6. CAPM and APT (Part2)
Table of Contents
A. Multifactor models
1. Factor Loadings & Risk Preiums
$$E(r_p) = r_f + \beta_{P,1}\times(E(r_1)r_f) + \beta_{P,2}\times(E(r_2)r_f)$$

Loading on the factor portfolios: $\beta_{P,1}$ and $\beta_{P,2}$
 How a security’s (P) returns “covary” with the returns of the factor portfolios (1 and 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 welldiversified portfolio of many securities:

Zero idiosyncratic risk

A beta of 1 related to one risk factor

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 welldiversified portfolio P must satisfy:
$$\frac{E(r_p)r_f}{\beta_p} = \frac{E(r_M)r_f}{\beta_M}$$
Hence we can rewrite 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?
 Requires no initial investment
 Earns a positive profit with complete certainty
No initial investment + No risk
C. 1Factor Arbitrage Pricing
Step 1: