same is valid for CVA and RAR. 4.4. The Capital Asset Pricing Model (CAPM) The assumptions of the CAPM are [page 194, 4, Copeland]: • Investors evaluate portfolios by looking at the expected

returns and standard deviation of the portfolios over one-period time horizon.

• Investors are never satiated, so when given a choice between two otherwise identical portfolios they will choose the one with the higher expected return.

• Investors are risk-averse, so when given a choice between two otherwise identical portfolios they will choose the one with lower standard deviation.

• Individual assets are infinitely divisible, meaning that an investor can buy a fraction of a share if he or she so desires.

• There is a risk-free rate at which the investor may either lend or borrow money.

• Taxes and transaction costs are irrelevant. In the following we considered one period tk = k, tk+1 = k + 1, k = 0,….,N − 1 and portfolios that consist of risk-free investments and risky investments (see Remark 4.1). As

209

risky securities, risk-free investments are also part of modern portfolio theory. We return to the return and risk considerations in the previous chapter. We distinguish between the active and the passive investors. In the previous chapter an active investor was considered as he could steer his return risk preference. A passive investor does see any opportunities in market place and clearing prices prevail. Every investor will hold a portfolio that consists of a share of the market portfolio defined as follows: Definition 4.9: The market portfolio MP with return rMP is a portfolio consisting of an investment in all securities like stocks, bonds, real properties and on where the proportion to be invested in each security corresponds to its relative market value. The relative market value of a security is simply equal to the aggregate market value of the security divided by the sum of the aggregate market value of all securities. In principle the market portfolio is a thought experiment and it is not possible to capture any single available asset in a portfolio. My and you belongings are part of the market portfolio. However, big universes like the MSCI world equity can serve as a proxy. A first market equilibrium principle yields based on two-fund separation principle described in [4, Copland, page 181]: Theorem 4.1: Each investor will have a utility maximizing portfolio that is combination of the risk-free asset and a portfolio (or fund) of risky assets that is determined by the line drawn from the risk-free rate of return tangent to the investor’s efficient set of risky assets. The portfolio that is on the tangent to efficient frontier and on the efficient frontier is the market portfolio. Proof: We consider a portfolio consisting of the risk-less asset RF and the market portfolio MP

210

Pr = w1rf + (1 − w1) rMP and for the standard deviation of the portfolio we have std(P)

= ( )( )21

2111

21 var(MP)wMP))cov(RF,w(1w2var(RF)w-1 +−− =

w1 std(MP). As var(RF) = 0 and cov(RF, MP) = 0 we have

w1 = )MP(std

)P(std

and

Pr = ⎟⎠

⎞⎜⎝

⎛ −)MP(std

)P(std1 rf + )MP(std

)P(std rMP,

hence

Pr = rf + )MP(std

)P(std (rMP − rf)

and in the risk return graph we have

Pr = rf + )MP(std

rr fMP − std(P).

◊ Definition 4.10: This line specified in Theorem 4.1 is called the capital market line (CML) (see Figure 4.6).

211

Remark 4.3: The proof of Theorem 4.1 does respect any constraints. Their impact on the CML is still subject to future research.

CML

Risk

Return

Market portfolio

Efficient Frontier

CML

Risk

Return

Market portfolio

Efficient Frontier

The modern portfolio theory Figure 4.6 Remark 4.4: As expected approximations of the Market portfolio are not efficient [5, Dalang R.,]. Remark 4.5: RAR are slopes on the efficient frontier and the SR is the slope of the CML. We note that the investors have the same Sharpe ratio

)MP(std

rr fMP − , (4.4.1a)

i.e. the investor is risk neutral along the CML. Theorem 4.2 (Capital asset pricing model (CAPM)): We consider a Portfolio P that consists of a risky asset C1 with return r1.Together with return of the risk-less asset rf and the Market portfolio MP with return rM we have: r1 − rf = β(C1,MP) (rMP − rf) (4.4.2a)

212

where

β( C1,MP) = )MPvar(

)MP,Ccov( 1 . (4.4.2b)

Proof: We consider a Portfolio P that consists of an asset C1 with return r1 and a weight w1 and the market portfolio MP considered as an asset. For the return of the portfolio we have Pr = w1r1 + (1 − w1) rMP and for the standard deviation of the portfolio we have std(P) =

( ) ( )( )21

211111

21 var(MP)w1MP),)cov(Cw(1w2)var(Cw −+−− .

We derive the return and the standard deviation of the portfolio with respect to w1:

1

p

wdrd

= r1 − rMP,

( ) 21

1)P(std

wd)P(stdd −

= (2w1var(C1) +

2(1 − w1) cov(C1, MP) − 2 w1cov(C1, MP) + 2(1 − w1) var(MP)). For the derivative in the return risk graph in the market portfolio, i.e. w1 = 0 we find

213

01w

p

)P(stddrd

=

=

01w1

1

p

wd)P(stdd

wdrd

=

=

)MP(std)MPvar()MP,Ccov(

rr1

MP1−

− .

As the slope has to be same the slope (1), hence

)MP(std)MPvar()MP,Ccov(

r r1

MP1−

− = )MP(std

rr fMP − ,

thus

r1 − rMP = (rMP − rf ) ⎥⎦

⎤⎢⎣

⎡ −)MPvar(

)MPvar()MP,Ccov( 1 =

rMP ⎥⎦

⎤⎢⎣

⎡ −1)MPvar(

)MP,Ccov( 1 − rf ⎥⎦

⎤⎢⎣

⎡ −1)MPvar(

)MP,Ccov( 1

which yields (2). ◊ Remark 4.6: β(P, MP) is called the CAPM beta. We summarize the CAPM as follows

The excess return of a risky investment is equal to the CAPM

beta times the excess return of the market portfolio return.

214

We conclude that the risk asset can be classified by the CAPM beta. The CAPM is one of the fundamental buildung blocks of MPT. For a qualititive description we refer to [9, Feibel, page 191] and for a more quantitive formulation we refer to [4, Copeland]. CAPM has the following properties: • (2) is linear, i.e. (2) is also valid for a portfolio P with return

rP = ∑ ==

n

1jjjrw w1 r1 + w2 r2 + ….. + wn rn,

i.e. rP − rf = β(P, MP) (rP − rMP) (4.3.3a) where

β(P,MP) = )MPvar()MP,Pcov( . (4.3.3b)

• The CAPM β(P,MP) is consistent with the least square β in

(3.3.9). • For the market portfolio we have

β(MP,MP) = )MPvar(

)MP,MPcov( = )MPvar()MPvar( = 1.

We note that there are also other portfolios that have β(P, MP) = 1. These are tracking portfolios of MP, i.e. Portfolios that are supposed to follow the market.

215

• An investment that has no risk will earn the risk-free return, where risk is measured by the volatility of returns.

• Two different types of risk causes the volatility in investment

returns. The first is the market risk of the investment, which reflects the degree to which investment values vary when the level of prices in the underlying market changes. Volatility that affects the market as a whole is assumed to correspond to changes in the underlying factors that influence market prices in general. For example, if we assume that stock valuations respond positively to increases in the growth rate of the economy as a whole, we can say that the economic growth factor is to some degree common to all stocks. Economic growth, and other factors which influence the value of market prices as a whole, are systematic risk factors.

• Systematic risk factor are reflected in the market returns,

therefore, we can isolate the influence of systemtic factors on an individual asset by observing market returns.

• All investment within the market are influenced by

systematic risk, but the degree of exposure to systematic risks varies from asset to asset.

• Factors specific to the asset can generate volatility, for

example, a change of mangement within a company. This type of volatility is unique to the asset, or unsystematic.

• Of the two components of the risk, those risks specific to

particular assets can be eliminated via portfolio diversification. We assume that the particular risks of different assets will offset each other in a diversified investment fund.

216

• Because the unique risk is diversifiable, the market does not reward the asset with a premium for this risk. The market rewards only the exposure to systematic risk. This is because systematic risk cannot be diversified away and is thus borne by every investor in the market.

• Investment are awared a degree of return over the risk-free

rate, or a risk premium, based on the degree of market risk. The reward to risk ratio is a linear function: For each unit of systematic risk the asset is exposed to the market, the market will reward the asset with one unit of excess return.

• An investment that has risk exposure similar to the market

taken as a whole will have returns that vary in line with the market. If an asset is less exposed to market factors, its returns will vary to a lesser degreee as the market as a whole. If the investment has greater exposure to systematic factors, returns will vary to a greater degreee than the market returns.

