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Expected Utility Regret Rule: Minimax and Bayes Optimal Portfolio Choice

Masahiro Kato

arXiv:2610.02290v1econ.EMcs.LGmath.STq-fin.MFstat.MLstat.TH

TL;DR

Portfolio choice methods that estimate return inputs before optimizing do not by themselves establish which portfolio is appropriate under expected utility regret. The paper proposes EUR to select a portfolio class and estimate its weights, and reports minimax and Bayes optimality with leading constants in regular parametric models.

  • Problem

    Estimating return parameters and then optimizing does not by itself establish which portfolio is appropriate, motivating a criterion that evaluates the chosen portfolio in utility terms.

    Source: §1 · pp. 1–2

  • Method

    The EUR rule jointly chooses among portfolio classes and estimates weights within the selected class.

    Source: §2.5 · p. 8, §10 · p. 52

  • Results

    In regular parametric models with fixed assets and finitely many classes, EUR attains minimax and Bayes expected utility regret bounds, including their leading constants, without using an evaluation prior.

    Source: §10 · p. 52

  • Takeaways & Limitations

    Near a symmetric model, a prescribed composition can improve precision while omitting an investment direction, and a binding money budget can change relative holdings.

    Source: §10 · p. 52

    A prescribed simple composition can omit an investment direction near symmetry, and a binding money budget can change relative holdings even when standardized returns remain symmetric.

    Source: §10 · p. 52

Abstract

from arXiv · show

This study considers the problem of portfolio choice, where we recommend a portfolio to an investor to maximize the expected utility of their wealth. Our goal is to construct an asymptotically optimal portfolio choice rule in terms of expected utility regret, the difference between the expected utility of an oracle investor and that achieved by a portfolio chosen from data. We propose the Expected Utility Regret (EUR) rule, which jointly selects a portfolio class and estimates its weights. In a regular parametric return model, a single EUR rule attains both the minimax and the Bayes lower bounds, including their leading constants, without using the prior distribution that defines the Bayes criterion. We then derive the mean--variance and risk-parity portfolios as special cases of this framework. Under smooth increasing and concave utility, the EUR rule and the sample mean--variance portfolio attain the same leading expected regret when expected excess returns approach zero sufficiently fast. When the returns divided by their volatilities have a joint distribution that does not depend on the order of the assets, the EUR rule and the risk-parity portfolio coincide.

1 Introduction

Portfolio rules built by estimating return parameters and then optimizing do not by themselves establish which portfolio is appropriate for an investor. The paper evaluates choices through expected utility regret and proposes the EUR rule, connecting general portfolio choice to familiar portfolios and optimality guarantees.

1.1 Contributions

  • Expected utility regret compares a data-chosen portfolio’s utility with that of an oracle who knows the return distribution, extending utility-loss analysis to general utility and choices among portfolio classes.It evaluates the resulting portfolio choice rather than parameter-estimation accuracy alone.Source: §1.1 · p. 2, §1 · p. 2
  • The EUR rule attains minimax and Bayes lower bounds, including their leading constants, in regular parametric models with fixed assets and finitely many classes, without using the evaluation prior.It uses preliminary data to screen classes and remaining observations to select a class and estimate its weights.Source: §1.1 · p. 2
  • When classes can tie, gaps of order n^-1/2 drive leading worst-case selection regret, while under Bayes evaluation both selection and weight estimation contribute at order n^-1.When there are no ties, weight estimation carries the worst-case constant; the variance of estimated utility differences governs class separation, with shared portfolio movements canceling.Source: §1.1 · pp. 2–3
  • When expected excess returns shrink faster than n^-1/4 but slower than n^-1/2, EUR and sample mean–variance portfolios attain the same minimax and Bayes constants; at order n^-1/4, skewness can make mean–variance incur excess regret.This comparison assumes nondegenerate return covariance.Source: §1.1 · p. 3
  • Under permutation symmetry of standardized returns and feasible holdings, equal standardized holdings yield risk parity as an exact EUR representation; relaxing the conditions can make simple compositions costly or change holdings under a binding budget.The composition can be recommended without knowing the utility function once either special-case route’s conditions hold.Source: §1.1 · p. 3

