11

Single Index Model

Simplifying the Markowitz model

• Problem in portfolio selection– What if the investment universe is large?– If 100 assets on the input list, how many

estimates does one need to prepare?– 100 E(ri), 100 i, and – how many?– Answer: (100 x 99)/2 = 4950 unique

• Advantage of the Single Index Model stems from its simplifying assumptions

Simplifying the Markowitz model

Single index modelAll assets derive only from the common factor

RM

ei is firm-specific, and hence uncorrelated across assets

Hence, if there are 100 assets in the investment universe, only need 100 beta estimates and the variance of RM to calculate all the covariance's

Simplifying the Markowitz model

Advantages:• Reduces the number of inputs for diversification• “A simplified model for portfolio analysis” by Sharpe (1963)• Easier for security analysts to specialize, e.g.,

communications, resources. • Everything is related only to the aggregate marketDrawback: • rules out other important risk sources (e.g., industry

factors)• Is the market index appropriate/representative?

Computational Advantages

• The single-index model compares all securities to a single benchmark

– An alternative to comparing a security to each of the others

– By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other

5

Single Index Model: Return Equations

iMiii eRR

PMPPP eRR

Individual asset Individual asset

PortfolioPortfolio

Beta

– A security’s beta is

7

2

2

( , )

where return on the market index

variance of the market returns

return on Security

i mi

m

m

m

i

COV R R

R

R i

Single Index Model: Risk Equations

• Beta of a portfolio:

• Covariance of two portfolio components:Cov AB= BA BB σ2

m

• Variance of a portfolio:

8

1

n

p i ii

x

2 2 2 2

2 2

p p m ep

p m

As the number of assets in portfolio increases, the second term becomes less and less importantAs the number of assets in portfolio increases, the second term becomes less and less important

Multi-Index Model

• A multi-index model considers independent variables other than the performance of an overall market index– Of particular interest are industry effects

• Factors associated with a particular line of business

9

Multi-Index Model• The general form of a multi-index model:

10

1 1 2 2 ...

where constant

return on the market index

return on an industry index

Security 's beta for industry index

Security 's market beta

retur

i i im m i i in n

i

m

j

ij

im

i

R a I I I I

a

I

I

i j

i

R

n on Security i

Capital Market Theory: An Overview• Capital market theory extends portfolio theory and

seeks to develops a model for pricing all risky assets based on their relevant risks

• Asset Pricing Models– Capital asset pricing model (CAPM) is a single factor model

allows for the calculation of the required rate of return (also Known as Model Return) for any risky asset based on the security’s beta

– Arbitrage Pricing Theory (APT) is a multi-factor model for determining the required rate of return

Capital Market Theory and a Risk-Free Asset

There are rather large implications for capital market theory when a risk-free asset exists.

• What is a risk-free asset?– An asset with zero variance– Provides the risk-free rate of return (RFR)– It will be an “intercept” value on a portfolio graph

between expected return and standard deviation.• Since it has zero variance, it will also have zero correlation with all

other risky assets

Combining a Risk-Free Asset with a Portfolio

Expected return is the weighted average of the two returns

))E(RW-(1(RFR)W)E(R iRFRFport

This is a linear relationship

Combining a Risk-Free Asset with a Portfolio

Standard deviation: The expected variance for a two-asset portfolio is

211,22122

22

21

21

2port rww2ww)E(

Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become

iRFiRF iRF,RFRF22

RF22

RF2port )rw-(1w2)w1(w)E(

Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

22RF

2port )w1()E( i

Combining a Risk-Free Asset with a Portfolio

Given the variance formula22

RF2port )w1()E( i

22RFport )w1()E( i The standard deviation is

i)w1( RF

Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

The Capital Market Line

Expected Return on the Portfolio

Standard Deviation of the Portfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free rate

Capital Market Line

The Capital Market Line and Utility Curves

Expected Return on the Portfolio

Standard Deviation of the Portfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free rate

Capital Market Line

Highly Risk

Averse Investor

A risk-taker

The Capital Market Line and Iso Utility Curves

Expected Return on the Portfolio

Standard Deviation of the Portfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free rate

Capital Market Line

A risk-taker’s utility curve

The risk-taker’s optimal portfolio

combination

Lending & Borrowing Under the CAPM

• Assumption of unlimited lending and borrowing at risk-free rate.

• Lending if portion of portfolio held in risk-free assets.

• Borrowing (leverage) if more than 100% of portfolio is invested in risky assets.

• Superior returns made possible with lending and borrowing; creates spectrum of risk preference for different investors.

Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port

RFR

M

CML

Borrowing

Lending

The Market Portfolio• Portfolio M lies at the point of tangency, so it has

the highest portfolio possibility line• This line of tangency is called the Capital Market

Line (CML)• Everybody will want to invest in Portfolio M and

borrow or lend to be somewhere on the CML (the CML is a new efficient frontier)– Therefore this portfolio must include all risky assets (or

else some assets would have no demand)

The Market Portfolio• Because the market is in equilibrium, all

assets are included in this portfolio in proportion to their market value.