The graphical representation of the CAPM in the security market line. In Figure 4.7 we have the β version and in Figure 4.8 the covariance version.

• Investors expect a risk premium in exchange for bearing this risk.

217

)MP,P(β

Pr

fr

1)MP,P( =β

MPrM

)MP,P(β

Pr

fr

1)MP,P( =β

MPrM

Figure 4.7

)MP,P(β

Pr

fr

1)MP,P( =β

MPrM

)MP,P(β

Pr

fr

1)MP,P( =β

MPrM

Figure 4.8 We mention here the dawn back of the CAPM in [4, Copeland, page 217], referring to [20, Ross]. 1. The only legimate test of the CAPM is whether or not the

market portfolio (which includes all assets) is mean - variance efficient.

2. If performance is measured relative to an index that is expost efficient, then from the mathematics of the efficient ser no security will have abnormal performance when when mearued as a departure from the security market line.

3. If performance Is measured relative to an ex post inefficient index, then any ranking of portfolio performance is possible, depending on which inefficient index has been chosen.

218

One application of the CAPM beta is the calculation of the

Treynor Ratio. The Treynor ratio can be used to rank the desirability of a particular asset in combination with other assets, where part of the total risk inherent in the standard deviation will be diversified. The Treynor Ratio is the return in excess of the risk-free rate divided by the beta. Definition 4.11: The Treynor ratio TR is the difference between the annual arithmetic mean return and annual arithmetic mean risk-free return divided by the annualized Beta of the fund return, i.e.

TR = )MP,P(

rr fPβ

− .

The CAPM can be applied for forecasting the return of a asset or a portfolio or measuring the deviation α(P,MB) of an asset or a portfolio to the relation (2), i.e. r1 = α(C1, MP) + rf + β(C1, MP) (r1 − rM) and by (3) rP = α(P, MP) + rf + β(P, MP) (rP − rM). This motivates the following definition: Definition 4.12: The CAPM Alpha or Jensen’s Alpha α(P, MB) is the factor that reconciles actual return to those predicted by the CAPM defined by α(P, MP) = rP − rf − β(P, MP) (rP − rM). In this regression rM is the independent variable and rP is the dependent variable. If α is greater than zero, the fund has a

219

return higher than expected by the CAPM. A negative α indicates that the fund performed worse than predicted given the market risk taken. 4.5. Relative adjusted measures Tracking error and down side risk is useful when measuring the active risk. In this section we present two measures that relate relative return to relative risk return. Definition 4.13: The Sortino ratio SR is the ratio between the annual average difference and the target return T provided by the benchmark and the annualized downside deviation, i.e.

SR = ( )∑≤=

⋅−

⋅−N

Tjr1k

2j

P

fT)(rN1

fTr

where f is the periodicity of the data per year. Definition 4.14: The Information ratio IR is a measure of the benchmark relative return gained for taking on benchmark relative risk

IR = ( )

)1(TE

rrN1 N

1k k,Bk,P∑ −=

and

annualized information ratio = ( )

f)1(TE

rrN1 N

1kk,Bk,P∑

=−

220

where f is the periodicity of the data per year. The information ratio has the same role for relative considerations as the Sharpe ratio has for absolute return consideration.

221

5. INVESTMENT CONTROLLING 5.1. Investment controlling and the investment process 5.1.1. Introduction Nowadays it is more and more important for an asset management company to have efficient and appropriate management information on the performance of their discretionary managed portfolios. Without decision-oriented information on the performance or quality of its products and/or asset managers a specific asset management company will find it more and more difficult to stand the increasing challenges of the asset management industry in future. Clients and consultants have similar needs where these often correspond to the asset manager one’s some years ago. Investment controlling deals with such needs and will be explained in detail in the following. 5.1.2. Definitions Investment controlling is an area of activity that is part of the overall controlling within the asset management and is an important component of the recurring investment decision making process. Definition 5.1 (general): From an asset management company point of view, investment controlling is defined as information management whose task it is to gather, to process, to check and to distribute information necessary to meet the overall objectives of the asset management company. In this respect the investment controlling objective consists in configuring the infrastructure – particularly within the framework of

222

the investment decision making process – in such a way that the processes (e.g. forecasting, decision making and implementation), the quality and the results (e.g. returns), the risks (e.g. of using derivatives) and the costs become more transparent and comprehensible. Definition 5.1’ (specific) : Considering the client perspective, in the following investment controlling is in general defined as independent monitoring of the performance of asset management products and/or accounts with the aim of ensuring that the client gets what was promised in the first place with respect to quality and performance. The investment decision making process illustrated schematically in Figure 4.1 may in reality be highly complex due to the large number of possible investment instruments, of decision makers involved (such as the research team, asset allocation team and specialists in the various investment categories) as well as of not always transparent levels of the decision making. The effect of this complexity is that the results of the investment process and its determining factors are not always apparent. In this context, investment controlling intents to visualise the contributions of the individual decisions of the investment process, especially with respect to return and risk, and to allocate them to the responsible decision makers. The results and conclusions of the different investment controlling activities are an important feedback and input into the investment process to enhance the quality or performance of the specific asset management product. Investment controlling is an important part of the investment process. It is the last step of the investment process and analyses the result of the overall process - the account performance - with respect to the sources of return as well as of risk. These results

223

are again input for the investment process and therefore the starting point for the ongoing investment process. Furthermore as illustrated in Figure 5.1 investment controlling may analyse all steps of the investment process starting from the definition of the investment target up to the re-balancing of the actual portfolio. As shown in Figure 5.1, depending on its setup, investment controlling not only encompasses pure controlling activities but also compliance and risk management tasks as well as focuses not only on asset managers but is also concerned with the impact of consultants and may be also of that of the client. In Switzerland, the most know and influential consultants are Ecofin, Complementa Investment - Controlling AG and PPCmetrics. Internationally we mention Russel and Wyson Watt. If the investment controlling concentrates on performance or quality related aspects and does not address financial aspects one could also speak about performance monitoring. In the following we use performance monitoring and investment controlling interchangeable.

Investmentcontrolling

Compliance

PerformancemonitoringFinance

costs/revenues

Risk management(non-investment)

Reporting(internal and external)

BehaviourProcessesForecasts Results

Investmentcontrolling

Compliance

PerformancemonitoringFinance

costs/revenues

Risk management(non-investment)

Reporting(internal and external)

BehaviourProcessesForecasts Results

Different facets of investment controlling Figure 5.1

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5.1.3. Purpose and objectives Form a general point of view investment controlling adds to the visibility, transparency and credibility of any asset management company. In detail investment controlling helps • Implementing best practice in performance measurement and

performance presentation, for example by implementing the GIPS standards

• Producing an independent performance analysis of the asset management accounts and/or products

• Enabling deep level analysis which is necessary to identify the real drivers of the account return and account risk and this from an ex-post as well as from an ex-ante point of view

• Monitoring risk and return of accounts and/or products against their designated benchmark and objectives, capturing performance dispersion

• Reducing unnecessary discussions by using more objective and less subjective information during the performance review

• Creating of or increasing the transparency and comparability of the asset management products and/or accounts

• Addressing performance issues on an regular basis and not leaving them running

• Creating a basis not only for ongoing analyses but also for structural changes in the investment process

• Reducing of unintended business risks through early addressing of potential performance issues

225

5.1.4. Investment controlling activities

Following the different aspects of the investment controlling one can easily imagine that investment controlling is very manifold and encompasses a lot of different activities like:

• Performance attribution or more precisely return attribution and/or risk attribution and this ex-post as well as ex-ante

• Market index and benchmark comparisons, composite dispersion analysis and peer group analysis with respect to the return and/or risk but also to characteristics like asset class or sector weights, duration, exposure to specific risk factors and so on

• Calculation of performance figures and statistics representing manager skills or the investment style and running style analysis

• Review of the set up of the specific asset management account with respect to benchmark, investment guidelines, transaction costs and management fees, etc.

