Proposition 3 gives a desk a perverse incentive: spread the posterior and averaged ES cannot rise. Van Oosten and Wang prove the result for ES at any level α. If one second-order model is more dispersed than another in convex order, its posterior-averaged ES is no higher. A desk booking capital on that measure gets a lower risk number for knowing less.
The authors put the incentive plainly: banks "would have an incentive to adopt a highly dispersed second-order model and compute the average expected shortfall under that model." The result extends to any concave statistic. Posterior-averaged entropic VaR, as defined by Shapiro, Zhou and Lin (2023), has the same problem (Proposition 11).
Their proposed fix retains ES as the pure-risk criterion. Its value in an actual portfolio remains unmeasured. Our first ETF build had constraints that prevented the fix from changing a single weight.
The paper is theory: no backtest, no Sharpe and no real-market data. Every illustration is synthetic. The performance figures below come from our run alone.
Where averaging goes wrong
The setup separates three choices. A statistic such as ES, VaR, entropic risk or expected loss scores a known distribution. A second-order model assigns plausibility to first-order models, typically through a parameter posterior; its average produces the predictive model. An attitude parameter governs how the decision maker responds to that spread.
The authors require four properties. With ambiguity absent, the rule gives the plain statistic. A neutral decision maker gets the statistic under the predictive model. Greater aversion cannot lower the risk number, and neither can a more dispersed posterior.
The smooth model of Klibanoff, Marinacci and Mukerji satisfies all four when its statistic is an expected loss. Theorem 2 pins down why, under an uncountable state space, for any rule that reduces a posterior to one model. Apply a loss function to each model's statistic, average over the posterior and invert. For the answer to equal the statistic under a reduced model for every posterior and every bounded act, the statistic must be that expected loss and the reduced model must be the predictive average. Substituting ES into the smooth model breaks neutrality.
The authors instead use an ambiguity iron, which turns a posterior and an attitude into a set of first-order models. They score each act by the worst value of the original statistic in that set.
For their concrete construction, choose a concave g. Retain each reweighted posterior ξ satisfying ξ(B) ≤ g(π(B)) for every set of models B, then take the predictive models generated by those ξ. The chosen distortion is g_β(x) = min(x/(1-β), 1). At β = 0, only the predictive model remains. As β grows, a reweighting can place up to 1/(1-β) times the original posterior mass on a slice of models. Proposition 4 establishes that the resulting sets are convex and compact.
With a convex loss ℓ, worst-case expected loss equals ℓ⁻¹ applied to a Choquet integral of the expected loss under each model. Under g_β, that integral averages the worst (1-β) share of posterior draws of those model-specific losses. For ES, Theorem 4 takes the infimum over m of m plus the Choquet integral of E[(X-m)+]/(1-α). Theorem 7 supplies a sampled estimate for any such integral. Sort the values, then give the k-th order statistic weight g((n-k+1)/n) - g((n-k)/n). The estimate converges almost surely with direct sampling or ergodic MCMC.
One posterior, several widths
A binary-state example makes the attitude parameter visible. Starting from a uniform Beta(1,1) prior, the ambiguity set for the probability of state one is [(1-β)/2, (1+β)/2]. At β = 0.6, it becomes [0.2, 0.8]. That interval exactly matches the 60% Bayesian credible interval, although the ambiguity set serves a decision and the credible interval serves coverage. Across 100 draws from a true p of 0.6, the authors say the intervals show "only minor visible differences."
On the balanced path, the predictive model does not change. Beta(1,1), Beta(3,3) after four observations and Beta(6,6) after ten all have predictive mean 1/2; the posteriors shrink in convex order. The iron's set shrinks with them. Theorem 8 and Proposition 9 describe the limit: if the posterior concentrates and the parametrization is continuous in total variation, the sets contract (Proposition 9). For a Lebesgue-continuous statistic on bounded acts, the worst-case value then converges to the pure-risk value (Theorem 8).
The paper gives misspecification a separate dial, a KL ball of radius ε around each model. Its consistency bound requires ε ≥ KL(μ0|μθ0), the divergence between the truth and the best model in the family. An allocator cannot observe that divergence.
What do the two-asset pictures establish?
The allocation examples use three hand-specified second-order models over a bivariate normal. Under π1, both positions have the same predictive distribution. X1 has point masses for its mean (-0.05) and volatility (0.01√5); X2's mean comes from N(-0.05, 0.02). At β = 0, w = 0.5 minimizes worst-case ES_0.95. The minimizer moves monotonically toward w = 1 as β passes through 0.2, 0.4, 0.6, 0.8 and 0.95.
Under π2, uncertainty affects both means and the equal split scarcely changes. Under π3, X2 draws its mean from N(-0.10, 0.03) and its volatility from a half-normal distribution; the minimizer shifts toward X1. With entropic risk at η = 10 under π3, the low-β position sits entirely in X2 before moving toward X1.
These illustrations have their answer built into the assumptions: π1 deliberately makes X1 unambiguous. We found the weights in plots only, without a table or a stated number of Monte Carlo draws. The authors leave convergence of optimal decisions for further work. Theorem 7 establishes convergence of the risk number for a fixed act, leaving the argmin open.
A 10% cap froze every ETF weight
We applied the idea to daily US stock and bond ETFs from 2020-01-01 to 2024-07-01. Each year, a dollar-volume ranking of the top 100 ETFs supplied five stock funds and five investment-grade or aggregate bond funds. We fitted a normal-inverse-Wishart posterior to 252 sessions of returns and took 2,048 parameter draws. ES at 0.95 covered the paper's β grid. We designated β = 0.6 as primary and β = 0 as the predictive benchmark. Orders filled at the close.
The constraints settled the allocation before the iron could. A long-only, fully invested book of ten names with a 10% cap has exactly one feasible portfolio: 10% in each name. Every β therefore held the same weights.
Our figures describe an equal-weight five-and-five ETF basket. It had 12.75% annualized volatility and a beta of 0.55 to SPY. Over the stated window, it returned 30.88%, with a Sharpe of 0.54 and a maximum drawdown of -25.22%. Those results include commissions of $0.004 a share, subject to a $1 minimum. The paper offers no performance figures for comparison. Our results cannot establish whether the iron helps or hurts.
Our run has two further faults. The annual ranking and fund classifications lack point-in-time timestamps, leaving possible look-ahead in the universe. It also logged 11,137 trades across 100 symbols for a ten-name book. We cannot reconcile that count with our rules.
A useful test would compare β = 0 and β = 0.6 on the same 2,048 draws, using a cap that cannot bind. β would have to be fixed beforehand because the paper supplies no selection rule. If the β = 0.6 book reduces realized tail loss by more than the cost of its extra turnover over 2020 to 2024, the iron earns a place in the allocator.
For now, Proposition 3 supplies its strongest case: a risk report should not reward a wider posterior.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.