Lecture 10Lecture 10

The Capital Asset Pricing ModelThe Capital Asset Pricing Model

Expectation, variance, standard error (deviation), covariance, and correlation of returns may be based on

(i) fundamental analysis

(ii) historical data

Preliminaries

Fundamental or Theoretical Analysis

S possible states

s probability of state s = 1,2,…,S

Rs likely return is state s

Notation

4 business cycle states (boom, normal, recession, depression)

3 industry demand states

2 firm demand share states

3 firm cost states

Then, there are 4*3*2*3 = 72 possible states (or situations)

Example: Suppose there are

S

1sssRRE Expectation (mean)

S

1s

2ss ERRRvar 2Variance

02 Standard error

BBs

S

1sAAssBAAB RRRRR,RCov ΕΕ

returns on stock A RAs s = 1,…,S

returns on stock B RBs s = 1,…,S

Covariance measures how two random variables are related

ABAB signsign

11122

22222 AB

BA

ABABBAAB

BA

ABBAAB R,Rcorr

Correlation is a normalized covariance

Note !

Example: Suppose we have a theoretical model that predicts the following returns on stocks A and B in 3 states.

States s RA RB

Boom 0.25 20% 5%

Normal 0.50 10% 10%

Recession 0.25 0% 15%

Expected returns

0.100.150.250.100.500.050.25

0.100.000.250.100.500.200.25

B

A

Variances

0.001250.100.150.250.100.100.500.100.050.25

0.0050.100.000.250.100.100.500.100.20.252222

B

2222A

Standard errors

0.03536

0.07071

B

A

2

2

B

A

Covariance 0.00250.10.150.100.25

0.10.10.10.10.50.10.050.10.20.252AB

Correlation 1.0

0.035360.07071

0.0025

BA

ABAB

Returns on stocks A and B are perfectly negatively correlated.

Stocks A can be used as a hedge against the risk in holding stock B

Historical Data Based Approach

From historical data, calculate the percentage returns R1, R2, …, RT

02 Sample standard deviation (or standard deviation)

Sample average percentage return

T

1tt

T1 RT

1

T

R... RR

Sample Variance

T

1t

2

t2 RR

1T

1

Historical Data Based Approach (continued)

Sample covariance of returns on stocks A and B, calculated from the historical samples of RA and RB

RA = (RA1, …, RAT) ; RB = (RB1, …, RBT)

T

1tBtB

T

1tAtA

BBt

T

1t

AAtAB

RT

1R ; R

T

1R

RRRR1T

1

Sample correlation of RA and RB

BA

ABAB

T

1t

2AAt RR

1T

122 ; AAA

T

1t

2BBt

2B RR

1T

1 ; 2BB

Expected Return and Variance of Returns on Portfolios

A portfolio is an investment in stocks.

Let be the proportion invested in stock n.

Then

2N

1xN

1nn

[0,1]xn

If the return on stock n is Rn, then the return on the portfolio is

N

1nnnNN11p RxRx...RxR

and the expected return on the portfolio is

N

1n

nn

N

1nnn

N

1nnnp RxΕRxRxΕ

N

2n

1n

1mnmmn

N

1n

2n

2

N

2nmm

1n

1mnnmn

2N

1nnn

2

2N

1nnnn

2N

1n

nn

N

1nnn

2pp

2p

xx2x

ΕRRΕRRxx2ΕΕRRxΕ

ΕRRxΕ

RxRxΕΕRRΕ

n

n

The variance of the returns on the portfolio is given by

Expected Return and Variance of Returns on Portfolios (continued)

Diversification

1. Variances are diversified away

2. Average covariance converges to covariance from economy-wide shocks affecting all stocks

Consider a special case with for each .

ThenN

1xn N1,...,n

2120

1NNN

1NN2

N

1

N

1

2

N

2n

1n

1mnm

2

N

1n

2n

2p

- In a diversified portfolio, only systematic risk affects returns.

- Diversifiable or unsystematic (idiosyncratic) risk is irrelevant to returns.

Diversification (continued)

Recall ;

Suppose you invest $100 in stock A and $200 in stock B.

0.10BA 0.035360.07071, BA

Returns on investment in assets A and B

States s RA RB Total return

Boom 0.25 120 (20%) 210 (5%) 330 (10%)

Normal 0.50 110 (10%) 220 (10%) 330 (10%)

Recession 0.25 100 ( 0%) 230 (15%) 330 (10%)

The mean return on the portfolio is 10%.

10%10%3

210%

3

1

RΕxRΕxRxRxΕRΕ BBAABBAAp

Diversification (continued)

The standard deviation of the return on the portfolio is zero.

No risk!

The mean return on the portfolio is a weighted average of

and AER BER

Recall that the correlation between the returns on A and

B is -1. This implies that the variation in returns on either

asset can be completely offset by holding the right proportion

of the other asset.