• Product review against client expectations and best practice

• Identifying of actual and potential performance issues and highlighting the serious ones to senior management

• Suggesting remedial action to solve performance issues

• Risk decomposition and risk budgeting

• Analyzing and identifying all steps of the investment process

• Review of the investment guidelines and benchmarks

• Checking whether risk levels and limits are appropriate

• (Aggregated) performance reporting to senior management

226

5.1.5. Necessary resources

To do an efficient and meaningful investment controlling a lot of requirements has to be fulfilled whereas it is very seldom that all of them are met:

• Investment controllers with both general controlling and asset management background or know-how (in theory and practice)

• Availability of all needed data in good quality within a reasonable timeframe

o Daily data (holdings and prices) o On stock level o For the account as well as for the benchmark o On forecasts as well as tactical asset allocations or actual

transactions

• Peer group information

• Availability of appropriate analytical tools to run performance attribution or simulations as well as tools to maintain composites or to do the performance reporting

• Asset management agreements or mutual funds prospects and investment guidelines

Without a critical amount of resources and without the relevant data an investment controlling can not be efficient and meaningful. For example, running a risk attribution for a Swiss fixed income portfolio with a risk analytics tool that does not incorporate a Swiss credit model may be risk decomposition but is still useless from an investment controlling point of view because it does not reflect the investment process.

227

5.2. Building blocs and set up of investment controlling 5.2.1. Introduction Following a general controlling setup the investment controlling process or in other words the performance monitoring process can be split up as illustrated in following figure 5.2.

Portfolio analytics / risk control (qualitative aspect)

Performancewatch list

Performancereporting

Monitoring of results (ex -post) and of Inputs (ex -ante)

Performancemeasurement

Performanceadministration

Performanceanalysis

Portfolioanalytics

Feed forward and feed back

Efficient controlling of forecast and of investment process outcome

Performancereview

Production / reporting ( quantitative aspect) Portfolio analytics / risk control (qualitative aspect)

Performancewatch list

Performancereporting

Monitoring of results (ex-post) snd of inputs (ex-ante)-

Performancemeasurement

Performanceadministration

Performanceanalysis

Portfolioanalytics

Feed forward and feed back

Performancereview

Production / reporting ( quantitative aspect)

The investment controlling or performance monitoring process Figure 5.2

The first four steps are quantitatively oriented and are used for calculating, maintaining or storing, visualising and analysing different performance figures of a specific account or a specific asset management product or composite. The last three steps are less production oriented and instead deliver qualitative statements on the investment process and its results. The performance watch

228

list determines problematical asset management accounts and/or products that are subsequently analysed by portfolio analytics and are discussed in depth in the performance review. In this respect, working out proposals for improvement and pointing out possible consequences for the investment process are the primary objectives of the qualitatively oriented analyses conducted in steps 5 to 7. In the following we illustrate the investment controlling or the performance monitoring process whereas for more details we refer to the other relevant chapter of the course. The purpose of this section is to get an overview of the different parts of the performance measurement and analysis and to see how to incorporate them to one general investment controlling framework. We proceed by the different steps of the performance monitoring process 5.2.2. Different steps of the performance monitoring process a. Performance measurement

This first step of the performance monitoring process deals with all aspects of return and risk measurement, i.e. the calculation of all necessary return and risk measures or figures like gross and net account returns, time weighted and money weighted rate of returns, risk figures such as volatility or tracking error and so on. Performance measurement normally focuses on the total account level and is a time series analysis. Figure 5.3 shows an example for a performance measurement report showing all the necessary input data to calculate a total return gross and net.

229

b. Performance administration Performance administration normally covers the

benchmark calculation and more importantly the composite construction and maintenance.

Sample performance measurement report Figure 5.3 Definition 5.2: A composite is an aggregation of one or more portfolio managed according to a similar investment mandate, objective, or strategy.

.

The composite return is the asset weighted average of the performance of all portfolios in the composite. Creating meaningful composites is essential to the fair presentation, consistency, and comparability of performance over time and among firms.

.

The composite return is the asset weighted average of the performance of all portfolios in the composite. Creating meaningful composites is essential to the fair presentation, consistency, and comparability of performance over time and among firms.

230

In order to assess the asset management skills of one or more asset managers or even of a whole firm it is necessary to classify the different accounts and assign accounts with same characteristics like same benchmarks, same investment strategies and/or styles to a composite as illustrated in Figure 5.4. Constructing and maintaining composites according to the GIPS standards may also be part of the performance administration. Being in compliance with the GIPS standards means best practice with respect to transparency and comparability in the area of performance presentation and is seen as the basis for an efficient investment controlling of an asset management company.

Idea of composite construction Figure 5.4 c. Performance reporting This step of the performance monitoring process includes the reporting of different performance figures for specific time periods - normally on a total account level. If an observer is interested in more detailed information on the sources of return and risk, he may start reviewing the performance figures on a total account

231

level and afterwards analyses a performance attribution which will be explained in the next section.

The following Figures 5.5 to 5.8 illustrate a sample performance reporting for a specific equity composite where the layout and the chosen analysis do not have to be composite or account specific. The report setup is normally client or user specific and varies from observer to observer. In practice depending on the size of the asset management company these kinds of reports are produced by a spreadsheet or an external software solution. Considering the external software solution it is often the case that the composite maintenance software and the reporting software is part of a single software solution. In setting up a performance report one has to define a lot of reporting characteristics like: • Which product, composite or account should be analysed? • Which time period should be considered? • Whether gross or net return should be reported? • Which return and risk measures should be shown? • Whether rolling and/or annual performance figures should be

presented?

It should be noted that it is crucial for getting the whole picture to analyse the account or composite performance from different aspects since performance figures are very sensitive to used methodology, time periods and input data. This means that varying the time period for example only for one month forward may lead to the fact that a very underperforming account transfers to a very outperforming account. Another example may be the choice of the risk free rate if calculating the Sharpe Ratio. In continental Europe it is common to use the relevant total return of the alternative risk free investment and in other countries it is more common to use the actual risk free rate. Depending on the chosen type of risk free

232

rate you may come up with totally different conclusions. Important is here especially whether you use the performance figure as a measure to compare past performance or to estimate and rank future performance or in other words whether you do an ex-post or a ex-ante performance analysis.

Sample performance report 1 Figure 5.5

233

Sample performance report 2 Figure 5.6

Sample performance report 3 Figure 5.7

E q u itie s W o rld B M M S C I ac tive M and a te s d irec t B e nch m ark M S C I W orld (r i) in C H F P e rfo rm ance Typ e A sse t W e igh te d G ro ss R e tu rn B as e C urrency C H F N o . o f A /C s < 5 Inceptio n D a te 01 Jan 19 97 M arke t V a lu e (m ) 84 .2 7 C o m p os ite C o de ZU -C O M P 25 0 D a te 31 D ec 2 00 3

-60

-40

-20

0

2 0

4 0

6 0

8 0

Jan

97

Nov

97

Sep

98

Jul 9

9

May

00

Mar

01

Jan

02

Nov

02

Sep

03

1 Ye

ar T

otal

Ret

urn

C om p o s ite

B en ch m ark

1 4

1 6

1 8

2 0

2 2

2 4

2 6

2 8

3 0

Jan

97

Nov

97

Sep

98

Jul 9

9

May

00

Mar

01

Jan

02

Nov

02

Sep

03

1 Ye

ar V

olat

ility

C o m po s ite

B e nch m ark

-10

-5

0

5

10

15

20

Jan

97

Nov

97

Sep

98

Jul 9

9

May

00

Mar

01

Jan

02

Nov

02

Sep

03

1 Ye

ar E

xces

s Re

turn

0

1

2

3

4

5

6

7

8

Jan

97

Nov

97

Sep

98

Jul 9

9

May

00

Mar

01

Jan

02

Nov

02

Sep

03

1 Ye

ar T

rack

ing

Erro

r

-4

-3

-2

-1

0

1

2

3

4

5

6

Jan

97

Nov

97

Sep

98

Jul 9

9

May

00

Mar

01

Jan

02

Nov

02

Sep

03

1 Ye

ar In

form

atio

n Ra

tio

Benchmark MSCI World (ri) in CHF Series Type Asset Weighted Gross Return Reporting Currency CHF No. of A/Cs 5 Inception Date 01 Jan 1997 Market Value (m) End of Period 84.27 Composite Code ZU-COMP250 Reporting Date 31 Dec 2003

Indexed Cumulative Relative Returns

90

95

100

105

110

115

120

125

130

135

Incep Oct 97 Aug 98 Jun 99 Apr 00 Feb 01 Dec 01 Oct 02 Aug 03

Inde

xed

Retu

rns

Composite Benchmark Relative

1 Month 1.67 1.69 -0.03 3 Months 6.60 7.11 -0.50 6 Months 8.43 9.58 -1.15 1 Year 17.98 19.64 -1.66 2 Years -12.65 -10.46 -2.18 3 Years -13.77 -11.82 -1.96 4 Years -13.70 -11.83 -1.87 5 Years -2.33 -2.46 0.13 Since Incep. 4.05 3.93 0.12

