Return and Risk: The Asset-Pricing Model: CAPM and APT

Return and Risk:

The Asset-Pricing Model:

CAPM and APT

Return and Risk:

The Asset-Pricing Model:

CAPM and APT

Portfolio TheoryPortfolio Theory

Ex1, Two portfolio Portfolio 1: a single investment A: with expected return :

1 and variance 25 Portfolio 2: equally weighted combination of two uncorrected

investments: with expected return of 1, and variances 25

Both have the same expected return : 1, but the variance of portfolio 2 is .52*25 + .52*25 = 12.5

Portfolio is preferred than portfolio 1.

It is because the zero covariance diversifies some of the return

volatility.

A portfolio of two risky assetsA portfolio of two risky assets

Ex2: Asset 1 has expected return of .22 and SD of .32 Asset 2 has expected return of .13 and SD of .23

Covariance is .01104

X1 X2 E(rp~) (rp

~)0 1 .13 .23

.2 .8 .148 .2037

.4 .6 .166 .2018

.6 .4 .184 .2250

.8 .2 .202 .2668 1 0 .22 .32

Efficient Portfolio FrontierEfficient Portfolio Frontier

E(R)

Efficient Sets and DiversificationEfficient Sets and Diversification

E(R)

-1 <

= 1

= -

1

What about portfolio of n assets?What about portfolio of n assets?

Expected returnof portfolio

Standarddeviation of

portfolio’s return.

Markowitz Portfolio TheoryMarkowitz Portfolio Theory

Combining stocks into portfolios can reduce

standard deviation below the level obtained from a

simple weighted average calculation.

Less than perfect correlation coefficients make this possible.

The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfoliosefficient portfolios.

Combination of risk-free and risky assetCombination of risk-free and risky asset

p2 = x1

212, Thus, : p = x11, x1 = p / 1

E(rp~) = x1r1 + x2 rf = p*(r1/1) + x2 rf

rf

.Asset 1




Which risky asset to choose?Which risky asset to choose?



Risk-freerate (Rf )

S.

Capital market line

.X

Y

Which risky asset to choose?Which risky asset to choose?




Risk-freerate (Rf )

4

S.5

..

Capital market line

.X

Y

Lending

Borrowing

The Chosen Portfolio, MThe Chosen Portfolio, M

Will different individual have different choice of different risky portfolio asset?

What if different individual holds the same expectation (homogeneous expectation), that is, .market reflects all the information?

What does portfolio S look like?

All investors will invest in portfolio S, regardless of their risk aversion. But they may NOT have the same portion of their wealth in the two assets.

Security Market LineSecurity Market Line

Expected returnon security (%)

Beta ofsecurity

Rm

Rf

0.8 1

S

M

T

..

.Security market line (SML)

Security Market LineSecurity Market Line

For a well diversified portfolio, the risk measure of individual stock is not the SD of return, it is beta.

Investors are not rewarded with any return for bearing any unsystematic risk.

Why should equilibrium prices of securities fall on SML?

If point A lies above the Security market line, then investors will bid up the price until the return goes back on line

If point B lies below the security market line, then investors will sell the security, push down the price until it goes back on line.

Capital Asset Pricing ModelCapital Asset Pricing Model

If investors hold market portfolio, how do they measure the risk of individual securities?

The covariance with the markets, that is Beta.

CAPM, for any security i,

E (ri~) = rf + i [E(rm

~ - rf)],

where, E(rm~ - rf) : expected market risk premium

i = COV(ri~, rm

~)/m2

Testing the CAPM

Avg Risk Premium

Portfolio Beta1.0

SML

30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk Premium

1931-65

Testing the CAPM

Avg Risk Premium

Portfolio Beta1.0

SML

30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk Premium

1966-91

Testing the CAPM

0

5

10

15

20

25

Average Return (%)

Company size

Smallest Largest

Company Size vs. Average Return

Testing the CAPM

0

5

10

15

20

25

Average Return (%)

Book-Market RatioHighest Lowest

Book-Market vs. Average Return

About CAPMAbout CAPM

Why does CAPM not hold ? Is CAPM dead? Expected return vs. real return Short term or long term effect?

The contribution of CAPM How the financial markets may price risky assets How to measure a risky asset’s risk How to calculate expected rate of return.

The measurement of betaThe measurement of beta

Choice of market proxy

The time period

Measurement error: the problem of overestimate for high

beta and underestimate for low beta stocks

Instability over time

Arbitrage Pricing Theory

Alternative to CAPM

Expected Risk

Premium = r - rf

= Bfactor1(rfactor1 - rf) + Bf2(rf2 - rf) + …

Return = a + bfactor1(rfactor1) + bf2(rf2) + …

Arbitrage Pricing Theory

Estimated risk premiums for taking on risk factors(1978-1990)

6.36Mrket

.83-Inflation

.49GNP Real

.59-rate Exchange

.61-rateInterest

5.10%spread Yield)(r

ium Risk PremEstimatedFactor

factor fr

Documents

Return and Risk: The Asset-Pricing Model: CAPM and APT