Stochastic Dominance

Compare two funds by mean and volatility alone and you're implicitly assuming both investors and returns are well described by that pair of numbers, false whenever returns are skewed. Two funds can have the same mean and volatility while one is a genuinely worse bet for literally every reasonable investor, because its downside is fatter. Stochastic dominance is the tool for saying "Fund A beats Fund B" in a way every risk-averse investor would agree with, without ever specifying whose utility function you're using.

The analogy: a menu two diners can agree on

Two diners, one who loves spice and one who doesn't, are choosing between two menus. If every dish on Menu A is at least as good as the corresponding dish on Menu B, and at least one dish is strictly better, then both diners prefer Menu A, you didn't need to know their specific tastes, just that A beats B dish-for-dish. Stochastic dominance asks the same question of return distributions: is one distribution "at least as good, everywhere, and strictly better somewhere" compared to another, a ranking every investor who prefers more money to less (or more money and less risk) would accept, regardless of their personal risk appetite.

Writing it down

First-order stochastic dominance (FSD). $A$ dominates $B$ if, for every outcome level $x$,

where $F$ is the cumulative distribution function. In words: at every wealth level, distribution A has accumulated no more probability of being below that level than B has, A's outcomes are systematically at least as good. Every investor who simply prefers more money to less (any increasing utility function) prefers $A$.

Second-order stochastic dominance (SSD) is weaker and more useful for risk-averse comparisons: $A$ dominates $B$ if

In words: the accumulated "shortfall area" under A's CDF never exceeds B's, at every cutoff, A carries no more downside risk in aggregate. Every risk-averse investor (concave utility, valuing a sure $100 over a coin flip for $0 or $200) prefers $A$, even if $A$ and $B$ share the same mean.

Compare two curves mentally, one shifted right of the other with the same shape, that shift is FSD in its purest form: the shifted curve's CDF sits below the other's everywhere, so it dominates outcome-by-outcome.

Worked example 1: FSD is easy to check with a table

Two strategies each with three equally likely outcomes:

A's CDF is always at or below B's CDF at every dollar level (A never has more accumulated probability of being at or below any given value): at $85, $F_A=0$ (nothing at or below $85) while $F_B = 1/3$; at $105, $F_A = 2/3$ while $F_B=2/3$; at $115, $F_A=2/3$, $F_B=1$. A dominates B by FSD, every value investor, regardless of risk tolerance, should strictly prefer A.

Worked example 2: same mean, SSD breaks the tie

Strategy C: 50% chance of $100, 50% chance of $100 (i.e., a sure $100). Strategy D: 50% chance of $50, 50% chance of $150. Both have mean $100, mean-variance analysis alone says nothing about preference without knowing risk aversion. FSD doesn't hold either way (C isn't uniformly above D: at $120, $F_C=0$ and $F_D=0.5$, so $F_C \le F_D$, but D can still exceed C so check the other tail: at $75, $F_C=0$ (nothing below), $F_D=0.5$, consistent so far, but at $100.01, $F_C=1$ while $F_D=0.5$, meaning $F_C > F_D$ there, so FSD fails). But SSD holds for C over D: the cumulative shortfall area under $F_C$ never exceeds that under $F_D$ at any cutoff, because C has zero spread. Every risk-averse investor prefers the sure $100, the textbook justification for a certainty-equivalent premium.

Compare this spread-out binomial shape against a tight spike at its mean (like Strategy C above), same mean, wildly different risk, exactly the gap FSD is blind to and SSD is built to detect.

What this means in practice

• Fund comparison without a utility assumption. SSD lets you say "Fund A is better for any risk-averse investor" without ever specifying a risk-aversion coefficient, a much more defensible claim than a single Sharpe ratio comparison.
• When dominance fails, say so. Most pairs of real strategies dominate neither way, one has better upside, the other better downside. That's a legitimate, honest conclusion: the choice genuinely depends on risk preference, and no single metric should paper over that.
• Options overlays. A covered call reduces upside and cuts downside relative to the naked stock, it is typically SSD-improving for a risk-averse holder over some region, but not FSD-dominant, since it truncates the best outcomes.