Periodical Returns in %

Composite Benchmark Relative

YTD 17.98 19.64 -1.66 2003 17.98 19.64 -1.66 2002 -35.32 -32.99 -2.33 2001 -15.99 -14.47 -1.52 2000 -13.46 -11.84 -1.62 1999 60.21 46.07 14.14 1998 21.29 17.51 3.78 1997 22.50 26.25 -3.75

Calendar Year Returns in %

Monthly Relative Returns

-4

-3

-2

-1

0

1

2

3

4

Jan 97 Nov 97 Sep 98 Jul 99 May 00 Mar 01 Jan 02 Nov 02 Sep 03

in %

Annual Risk Figures in % Composite Benchmark

Volatility over 1 Year 16.90 15.80 Volatility Since Inception 22.98 20.80 Sharpe Ratio over 1 Year 1.06 1.24 Sharpe Ratio Since Inception 0.10 0.10 Tracking Error over 1 Year 2.12 N/A Tracking Error Since Inception 4.69 N/A Information Ratio over 1 Year -0.78 N/A Information Ratio Since Inception 0.03 N/A Correlation over 1 Year 0.99 N/A Correlation Since Inception 0.98 N/A

Equities World BM MSCI active Mandates direct

234

Sample performance report 4 Figure 5.8 d. Performance contribution and attribution

Performance contribution and attribution is introduced and explained by model example in Chapter 2 and 3. Is a central component of the performance monitoring process and it is defined as a process that determines the return and risk contributions of the individual decision making steps within an investment process. Thus, performance attribution is concerned not only with the past but also with the future, and determines which return and risk contributions are due to which decisions (regarding investment category and instruments) and to which decision makers on an ex-post as well as ex-ante basis. Figure 5.9 illustrates the various levels of analysis of performance attribution as well as possible allocation criteria of return and risk contributions. It is evident that performance attribution can be

Equities World BM MSCI active Mandates direct Benchmark MSCI World (ri) in CHF Performance Type Asset Weighted Gross Return Base Currency CHF No. of A/Cs <5 Inception Date 01 Jan 1997 Market Value (m) 84.27

Composite Code ZU-COMP250 Date 31 Dec 2003

-40

-30

-20

-10

0

10

20

30

1 Month 3 Months 6 Months 1 Year 3 Years 2003 2002

Tota

l Ret

urn

(ann

. if >

1 y

ear)

Composite

Benchmark

0

5

10

15

20

25

30

1 Year 2 Years 3 Years 2003 2002 2001

Vola

tility

(ann

.)

Composite

Benchmark

-2.5

-2

-1.5

-1

-0.5

0

1 Month 3Months

6Months

1 Year 3 Years 2003 2002

Exce

ss R

etur

n (a

nn. i

f > 1

yea

r)

0

1

2

3

4

5

6

1 Year 2 Years 3 Years 2003 2002 2001

Trac

king

Erro

r (an

n.)

-0.9

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-0.7

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-0.2

-0.1

0

1 Year 2 Years 3 Years 2003 2002 2001In

form

atio

n Ra

tio (a

nn.)

235

carried out in a variety of different ways. On the one hand, return and risk contributions may be calculated on an absolute basis, i.e. isolated for an asset management account or for a specific benchmark, or on a relative basis, i.e. for an asset management account in comparison to a benchmark. On the other hand, the performance attribution may be focused on the past (ex post) or the future (ex ante). In summary, performance attribution is defined as the decomposition of historical or expected absolute or relative return and/or historical and expected absolute or relative risk.

Levels of analysis and allocation criteria of performance analysis Figure 5.9 As seen in Figure 5.9, in general performance attribution can be used to measure the return and risk contributions of categories, sectors and instruments (e.g. asset categories, countries, currencies or securities), of decision makers such as the client

236

himself, portfolio managers or consultants, and finally of asset management activities like the definition of the benchmark, the strategic or tactical asset allocation or the stock picking. Performance attribution has been developed since 1980’s. In the early days, the return and risk contributions were at first considered at an aggregated level – following a top-down approach. Particularly in the recent years, more detailed and more profound analyses were developed. It is increasingly possible to conduct performance attribution on a stock level and on a daily basis. In parallel with this development, the focus of performance attribution has shifted away from returns towards risk and risk-adjusted returns. However, depending on the investment category considered and the investment instruments used, risk analysis and risk-adjusted return analysis still require considerable further development. The development of the performance measurement and attribution is illustrated in Figure 5.10. In the following we present a sample performance attribution report which is split up into 6 parts (Figures 11 to 16): • Overview page covering aggregated return and risk information • Ex-post return decomposition according to sectors • Ex-post management effects according to sectors • Ex-post return decomposition according to countries • Ex-post management effects according to countries • Ex-ante risk decomposition

237

Portfolio view Detailed analysis(Return and risk decomposition)

Status quo /best practice

time

Performance measurement Performance attribution

Absoluteprofit

Return

"correct" returnmethodology

Benchmark

Risk

Asset classes/ countries

Sectors

Stocks

Derivatives

???

Fromquarterly to

monthly

From monthlyto daily to real

time

from ex post to ex ante

Status quo of performance attribution Figure 5.10

NAME (ID) Currency CHF Asset Allocation -1.01%

PM Return Portfolio 18.28% Stock Selection -0.59%

BENCHMARK Return Benchmark 19.60% Interaction 0.28%

PERIOD Return Relative -1.32% Total -1.32%

(Average over-/underweight) Attribution Analysis - by MSCI Sector

Risk Analysis (end period) Portfolio Benchmark Attribution Analysis - by 5 World RegionsNumber of Securities 67 1'550Number of Currencies 8 0Portfolio Value 84'334'091Total Risk (ex-ante) 18.81% 18.21%- Factor Specific Risk 18.66% 18.18%- Stock Specific Risk 2.39% 1.02%Tracking Error (ex-ante) 2.57%Value at Risk (at 95%) 3'570'469R-squared 0.98Beta-adjusted Risk 18.64% 18.21%Predicted Beta 1.02Predicted Dividend Yield 1.86 2.01P/E Ratio (E: 12 months) 28.39 26.00P/B Ratio (B: year-end) 2.58 2.56

Important Remark: Differences between attribution returns and the returns of the official performance measurement tool are usual. They can be explained by the two systems using two different methodologies and by intraday trading gains or losses. Above figures are subject to future changes.

-0.96%

0.13%

-0.49%

-1.32%

Composite World MSCI, active, mandates AMPE

MSCI World in CHF

31.12.2002 - 31.12.2003

PortfolioAttribution by MSCI

SectorAttribution by 5 World

Regions Return Attribution Effects

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

Consu

mer Disc

retionary

Consu

mer Staple

s

Energy

Financ

ials

Health

Care

Indus

trials

Inform

ation

Techn

ology

Materia

ls

Teleco

mm Serv

ices

Utilitie

sCas

h-1.5%

-1.0%

-0.5%

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mer Disc

retiona

ry

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Indus

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ation

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s

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mm Serv

ices

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sCas

hTota

l

Asset Allocation Stock Selection

-1.2%

-1.0%

-0.8%

-0.6%

-0.4%

-0.2%

0.0%

0.2%

0.4%

0.6%

Asia ex Japan Europe Japan North America Cash Total

Asset Allocation Stock Selection

Sample performance attribution report part 1 Figure 5.11

238

Average Average Relative DifferencePF Weight BM Weight Weight PF-BM

Consumer Discretionary 12.03% 12.18% -0.15% -7.49% Consumer Staples 8.83% 9.26% -0.43% 2.65% Energy 8.29% 7.41% 0.88% -2.79% Financials 21.70% 22.94% -1.24% -1.04% Health Care 11.73% 12.53% -0.80% 2.29% Industrials 10.45% 9.71% 0.74% 10.85% Information Technology 12.33% 12.49% -0.16% -7.73% Materials 4.84% 4.53% 0.31% -7.22% Telecomm Services 6.15% 5.24% 0.91% 2.38% Utilities 2.44% 3.72% -1.28% 18.64% Cash 1.21% 0.00% 1.21% 8.94% Total 100.00% 100.00% 0.00% -1.32%

Import remarks:- The weight numbers gives an overview of the average weight invested in the different groups (e.g. sectors) with daily weights averaged over the chosen period.- The absolute return numbers give an overview of the group's absolute performance (e.g. sector) within thc chosen time period (portfolio and bechmark)- The difference PF-BM compares the group's performance of the portfolio to the group's performance in the benchmark within the chosen time period.- The difference BM-BM Tot compares the group's performance in the benchmark with the performance of the total benchmark within the chosen time period.

