School

University of WaterlooDepartment

Actuarial ScienceCourse Code

ACTSC371Professor

Ken Seng TanChapter

5This

**preview**shows page 1. to view the full**4 pages of the document.**Chapter 5 Capital Allocation to Risky Assets

5.1 Risk and Risk Aversion

Risk, Speculation, and Gambling

Speculation (投机): the assumption of considerable business risk in obtaining commensurate gain.

Commensurate gain: a positive expected proﬁt beyond the risk-free alternative. This is the risk premium,

the incremental expected gain from taking on the risk.

Considerable risk: the risk is sufﬁcient to affect the decision. An individual might reject a prospect that

has a positive risk premium because the added gain is insufﬁcient to make up for the risk involved.

Risk Aversion and Utility Values

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 consider only risk-free or

speculative prospects with positive risk premiums. 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.

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 value 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.

-Where U is the utility value and “A” is an index of the investor’s aversion to taking on risk. (The factor of

1⁄2 is a scaling convention that will simplify calculations in later chapters. It has no economic

signiﬁcance, and we could eliminate it simply by deﬁning a “new” A with half the value of the A used

here.)

-Utility is enhanced by high expected returns and diminished by high risk

-More risk-averse investors (who have the larger A’s) penalize risky investments more severely

-The value of A has inﬂuence on the U

Because we can compare utility values to the rate offered on risk-free investments when choosing

between a risky portfolio and a safe one, we may interpret a portfolio’s utility value as its “certainty

equivalent” rate of return to an investor. The certainty equivalent rate of a portfolio is the rate that risk-

free investments would need to offer with certainty to be considered equally attractive to the risky

portfolio.

Risk-neutral investors judge risky prospects solely by their expected rates of return. The level of risk is

irrelevant to the risk-neutral investor, meaning that there is no penalization for risk. For this investor, a

portfolio’s certainty equivalent rate is simply its expected rate of return.

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