| Order | Condition | Applies To |
|---|---|---|
| First-Order (FOSD) | A's CDF lies everywhere below B's: every percentile of A is at least as high as the corresponding percentile of B | Every investor who prefers more wealth to less, regardless of risk attitude |
| Second-Order (SOSD) | A dominates B for all risk-averse investors: integrals of CDFs satisfy a specific inequality, often arising when B is a mean-preserving spread of A | Every risk-averse (concave-utility) investor |
| Third-Order (TOSD) | A dominates B for all risk-averse, prudent investors (positive third derivative of utility) | Investors who additionally exhibit precautionary saving behavior |
Worked example with round fictional numbers. Fund A returns 8 percent in good years and 4 percent in bad years, with equal probability. Fund B returns 12 percent in good years and 0 percent in bad years, with equal probability. Both have an expected return of 6 percent. B is a mean-preserving spread of A -- same mean, wider dispersion. Any risk-averse investor will prefer A under SOSD because the utility loss from B's 0 percent outcomes exceeds the utility gain from B's 12 percent outcomes (Jensen's inequality at work). The choice does not depend on whether you use square-root utility, log utility, or any other concave form -- it holds for all of them.
Common mistake: comparing two funds only by expected return and overlooking the shape of the distribution. A fund with higher expected return but a wider, more skewed return profile may be second-order dominated by a lower-expected-return fund with a tighter, more symmetric profile. The lower-expected-return fund is unambiguously preferred by every risk-averse investor -- a result the headline expected-return comparison hides entirely.
Sit with the ideas.
Investment A has returns drawn from a distribution that is identical to Investment B's distribution, but every outcome is shifted upward by 2 percentage points. A second pair: Investment C and Investment D have the same expected return, but D's return distribution is a mean-preserving spread of C's (same mean, wider variance). Which ranking holds without needing to assume a specific utility function?