AB

Deriving an appropriate discount rate for risky cash flows

1. The opportunity set for two assets

3. The efficient set with a riskless asset

2. The opportunity set and efficient set with many securities

4. The CAPM (capital asset pricing model) equation

5. A risk-return separation theorem

The opportunity set for two assets

Suppose there are two assets A and B in proportions and .

Then, since .

Ax Bx

AB x1x 1xx BA

2B

2BABBA

2A

2A

2BBBABBAA

2BBAABBAA

2pp

2p

xx2xx

RxRxRxRx

RxRxRxRx

RR

BAAB

BAAA

BAAApp

RRxR

Rx1Rx

Rx1RxR

The opportunity set for two assets (continued)

From , we have . BAABp RRxR BA

BpA

RR

Rx

Then we have

2B

2AABBA

2A

2B

p2BAABBA

2AB

2p2

BA

2BAB

2A2

p

RRR2R

RRRR2

RR

2

Using the above equation, we can trace a feasible (or opportunity) set of attainable and for given BAABABABBABA RR ,,,,,

p p

Example

We are given the following parameter values,

0.1639 ; 11.50% ; 25.86%

5.5%R ; 17.5%R

ABBA

BA

For these values, the above equation becomes approximately

0.04970.98806.2394 p2p

2p

which looks like the following in space. pp ,

Example (continued)

Opportunity set for assets A and B

Portfolio MV (minimum variance) has the lowest risk obtainable with assets A and B.

Between B and MV, replacement of B by A increases and

reduces . This always happens if and may

happen for .

p

p 0AB0AB

When , a riskless portfolio can be obtained by holding

A and B in right proportions.

1AB

The opportunity set and efficient set with many securities

Each pair of securities ((A,B),(A,C),(B,C)) gives an opportunity set

Except for portfolios close to MV, the efficient set is very close to a straight line. Also as the variance of the MV portfolio decreases, the efficient set gets closer to a straight line.

Suppose we add asset C, to the previous example, with the parameter values

0.05 ; 0.20 ; 15.0% ; 10.5%R CBCACC

Linear combination of portfolios in any of these opportunity set will

lead to additional curve in s space. ,

It can be shown that the opportunity set for assets is an area bounded by a rectangular hyperbola.

3N

The efficient set with a riskless asset

If one asset is riskless, the variance of returns on that asset, and the covariance with returns on all other assets will be zero.

In equilibrium, the riskless rate < return on MV. Hence, the opportunity set will be the tangent line from the riskless asset to the efficient set.

In the two security case discussed earlier, suppose B is riskless, I.e., . Then from the above equation, we have 0B

BA

AB

pBA

AA

BA

Bpp

RR

R

RRRR

R

The efficient set with a riskless asset (continued)

Homogeneous expectations assumption

All investors have the same estimates on expectations, variances and covariances.

Under homogeneous expectations, all investors would hold the portfolio of risky assets represented by the tangency portfolio.

It is a market-valued weighted portfolio of all existing securities, I.e. market portfolio. A proxy commonly used is S&P 500.

What is the tangency portfolio?

Use of such a broad-based index as a proxy is justified since most investors hold diversified portfolios.

m

2mi

i R

R,RCov

Formula for beta

mi R,RCov

m2 R

covariance between the return on asset i and the return on market

variance of market portfolio

The efficient set with a riskless asset (continued)

The best measure of the risk of a security in a large portfolio is the beta of the security, which measures the responsiveness of the security to the movements in the market portfolio.

Example

States Probability Economyshock

Firmshock

Marketreturn (%)

Firm return(%)

1234

0.250.250.250.25

RecessionRecession

BoomBoom

DownUp

DownUp

-5-51515

-15-51525

5%150.25150.2550.2550.25RR4

1smssm

5%25155150.25RR4

1sfssf

0.01RR4

1s

2mmss

2M

0.015RRRRR,RCov4

1s

mmsffssmf

Example (continued)

The beta coefficient for this firm is

1.5

0.01

0.015

R

R,RCov

m2

mf

Returns on this firm’s stock magnify market returns.

The CAPM equation

Relationship between risk and expected return

If there is a riskless asset with return r, there is a straight line trade off between risk and expected return for a security.

sloperR

is the contribution of this security to the portfolio risk.

If the tangency portfolio is the market portfolio with expected

return and standard deviation , then mR mR

m

m

R

rRslope

The CAPM equation (continued)

Equilibrium expected return on asset j :

jm

mj

R

rRrR

It can be shown that m

mjj R

R,RCov

Then we have

rRrRR

R,RCov

R

R,RCov

R

rRrR

mjm

m2

mj

m

mj

m

mj

rRrR mjj CAPM equation

The CAPM equation (continued)

(Expected return on a security)

= (current risk free interest rate)

+ (beta coefficient of the security)*(historical market risk premium)

rRrR mjj CAPM equation

Finally, we established a way of determining appropriate discount rate for risky cash flows. We first measure its risk by its beta coefficient, and then obtain the required return from the CAPM equation.