1.2 Related Literature

  • Unlike best-arm choice, portfolio choice does not allocate observations across actions and recommends a point in a continuum whose weights also require estimation; Kato (2025b,a) obtain simultaneous minimax and Bayes optimality with exact constants.Source: §1.2 · p. 4

2 Setup

The setup frames portfolio choice as selecting feasible investments from observed returns when the return distribution is unknown. It evaluates data-based choices by expected utility regret and represents feasible portfolios as convex classes, allowing nonconvex restrictions to be handled as choices among classes.

2 Setup

  • Investors use observed returns to choose assets and weights for a new investment period, while returns are modeled as independent draws from an unknown distribution.The number of assets and return model are held fixed as sample size grows, unless otherwise stated.Source: §2 · p. 5, §2.1 · p. 6

2.2 Portfolio Choice Problem

  • A portfolio choice rule maps training observations to a feasible portfolio, while the oracle portfolio maximizes expected utility when the return distribution is known.The rule may randomize conditional on the observations.Source: §2.2 · p. 6

2.3 Expected Utility Regret

  • Expected utility regret measures the utility lost by choosing a portfolio from data rather than an oracle portfolio, averaging over training data and any rule randomization.Portfolio utility is evaluated on a new return; adding a constant to utility leaves regret unchanged, while positive rescaling scales regret and its leading constants.Source: §2.3 · pp. 6–7

2.4 Uncertainty

  • Rules are evaluated either by their largest expected regret over the return model or by expected regret integrated under a specified prior.The model represents distributions the investor cannot distinguish before observing data.Source: §2.4 · p. 7

Minimax optimality.

  • Asymptotic minimax optimality compares a rule’s worst-case regret with the smallest achievable worst-case regret using the same normalization.The supremum is taken anew at each sample size, including distributions approaching ties between optimal classes.Source: Minimax optimality · p. 7

Bayes optimality.

  • Bayes evaluation integrates regret over a prior with a continuous density on the compact parameter set, and asymptotic optimality compares the rule with the prior-specific infimum.The EUR construction is the same for every such prior, although the competing rule in the infimum may use it.Source: Bayes optimality · p. 7

2.5 Portfolio Class

  • Writing the feasible set as a union of nonempty closed convex portfolio classes separates choosing a class from choosing weights within it.This representation accommodates complicated, including nonconvex, feasible sets while retaining convexity within each class.Source: §2.5 · p. 8
  • When classes can tie, choosing the wrong class can dominate worst-case regret because it loses the utility gap, whereas weight-estimation loss within a class is of squared-error order.For two singleton classes, a gap of order n^-1/2 can make the wrong portfolio be selected with probability bounded away from zero, yielding regret of order n^-1/2.Source: §2.5 · pp. 8–9

3 Analyses of Estimation Error and Expected Utility Regret

This section analyzes expected utility regret by separating class-selection and within-class allocation losses, and examines how estimation accuracy, class comparisons, and portfolio geometry shape their bounds. It frames approximation accuracy in terms of the utility consequences of the resulting allocation rather than return normality.