5.09% 13.30%

by MSCI SectorAbsolute

BM Return23.07%

AbsolutePF Return

15.58% 7.74%

24.94% 7.31%

24.08% 32.79% 13.19% 30.33% 12.64% 15.45% 0.00%

-6.30% 5.34%

-12.30% 4.48%

DifferenceBM-BM Tot

3.47% -14.51%

10.73% -6.96% -4.15%

-19.60%

25.06%

19.60% 0.00%

10.51% 23.90% 9.60%

34.93%

23.11% 15.02% 34.09% 8.94%

18.28%

Please note that cash is included in the relevant country or regional groups and shown as "other assets" else (e.g. in sectors). Concerning derivatives, only futures are split up at the moment; derivatives (call or put options) are not yet included. Moreover, illiquid securities as private placements are not yet taken into account.

Sample performance attribution report part 2 Figure 5.12

A s s e t A llo c a tio n

C o n su m e r D isc re tio n a ry 0 .0 0 % C o n su m e r S tap le s 0 .0 5 % E n e rg y 0 .0 0 % F in an c ia ls -0 .1 0 % H e a lth C are 0 .2 0 % In d us tr ia ls -0 .0 4 % In fo rm a tio n T e c h no lo g y -0 .2 1 % M a te ria ls 0 .0 1 % T e le c om m S e rv ic e s -0 .1 1 % U tilit ie s 0 .0 2 % C ash -0 .7 8 % T o ta l -0 .9 6 %

Im p ort rem arks :A s se t A llo c a tio n E ffe c t is th e p o rtio n o f p o rtfo lio e xce ss re tu rn th a t is a ttrib u tab le to tak ing d iffe re n t g ro up be ts fro m th e b e nc h m ark . A n o ve rw e ig h t o f a g ro u p (e .g . S P I se c to r "C he m ic a ls" ) th a t o u tpe rfo rm s th e w ho le b en c hm a rk (e .g . S P I) w ill g e ne ra te a p o s it ive asse t a llo c a tio n e ffe c t.S e c u rity S e le c tio n E ffe c t is the p o rtio n o f p o rtfo lio e xce ss re tu rn a ttr ib u tab le to ch o o s ing d iffe re n t se c u r it ie s w ith in g ro u p s fro m th e b e n c hm ark . A n o ve rw e ig h to f a w e ll-p e rfo rm in g sec u r ity (e .g . N o vartis ) in c o m p a riso n to its g ro up be n c hm ark (e .g . S P I s ec to r "C h em ic a ls " ) w ill g en e ra te a p o s it ive s to ck se le c tio n e ffe c t.In te ra c tio n E ffe c t is th e p o rtio n o f th e p o rtfo lio e xce ss re tu rn w h ic h is no t a ttr ib u tab le to a sse t a llo c a tio n n o t s to c k se lec tio n .

-1 .3 2 %

-0 .3 5 % 0 .0 2 % 0 .3 5 %

-0 .7 8 %

-0 .3 0 % 0 .5 2 % 1 .0 7 %

-1 .1 1 %

T o ta l

-0 .8 9 % 0 .3 6 %

-0 .2 1 %

b y M S C I S e c to rS to c k

S e le c tio nIn te ra c tio n

-0 .8 6 % 0 .2 8 %

-0 .1 9 % -0 .2 1 % 0 .3 5 % 0 .9 9 %

-0 .9 1 % -0 .2 8 % 0 .2 1 % 0 .7 4 % 0 .0 0 % 0 .1 3 %

-0 .0 3 % 0 .0 3 %

-0 .0 2 % 0 .0 1 %

-0 .0 3 % 0 .1 2 % 0 .0 1 %

-0 .0 8 % -0 .0 8 % -0 .4 1 % 0 .0 0 %

-0 .4 9 %

Sample performance attribution report part 3 Figure 5.13

239

Average Average Relative DifferencePF WeightBM Weight Weight PF-BM

Asia ex Japan 1.86% 3.16% -1.30% -18.99% Europe 27.62% 28.92% -1.30% 1.04% Japan 9.83% 8.74% 1.09% -1.59% North America 59.48% 59.17% 0.31% -0.39% Cash 1.21% 0.00% 1.21% 8.94% Total 100.00% 100.00% 0.00% -1.32%

Import remarks:- The weight numbers gives an overview of the average weight invested in the different groups (e.g. sectors) with daily weights averaged over the chosen period.- The absolute return numbers give an overview of the group's absolute performance (e.g. sector) within the chosen time period (portfolio and bechmark)- The difference PF-BM compares the group's performance of the portfolio to the group's performance in the benchmark within the chosen time period.- The difference BM-BM Tot compares the group's performance in the benchmark with the performance of the total benchmark within the chosen time period.

18.28%

25.40% 20.16% 15.99% 8.94%

2.15% -3.23%

-19.60% 0.00%

DifferenceBM-BM Tot

11.91% 4.76%

21.75% 16.38% 0.00%

19.60%

Please note that cash is included in the relevant country or regional groups and shown as "other assets" else (e.g. in sectors). Concerning derivatives, only futures are split up at the moment; derivatives (call or put options) are not yet included. Moreover, illiquid securities as private placements are not yet taken into account.

by 5 World RegionsAbsolute

BM Return31.51%

AbsolutePF Return

12.52% 24.36%

Sample performance attribution report part 4 Figure 5.14

A ssetA llocation

Asia ex Japan -0.10% Europe -0.04% Japan -0.09% North Am erica 0.01% C ash -0.78% Total -1.01%

Im port rem arks:A sset A llocation Effect is the portion of portfo lio excess return that is attribu table to taking d ifferent group bets from the benchmark. An overweight of a group (e.g . SPI sector "Chem icals" ) that outperforms the whole benchmark (e.g . SP I) w ill generate a positive asset a llocation effect.Security Selection Effect is the portion of portfo lio excess return a ttribu tab le to choosing d ifferent securities w ith in groups from the benchmark. A n overweightof a w ell-perform ing security (e.g. Novartis) in comparison to its group benchmark (e.g. SPI sector "Chem icals") w ill generate a positive stock selec tion effect.Interaction Effect is the portion of the portfo lio excess re turn wh ich is not attributable to asset a llocation not stock selection.

0 .23% -0 .33% -0 .05% -0 .78% -1.32%

-0 .20% 0 .00%

-0.59%

S tockS election

Total

-0 .56% 0 .30%

Interaction

-0 .36%

by 5 W orld Regions

0.34% -0 .16%

-0 .07% -0 .08% 0 .14% 0 .00% 0.28%

Sample performance attribution report part 5 Figure 5.15

240

Risk Model: Global Portfolio Benchmark

Number of Securities 67 1'550

Number of Currencies 8 0

Portfolio Value

Total Risk (ex-ante) 18.81% 18.21%

- Factor Specific Risk 18.66% 18.18%

- Stock Specific Risk 2.39% 1.02%

Tracking Error (ex-ante) 2.57%

Relative Value at Risk

R-squared 0.98

Beta-adjusted Risk 18.64% 18.21%

Predicted Beta 1.02

Predicted Dividend Yield 1.86 2.01

P/E Ratio (E: 12 months) 28.39 26.00

P/B Ratio (B: year-end) 2.58 2.56

3'570'469 0.52%

2.08%

9.35%

2.39% Stock Specific Risk

- Covariance (+/-)

Explication of risk model: Factor risk is a standard deviation that ismeasured by multiplying the 5-year exposure of the components of a portfolioto each risk factor and by multiplying these figures by the externally determinedrisk of each factor. The Tracking Error is measured similarly except that it is thedifference between portfolio and benchmark exposure that is multiplied.Specific Risk is the standard deviation that measures the volatility of the risk notcaptured by the factor model. The model consists of 3 regional, 21 country, 38industry and 8 fundamental factors (market cap, 4-year E/P growth, E/P, B/P, 5-year yield, long term debt, 5-year ROE variablity and 5-year earnings variability).

