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
Expected returnof portfolio
Standarddeviation of
portfolio’s return.
Which risky asset to choose?Which risky asset to choose?
Standarddeviation of
portfolio’s return.
Risk-freerate (Rf )
S.
Capital market line
.X
Y
Which risky asset to choose?Which risky asset to choose?
Expected returnof portfolio
Standarddeviation of
portfolio’s return.
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