3.1 Return Model

  • The return-model analysis assumes integrable increasing concave utility and regular parametric returns, with smoothness, positive Fisher information, and nondegenerate class ties for comparisons.The regular model permits finite-support returns and continuous returns with common bounded support and a smooth positive density.Source: §3.1 · pp. 11–12

3.2 Regret Decomposition

  • Total regret decomposes into loss from selecting a suboptimal class and loss from allocating weights within the selected class.The two components depend on the portfolio-class collection, although their sum does not.Source: §3.2 · p. 13

3.3 Estimation Error and Regret

  • Expected within-class allocation regret is O(n^-q/(2(q−1))), reducing to O(n^-1) when utility loss is locally quadratic (q = 2).The bound follows from the sample-average gradient error under the stated accuracy and utility-loss assumptions.Source: §3.3 · p. 14
  • A split-sample rule fits portfolios on one sample and compares their average utilities on another, with a regret bound that also applies to infinitely many classes.The selection bound uses the complexity of utility differences between fitted portfolios.Source: §3.3 · pp. 15–16
  • Only gaps comparable to n^-1/2 can yield a positive leading selection-regret constant; for two classes, the sign rule’s scaled expected regret has maximum σ sup_{t≥0} tΦ(-t) when corresponding alternatives are in the model.For gn = σt/√n and bounded t ≥ 0, √n times the sign rule’s expected selection regret converges to σtΦ(-t), where σ > 0 is the limiting standard deviation.Source: §3.3 · p. 16
  • Portfolio complexity depends on feasible directions: squared Gaussian width can be order K for a dense interior optimizer but at most order s for support size s.The relevant dimension is determined by the tangent cone of the feasible set.Source: §3.3 · pp. 16–17
  • A portfolio approximation with mean squared error O(n^-1) has expected regret differing by o(n^-1) from an estimator whose approximation error is o(n^-1).The conclusion also holds after taking a supremum or integrating over distributions when the required bounds and remainders hold uniformly.Source: §3.3 · p. 17
  • Mean–variance approximations should be judged by utility lost at their resulting allocation, not return normality; equation (48) also gives a sufficient accuracy requirement for EUR-rule approximations within a class.The proof is in Appendix J.Source: §3.3 · p. 18

4 The Expected Utility Regret Rule

The EUR rule estimates portfolio weights and selects among available portfolio classes using maximum-likelihood estimates of the return parameter. With multiple classes, it screens out clearly inferior options using a preliminary sample, then uses separate observations to compare remaining classes and estimate their weights.

4 The Expected Utility Regret Rule

  • The EUR rule uses maximum-likelihood estimation and estimates each class’s optimized expected utility and portfolio weights under the fitted return distribution.With one class, it uses all observations to estimate that class’s weights; with multiple classes, it uses separate preliminary and second samples.Source: §4 · p. 18

4.1 EUR Rule with One Portfolio Class

  • With one portfolio class, the rule estimates its portfolio weights from all available observations, without comparing classes.For one observation, it instead holds a fixed feasible portfolio.Source: §4.1 · p. 18

4.2 EUR Rule with Multiple Portfolio Classes

  • With multiple classes, a preliminary sample screens for classes whose estimated optimized expected utility is within a threshold of the maximum, and a second sample chooses among candidates and estimates their weights.For three or more candidates, the choice uses a finite Gaussian comparison and a covariance estimate from the preliminary sample.Source: §4.2 · p. 19

4.3 Statement of the Rule

  • The complete EUR rule uses neither the true return parameter nor an evaluation prior, and its stages use the designated subsamples when multiple classes are available.For three or more candidate classes, it compares estimated utility differences using a finite linear program to determine selection probabilities.Source: §4.3 · pp. 19–20

5 Minimax Optimality

The EUR rule screens portfolio classes and chooses a class and its weights from data. Its leading worst-case expected regret matches the lower bound, with the constant shaped by class ties and weight estimation.

