复旦大学：《投资学讲义》（英文版） Chapter 14 The CAPM ——-Applications and tests

Chapter 14 The CaPm---applications and tests an longzhen

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

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(R)=R+B(E[Rm]-RD) R=+y1月+e,i=12,,n Yi=R-R

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) m f f i i i i f i m f R R R R e i n E R R E R R − = + + = = + − = = ? 1 ? 0 0 1 ˆ ˆ , 1,2,..., ˆ ( ) ( [ ] ) γ γ γ γ β β

continued Douglas(1969) Adds own-variance to regression significant Linter adds(e to regression - significant Miller and Scholes(1972) Measurement error inbs Correlation between measurement error and a(e) Skewness of returns Black, Jensen, and Scholes (1972) Use portfolio to maximize dispersion of beta s isa+ RI -, B (R -R)+e Time-series test Low B stocks - positive a,'s High B stoCkS -> negative d,'s

continued • Douglas (1969) • Adds own-variance to regression significant; • Linter adds to regression significant; • Miller and Scholes (1972) • Measurement error in ‘s; • Correlation between measurement error and • Skewness of returns . • Black, Jensen, and Scholes (1972) • Time-series test • Use portfolio to maximize dispersion of beta’s • Low stocks positive • High stocks negative ˆ ( ) 2 i σ e βi ˆ ˆ ( ) 2 i σ e 0 ( ) ? = − = + − + i it f i i mt f it R R R R e α α β β ˆ s i α ˆ ' β ˆ s i α ˆ

Hypothesis testing Definition of size and power H true H false Accept correct Type I error Reject type i error correct Size=Pr(Type I error) Power=l-Pr(type II error) Tradeoff between size and power Fix size find most powerful test

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 B Xm+ R CAPM holds In→少>a=0 Vara I ( H N(0,V)

CAPM test • CAPM holds • • H: it it ft mt mt ft it i i mt it X R R X R R X X e ≡ − ≡ − = + + , α β αi = 0 [ ˆ ] (1 ) 2 2 2 m e m i T Var V σ σ µ α ≡ α = + ˆ ~ (0, ) αi N Vα

Some numbers for monthly u. s. data 1985-1989 ·S&P500T-bls:T=60.Am=0.01.7m=0051 n=9(1+0465:=001470 60 What is 02? R, =a+BRute market model a2=B3o2+2 for typical Nyse stock p=0.25 for typical NYSE stock a=(1-0.250.10)=0.0075 001144

Some numbers for monthly U.s. data,1985-1989 • S&P500 T-bills: T=60, • What is ? • market model • • for typical NYSE stock • for typical NYSE stock 2 2 ( 1 0.0465 ) 0.01744 60 ˆ ˆ 0.011, ˆ 0.051 e e m m V σ σ µ σ α = + = = = 2 σ e it i i mt it R = α + β R + e 2 2 2 2 σ i = βi σ m + σ e 2 2 ˆ = 0.10 σ i ρ = 0.25 ˆ ( 1 0.25)( 0.10 ) 0.0075 2 2 σ e = − = Vˆ α = 0.01144

continued Alternative Hypothesis K: a=0.01 12% Annually Does v change

continued • Alternative Hypothesis: • Does change? : = 0.01 K αi 12% Annually! Vα

Testability of CaPM The wide acceptance of the capm and apt makes it all more important to test their predictions empirically How does a product of abstract reasoning hold in reality? Unfortunately, the predictions of the Capm and aptare hard to test empirically -neither the market portfolio in CaPm nor the risk factor in apt is observable expected returns are unobservable. and could be time varying, volatility is not directly observable, and is time-varying

Testability of CAPM • The wide acceptance of the CAPM and APT makes it all more important to test their predictions empirically. • How does a product of abstract reasoning hold in reality? • Unfortunately, the predictions of the CAPM and APT are hard to test empirically • ---neither the market portfolio in CAPM nor the risk factor in APT is observable; • ---expected returns are unobservable, and could be time￾varying; • ---volatility is not directly observable, and is time-varying

An ideal test of the capm In an idea situation, we have the following input 1. Risk-free borrowing/lending rate R, 2. Expected returns on the market E(R )and on the risky assetE(R') 3. The exposure to market risk B =coy(RM, R)/var(R") These input allow us to examine the relation between reward (E(R )-Rd and risk B More risk. more reward? 2. Do they line up? 3. What is the reward for a risk exposure of 1? 4. Zero risk. zero reward

An ideal test of the CAPM • In an idea situation, we have the following input: • 1. Risk-free borrowing/lending rate ; • 2. Expected returns on the market and on the risky asset ; • 3. The exposure to market risk ; • These input allow us to examine the relation between reward • and risk : • 1. More risk, more reward? • 2. Do they line up? • 3. What is the reward for a risk exposure of 1? • 4. Zero risk, zero reward? R f ( ) M E R ( ) i E R cov( , ) / var( ) M i M β i = R R R ( ( ) )f i E R − R β i