Using Stochastic Dominance to Compare Return Distributions
Summary
The exchange asks how to compare modeled strategy returns when mean and standard deviation fail to capture important differences in the shape of their distributions. It proposes stochastic dominance as a framework for identifying when one return distribution is preferable to another without committing to a single utility function.
First-order stochastic dominance holds when one distribution offers at least as high a probability of reaching or exceeding every outcome, with a strict advantage somewhere. It corresponds to preference by all expected-utility maximizers with increasing utility. Second-order dominance applies when preferences are both increasing and concave, capturing risk aversion; it can be checked by comparing integrated differences between the cumulative distribution functions. These criteria can eliminate dominated choices, but they may leave multiple distributions incomparable. The method therefore narrows the selection set without specifying an individual investor's utility or making a final choice among all remaining strategies.
Key ideas
- Mean and standard deviation can miss meaningful differences between return distributions.
- First-order stochastic dominance means one distribution has at least as favorable an exceedance probability at every outcome.
- First-order dominance is consistent with all investors who prefer higher outcomes.
- Second-order dominance incorporates nondecreasing, concave utility and thus risk aversion.
- Cumulative distribution functions can be integrated to check the stated second-order dominance condition.
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Full text
# Pareto comparison of return distributions
# Pareto comparison of return distributions
In making a choice among financial strategies, each of which has some estimated return distribution, some strategies will clearly be better than others. But many times, the choice is a question of risk tolerance. That said, there's still a problem: the risk/reward tradeoff is often too coarse -- because different distributions (e.g., outputs from a monte carlo simulation) may have the same mean and standard deviation while being qualitatively different (e.g., see image below).
Is there any more general, principled way to compare arbitrary probability distributions of expected returns? Like defining some kind of (maybe multi-dimensional) pareto curve over PDF space? Or another quantitative technique to make this selection process more rigorously grounded?
Given a set of modeled strategies, and return distributions like these ones, how can we identify the set of "dominating" distributions that are "best" in some sense? How can we rigorously guide the selection among them?
## Answer by Kermittfrog (score 1)
https://quant.stackexchange.com/a/70321
To expand a bit on my comment: The idea of stochastic dominance may help you here.
Comparing two distributions $A$ and $B$, we say that $A$ stochastically dominates $B$ ($A\succcurlyeq B$)if certain conditions (to be made preice) regarding the distributions hold. The nice thing is that these conditions relate to general aspects of preferences and utility, without specifying a fixed utility function.
So for example, distribution $A$ first order stochastically dominates $B$, ,$A\succcurlyeq_{\mathrm{1st.\ ord.}} B$ if for any outcome $x$, the probability of reaching $x$ or more is equal or higher under $A$ than under $B$ (with at least one strict inequality). This is the case
> if and only if every expected utility maximizer with an increasing utility function prefers gamble A over gamble B
For second-order stochastic dominance, $A$ will dominate $B$ if and only if
> $E[u(A)]\geq E[u(B)]$ for all nondecreasing and concave utility functions $u(x)$.
Conveniently, this directly translates to the condition
$$ \int_{-\infty}^x[F_B(t)-F_A(t)]\mathrm{d}t\geq 0 $$ which is easily calculated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.