5 Minimax Optimality

  • The rule uses a preliminary sample to screen candidate classes, then uses the remaining observations to compare them and estimate portfolio weights.With two candidates it uses the sign rule; with three or more, it uses a finite Gaussian comparison.Source: §5 · pp. 20–21, §5.5 · p. 24

5.1 Minimax Constant

  • Near a tie between optimal classes, selection contributes at order n^-1/2; without ties, weight estimation contributes at order n^-1.The two-class comparison has a closed-form value, attained by the sign rule.Source: §5.1 · pp. 21–22, §5.4 · p. 24

5.2 Minimax Lower Bound

  • The minimax lower bound reflects Gaussian comparison error near a tie, or information limits on estimating optimal weights when no tie occurs.The bound applies to all rules based on the original n return observations.Source: §5.2 · p. 23

5.3 Worst-Case Upper Bound

  • The EUR rule’s worst-case regret is bounded uniformly, with tie parameters contributing the selection constant and within-class estimation contributing at order n^-1.When there are no ties, only the allocation term remains.Source: §5.3 · p. 23

5.4 Minimax Optimality

  • The EUR rule attains the smallest possible leading worst-case expected regret in the specified return model.The lower and upper bounds share the same constant.Source: §5.4 · p. 23

5.5 Tuning Parameters and Approximations

  • The preliminary screen removes classes with sufficiently large utility gaps, while the comparison procedure accounts for correlated estimated utility differences among three or more candidates.A vanishing perturbation supports Gaussian comparison even for finite-support returns without changing the limiting constant.Source: §5.5 · p. 24

5.6 Discussion

  • Halving leading worst-case regret requires about four times as many observations when selection dominates, versus twice as many when only weight estimation matters.In the illustrated model, near-boundary wrong-class choices contribute at order n^-1/2, while estimating the third-asset position contributes at order n^-1.Source: §5.6 · pp. 24–25

6 Bayes Optimality

The EUR rule’s prior-integrated regret has contributions from class selection near ties and weight estimation. Its integrated lower and upper bounds agree, so the same rule is Bayes optimal without using the evaluation prior.

6.1 Bayes Constant

  • The Bayes constant sums class-selection costs near pairwise ties and weight-estimation costs where a unique class is optimal.A tie neighborhood of width n^-1/2 has prior probability of order n^-1/2 when the density is positive on the tie surface.Source: §6.1 · pp. 25–26

6.2 Bayes Lower Bound

  • The integrated lower bound applies even to rules that use the prior, because prior knowledge cannot remove local comparison or weight-estimation error.Near a regular pairwise tie, posterior comparison error is, to leading order, the Gaussian error of an efficient utility-difference estimate.Source: §6.2 · p. 26

6.3 Worst-Case Upper Bound

  • The EUR rule’s integrated upper bound matches the sum of the selection and allocation constants for each specified prior.The upper-bound argument controls triple-tie regions and integrates the local comparison and within-class estimation costs.Source: §6.3 · pp. 26–27

6.4 Bayes Optimality

  • The EUR rule attains the Bayes lower bound for every specified prior, and using the evaluation prior cannot improve the leading constant.The same rule is also asymptotically minimax and is independent of the prior.Source: §6.4 · p. 27
  • Under a single-class model, the selection constant is zero and the allocation constant is the prior integral of the within-class cost.The EUR rule is independent of the evaluation prior.Source: §6.4 · p. 27

6.5 Discussion

  • Allocation costs remain even without ties when estimating the return parameter changes optimal weights in a direction with positive utility curvature.In the example with two opposite positions and a third-asset position, a prior positive near θ = 0 assigns leading cost to both selection and allocation.Source: §6.5 · p. 28

7 Relationship to the Existing Portfolios and Their Extensions

This section relates the EUR rule to sample mean–variance and risk-parity portfolios under distinct conditions, and extends the minimax and Bayes conclusions to a derivative for a restricted portfolio form. It also examines what is lost when simplifying conditions fail, as well as the costs of unknown asset scales and money-budget constraints.

