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 $x$,

$$F_A(x) \le F_B(x) \quad \text{for all } x,$$

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

$$\int_{-\infty}^{x} F_A(t)\, dt \le \int_{-\infty}^{x} F_B(t) \, dt \quad \text{for all } x.$$

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:

outcome	Strategy A	Strategy B
bad	$90	$80
mid	$100	$100
good	$120	$110

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.