A Mathematical Framework for Sequence Risk

Sequence-of-returns risk is the dependence of terminal wealth on the order in which returns arrive. It is present whenever an investor contributes to or withdraws from a portfolio and is well documented, yet its effect has been difficult to isolate analytically.

Terminal-wealth dispersion reflects both the returns realized and their sequence. We separate these effects by conditioning on the realized multiset of returns and applying the law of total variance, which decomposes terminal-wealth variance into return-level and sequencing components. The decomposition extends to any square-integrable outcome and holds under any joint law of returns. When returns are independent and identically distributed, both components admit closed forms in terms of normalized elementary symmetric polynomials. Under mild conditions, the sequencing component is zero if and only if no cash flow precedes the final period, and its share in the small-volatility limit is governed by the coefficient of variation of the exposure weights.

We then apply the framework to objective-based outcomes, such as whether an accumulation reaches its target or a decumulation avoids depletion, and find sequencing modest as a share of terminal-wealth variance but substantially larger for objective-outcome variance. How much sequence risk matters therefore depends on the metric used. We further identify where along the horizon risk is most costly, recovering the volatility orderings associated with rising-equity and declining glide paths from the sign of the cash flows, and confirm these findings across a century of realized U.S. returns. Finally, we assess how far these results extend under serial dependence.