02-04 The Capital Assets Pricing Model (CAPM)

资本资产定价模型
$\omega_i$ :portion of funds in asset i
$\sum_i \mathrm{abs}(\omega_i)=1.0$
$\gamma_p(t)=\sum_i \omega_i \gamma_i(t)$

Assume:
stock A +1%, $\omega_A$ =75%, stock B - 2%, $\omega_B$ =-25%
$\gamma_p$ =1%75% + (-2%)(-25%)=1.25%

the market portfolio
Cap weighted
$\omega_i=\frac{\mathrm{mktcap}_i}{ \sum_j \mathrm{mktcap}_j}$

The CAPM equation:

$\gamma_i(t)=\beta_i \gamma_m(t) + \alpha_i(t)$
particular stock return = market + residual

CAPM says:
$E(\alpha)=0$

CAPM vs Active managment

passive: buy index and hold
active: pick stocks, buy underweight, sell overweight
$\gamma_i(t)=\beta_i \gamma_m(t) + \alpha_i(t)$
for $\alpha_i(t)$ :

CAPM says $\alpha$ is random, and $E(\alpha)=0$
however, active managers believe they can predict $\alpha$

CAPM for portfolios

$\gamma_p(t)=\sum_i \omega_i (\beta_i \gamma_m(t) + \alpha_i (t)$
$\beta_p = \sum_i \omega_i \beta_i$
$\gamma_p(t)=\beta_p \gamma_m (t) + \alpha_p (t)$ [CAPM]
$= \beta_p \gamma_m (t) + \sum_i \omega_i \alpha_i (t)$ [Active]

Implications of CAPM

$\gamma_p=\beta_p \gamma_m + \alpha_p$

Expected value of $\alpha = 0$
Only way to beat market is choose $\beta$
Choose high $\beta$ in up markets
choose low $beta$ in down markets
Efficent Market Hypothesis(EMH)
says you can't predict the market, so CAPM says you can't beat market.

Arbitrage Pricing Theory (APT, 套利定价理论)

Stephen Rors, 1976
$\gamma_i=\beta_i \gamma_m + \alpha_i$
CAPM = Ocean
$\beta$ 项可以分解为finance, tech等
$\gamma_i=\beta_{iF} \gamma_F + \beta_{iT} \gamma_T + \beta_{iM} \gamma_M...+\alpha_i$