Tracking Error

2.57%

1.50%

0.18%

0.83%

0.77%

0.78%

0.27%

Portfolio

18.81%

18.66%

11.50%

6.98%

2.64%

1.44%

8.42%

- Country

- Industry

- Fundamental

- Currency

84'334'091

Risk Model: Global

Total Risk (ex-ante)

Factor Specific Risk

- Region

Sample performance attribution report part 6 Figure 5.16 e. Performance watch list

As the name says the performance watch list consists of accounts or composites that should be watched and may be reviewed. The reason why accounts or composites are put on the watch list may be manifold and may be of quantitative but also of qualitative nature such as a underperformance versus a benchmark or a peer group, too much or too less risk, client complaints and so on. The determination of the watch list follows the process illustrated in Figure 5.17. It starts normally with a mechanical filter focusing on the historical and forward looking characteristics of the products and of the individual accounts, ex-ante risk limits or risk budgeting constraints as well as on the clients feedback. The investment controlling committee decides in the next step on the accounts or products that seem to be problematic and that go onto

241

the performance watch list. Afterwards within the performance review meeting each watch list account or product is analysed in detail considering all kind of information from investment guidelines up to an ex-ante risk break down. As a result of this performance review corrective steps may be defined and implemented. If the performance does not improve over longer time horizons serious performance problems are reported to the senior management via an escalation process.

Performance watch list process Figure 5.17 f. Portfolio analytics

Portfolio analytics provides deeper insight into the accounts or products on the watch list by taking all the available quantitative information produced in one of the preceding steps of the performance monitoring process. Portfolio analytics is very often also referred to as performance attribution.

242

We distinguish between these two steps to highlight that the performance attribution analysis on its own and without any qualitative judgement is nothing else as a detailed performance reporting. Through interpreting the performance figures by considering all circumstances the performance attribution becomes a meaningful management information tool. For the definition of portfolio analytics we refer to the Introduction 1 In comparison performance attribution is the quantitative and portfolio analytics the qualitative assessment of the quality of an asset management portfolio or product. What problems or pitfalls may occur during this step of the performance monitoring process is explained in a later chapter. At this point it should only be mentioned that performance attribution or portfolio analytics bear quite a high risk of misinterpretations and should be handled carefully. g. Performance review The performance review deals with the accounts and the products which are on the performance watch list and tries to analyse where the performance problems came from and whether any corrective action is necessary or would help to bring the product again on the right track. Within this step of the performance monitoring process the portfolios or products are analysed in detail considering all kind of information from investment guidelines up to an ex-ante risk break down. The following questions may be addressed within a typical performance review (The figures in brackets refers to Figure 5.18): • Where does the return come from and from which decisions do

they originate (1)? • What risks have been taken (relative/absolute)? • Is the choice of the benchmark still sensible? Are the general

circumstances still valid or reasonable (6)?

243

• Are the investment guidelines still reasonable and have they been respected (4)?

• What was the impact of the costs on the overall return (5)? • Is the risk profile and risk budget still appropriate ((2) and (3))? • What was the performance of the competitors? 5.2.3. Example of a performance review In the following we present an example for a performance review incorporating most of what was said or explained in the above chapters. A performance review follows normally the investment process of the specific portfolio or product and analyses and reviews all steps of the investment process as illustrated in Figure 5.18.

In this example the above mentioned composite “equities world BM MSCI active direct” or as mentioned in the performance attribution section the “composite world MSCI, active, mandates” was put on the performance watch list due to it negative excess return of -1.32% over the last 12 months, as of end of December 2003. The investment controller got the task to run a performance review and to analyse why this product underperformed quite substantially.

244

Portfolio construction

Investmentguidelines

Portfolio analytics /Performance analysis

Reb

alan

cing

Ass

et a

lloca

tion

proc

ess

Investment target, risk profile and benchmark

Actual market data andforecasts

15

4

3

2

6

Portfolio construction

Investmentguidelines

Portfolio analytics /Performance analysis

Reb

alan

cing

Ass

et a

lloca

tion

proc

ess

Investment target, risk profile and benchmark

Actual market data andforecasts

15

4

3

2

6

Aspects of the performance review Figure 5.18 After analysing the different performance reports and performance analysis the investment controller came up with the following facts and conclusions (see Figures 5.14 to 5.16): • The asset manager did not do big sector bets => maximum overweight 1.28% • The asset manager did not do big country bets => maximum

overweight 1.30% • The asset manager did big security specific bets by investing

“only” in 67 out of 1550 securities in the benchmark • The asset manager took quite a lot of security specific risk =>

stock specific risk of 2.08% versus factor specific risk of 1.50%

• Conclusion 1: the asset manager pursued a stock picking approach with neutral sector and country bets!

245

• The biggest return contributions come from Overweight in cash from March to May => - 0.78% Stock selection

Consumer discretionary => - 0.86% Information technology => - 0.91% Industrials => + 0.99% Utilities => + 0.74%

Interaction utilities => - 0.41% • The biggest relative risk contribution come from the stock

specific risk => 2.08% • Conclusion 2: Stock picking that did not pay out! As described, the negative excess return comes mainly from the cash bet in March until May which explained nearly 100% of the asset allocation effect. In addition the underweight of utilities - an outperforming sector - results in a quite high negative interaction effect. One could conclude that the asset manager had quite bad luck with his asset allocation decisions over the last 12 month. As a result of this analysis the investment controller could conclude that no remedial action is required and could decide to wait for another period seeing whether the product recovers or not. If the investment controller decides that remedial action is required his suggestions could range from a change in the investment strategy up to the replacement of the asset manager.

246

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

Total Monthly 0.04% 0.53% -0.57% 0.49% -0.99% -0.08% 0.49% -0.06% -0.83% -0.15% -0.14% -0.14%

Asset Allocation Monthly -0.12% 0.02% -0.11% -0.63% -0.12% 0.01% 0.05% -0.15% -0.17% -0.07% -0.02% 0.08%

Stock Picking Monthly 0.47% 0.47% -0.51% 1.23% -1.00% -0.05% 0.41% 0.03% -0.67% 0.04% -0.05% -0.18%

Interaction Monthly -0.31% 0.04% 0.05% -0.11% 0.13% -0.04% 0.03% 0.06% 0.01% -0.12% -0.07% -0.04%

Total Cummulated 0.04% 0.53% -0.04% 0.49% -0.17% -0.25% 0.26% 0.21% -0.80% -0.98% -1.12% -1.32%

Asset Allocation Cummulated -0.12% -0.10% -0.21% -0.80% -0.52% -0.54% -0.51% -0.70% -0.90% -1.03% -1.05% -0.96%

Stock Picking Cummulated 0.47% 0.91% 0.40% 1.66% 0.63% 0.62% 1.08% 1.16% 0.32% 0.41% 0.36% 0.13%

Interaction Cummulated -0.31% -0.28% -0.23% -0.37% -0.28% -0.33% -0.31% -0.25% -0.22% -0.36% -0.43% -0.49%

31.01.2003 28.02.2003 31.03.2003 30.04.2003 31.05.2003 30.06.2003 31.07.2003 31.08.2003 30.09.2003 31.10.2003 30.11.2003 31.12.2003

Management effect over time Figure 5.19

0%

5%

10%

15%

20%

25%

Jan 03 Feb 03 Mrz 03 Apr 03 Mai 03 Jun 03 Jul 03 Aug 03 Sep 03 Okt 03 Nov 03 Dez 03

Consumer Discretionary Consumer Staples Energy FinancialsHealth Care Industrials Information Technology MaterialsTelecomm Services Utilities Other Assets

Sector weights over time Figure 5.20

247

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

Jan 03 Feb 03 Mrz 03 Apr 03 Mai 03 Jun 03 Jul 03 Aug 03 Sep 03 Okt 03 Nov 03 Dez 03

Total Tracking Error Factor Specific a) Region b) Country c) Industryd) Fundamental e) Currency f) Covariance Stock Specific

Ex-ante risk contributions over time Figure 5.21

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5.3. The benefits of GIPS reporting As mentioned in the previous chapter GIPS composites are the basis for an efficient investment controlling of an asset management company. In this chapter we will explain the need for being in compliance with the GIPS standards (Global Investment Performance Standards) and show that the value added is not only limited to the asset manager but is also to the client’s benefit. In other words we will discuss the performance of an asset manager from the senior management’s as well as from a client’s point of view. For details on the GIPS standards or on specific country versions like SPPS (Swiss Performance Presentation Standards), please see the relevant chapters beforehand [2, CFA]. The case for the GIPS standards is best explained by looking at Figure 6.1. Assuming that a client is searching for an asset manager to manage a specific discretionary account and got just a performance number back from the different marketing officers. In the sample below asset manager A and B show the same historical performance track record for the specific asset management product. Looking at the two annualized return figures – which are identical – it is difficult for the client to decide on which asset manager to take. To come to a final decision a lot of additional information is needed. Considering just the return figures the client can not come to a meaningful decision because he does not know: • Where do the return figures come from: a model portfolio, a

GIPS composite, a representative account, the best performing account, the largest account, etc.