The CAPM equation (continued)

Interpretation

Recall that the variance of return on a diversified portfolio is basically the “average covariance”. The beta coefficient for asset j can be considered as the share of overall market risk contributed by asset j. Then CAPM equation says that an asset shares the market excess return to the extent that it contributes to the total market risk.

j

rRm

In practice, we usually estimate using linear regression using historical returns data on and

Regression

j

iR mR

terrorRR mtiiit

The CAPM equation (continued)

m

miT

t

mmt

T

t

mmtiit

i R

RRCov

RR

RRRR

2

1

2

1 ,ˆ

statistical (least squares) estimator for j

The Security Market Line (SML)

When

rR0

RR1

RR1

jj

mjj

mjj

The Security Market Line (SML) below graphs

expected return against beta, using the CAPM equation.

rRrR mjj

Slope of the SML is the risk premium. For the S&P500 and US treasury bills, the risk premium is about 8.5%. (The book uses 9.2%, which is based on Ibbotson et. al study). This estimate is often used as a forecast for the risk premium on stocks in the future.

SML (continued)

The SML applies to portfolios as well as individual securities. For a portfolio with of A and of B, with beta coefficients and the expected return on the portfolio is

Ax Bx AB

BBAAp RxRx

Note that

implying

mBBmAA

mBBAAmp

R,RCovxR,RCovx

R,RxRxCovR,RCov

BBAAp xx

Hence, the portfolio also will be on the SML.

The SML should not be confused with the efficient set.

A Risk-Return Separation Theorem

An investment will be worth taking only if it is at least as desirable as what is already available in the financial markets.

A new investment will be worthwhile if and only if it is outside (above) the efficient set (or the risk-return budget constraint).

No matter where individual would choose to be on the efficient set, an investment can only make them better off if it is above the efficient set.

If the two financial separation theorems did not hold, then the firms would need to know the inter-temporal and risk-return preferences of each owner to decide desirable investments.

State Prob. Returnon A

Returnon B

Returnon C

1234

0.100.400.400.10

0.250.200.150.10

0.250.150.200.10

0.100.150.200.25

There are 3 securities in the market with the following payoffs:

What are expected returns and standard deviations of the returns?

4

1sissi RR AR

BR

CR

0.1750.1750.175

2A2B2C

0.04030.04030.0403

4

1s

2iissi RR

Problem 10.13 from the text

What are covariances and correlations between the returns? For j = A,B,C and k = A,B,C

kks

4

1s

jjssjk RRRR

kj

jkjk

ABACBC

0.000625-0.001625-0.000625

ABACBC

0.385-1.000-0.385

Problem 10.13 from the text (continued)

Problem 10.13 from the text (continued)

What are expected returns and standard deviations of the portfolios?

ABP

ACP

BCP

5.0;5.0 BA xx

5.0;5.0 CA xx

5.0;5.0 CB xx

BABBAAP RRRxRxRAB

5.05.0

CAP RRRAC

5.05.0

CBP RRRBC

5.05.0

0.1750.1750.50.1750.5RxRxR BBAAPAB

0.175RRR ABBCAC PPP

2BAB

2A

B2BBABAA

2A

BBAA2P

0.250.2520.25

RVarxR,RCovx2xRVarx

RxRxVarAB

0.0335ABP 0

ACP 0.0224BCP

Problem 10.39 from the text

Suppose you have invested $30,000 in the following 4 stocks

S e c u r ity A m o u n tin v e s te d

B e ta ix rRrR mii

S to c k AS to c k BS to c k CS to c k D

5 ,0 0 01 0 ,0 0 08 ,0 0 07 ,0 0 0

0 .7 51 .1 01 .3 61 .8 8

5 /3 01 0 /3 0 8 /3 0 7 /3 0

0 .1 2 2 50 .1 6 1 00 .1 8 9 60 .2 4 6 8

The risk free rate is 4% and the expected return on the market portfolio is 15%. Based on the CAPM, what is the expected return on the above portfolio?

Let denote the proportion invested in stock i (I=A,B,C,D)

and the beta coefficient of the stock i.i

ix

Problem 10.39 from the text (continued)

There are two ways to answer the question.

1. Calculate the beta coefficient for the portfolio, and get the expected return on the portfolio directly from CAPM equation.

P

1822.011.004.0 PmPP rRrR

293.1 DDCCBBAAP xxxx

2. Calculate the expected return individually for I = A,B,C,D and obtain the expected return on the portfolio as

1822.0 DDCCBBAAP RxRxRxRxR

iR

PR