ACTSC371 Chapter Notes - Chapter 5: Risk Premium, Risk Aversion, Standard Deviation
SchoolUniversity of Waterloo
ProfessorKen Seng Tan
This 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
-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
-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
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.
You're Reading a Preview
Unlock to view full version