• Whether the return figures belong to accounts managed by the relevant asset management company or belong to a track record of a specific asset manager generated with one of his former employer

249

• Whether the return figures are gross or net returns and how the different fees are reflected within the calculation; for example in continental Europe the net return may be net of custody fees and in North America vice versa

• What is the underlying reporting period: since inception, last 12 months, the best looking period, etc.

• How the return figures where calculated: as a money weighted rate of return, as a time weighted rate of return and if the latter what approximation method is used

• What was the investment strategy or the investment objective? • Which benchmark is representing the investment strategy best?

Who is the better portfolio manager? Figure 5.22 Without an answer to the above questions a client is hardly able to judge and to compare the different asset management companies respectively their performance track records. „Cherry Picking“, i.e. the intentional selection of a specific account or observation period, the use of sample accounts, model portfolios or simulations, the transfer of performance track records as well as the not standardized methodologies for calculation performance figures lead to the issue of missing comparability of performance figures of different asset management companies. The issue of

asset manager A asset manager B

5 % p.a. 5 % p.a.

Two asset managers that manage a comparable account

250

the impossibility to compare performance figures and to assess the performance of asset managers lead to the necessity for a unique set of rules regulating the performance presentation and its calculation. In other words – as illustrated in figure 2 – the issue of performance presentations can be summarized as the case where the client and also the senior management have to ask a lot of unnecessary questions to get an objective and correct picture of the performance or of the quality of the asset manager. Such unnecessary questions are for instance

• Which accounts are presented? • How has the return been calculated? • Which accounts are behind the track record?

To illustrate the main issue in presenting performance – the determination of the performance track record – we use a fictitious case for a sample asset management company XYZ. In our example a prospective client asked the asset manager to present his historical performance track record for a specific product ABC. As shown in figure 6.2 the asset manager XYZ managed 3 accounts in this product category for the period n until n+2. Account C was terminated at the end of n and account B closed at the end of n+2. Only account A was managed for the whole period from n to n+2. Beside the account returns figure 3 also encompasses the asset under management as of the beginning of each year. The question is now which performance track record should be shown to a prospective client.

251

Which track record is the most representative to be shown? Figure 5.23 In determining a performance track record an asset manager has several different alternatives to chose from such as: • Alternative 1: Sample or representative account • Alternative 2: Account with the longest performance history • Alternative 3: Equally weighted average of the returns for all

actual accounts • Alternative 4: Equally weighted average of the returns for all

accounts ever managed • Alternative 5: Capital weighted average of the returns for all

actual accounts • Alternative 6: Capital weighted average of the returns for all

accounts ever managed • Alternative 7: Model portfolio or model strategy • Alternative 8: Account with the best performance history

Year n n+1 n+2

Account A

Return 6.2% -2.0% 4.2%

Assets on the 1 st of January 10.0 10.6 10.4

Account B

Return 5.0% -3.3%

Assets on the 1 st of January 100.0 105.0

Account C

Return 4.1%Assets on the 1 st of January 500.0

252

In our sample case, after some discussions the asset manager decided to follow alternative 3 which seems to be reasonable at first sight. Figure 4 illustrate this performance track record by the indexed cumulative return which is in our case identical to the performance track record of account A.

95

100

105

110

2000 2001 2002 2003

Selection of alternative 3, as all actual accounts are reflected

Is it possible to asses the performance with this track record? Figure 5.24 The question which arises now is whether a prospective client can asses the quality of the asset manager by using such a performance track record. The answer is clearly no because the performance of an asset manager is normally not identical with the performance of the actual managed accounts but with the performance of all ever managed accounts. Neglecting the terminated accounts in determining the performance track record results in an effect called “survivor of the best”. The effect “survivor of the best” means in this context that neglecting the terminated accounts is normally the same as neglecting the bad performing accounts because it is normally a bad performing account which is

253

terminated – and it are the good performing accounts which normally survive. In our sample case the performance track record – following alternative 3 – is becoming better over time because the on average bad performing accounts C and B were terminated at the end of n and n+1 and the performance track record was adjusted accordingly. Incorporating the terminated accounts – means alternative 4 – would result in a cumulative return for the period n to n+2 of 6.61% in comparison of 8.45% for alternative 3. Alternative 4 seems to be a good way of determining a performance track record but is not considering the assets under management of the different accounts A, B, and C. This means that the returns of the different accounts are equally weighted which results in a bias towards the smaller accounts. In our example in 2001 the performance track record profit from the equal weighting of the large account C with a relatively bad return of 4.1%. Considering the assets under management of the different accounts over time – means alternative 6 – would result in a performance track record shown in figure 5. The performance track record of alternative 6 is not so good than that of alternative 4 because the bad performing accounts C and B had quiet a high weight to the overall performance in the first two years. Determining a performance track record following alternative 6 is best practice because it follows the spirit of the GIPS standards (Global Investment Performance Standards or in Switzerland the country version of GIPS the Swiss Performance Presentation Standards – SPPS) which are explained in detail in one of the former chapters. The question which arises is now whether such a performance track record is sufficient to asses the quality of an asset manager. The answer is again clearly no. An observer needs more information on the asset manager and its product which ranges from the benchmark return, the number of accounts

254

managed in such a product, the calculation method used, and so on. Such a performance presentation is illustrated in figure 6 which includes the basic information needed to get a good starting picture in evaluating the quality of an asset manager.

95

100

105

110

2000 2001 2002 2003

Selection of alternative 6, as all accounts ever managed are reflected Alternative 6

Alternative 3

In order to asses the performance do we need further information? Figure 5.25

Simple composite performance presentation Figure 5.26

Composite Benchmark Number of Composite

return return accounts asset size

n+2 4.2% 4.8% 1 10.4

n+1 - 3.2% -1.5% 2 115.6

n 4.3% 3.3% 3 610.0

=> The basis of the performance discussion and assessment should always be a GIPS compliant report!

255

Figure 6 shows a sample performance presentation for a global equity composite which follows the GIPS standards and includes the minimum information required by the GIPS standards. This basic information on the performance for a specific product or composite enables the prospective client to concentrate on meaningful questions and to get ride of unnecessary questions. Assessing the quality of an asset management company or a specific product or composite is quiet easy if one considers a GIPS compliant performance presentation – and this is not only true for prospective clients but also for the senior management of the asset management company itself.

Sample GIPS compliant performance presentation Figure 5.27

256

In summary the starting point of a performance evaluation from a client’s as well as from a senior management’s perspective should always be a GIPS compliant performance presentation because • It enhances transparency and especially enhances the

understanding of the used performance measurement methods and the performance presentation itself,

• It avoids cherry picking of accounts, time periods, and so on, • It increases the comparability of different products and asset

managers, • It improves the ability to judge the quality of the asset manager

or a specific product or composite and • It enables objective discussions on the performance and

therefore enables to focus on the essential issues. At the end we would like to stress that investment controlling can only be done in an efficient and effective way with composite information which preferably was constructed and maintained according to the GIPS standards. Enabling a meaningful evaluation of the performance and the quality of an asset manager is the main benefit for the client and for the asset management company from GIPS compliant performance presentations.

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5.4. Some critical issues in performance attribution In this chapter we address certain issues in performance attribution which people usually are not aware of. Performance attribution is very complex not only to calculate but also to set up. Setting up the performance attribution in a not suitable or even wrong way results in misleading information or misinterpretation and with this leads to wrong feedback into the investment process. In the following we explain some of these issues but be aware that there are other issues left.

As said above performance attribution is very sensitive to misinterpretation because performance

• Is a result from several obvious and less obvious decisions • Originates from several obvious and less obvious decision

makers • Can vary over time

Misinterpretation of the historical performance may lead to erroneous or even wrong decisions as they • Asses the quality of the decision makers • Give feedback into the decision making process Furthermore the investment universe as well as the implementation of the investment strategy has to be considered by setting up the performance attribution. As shown in figure 6.12, in this situation one especially has to define whether certain decisions are stock picking or asset allocation decisions. For case A) it has to be defined whether the 20% investments in small and mid caps is a stock picking or an asset allocation decision? If the performance analyst sets up the performance attribution wrongly than of course

258

he get misleading figures for the management effects. In case B) the performance analyst has to specify whether the investments in (US) biotech stocks is a stock picking decision versus the US equity index or whether its is an asset allocation decision versus the overall benchmark because biotech stocks are not an explicit part of the overall benchmark.

