金融学投资经管课件教案-Lecture05

Building Block I: One-factor /Multifactor Models
The universe of assets is large There are infinite numbers of securities These securities differ from each other in nontrivial ways
E rp rf Erm rf pm
rf pm
Extensions
The well diversified portfolio almost can be
relaxed to any security
Eri rf Erm rf i
mF
pF
If we use market return to represent the risk factor,
we
have F

rm
Erm , then
mF

Covrm, rm Varrm
1
We then have a SML equivalent to that of CAPM,
2 ep Var
wiei

1 n

2


2
ei


1 n

2
ei

rp E rp pF
For equally weighted portfolio, the nonsystematic variance approaches zero as n becomes larger
In another word, to preclude arbitrage opportunities, the expected return on all well diversified portfolios must lie on a straight line
Proof of One-Factor SML
Eri rf ikk
k is the risk price for bearing the risk from the
kth factor
Three Building Blocks
One-factor /Multifactor Models Well Diversified Portfolio Non-arbitrage Assumptions
Exercise
Important differences between CAPM and APT are A) CAPM depends on risk-return dominance; APT depends on a no arbitrage condition. B) CAPM assumes many small changes are required to bring the market back to equilibrium; APT assumes a few large changes are required to bring the market back to equilibrium. C) Implications for prices derived from CAPM arguments are more general than prices from APT
Eri rf i1 Er1 rf i2 Er2 rf
or
Eri rf ikk
Where do we find the factors?
ri Eri i1F1 i2F2 ... ei
The guidance to look for factors.
Problems with CAPM
Failure of empirical tests The information required by mean-
variance approach grows substantially
N=100 stocks
100 variance 100(100-1)/2=1225 covariance
The derivation of APT model is based on three building blPT Theorem
In the absence of arbitrage, there must exist constants k that, together with the factor loadings, describe every asset’s mean return
Cov(Fi,Fj) = 0, Cov(ei,ej) = 0, and Cov(Fi,ei) = 0

ij
ei is
Cov ri , Fj raVanrdFojm
is the return’s error with mean
sensitivity to the 0 and variance
It holds other than the equally weighted one
Building Block III: Non-arbitrage Assumptions
Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit.
Economic reason Explanation power
Estimation
Beta maintains the economic meaning: it still measures the sensitivity to risk sources.
It can be estimated by running regressions for each asset
Arbitrage opportunity?
Proof of One-Factor SML
Proposition II: For well diversified portfolios with different betas, their expected return must be proportional to beta.
If n=3000 stocks, more than 4 million covariance!
An Alternative: APT
Arbitrage Pricing Theory
Ross (1976,JET) developed Arbitrage Pricing Theory (APT) which derives asset prices from arbitrage arguments.
E rp a b pF
Risk free asset is a special well diversified portfolio
Market index portfolio M should be on the line,
therefore
E rp

rf

Erm rf
factor
2 ei
Building Bock II: Well Diversified Portfolio
If a portfolio is well diversified, only systematic risk
remains
rp E rp p F ep where p wii
Principle of capital market: The equilibrium market prices are rational in that they rule out arbitrage opportunities
Security prices satisfy a “no-arbitrage” condition.
rit i i Ft eit
rit i i1F1t i2F2t ... eit
Summary
APT’s three building blocks Derivation of SML with APT Extensions Selecting factors Comparison with CAPM
The one-factor SML can be extended to multifactor SML with the assumption
ri E ri i1F1 i2F2 ei
with Fi fi E fi
The multifactor SML is then
Portfolio on F1 on F2
Expected Return
A
1.0
2.0
19%
B
2.0
0.0
12%
Assuming no arbitrage opportunities exist, what are the risk premium on the factor F1 portfolio and the F2 portfolio?
Proof of One-Factor SML
Proposition I: Well diversified portfolios with equal betas must have equal expected returns in market equilibrium
rA ErA AF rB ErB B F