Capital Asset Pricing Model

Applied covariance:Project Part 1

Review question Asset A has an expected rate of return

of .15. Asset B has an expected rate of return

of .25. Consider a portfolio consisting 30% asset

A and 70% asset B. What is the expected rate of return on the

portfolio?

Answer Expected rate of return is .3*.15+.7*.25 = .22

Review variance, covariance Variance: square the deviations and

take expectation. Covariance: multiply the deviations and

take expectation.

Notation Variance

Covariance

Portfolio weights

AB

22 , BA

BA XX ,

1 BA XX

Portfolio variance The role of covariance. Equation 9

ABBABBAAP XXXX 222222

It all happens because

( )x y x y xy 2 2 2 2

Portfolio risk and return,

BBAAP RXRXR

BBAAP DevXDevXDev

BBAAP RXRXR

BBAAP DevXDevXDev

22 )( BBAAP DevXDevXDev

Portfolio deviation

Deviation squared

DevP2

( )x y x y xy 2 2 2 2

BABABBAA DevDevXXDevXDevX 22222

Portfolio variance

E DevP[ ]2

][2][][ 2222

BABA

BBAA

DevDevEXXDevEXDevEX

Var RP( ) ),(2)())( 22

BABA

BBAA

RRCovXXRVarXRVarX

Portfolio variance depends oncovariance of the assets.

Positive covariance raises thevariance of the portfolio.

ABBABBAAP XXXX 222222

Correlation coefficient

BA

ABABBA RRCorr

),(

BAABBABBAAP XXXX 222222

Application Asset B is the market portfolio Call it asset M. Everyone prefers to hold M, in theory Asset A is any asset. Think of adding a little A to the market

portfolio.

Question does adding a little of asset A to the

market portfolio increase the risk? Yes if

No if

2MAM

2MAM

Derivation

AMAAMA

MAAAPA

XX

XXdXd

2)1(2

)1(22 222

AMAAMAAAP XXXX )1(2)1( 22222

AMMPA

A dXdXat 22,0 22

Beta measures risk How much risk is added depends on

the relation of sigma AM and sigma squared M

Define beta

2M

AMA

Beta item Download price data for your stock and

the market (S&P 500). Construct rates of return. Compute variances and covariances. Compute beta for the stock. Don’t use the financial formulas, except

as a check on your work

Another check on your work Regression Idea: take some points in (Dev M,Dev

A) space and fit a line to them. Let b*Dev M be an estimate of Dev A. Minimize sum of squared errors.

Sum of squared errors2

,1

, )()( tM

Tt

ttA bDevDevbF

Minimize it

0)(2)('

0)('

,,1

,

tMtM

Tt

ttA DevbDevDevbF

bF

01

2,

1,,

Tt

ttM

Tt

ttMtA bDevDevDev

Divide by T-1

Tt

ttM

Tt

ttMtA

DevT

DevDevTb

1

2,

1,,

11

11

The estimate of Is the ratio of sample covariance over

variance of the market. It’s beta, except for using sample

statistics instead of population values.

Problem 8.1; read Ch 8.2 If the product is marketed now, its

chance of success is .5 and the payoff is 20M in present value. Failure = 5M

If the product is tested and improved, launch is delayed one year. The cost is 2M and the chance of success is .75.

Discount at 15%. Question: Launch now or later?

The story of CAPM Investors prefer higher expected return

and dislike risk. All have the same information. Two (mutual) funds are sufficient to

satisfy all such investors:

The two funds: 1) The "risk-free" asset, i.e., Treasury

Bills 2) The market portfolio consisting of all

risky assets held in proportion to their market value.

The market portfolio Its expected return is 8.5% over the T-

Bill rate It bears the market risk Its beta is unity by definition.

Capital asset pricing model

jfMfj RRERRE ][)(

T-bill rate is known.Market premium is known, approximately 8.5%.Estimate beta as in the project

Security market line It’s straight. Risk-return relation is a straight line.

Why is it a straight line? Beta is the measure of risk that matters. Given beta construct a portfolio with the

same beta by a mix of T-Bills (beta = 0) and the market portfolio (beta = 1)

Expected return on the portfolio is on the SML.

So any asset with the same beta must also be on the SML.

beta

Rate of returnexpected by the market

Rf

E[RM] Security m

arket lin

e

1

Examinations Samples on the web page. 1. A midterm from the past. 2. Sample questions for midterm and

final. Practice the technique of answering in

short essays.

Review item Return on asset A has a std dev of .05 Return on asset B has a std dev of .07 Correlation of return on asset A with

return on asset B is 1. What is the covariance of the returns?

Answer: AB = AB*A*B=.0035