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Chapter 5

ACTSC371 Chapter Notes - Chapter 5: Risk Premium, Risk Aversion, Standard Deviation


Department
Actuarial Science
Course Code
ACTSC371
Professor
Ken Seng Tan
Chapter
5

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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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