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課程三：風險與報酬Risk and Return

This note is for lecture use only.

本講義僅供上課教學之用。

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Holding period return (HPR)

price Beginning

dividendCash price Beginning - price EndingHPR

報酬 Returns

0

101 )(

p

Dpp

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算術平均 Arithmetic averageSum of returns in each period divided by number of periods

幾何平均 Geometric averageThe single per-period return that gives the same cumulative performance as actual returnsRequired for mutual fund literature

Dollar-weighted returnTreat cash flows like capital budgeting problem and calculate the internal rate of return (IRR)

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例題Year Price

(End)Div

(Beg)Holding Period Return

1996 100 4 (Pt – Pt-1 + Dt) / Pt-1

1997 110 4 (110 – 100 + 4) / 100 = 0.14001998 90 4 (90 – 110 + 4) / 110 = -0.14551999 95 4 (95 – 90 + 4) / 90 = 0.1000

算術平均 : (0.14 -0.1455 + 0.10) / 3 = 0.0315 = 3.15%

幾何平均 : (1 + RG) = (1 + 0.14) x (1 - 0.1455) x (1 + 0.10)

RG = [(1 + 0.14) x (1 - 0.1455) x (1 + 0.10)]1/3 - 1

RG = 0.0233 = 2.33%

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Annual percentage rate - APRUsually what most people imagine when they think of returns.Ignores compounding of interest on interestSimply (rate per period) x (number of periods)

Effective annual rate - EARCorrects APR for interest on interest compoundingEAR= (1 + APR/n)n - 1當 n 趨向無窮大 EAR = eAPR - 1

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Risk 風險Returns are important, but they can’t be the sole driver of investment decisionsReturns are uncertain and we need a way to quantitatively measure the uncertaintyAn intuitive measure should take into account how likely are the returnsProbability distributions of return

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重要統計量Mean: What is the expected value?Median: What is the middle value?Mode: Which value occurs most frequently?Variance: How compact is the distribution?偏度 Skewness: Is the distribution symmetric?峰度 Kurtosis: What do the tails look like?

常態分配只需均數及變異數即夠描述整個分配。

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Expected Returns & RiskExpected Return

Variance

E r p s r ss

( ) ( ) ( )

Var r p s r s E rs

( ) ( ) ( ) ( ) 2 2

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資產配置 Asset Allocation存在兩資產，一為風險性，一為無風險性。

Risky assetE(ra) = 15% 2(ra) = 22%

Risk-free assetrf = 7%

What portfolios can we hold?We can invest y(%) in the risky asset and (1-y) in the risk-free assetE(rp)= E(ra) y + rf (1-y) p = y a ( 特例 )

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0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Standard Deviation

Ex

pe

cte

d R

etu

rn

aP

r f = 7%

p =

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資產配置線 Capital Allocation Line (CAL)Both the risk premium and the standard deviation of the portfolio increase with weight in the risky assetVarying the weights gives us all portfolio combinations, which fall on a single line - the Capital Allocation Line (CAL)The slope of the (CAL) is the Reward to Variability Ratio

SE r rp f

p

( ).

15 7

2236

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Optimal Portfolio Selection

What happened to concept of risk aversion?Investors are assumed to be risk averse so that they only accept risky security if it provides compensation via risk premium. How does that impact our CAL approach?How do we pick the risky portfolio?

Active versus passive managementCapital Market Line (CML) 資本市場線

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多角化 Efficient DiversificationWe showed that with an optimal risky portfolio, all investment will be on the CALHow do we select the optimal risky portfolio?Why are portfolios of securities better than single securities?What do we mean by diversification?Why diversify?

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Covariance and CorrelationWhat is covariance?Is this important?Is there a difference between covariance and correlation?

s

yyxx rEsrrEsr )]()([)]()([y)Cov(x,

yxxy

yxCov

),(tCoefficienn Correlatio

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Portfolios of SecuritiesInvestors’ opportunity set is comprised not only of sets of individual securities but also combinations, or portfolios, of securitiesThe return on a portfolio is the weighted average of returns on component securities:

The expected return is also a weighted average

R w Rpt i iti

N

1

E R w E Rpt i iti

N( ) ( )

