# 5. CAPM and APT (Part-1)

## 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 ($\sigma^2[\beta_iR_m]$) + Firm-specific 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:

1. Individual investors are “price takers”
• The trading does not affect the price
• Accept the price as given
2. Investments are limited to traded finanicial assets
• No real estate, art, etc.
3. No taxes or transaction costs
• Top bracket (22% for income > 320k)
4. People only care about mean and variance of returns
5. 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 risk-rate"

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 risk-free asset and the market portfolio.

The CML is superior to Efficient Frontier because it combines risky assets with risk-free 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 risk-free 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 security’s excess return to the systematic factor (mkt risk premium).

### III. Understanding alpha

Alpha measures how much expected returns differ from CAPM-implied 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:

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

2. Undiversifiable - Source of risk can not be diversified away in a large portfolio.

### II. APT: Assumptions

1. No taxes, no transaction costs
2. Investors can form well-diversified portfolios
• Well-diversified portfolios only contain systematic risk
3. 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
• 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 week-2, you will see more of the differences with CAPM.

• To test APT, we will introduce multi-factor models (empirical models).

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