7.1 Portfolios from a Small Optimal Risky Investment

  • When expected excess returns shrink faster than n^-1/4 but more slowly than n^-1/2, the sample mean–variance portfolio and one-class EUR rule attain the same local minimax and Bayes constant C0 without using the prior density.The constant is the same for every admissible density f; neither rule uses f or H_n.Source: §7 · p. 28, §7.1 · p. 30

7.2 Higher Moments in the Portfolio Rule

  • At mean magnitudes of order n^-1/4, third-moment effects can shift optimal weights at the estimation-error scale, and the sample mean–variance portfolio incurs additional worst-case and Bayes regret constants.When the third-moment displacement is nonzero in some direction, both additional constants are positive.Source: §7.2 · p. 31
  • If utility is not a polynomial of degree at most two, two bounded finite-support return models can share means and covariances but have different oracle portfolios, causing positive regret for rules converging to a decision based only on those moments.Thus, means and covariances do not generally determine the oracle portfolio.Source: §7.2 · p. 32

7.3 Portfolios from Symmetry

  • Under transitive permutation symmetry of standardized returns and feasible holdings, the EUR rule uses equal standardized positions, yielding inverse-volatility weights with equal contributions to volatility.This result does not require expected excess returns to shrink with n; utility affects the total amount invested, not the composition.Source: §7.3 · pp. 32–33

7.4 The Cost of Imposing a Simple Form

  • Fixing portfolio composition at v removes the perpendicular estimation cost c_perp but adds loss L(h), so the restriction lowers local expected regret when L(h) < c_perp and raises it when L(h) > c_perp.The comparison concerns local perturbations around an interior parameter value.Source: §7.4 · p. 34
  • When relative asset volatilities are unknown, the EUR rule estimates the composition under conditions where returns standardized by those scales remain invariant under a transitive permutation group.Source: §7.4 · p. 35
  • The minimax and Bayes conclusions continue to hold when the general allocation constant uses the derivative for the imposed simple form.At equal standardized holdings, symmetry and strict concavity imply equal fitted-covariance contributions.Source: §7.4 · p. 35
  • In a sign-and-gamma-magnitude return family with unknown asset scales, the regret constant separates investment-amount estimation from scale estimation, which has common-scale and composition costs.The scale-estimation terms are j_π α_π^2/(2k_G) for common proportional changes and j_π(K − 1)α_π^2/(2k_G) for composition changes.Source: §7.4 · pp. 35–36
  • A money budget at least as large as the unconstrained investment leaves the optimum unchanged; with a binding budget, equal standardized holdings require equal entries of c.Just below the unconstrained budget, the investor reallocates across assets rather than reducing all standardized holdings equally; the cost of preserving the symmetric composition has a positive coefficient when relative volatilities differ.Source: §7.4 · p. 36

8 EUR Rule with State-Space Modeling

The state-space extension applies the EUR framework to portfolios that depend on past returns by estimating an allocation function of the observed state. It develops conditional portfolio estimation and bounds that account for estimation, approximation, and selection.

8.1 State-Space Model

  • A state is formed from the past L returns, and an allocation function maps that state to a portfolio; expected utility is evaluated over the stationary joint distribution of the state and next return.The estimated allocation function is held fixed in that expectation.Source: §8.1 · p. 37

8.2 Conditional Portfolio Estimation

  • Conditional portfolio estimation chooses, at each state, a portfolio maximizing an estimated conditional objective, using its estimated gradient to control estimation error.The analysis bounds integrated gradient error over the stationary state distribution rather than its supremum over states.Source: §8.2 · p. 38
  • Under the stated assumptions, Theorem 8.1 establishes a minimax rate for estimating a smooth state-dependent allocation function.The expected-regret rate is the estimation rate raised to q/(q −1), reflecting the utility-loss condition.Source: §8.2 · p. 39

8.3 History Length and Approximation

  • Longer histories can reduce approximation regret but increase state dimension and estimation cost, while selecting history length from data adds a selection error.The data-selected rule attains the same order under the stated conditions on the candidate complexities and error allowances.Source: §8.3 · pp. 39, 41
  • The approximation-class oracle bound separates approximation loss, estimation error, and selection error.Expected bounds can add a term for the probability that the event underlying the bound fails.Source: §8.3 · pp. 40–41

9 Experiments

The experiments compare EUR regret with theoretical constants and empirical utility-maximizing portfolios, and evaluate portfolio objectives on industry returns through simulations of class selection, weight estimation, and risk parity. They report uncertainty in estimated contributions and class-selection reproducibility, sensitivity to portfolio-design specifications, and statistically insignificant certainty-equivalent differences between minimum-variance and equal-weight portfolios in the reported cases.