1

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Portfolio Risk

HOWEVER, the standard deviation of a portfolio is NOT just a weighted average of securities standard deviations.We also need to account for their covariances. Example with 2 securities: X and Y

y)Cov(x,2

222

yx

yyxxp

ww

ww

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Portfolio Risk

xyyxyxyyxxp wwww 222222

Will portfolio standard deviation be higher or lower than simple weig

hted average of standard deviations?

y)Cov(x,222222yxyyxxp wwww

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Portfolio RiskVariance of portfolio of TWO securities:

What happens to risk if two securities are perfectly positively correlated? Perfectly negatively? What about general case? Intuitively, what implications can we infer for efficient portfolio selection strategies?

y)Cov(x,

222222

xyyxxy

xyyxyyxxp

where

wwww

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For N securities, in general, the formula is:

The first term is a complex average of securities variances; the second term captures N(N-1) covariance terms.Intuitively, what happens to the portfolio’s variance as N gets large?

N

i

N

i

N

jij

ijjiiip www1 1 1

222

ijjiCov ),(

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Risk DiversificationAs N gets large, the number of covariances outnumber variancesCovariances among US stocks tend to be lower on average than stock’s own variances (Fisher & Lorie, 1966) Problem: Derive risk formula of an equally-weighted portfolio, i.e.

Naive Diversification

020406080

100120

Number of StocksR

isk

as %

of

Ave

Sto

ck R

isk

( )w Ni 1

Limit of 31%

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What Affects Risk?Market risk 市場風險，系統風險，不可分散的風險

Risk factors common to the whole economySystematic or non-diversifiable

Firm specific risk 公司個別風險，非系統風險，可分散的風險

Risk that can be eliminated by diversificationUnique riskNonsystematic or diversifiable

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效率前緣Understanding the return and risk attributes of portfolios of individual securities allows us to construct more efficient combinations which strategically attempt to “reduce risk as much as possible for a given level of expected return.” How do we do it?We work in a mean-variance framework

Assumes all investors prefer higher returns, all else equalAssumes all investors prefer lower risk, all else equal

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Efficient Portfolios

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5

Case A

Case B

Case C

1Security

Security 2

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For N securities, the problem of identifying efficient portfolios is similar, except that we need special skills in linear programming

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5

Standard Deviation

Expe

cted

Ret

urn

MIN w w w

subject to E R R

p i i i j ijji j

N

i

N

i

N

p p

2 2 2

111

( )

Efficient Frontier

效率前緣

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Optimal Risky Portfolio

0.00

0.10

0.20

0.30

0.40

0.50

Standard Deviation

Exp

ected

Retu

rn

Tangency Portfolio “T”

Efficient FrontierUtility 1

Assumption: No Riskless Asset Available

Preferred Direction

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0.00

0.10

0.20

0.30

0.40

0.50

Standard Deviation

Exp

ected

Retu

rn

Efficient Frontier

Capital Allocation Line “M”

Optimal Risky Portfolio “M”

Assumption: Riskless Asset Now Available

Capital Allocation Line “A”A

Preferred Direction

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0.00

0.10

0.20

0.30

0.40

0.50

Standard Deviation

Exp

ected

Retu

rn

Efficient FrontierUtility 1

Capital Allocation LineUtility 2

T

Optimal Risky Portfolio “M”

Optimal Allocation “C”

Assumption: Riskless Asset Now Available

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涵義Optimal Portfolio Selection requires 2 steps:

Optimal Risky Portfolio DeterminationCapital Allocation Decision

All rational risk-averse investors will passively index holdings to the market portfolio and the risk free asset.

“Two fund separation” Principle

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0.000.100.200.300.400.50

Standard Deviation

Exp

ected

Retu

rn

Efficient Frontier

T

Optimal Risky Portfolio “M”

Assumption: rf does not equal rb

rf

rb

CAL借貸利率不一樣時

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資本資產定價模式 Capital Asset Pricing Model

The CAPM is a centerpiece of modern finance that gives predictions about the relationship between risk & expected returnBased on original work on portfolio theory of Harry Markowitz by William Sharpe & John Lintner in 1965-66.Begins with simplistic assumptions for hypothetical world of investors and builds into reasonable & comprehensive model

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AssumptionsInvestors are price takersOne-period investment horizon (“myopic”)Fixed quantities of assets and all marketableNo taxes, transactions costs, regulations, etcInvestors are mean-variance optimizersAll investors analyze securities in same way with same probabilistic forecasts for each - homogenous expectations

