## Example 67 The Factor Equations of Pure Factor Portfolios

Write out the factor equations for the two factor portfolios in Example 6.6 and determine their risk premiums if the risk-free rate is 4 percent.

Answer: The expected returns for factor portfolios 1 and 2 are their respective portfolio-weighted averages of the expected returns of the individual securities, implying

Thus, the factor equation for factor portfolio 1 is

Rp1 = .06 + F1 + 0F2 For factor portfolio 2, the factor equation is

Rp2 = .04 + 0F, + F2 The risk premiums are, respectively

(Factor Portfolio 1) A1 = .06 - .04 = .02 (Factor Portfolio 2) A2 = .04 - .04 = 0

What Determines the Risk Premiums of Pure Factor Portfolios? Pure factor portfolios, being risky, generally have expected returns that differ from the risk-free return. Some factors may carry a positive risk premium; others, such as factor portfolio 2 in Example 6.7, may have a zero or negative risk premium. Whether a factor portfolio has a positive or a negative risk premium depends on the aggregate supply of the factor in the financial markets and the tastes of investors. If the assumptions of the Capital Asset Pricing Model are true, then the risk premiums of the factor portfolios are proportional to their covariances with the return of the market portfolio.20

20If the factors are uncorrelated with each other, the covariance of a factor with the return of the market portfolio is determined by the market portfolio's factor beta on that factor. In this case, the factor will have a positive risk premium if the market portfolio has a positive factor beta on a factor and vice versa. The economic intuition for this is straightforward: Under the assumptions of the CAPM, investors must be induced to hold the market portfolio so that supply is equal to demand. This is the same as inducing them to hold the factors in exactly the same proportions as they are contained in the market portfolio. Hence, if the market portfolio has a negative beta on a factor, implying that the factor is in negative supply, investors must be induced to short the factor so that the supply of the factor is equal to its demand. To induce an investor to short a factor, the action must carry a reward. If the factor itself has a negative risk premium, short positions in it earn a positive reward. The opposite holds true for factors on which the market portfolio has a positive factor beta.

Why the Interest in Pure Factor Portfolios? From a computational standpoint, it is easier to track an investment with a portfolio of the factor portfolios than with a portfolio of more basic investments, such as individual stocks. For example, an investment that has a beta of .25 on factor 1 and .5 on factor 2 is tracked by a portfolio with weights of .25 on factor portfolio 1 and .5 on factor portfolio 2. A .25 weight on the risk-free asset is also needed to make the weights sum to one.

The construction of this tracking portfolio is easy because each of the building blocks has only one function: Only the weight on factor portfolio 1 affects the tracking portfolio's factor 1 beta. Only the weight on factor portfolio 2 affects the tracking portfolio's factor 2 beta. The risk-free asset is used only to make the portfolio weights sum to 1, after the other two weights are determined. Thus, it is particularly simple to construct tracking portfolios after first taking the intermediate step of forming factor portfolios. The next section uses this insight.

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### Responses

• kimmo
What is pure factor portfolio?
8 years ago