CHPT14- The CAPM ---test.pdf

Chapter 14 The CAPM ---Applications and
tests
Fan Longzhen
Predictions and applications
• CAPM: in market equilibrium, investors are only rewarded for bearing
the market risk;
• APT: in the absence of arbitrage, investors are only rewarded for
bearing the factor risk;
• Applications:
•---professional portfolio managers: evaluating security returns and
fund performance
•---corporate manager: capital budgeting decisions.
Early tests of CAPM
• Cross-sectional test of the model:
• Douglas (1969);
• Miller and Scholes (1972);
• Black, Jensen and Scholes (1972);
• Fama and Macbeth (1973)
E(Ri ) = R f + βi (E[Rm ] − R f )
Rγ= + ˆ+ e , i = 1,2,...,n
i γ0 1 i i
? β
γˆ R
0 = f
?
γˆ R − R
1 = m f
continued
• Douglas (1969)
• Adds own-variance to regression significant;
2
• Linter addsσˆ( e i ) to regression significant;
• Miller and Scholes (1972)
• Measurement error in ˆ‘s;
βi
2
• Correlation between measurement error and σˆ(ei )
• Skewness of returns .
• Black, Jensen, and Scholes (1972)
Rit − R f = αi + βi (Rmt − R f ) + eit
• Time-series test ?
α=0
• Use portfolio to maximize dispersion of beta’s i
• Low βˆ stocks positive αˆi ' s
αˆ' s
• High βˆ stocks negative i
Hypothesis testing
• Definition of size and power
• H true H false
• Accept correct Type II error
• Reject type I error correct
• Size=Pr(Type I error);
• Power=1-Pr(type II error);
• Tradeoff between size and power;
• Fix size, find most powerful test.
CAPM test
X it = αi + βi X mt + eit
X it ≡ Rit − R ft , X mt ≡ Rmt − R ft
• CAPM holds αα= 0
σi
2 2
• e µ m
Var[ ˆi ] ≡ Vα= (1+ 2 )
T σ m
•H: αˆi ~ N(0,Vα)
Some numbers for monthly . data,1985-1989
σ
ˆˆ
• S&P500 T-bills: T=60, µm = ,σ m =
2
Vˆ= e (1+ ) = 2
α 60 e
2
• What is σ e ?
Rit =σαi + βi Rmt + eit
•βσ market model
2 = 2 2 + σ 2
• i i m e
ˆ 2 2
•σ i = for typical NYSE stock
•ρ= for typical NYSE stock