Risk Aversion
Lecture 1: Risk and Risk Aversion • This should mostly be review given your Microeconomics courses
• Readings: ▪ Ingersoll – Chapter 1 ▪ Leroy and Werner Chapters 8 & 9 ▪ Ross – “Stronger Measures of Risk Aversion”
The most interesting aspect of Asset Pricing, the focus of this course, considers how securities markets price risk (the time dimension alone is largely mechanical although there are interesting interactions between the two). For this question to be interesting, it must be that there is a positive price for risk – i.e. investors require some compensation for exposing their portfolios to risk (this certainly appears to be true from the data). Theoretically, this in turn requires that investors dislike risk or that they are risk averse. For intuition’s sake, we will review some of the relevant concepts.
Definition: Let [pic]be a preference relation with an expected utility representation. [pic] is said to exhibit or display risk aversion if for any simple gamble [pic] with expected value g, denoted [pic], the relation weakly prefers the fixed value g to the simple gamble → g [pic] [pic] [pic]g, [pic]. The weak preference allows for indifference so “weak risk aversion” includes risk neutrality.
(Strict risk aversion, risk neutrality, and risk seeking (weak or strict) are defined analogously.)
Example: A simple gamble: Consider a random payoff [pic] which pays [pic] > 0 with probability 1 ≥ p ≥ 0 or [pic] ≠ [pic] with probability 1 - p. The expected value of [pic] is p[pic]+ (1-p)[pic] = E([pic]) = g. This gamble is said to be ‘fair’ if E[[pic]] = g = 0. We can alternatively define a risk averse agent as one who is unwilling or indifferent to taking any fair gamble, and strictly risk averse if unwilling to accept any fair gamble. In the above definition, a risk averse individual (weakly) prefers to receive the amount E([pic]) = g rather than face the bet [pic].
Definition: A function f( ):
• Readings: ▪ Ingersoll – Chapter 1 ▪ Leroy and Werner Chapters 8 & 9 ▪ Ross – “Stronger Measures of Risk Aversion”
The most interesting aspect of Asset Pricing, the focus of this course, considers how securities markets price risk (the time dimension alone is largely mechanical although there are interesting interactions between the two). For this question to be interesting, it must be that there is a positive price for risk – i.e. investors require some compensation for exposing their portfolios to risk (this certainly appears to be true from the data). Theoretically, this in turn requires that investors dislike risk or that they are risk averse. For intuition’s sake, we will review some of the relevant concepts.
Definition: Let [pic]be a preference relation with an expected utility representation. [pic] is said to exhibit or display risk aversion if for any simple gamble [pic] with expected value g, denoted [pic], the relation weakly prefers the fixed value g to the simple gamble → g [pic] [pic] [pic]g, [pic]. The weak preference allows for indifference so “weak risk aversion” includes risk neutrality.
(Strict risk aversion, risk neutrality, and risk seeking (weak or strict) are defined analogously.)
Example: A simple gamble: Consider a random payoff [pic] which pays [pic] > 0 with probability 1 ≥ p ≥ 0 or [pic] ≠ [pic] with probability 1 - p. The expected value of [pic] is p[pic]+ (1-p)[pic] = E([pic]) = g. This gamble is said to be ‘fair’ if E[[pic]] = g = 0. We can alternatively define a risk averse agent as one who is unwilling or indifferent to taking any fair gamble, and strictly risk averse if unwilling to accept any fair gamble. In the above definition, a risk averse individual (weakly) prefers to receive the amount E([pic]) = g rather than face the bet [pic].
Definition: A function f( ):