• Because it contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away

The CML and the Separation Theorem

• The CML leads all investors to invest in the M portfolio (The Investment Decision)

• Individual investors should differ in position on the CML depending on risk preferences (which leads to the Financing Decision)– Risk averse investors will lend part of the portfolio at the

risk-free rate and invest the remainder in the market portfolio (points left of M)

– Aggressive investors would borrow funds at the RFR and invest everything in the market portfolio (points to the right of M)

Fund Separation

Rf

A

(M) Market Portfolio

B

CML

σM

E(RM)

E(Rp)

σ(Rp)

CML Equation: E(Rp) = Rf + [(E(RM)- Rf)/σM]σ(Rp)

Everyone’s U-maximizing portfolio consists of a combination of 2 assets only: Risk-free asset and the market portfolio. This is true irrespective of the difference of their risk-preferences

Capital Asset Pricing Model

• CAPM indicates what should be the required rates of return on risky portfolios

• This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models

• You can compare an expected rate of return to the required rate of return implied by CAPM –

• over/ under valued? If required Return (Model Return is less than expected, Security is under valued)

Determining the Expected Return

• The required rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset

• The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (RM-RFR)

RFR)-(RRFR)R(R Mi i

Determining the Expected Return

• In equilibrium, all assets and all portfolios of assets should plot on the SML– The SML gives the market “going rate of return” or what

you should earn as a return for a security– Any security with an expected return that plots above the

SML is underpriced– Any security with an expected return that plots below the

SML is overpriced

Over/Under Valuation (Alpha)

If the market portfolio is not efficient, then stocks will not all lie on the security market line. The distance of a stock above or below the security market line is the stock’s alpha (α). We can improve upon the market portfolio by buying stocks with positive alphas and selling stocks with negative alphas.

28

Graph of SML)R(R i

)Beta(Cov 2Mim/0.1

mR

SML

0

Negative Beta

RFR

CML versus SML

• Please notice that the CML is used to illustrate all of the efficient portfolio combinations available to investors.

• It differs significantly from the SML that is used to predict the required return that investors should demand given the riskiness (beta) of the investment.

Real World Popularity

After its discovery, the CAPM was immediately applied in the real world.To measure the “correct” excess expected return for any security all one has to know is:(1) the market risk premium (E(RM) – r)

(2) security beta (βj)

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Issues in Beta Estimation

• The Impact of the Time Interval– Number of observations and time interval used in

regression vary– weekly rates of return Vs. monthly return– There is no “correct” interval for analysis

Issues in Beta Estimation• The Effect of the Market Proxy

– A measure of the market portfolio is needed– Nifty/Sensex Composite Index is most often used

• Includes a large proportion of the total market value of Indian stocks

– Weaknesses of Using Nifty/Sensex as the Market Proxy• Includes only Indian stocks • The theoretical market portfolio should include all types of assets

from all around the world

Empirical Criticisms of BetaNonstationary Beta Problems

• Nonstationary Beta Problem: Difficulty tied to the fact that betas are inherently unstable

Testable Limitations Of CAPM

• ß, the slope of the regression of a security’s return on the market return, is the only risk factor needed to explain expected return.

• ß captures a positive expected return premium for risk.

• Other risk factors emerge:

– firm size

– low P/E, price/cash flow, P/B, and sales growth

Other Problems: (1926-2004) US Market…

36

Small Stocks

• Looking at that plot, small stocks appear to have higher returns.

• Do these stocks correctly plot on the SML?

37

So what should we conclude?

• It is difficult to say decisively, whether the Single-factor model or the standard CAPM are good or bad. – There is much empirical support for both

• Even so, there are very cogent arguments questioning this evidence

So what should we conclude?

• Return and risk appear to be linearly related over long periods of time (when risk is defined as systematic risk; that is, the risk measured by beta) is important

• The fact that return is not related to residual risk is also important– While these facts certainly do not constitute tests,

per se, they have important implications for behavior

So what should we conclude?

• Within the CAPM investors are not rewarded for taking nonmarket risk– They are, however, rewarded for bearing added

market risk• Regardless of the model being explored, these facts

seem to hold in the CAPM

Arbitrage Pricing Theory (APT)

• CAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmark

• An alternative pricing theory with fewer assumptions was developed:– Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT)

• Developed by Stephen Ross (1976)• Basic idea: Calculate relations among expected returns that

will rule out arbitrage by investors• Arbitrage: Creation of riskless profits made possible by

relative mispricing among securities.

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Arbitrage Pricing Theory

Multiple factors expected to have an impact on all assets:

• Inflation• Growth in GNP• Major political upheavals• Changes in interest rates• And many more….Contrast with CAPM assumption that only beta is

relevant

Arbitrage Pricing Theory (APT)

• The expected return on any asset i (Ei) can be expressed as:

ikkiii bbbE ...22110

Example of Two Stocks and a Two-Factor Model

= changes in the rate of inflation. The risk premium related to this factor is 1 percent for every 1 percent change in the rate

1

)01.( 1 = percent growth in real GNP. The average risk premium related to this factor is 2 percent for every 1 percent change in the rate

= the rate of return on a zero-systematic-risk asset (zero beta: boj=0) is 3 percent

2)02.( 2

)03.( 3 3

Multifactor Models and Risk Estimation

Multifactor Models in Practice• Macroeconomic-Based Risk Factor Models• Microeconomic-Based Risk Factor Models• Extensions of Characteristic-Based Risk

Factor Models

Fama-French Three-Factor Model

• Fama-French found that size and B/M do a better job of explaining returns, so they said the model should be:

47

( ) ( ) ( )i f i i M f i i ir r r r s SMB h HML e

SMB = small minus big =

HML = high B/M minus low B/M =

small big

value growth

r r

r r

• F&F does a better job alpha very close to zero• Main criticism: No theory justifying why size and B/M should be risk factors.