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Investors hold Market PortfolioAll investors will identi

fy same optimal risky portfolio, “M” to combine with riskless asset For supply/demand to clear, the holdings of each security will be by relative market value outstandingM =“Market portfolio”

0.00

0.10

0.20

0.30

0.40

0.50

Standard Deviation

Exp

ecte

d R

etu

rn

M=“Market”

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Passive Indexing is Efficient

Market portfolio must be on efficient frontier and it is tangent point for the best feasible capital allocation line Rational investors will passively hold an equity index fund & a money market fundIn 1991 75% of $275b of institutional funds were “indexed”

0.00

0.10

0.20

0.30

0.40

0.50

Standard Deviation

Exp

ecte

d R

etu

rn

Capital Market Line

M

E RM( )

Mr f

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Equilibrium Expected Returns

CAPM is built on insight that appropriate risk premium on an asset is determined by contribution to risk of investor’s overall portfolio. Portfolio risk is what matters“Market price of risk” or is the benchmark tradeoff for risk & return, because all investors holdings are on CMLHow does any individual security contribute to the risk of a well-diversified portfolio like the market portfolio?

[ ( ) ] /E R rM f M

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Equilibrium Expected ReturnsSince we can diversify away firm-specific risk, should it be rewarded?We use beta as a measure of systematic risk. What about the betas of portfolios?

2)(

),(

m

im

m

mii rVar

rrCov

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Equilibrium Expected ReturnsIn equilibrium, all assets (and all portfolios) should have the same reward to risk tradeoff.

However, this implies that this should hold for the market portfolio as well.

We have a simple expression for expected returns on any asset or portfolio.

m

fm

i

fi rRErRE

)()(

])([)( fmifi rrErrE

j

fj

i

fi rRErRE

)()(

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Security Market Line 證券市場線

0.00

0.10

0.20

0.30

0.40

0.50

-1 -0.5 0 0.5 1 1.5 2 2.5

Beta

Exp

ecte

d R

etu

rn

SML

M

A

A 0

If a security plots off the Security Market Line, its expectedreturn is different from its “fair” return, or it is “mispriced.”

A

E RA( )

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SML Versus CMLSecurity Market LineExamines individual asset risk premiums against risk measure appropriate for individual assets. With individual assets, the only relevant risk is systematic risk, hence we examine beta.

Capital Market LineExamines efficient portfolio risk premiums against appropriate risk measure.With well diversified portfolios, the relevant measure of risk is total risk, hence we examine standard deviation.

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Application of CAPM

Two professional money managers are being evaluated. One averaged 19% last year and the other only 16%. However, the first manager’s beta was 1.5 and the second manager had a beta of 1.0.Which manager performed better?If the market risk premium were 8% and T-bills were yielding 6%, which is better?What if market risk premium is 12 % and T-bills yield 3%?

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The Market Model

Alphas and betas are measured statistically using historical returns on the security and the market portfolio proxy, e.g. S&P 500Simple regression model, known as Market Model, is used (in excess returns):

Can we test if CAPM is true doing this? Are there testable implications?

itftmtiiftit rrrr ][

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華航 加權指數年月日 報酬率 報酬率 超常報酬 ( )％ ( )％

Aug-87 1.85 0 1.943731Jul-87 -1.37 0.26 -1.53674Jun-87 -0.45 -1.02 0.66557

May-87 -0.9 0.07 -0.87639Apr-87 -1.33 -1.68 0.446759Mar-87 2.27 2.7 -0.34113Feb-87 -1.35 -0.55 -0.70528Jan-87 -0.45 0.4 -0.75699Dec-86 3.23 1.67 1.650722Nov-86 -1.36 -0.03 -1.23621Oct-86 1.85 0.27 1.673245Sep-86 -0.46 0.56 -0.92728

y=a+bx beta 1.001802alpha -0.09373

市場模式的應用

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Arbitrage Pricing Theory 套利模式Steve Ross in 1977 An arbitrage opportunity arises when an investor can construct a zero-investment portfolio that will yield sure profits in futureA zero-investment portfolio is one in which some securities are long, others short with no commitment of investor’s money

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CAPM vs APTExpected returns are related to multiple sources of risk (APT) vs only market (beta) risk (CAPM)No special role for market portfolio in APTEquilibrium achieved by arbitrage in APT; CAPM requires rational risk-averse, mean-variance optimizing investors.