and Risk Risk Aversion
Chapter 6
Why the Need for a New Theory?
economic decisions under uncertainty are not based only on monetary outcomes St. Petersburg Paradox (Bernoulli, 1738)
a coin is tossed until "head" appears (toss n) payoff from participating: R(n) = 2n how much would you pay as entry fee?
people usually exhibit decreasing marginal utility (e.g., log utility) risk aversion
1-2
Risky Investments
Lotteries
simple lotteries = investment opportunities where a certain wealth is put at risk and only two outcomes are possible compound lotteries allow for more than two outcomes and can be interpreted as combinations of simple lotteries elements of a lottery:
final wealth for each possible outcome probabilities associated to each possible outcome
1-3
Risk - Uncertain Outcomes
p = .6
W1 = 150 (profit = 50)
W = 100 1 p = .4 W2 = 80 (profit = 20)
E(W) = pW1 + (1 p)W2 = .6 (150) + .4(80) = 122
2 = p[W1 E(W)]2 + (1 p) [W2 E(W)]2 =
.6 (150 122)2 + .4(80 122)2 = 1,176
= 34.293
1-4
Risky Investments with Risk-Free
p = .6
W1 = 150 (profit = 50) W2 = 80 (profit = 20) W1 = 105 (profit = 5)
Risky Project W = 100
1 p = .4
Risk Free T-bills
Risk Premium = E(W) Risk-free return = 17
1-5
Risk Aversion
fair game = lottery with zero risk premium investor's view of risk
risk averse = reject investment projects that are fair games or worse
require a risk premium risk premium increases with risk
risk neutral = evaluate investment projects based only on expected returns (ignore risk) risk lover = prefer higher risk (similar to requiring a negative risk premium) most individuals are risk averse
1-6
Risk Aversion & Utility
Utility Function
mean-variance criterion = individuals compare investment opportunities based on expected return and risk (variance) example of a utility function for risk averse individuals:
U = E(r) 0.005 A 2
A measures the degree of risk aversion (higher A corresponds to more risk-averse individuals) risk aversion: U increases E(r) with and falls with 2
1-7
Risk Aversion & Utility (cont.)
mean-variance criterion
certainty equivalent (rate) = risk-free rate that gives the same utility as the risky portfolio an individual always rejects an investment portfolio with certainty equivalent rate less than the risk-free rate dominance principle = investment A dominates investment B if it offers higher expected return and lower risk, at least one strictly indifference curve = set of investment opportunities that give the same utility
1-8
Indifference Curves
Expected Return
Increasing Utility
Standard Deviation
1-9
Asset Risk vs. Portfolio Risk
investment projects or portfolios are composed of many different assets hedging = investing in an asset that tends to offset exposure to a certain kind of risk diversification = strategy based on investing in a variety of assets so that exposure to any kind of particular risk is limited
1-10
Asset Expected Return and Variance
expected return of an asset = probability weighted average return in all scenarios
E (r ) = P( s )r ( s )
s
variance of an asset's return = expected value of the squared deviations from the expected return
2 = P ( s )[r ( s ) - E (...