9.1 Simulation Studies

  • As sample size grows, two-class EUR regret shifts from weight estimation toward class selection, while the largest normalized regret approaches 0.01253.At n = 1,024,000, the integrated regrets, rescaled by the effective estimation sample size, were 0.25630 and 0.32162, matching their theoretical limits and showing the cost of reserving observations for preliminary comparison.Source: Table 1 · p. 42, §9.1 · p. 42
  • Regular-allocation regret ratios rose from 0.411 at n = 50 to 0.984 at n = 1,000 and 1.015 at n = 2,000 relative to the 0.3832 limit constant.At n = 50, at least one estimated weight was zero in 86.7% of samples, versus 0.13% at n = 2,000. For v = 1, the sample mean–variance portfolio had lower regret than EUR through n = 65,536, but higher regret from n = 262,144.Source: Figure 2 · p. 43, §9.1 · pp. 43, 45, Table 3 · p. 45
  • For negatively correlated utility differences, the finer-grid Gaussian selector had lower maximum regret than selecting the largest observation; increased dependence also reduced normalized selection regret from 0.0432 to less than 0.0001.For the first two covariance matrices, selecting the largest observation did better, although refining the grids reduced the gap. As ρ rose from 0 to 0.99, normalized complexity declined from 0.2195 to 0.0238.Source: Table 2 · p. 42, §9.1 · p. 43, Figure 2 · p. 43
  • At n = 5,000, normalized integrated regret was 0.03999, 0.05992, and 0.07481 under uniform, quadratic, and quartic priors, while second-order selection incurred 0.0514 worst-case regret.The prior-specific limits were 0.040, 0.060, and 0.075, and √n times worst-case regret was 0.01385 versus the Gaussian limit 0.01360. With identical means and second moments but different expected utilities, third- and fourth-order criteria had 0.0001 worst-case regret, versus zero for full expected utility.Source: §9.1 · p. 44, Figure 3 · p. 44, Figure 4 · p. 45
  • The risk-parity EUR rule assigned 25% of portfolio volatility to each asset, and its normalized expected regret approached one for both tested utilities.With estimated relative scales, the integrated leading constant 0.10005 comprised 0.08781 for estimating investment amount and 0.01224 for scales; the regret and its difference from the known-scale comparator stayed close to their limits.Source: §9.1 · p. 46, Table 4 · p. 46, Table 5 · p. 47
  • Under local asymmetry, imposing symmetric composition was better below |h| ≈ 0.82607 and EUR was better above it; the reported rule ordering reversed at h = 2.The three budget-comparison columns agreed more closely as the shortfall decreased.Source: §9.1 · pp. 47–48, Table 6 · p. 47, Table 7 · p. 48

