6. CAPM and APT (Part-2)

A. Multifactor models

1. Factor Loadings & Risk Preiums

$$E(r_p) = r_f + \beta_{P,1}\times(E(r_1)-r_f) + \beta_{P,2}\times(E(r_2)-r_f)$$

1. Loading on the factor portfolios: $\beta_{P,1}$ and $\beta_{P,2}$

   • How a security’s (P) returns “co-vary” with the returns of the factor portfolios (1 and 2)

2. Risk premium on the factor portfolios: $(E(r_1)-r_f)$ and $(E(r_2)-r_f)$

   • Risk premium associated with the risk factor portfolio

2. What is Factor Portfolio?

A well-diversified portfolio of many securities:

1. Zero idiosyncratic risk

2. A beta of 1 related to one risk factor

3. A beta of 0 w.r.t other risk factors

   • Market portfolio with respect to the interest rate risk factor portfolio is:

   $$\beta_{M, IR} = 0$$

3. Understanding equilibrium

In equilibrium, a well-diversified portfolio P must satisfy:

$$\frac{E(r_p)-r_f}{\beta_p} = \frac{E(r_M)-r_f}{\beta_M}$$

Hence we can re-write the equation as:

$$E(r_p) = r_f + \beta_p\times(E(r_M)-r_f)$$

This is very similar to CAPM.

$$E(r_i) = r_f + \beta_i\times(E(r_M)-r_f)$$

If $\alpha_p \ne 0$, an arbitrage opportunity exists.

B. Arbitrage Pricing Theory (APT)

INTUITION for arbitrage pricing:

How to take advantage of arbitrage opportunity?

1. Requires no initial investment
2. Earns a positive profit with complete certainty

No initial investment + No risk

C. 1-Factor Arbitrage Pricing

Step 1:

D. 2-Factor Arbitrage Pricing