This section provides a reminder of the Capital Asset Pricing Model (CAPM) as an example of a one-factor model designed for explaining individual stock returns. Next, we introduce the generic expressions of single-factor models and of multiple-factor models.

Example of a One-factor Model

A traditional example of a single-factor model is the Sharpe's CAPM, where the return of a single stock is explained by the return on the stock index. Under the CAPM, stock returns are driven by the equity index return. The coefficient of the equity index is the P, which measures the relative sensitivity of the stock return to the index return. The expected return of a stock should be related to the expected equity index return and the coefficient is the P which depends on the particular stock. The conceptual equation of the CAPM for a single stock "i" is:

The equation says that the expected stock return should be equal to the risk-free rate plus a "risk premium," that is the difference between the expected index return and the risk-free rate, times p., which is specific to the stock of interest. The P is the sensitivity of the stock return to the stock index return. For the index itself, the P has to be equal to 1. Some stocks have high P and others low p. The response to a standard shock on the equity index return, for example +1%, is the P x 1%. If P is higher than 1, for example 2, the stock return should, on average, respond with an increase of +2%. If the P is 0.5, the response to the same shock on equity index would be, on average, +0.5%.

The statistical version of the theoretical equation is a one-factor model:

Since a factor model is based on a regression analysis, in all factor models, all cross covariances between the factors and the residual are zero and the expectation of the residual term is zero. According to the theory, the constant should be P0 — ril - p.) and the factor coefficient should be p. — P .. The residual in the linear statistical relation is the random variation of the stock return that is independent of the equity index return. Both the residual and the coefficient depend on the stock.

This single-factor model does not fully explain the variations of the stock return. The volatility of the first term is the general or "systematic risk." The volatility of the residual is the specific risk of the individual stock. The systematic risk cannot be diversified away in portfolios since all stocks are more or less sensitive to index return, meaning that they all move with the Index, although with various sensitivities. The total risk of the stock return sums up the specific and the general risk. Statistically, since the residual is independent of the common factor, the variance of the return is the sum of the variance of the first term and of the variance of the residual. Such a model has been extended, notably with Ross's APT [65], to several variables. Extending the number of factors allows one to explain a grater fraction of the total risk with factors.

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