9.2 Empirical Studies

  • Minimum variance had the highest test-period certainty equivalent, 1.022 percentage points above equal weighting; full utility exceeded its second-order approximation by 0.052 points.Full utility exceeded ten-asset-simplex full-utility maximization by 1.073 points. Under the common three-industry restriction, full utility had the highest point estimate, but differences were imprecise and did not reproduce the ten-industry minimum-variance performance.Source: Table 8 · p. 49, §9.2 · p. 49, Table 9 · p. 50
  • The empirical analysis separates certainty-equivalent contributions from class selection and final weight estimation across four combinations of full-utility and second-order objectives.The two contributions sum to the certainty-equivalent difference between the full-utility and second-order rules; their variances are estimated using Newey–West long-run variance.Source: §9.2 · p. 50
  • Both class-selection and within-class allocation contributions have intervals that include zero.The underlying certainty equivalents CF,F, CF,Q, CQ,F, and CQ,Q are 1.903%, 2.024%, 1.728%, and 1.851%, respectively.Source: Table 10 · p. 50, §9.2 · p. 51, Figure 6 · p. 51
  • The objectives select the same three-industry class in 22 of 25 annual decisions, but mean resampled same-class probabilities are 0.318 for full utility and 0.343 for the second-order criterion.In disagreement years, the smaller probability is on average 0.108 larger than in agreement years (permutation p-value 0.496); validation-block Taylor-remainder differences correlate 0.905 with out-of-sample utility differences.Source: §9.2 · p. 51
  • Across design specifications, certainty-equivalent differences range from −0.554 to 0.311 percentage points, and every bootstrap interval contains zero.Weight-estimation contributions are negative in eight of nine specifications and essentially zero at CRRA 2; class-selection contributions are negative for four-industry classes and both alternative training windows, and positive in the other six specifications.Source: §9.2 · p. 51, Figure 6 · p. 51
  • The minimum-variance portfolio has a positive but statistically insignificant certainty-equivalent difference from equal weighting in the primary specification and the reported alternative cases.The cases include high risk aversion, a longer training window, and either alternative transaction cost.Source: §9.2 · p. 52

10 Conclusion

The EUR rule attains the minimax and Bayes lower bounds for expected utility regret, including their leading constants, in the stated regular parametric setting. The conclusion also identifies when familiar portfolio compositions arise and where imposing a simple form can incur costs.

10 Conclusion

  • In a regular parametric return model with fixed assets and finitely many classes, EUR attains both regret lower bounds, including their leading constants, without using the evaluation prior.Class uncertainty can dominate the worst-case leading order, while both class selection and weight estimation contribute under the Bayes criterion.Source: §10 · p. 52
  • When excess returns shrink faster than n^-1/4, estimated means and covariances suffice at leading regret order; at order n^-1/4, third moments change the regret constant.Under permutation symmetry of standardized returns and feasible holdings, the second route gives an exact risk-parity composition; unknown relative volatilities add composition-estimation loss.Source: §10 · p. 52
  • In both routes, optimal composition does not require knowing utility: utility affects only the amount invested in risky assets, to first order in the first route and exactly in the second.Source: §10 · p. 52
  • Near a symmetric model, prescribing a composition can gain precision while omitting an investment direction, and a binding money budget can change relative holdings despite symmetric standardized returns.Source: §10 · p. 52

A Bounds from Estimation Error

The bounds analysis separates regret from estimating weights within a class from regret due to selecting among fitted classes. The selection step uses independent observations to compare fitted portfolios.

A Bounds from Estimation Error

B Lower Bounds from Finite-Support Return Models

The appendix establishes lower bounds using bounded finite-support return distributions. Separate constructions make either the optimal allocation or the class containing it difficult to identify from data.

B Lower Bounds from Finite-Support Return Models

  • The allocation lower bound varies the population-optimal weights within an r-dimensional class, with KL divergence measuring how difficult incorrect choices are to avoid.For q = 2, the stated rate is r/n; for fixed r, its power of n matches the upper bound in Section 3.Source: B Lower Bounds from Finite-Support… · p. 59
  • The allocation lower bound assumes a fixed regular r-dimensional submodel with a bounded finite-support family and locally feasible allocation coordinates.Source: B Lower Bounds from Finite-Support… · p. 59
  • The class-selection construction changes which class contains the optimum while keeping KL divergences small, so choosing the wrong class incurs utility loss even with accurate weights.Source: B Lower Bounds from Finite-Support… · p. 60
  • The class-selection bound applies when a separated near-tie packing is available at δ ≍ log M/n and log M ≤ c′n; M counts constructed return distributions, not portfolios or classes.Source: B Lower Bounds from Finite-Support… · p. 60

C Efficient Estimation of Expected Utility and Portfolio Weights

This section derives efficient influence functions for expected utility comparisons and portfolio weights, then relates estimation covariance to expected regret. It also shows that correctly specified parametric information can improve leading constants without changing the powers of n.

