## Risk Aversion and Utility Values

We have discussed risk with simple prospects and how risk premiums bear on speculation. A prospect that has a zero risk premium is called a fair game. Investors who are risk averse reject investment portfolios that are fair games or worse. Risk-averse investors are willing to consider only risk-free or speculative prospects with positive risk premia. Loosely speaking, a risk-averse investor "penalizes" the expected rate of return of a risky portfolio by a certain percentage (or penalizes the expected profit by a dollar amount) to account for the risk involved. The greater the risk, the larger the penalty. One might wonder why we assume risk aversion as fundamental. We believe that most investors would accept this view from simple introspection, but we discuss the question more fully in Appendix B of this chapter.

We can formalize the notion of a risk-penalty system. To do so, we will assume that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios. The utility score may be viewed as a means of ranking portfolios. Higher utility values are assigned to portfolios with more attractive risk-return profiles. Portfolios receive higher utility scores for higher expected returns and lower scores for higher volatility. Many particular "scoring" systems are legitimate. One reasonable function that is commonly employed by financial theorists and the AIMR (Association of Investment Management and Research) assigns a portfolio with expected return E(r) and variance of returns ct2 the following utility score:

where U is the utility value and A is an index of the investor's risk aversion. The factor of .005 is a scaling convention that allows us to express the expected return and standard deviation in equation 6.1 as percentages rather than decimals.

CONCEPT CHECK ^ QUESTION 2

Bodie-Kane-Marcus: |
II. Portfolio Theory |
6. Risk and Risk Aversion |
© The McGraw-Hill |

Investments, Fifth Edition Companies, 2001

Investments, Fifth Edition Companies, 2001

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