Question

2. (a) Explain carefully how stock returns are derived by arbitrage pricing
theory (APT).
(b) Briefly evaluate the empirical performance of APT.

Answer 1

(a) Arbitrage pricing theory is an asset pricing model in which expected returns of an asset are modeled linearly as a function of various factors or market indices. Below is the relevant equation

$r_i = \beta_0 + \beta_1X_1 + \beta_2X_2 + ..+ \beta_nX_n$

All Xi refer to the different factors that explain the returns. The coefficients of the above equations are estimated by Ordinary least squares. $\beta_i$ slope coefficient indicate the sensitivity of each factor to the return on the asset. Once the model is estimated, the returns are derived with the help of the estimated model given the values of the above factors.

Under APT, if a stock if mispriced then its predicted price diverges from the actual price. Thus there are two possibilities:- either the stock is underpriced or overpriced. I show below the investment strategy when the stock is underpriced:
1. If the current price of the asset is too low then the investor can short-sell the portfolio and buy the underpriced asset.
2. At the end of the period, the mispriced asset is sold and the portfolio is bought back.
3. The difference between the two price is the profit.

(b) Roll and Ross (1980) conducted an empirical performance of APT. Over a period of 10 year, they studied daily returns for a sample of 1269 securities from NYSE and AMEX. They arranged them into 42 groups with 30 securities in each group. They then did maximum-likelihood factor analysis and used the estimated factor loading as explanatory variable in stage two of cross sectional regression test. They found that at least three factors constitute significant explanatory variable in the regression model. Huges (1981) and Chen (1983) identify five significant factors in the model structure.