Stochastic Dominance Fsd Ssd

Lecture Notes 1

In these notes, we extend our discussions on risk and return.

1. Stochastic Dominance

We have mostly focused on variance or standard deviation as a measure of risk. It turns out that in general, relying entirely on variance alone does not provide produce reasonable ranking of the riskiness among alternative risky investment. Consider the following example in which a risk averse investor has a utility function of u=lnW, where W is the wealth.

|State |X |Probability |Y |Probability |

|1 |1 |0.8 |10 |0.99 |

|2 |100 |0.2 |1090 |0.01 |

One finds that [pic]; and [pic]. According to the variance criterion, one would conclude that investment X is better. But, if we calculate the expected utility for this investor, we find that [pic]. In other words, the risk averse investor prefers the second investment.

2. First-Order Stochastic Dominance

Suppose that [pic] and [pic] are the cumulative probability for payoff under the first and the second investments. Then, [pic] is said to dominate [pic] according to the first order stochastic dominance, if for any t [pic]. If [pic] dominates [pic] according to the first-order stochastic dominance, any investor who prefers more money than less will press investment X over investment Y.

The first-order stochastic dominance ranks an investment opportunities for any investors who prefer more to less (non-satiated). It puts no restrictions on how risk averse investors can be. Thus, it is not a ranking of riskiness per se.

3. Second-Order Stochastic Dominance

Again, suppose that [pic] and [pic] are the cumulative probability for payoff under the first...