The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition

CAPM INVESTORS DO NOT GET PAID FOR BEARING RISK

Recall that if the SL-CAPM assumptions are made then a stock’s beta (regression against the market portfolio) is proportional to its excess return, as shown in equation (4.5). Markowitz shows that this does not imply that CAPM investors are paid to bear risk.⁴⁸

This is most easily seen if we assume that risks are uncorrelated. (CAPM should cover this case too.) In this case, we show that two securities can have the same variance but different expected returns, or the same expected returns and different variances. Therefore, it cannot be true that the investor is paid for bearing risk!

According to equation (4.10), the beta of r_i against r_M is

Therefore, equation (4.5) holds if and only if we also have

(4.11)

where

In other words, excess return is proportional to β_i if and only if it is proportional to the covariance between r_i and r^M.

As a calculus exercise one can show that, in the uncorrelated case, the SL-CAPM investor minimizes portfolio variance for given portfolio mean if and only if the investor choose a portfolio such that

(4.12a)

where V_i is the variance of r_i and k^I depends on the investor’s risk aversion.

Equation ...