5. CAPM and APT (Part1)
Table of Contents
A. Index Model & Capital Asset Pricing
I. Two Components of Stock Risk
$$\sigma_i^2 = \sigma^2(\alpha_i) + \sigma^2(\beta_iR_m) + \sigma^2(\epsilon_i)$$
Total risk = Systematic risk ($\sigma^2[\beta_iR_m]$) + Firmspecific risk ($\sigma^2[\epsilon_i]$)

$\sigma^2(\alpha_i) = 0$ Why? As $\alpha_i$ is the intercept hence does not change;

$var(\epsilon_i) = 0$? Not necessarily in index model, but yes in CAPM.
 In CAPM, we only invest in the market portfolio and it is fully diversified.
$R^2$ measures how much of the variation in $R_i$ is explained by $R_m$.
$R^2$ is fitness of the model.
$$R^2 = \frac{\beta_i^2 \sigma_m^2}{\sigma_i^2}$$
B. Capital Asset Pricing Model (CAPM)
The CAPM shows how risk and expected returns relate in equilibrium, and what type of investment risk matters.
Assumptions:
 Individual investors are “price takers”
 The trading does not affect the price
 Accept the price as given
 Investments are limited to traded finanicial assets
 No real estate, art, etc.
 No taxes or transaction costs
 Top bracket (22% for income > 320k)
 People only care about mean and variance of returns
 Strong assumption: People all have the same expectations, and the mean and variance of returns are known (Homogeneous expectations).
I. Overview of Implications
"All investors will hold some combination of the market portfolio (M) and the riskrate"
Market portfolio:
 All assets of the security universe
 Market value weighted
 Optimal risky portfolio and on efficient frontier
CAPM, Capital Market Line
$$E[r_{C,M}] = r_f + y_M(E[r_M]r_f)$$
Quick Recap
A theoretical concept that gives optimal combinations of a riskfree asset and the market portfolio.
The CML is superior to Efficient Frontier because it combines risky assets with riskfree assets.
Given all the assets have been invested in the risky portfolio (y=1), the market risk premium is:
$$E[r_M]r_f = A\times\sigma_M^2$$
According to capital asset pricing model, the expected return of an asset is a linear function of the expected return of the market portfolio and riskfree rate.
$$E[r_{i}] = r_f + \beta_i(E[r_M]r_f)$$
II. Understanding Beta
Beta is a measure of the systematic risk of a security relative to the market.
$$\beta_i = \frac{cov(r_i,r_M)}{\sigma_M^2}$$
 Beta is the sensitivity of a securityβs excess return to the systematic factor (mkt risk premium).
III. Understanding alpha
Alpha measures how much expected returns differ from CAPMimplied expected returns.
$$\alpha = E(r_i)  r_f  \beta_i(E(r_M)r_f)$$
 If $\alpha_i > 0$, then the security is underpriced and should be bought.
 If $\alpha_i < 0$, then the security is overpriced and should be short.
III. CAPM: Security Market Line
SML Slope is given by:
$$\frac{E[r_i]r_f}{\beta_i} = \frac{E[r_m]r_f}{\beta_m}$$
And $\beta_m = 1$, which gives us the CAPM equation.
C. Arbitrage Pricing Theory (APT)
I. Overview of APT
"Describes the relation between expected returns and risk when there are ONE or MORE sources of systematic risk."
A systematic risk must be:

Pervasive  Source of risk must potentially impact most companies, leading the stock prices to unexpcetedly change.

Undiversifiable  Source of risk can not be diversified away in a large portfolio.
II. APT: Assumptions
 No taxes, no transaction costs
 Investors can form welldiversified portfolios
 Welldiversified portfolios only contain systematic risk
 No arbitrage opportunities exist
Arbitrage: An investment strategy in which an investor simultaneously buys and sells an asset in different markets to profit from a difference in the price.
 An arbitrage opportunity exists when two securities always have the same payoff but DO NOT have the same price
D. How CML, CAPM, single index model and APT are related?

CML and CAPM: If the assumptions for cAP are true, the optimal risky portfolio becomes the market portfolio, and then we have CAPM.

Single index model is used to test the theories, for example, here CAPM, to see whether CAPM receives empirical support using data.

APT is another theoretical model to price the financial assets. It’s based on different assumptions. In week2, you will see more of the differences with CAPM.

To test APT, we will introduce multifactor models (empirical models).