Benchmark SMIInvestment universe SPI

20% Investments in Small&Mid Cap

Relative return +2.0%Asset allocation -0.5%Stock picking +2.7%Interaction -0.2%

Benchmark MSCI Worldwith 10 sub accounts

with sub account US-Biotech

Stock pickingor

Asset allocation ?

A)

B)

Benchmark SMIInvestment universe SPI

20% Investments in Small&Mid Cap

Relative return +2.0%Asset allocation -0.5%Stock picking +2.7%Interaction -0.2%

Benchmark MSCI Worldwith 10 sub accounts

with sub account US-Biotech

Stock pickingor

Asset allocation ?

A)

B)

Is each decision in the investment process reflected? Figure 5.28

. Until now we just addressed issues in setting up a return attribution but there are similar problems if setting up a risk attribution. The issues with risk attribution are a bit bigger because the risk attribution software available is not as flexible as the return attribution software. Normally the risk attribution software decomposes the absolute or relative risk following a specific risk

259

model which may not represent the actual investment process. Therefore the figures of a risk attribution have be handled with care because the different risk factors and their contributions to the overall absolute or relative risk are normally not linked to the steps of the investment process or the different decisions taken. Figure 6.14 illustrates this issue by indicating that in an ideal world the risk attribution have to be linked to the return attribution.

0.52%

2.08%

9.35%

2.39% Stock Specific Risk

- Covariance (+/-)

Tracking Error

2.57%

1.50%

0.18%

0.83%

0.77%

0.78%

0.27%

Portfolio

18.81%

18.66%

11.50%

6.98%

2.64%

1.44%

8.42%

- Country

- Industry

- Fundamental

- Currency

Risk Model: Global

Total Risk (ex-ante)

Factor Specific Risk

- Region

AssetAllocation

Consumer Discretionary 0.00% Consumer Staples 0.05% Energy 0.00% Financials -0.10% Health Care 0.20% Industrials -0.04% Information Technology -0.21% Materials 0.01% Telecomm Services -0.11% Utilities 0.02% Cash -0.78% Total -0.96% -1.32%

-0.35%0.02%0.35%

-0.78%

-0.30%0.52%1.07%

-1.11%

Total

-0.89%0.36%

-0.21%

by MSCI SectorStock

SelectionInteraction

-0.86%0.28%

-0.19%-0.21%0.35%0.99%

-0.91%-0.28%0.21%0.74%0.00%0.13%

-0.03%0.03%

-0.02%0.01%

-0.03%0.12%0.01%

-0.08%-0.08%-0.41%0.00%-0.49%

Is the Risk contribution consistent with the return attribution? Figure 5.29 After explaining some of the pitfall with setting up and running a performance attribution it would be useful to have some rules to hinder these kind of misleading performance analyses such as:

260

• Reflect each step of the investment process • Reflect each decision in the investment process • Mirror the investment style correctly • Analyse the impact of the investment guidelines • Define whether a certain product or an asset manager has to

be assessed • Analyse the variation of the management effect over time • Analyse the impact of leverage As you may be expect there is nothing like a complete list of rules to follow to prevent such pitfalls. The only thing what helps in general is transparency and here there is a good example for such general rule or standard on transparency – the GIPS standards. We argue that there is also a need for performance attribution presentation standards which would help in solving this issue.

At the end we conclude with summarizing some of the main points on investment controlling: • Investment controlling is in its closer but also in its wider

meaning an absolute must • Complying with GIPS or using GIPS composites is imperative

for a sound and convincing investment controlling • The client perspective has to be reflected • Transparency is the key for a meaningful feedback into the

decision making or investment process • Investment controlling enables:

o Realistic and unbiased performance discussion o Controlling of the consistent implementation of investment

strategies o Reduction of unintended or not obvious investment and

non-investment risks o Independent performance reviews

261

o Focus on ex-ante risks and the investment process o Early awareness of potential performance or process

oriented problems

Index Annualized Return, 48

Asset class, 170

Attribution, 27

Benchmark, 22

Beta, 212

Compounding, 9

Contribution, 18

Correlation, 129

Covariance, 128

Excess Return, 22

Geometrical Attribution, 38

Money weighted rate of return (MWR), 68

Time weighted rate of return (TWR), 52

Portfolio Analytics, 5

Return, 8

Segment, 20

Standard deviation, 119

Tracking Error, 130

Variance, 119

Volatility, 120

Literature [1] Bonafede Julia K., Foresti Steven J., Matheos P., Fall 2002 A multi-period Linking Algorithm that has stood the test of

time Journal of Performance Measurement [2] Brian R. Bruce, 1990 Quantitative International Investing McGraw-Hill Book Company [3] CFA Institute, May 2006 Global Investment Performance Standards (GIPS) [4] Copeland T., Weston F, 1988 Financial Theory and Corporate Policy Addison – Wesley, 3rd edition [5] Dalang R., Marty W., Osinski Ch., Winter 2001/2002 Performance of quantitative versus passive investing Journal of Performance Measurement [6] Diderich C., Marty W., June 2000, pp. 238-245 The Min-Max Portfolio Optimisation Strategy: An empirical

study on Balanced Portfolios Lecture Notes in Computer Science Numerical Analysis and its Applications 2000, pp. 238-245 [7] Dougherty Ch., 2002 Introduction to Econometrics Oxford university press

[8] Elton, E. J., Gruber M., 2007 Modern portfolio theory and investment analysis Wiley, 7th Edition [9] Feibel B. J., 2003 Investment performance measurement Whiley, Finance [10] Gabisch G., Lorenz H.-W., 1989

Business Cycle Theory 2nd edition Springer Verlag [11] Kleeberg, Jochen M., 1995 Der Anlageerfolg des Minimum-Varianz Portfolios. Schriftenreihe “Portfoliomanagement”. Uhlenbruch Verlag [12] Kreyszig E., 1978 Introductory Functional Analysis with Applications John Wiley & Sons [13] Lorenz H.-W., 1993 Nonlinear Dynamical Economics and Chaotic Motion

Springer-Verlag Heidelberg, Berlin, New York [14] Illmer S., Marty W., Summer 2003 Decomposing the Money-Weighted Rate of Return, Journal of Performance Measurement [15] Neumaier A., 2001, page 61-90 Introduction to Numerical Analysis Cambridge University Press

[16] Malkiel G. Burton, 2007 A Random Walk Down Wall Street W.W. Norton & Company [17] Marty W., 6.10.2006 Stability is the Enemy of Optimality unpublished Translated from NZZ [18] Miller and Modigliani Dividend Policy, Growth, and the Valuation of Shares Journal of Business 34 (Oct. 1961) pp. 411 - 433 [19] Rustem B., Marty W., Becker R., 2000, pp. 1591 – 1621 Robust min max portfolio strategies for rival forecast and risk

scenarios Journal of Economic Dynamic and Control Vol. 24 (11-12) [20] Ross S.A. March 1977, page 177 − 184 The Capital Asset Pricing Model Short sales restriction and

related issues Journal of Financial March Ross, S.A. [21] Sharpe W.F., 1995 Investments Prentice Hall International Editions, 5rd edition [22] Shestopaloff Y, 2009 Science of Inexact Mathematics Akvy Press, 1st edition [23] Taleb N. N., 2007 The Black Swan The impact of the highly improbable Random House

[24] Varga R., 2000, page 22 - 53 Matrix Iterative Analysis Springer Verlag, 2nd edition [25] Vörös J., 1987 page 302 - 310 The explicit derivation of the efficient portfolio frontier in the

case of degeneracy and general singularity Eur. J. Operat. Res. 32

The author Dr. Wolfgang Marty is vice president at Credit Suisse. He joined Credit Suisse Asset Management in 1998 as Head Product Engineering. He specializes in Performance Attribution, Portfolio Optimisation and Fixed Income in general. Prior to joining Credit Suisse Asset Management, Marty worked for UBS AG in London, Chicago and Zurich. He started his career as assistant for applied mathematics at the Swiss Federal Institute of Technology. Marty holds a university degree in Mathematics from the Swiss Federal Institute of Technology in Zurich and a doctorate from the University of Zurich. He chairs the method and measure sub committee of the European Bond Commission (EBC) and he is president of the Swiss Bond Commission (OKS). Furthermore he is a member of the Fixed Income Index Commission at the Swiss Stock Exchange and a member of Index team that monitors the Liquid Swiss Index (LSI).