C Efficient Estimation of Expected Utility and Portfolio Weights

D Proof of the Mean–Variance Result

The proof localizes the population optimum near zero, establishes regret upper bounds for the sample mean–variance portfolio and EUR, and shows that no rule improves the lower-bound constant.

D.1 Localization of the Population Optimum

  • At zero expected returns, cash uniquely maximizes expected utility; nearby population-optimal weights are interior and satisfy ∥w⋆(µ)∥2 ≤ C∥µ∥2.Strict concavity and nondegenerate covariance give uniqueness at zero; continuity and the implicit function theorem establish the local behavior.Source: §D.1 · p. 65

D.2 Expected Regret of the Two Estimated Portfolios

  • The sample mean–variance portfolio and EUR attain upper bounds with constant C0 under the stated uniform and prior-integrated criteria.The proof obtains the sample mean–variance upper bounds in (82) and (83), and the EUR calculation gives C0 uniformly and after integration under Hn.Source: §D.2 · p. 66

D.3 Lower Bound for Arbitrary Rules

  • For the lower bound, concavity permits projecting every rule into a shrinking neighborhood of cash, where integration by parts applies to arbitrary measurable rules.The projection weakly increases expected utility for every parameter value and does not depend on the prior.Source: §D.3 · p. 67
  • The lower-bound argument proves both equalities in Theorem 7.2, so no rule improves the attained constants.The bound applies to every measurable rule, and the worst-case regret is at least the prior-integrated regret.Source: §D.3 · p. 67

E Proof of the Third-Moment Result

The proof identifies a third-moment contribution to mean–variance regret at the n^-1/4 scale, while EUR attains the lower bound. It also shows that matching low-order moments need not determine portfolio rankings.

E Proof of the Third-Moment Result

  • At µ = n^-1/4v, mean–variance regret includes a third-moment contribution, while EUR’s regret has no deterministic β(v) term.The regret expansion gives C0 + M3(v) + o(1) for mean–variance; for EUR, the fitted-distribution optimizer removes the deterministic β(v) term.Source: E Proof of the Third-Moment Result · p. 68
  • At the n^-1/4 scale, the lower bound remains C0 for both infima and is attained by EUR; the proof also establishes the additional constants in Theorem 7.3.The result holds for both the worst-case and prior-integrated criteria.Source: E Proof of the Third-Moment Result · p. 68

F Moment Separation

  • Two models with identical mean vectors and covariance matrices can reverse the expected-utility ranking of their pure-asset portfolios.The construction assigns distributions with matching moments through order two to the assets, then reverses their assignments across models.Source: F Moment Separation · p. 69
  • Any rule using only the common mean and covariance has worst-case regret of at least c/2 in the two-model construction.The preferred pure-asset neighborhoods differ across the models, although a moment-only rule selects the same neighborhood with the same probability.Source: F Moment Separation · p. 69

G The Taylor Criteria Used in the Experiments

The experiments compare empirical expected utility with criteria based on Taylor polynomials of utility. The proof distinguishes objective approximation from the portfolio-concentration and regret results.

G The Taylor Criteria Used in the Experiments

  • The empirical criterion replaces expected utility with its sample average, and its second-order version includes squared mean, unlike the covariance-based criterion.For L = 2, the population criterion includes the squared mean through E[(w⊤R)2].Source: G The Taylor Criteria Used in the… · p. 69
  • A finite bound on the (L+1)th derivative of utility supports Taylor approximation of the objective on the attainable-wealth interval.The stated condition is that U has L+1 continuous derivatives and MU,L+1 is finite.Source: G The Taylor Criteria Used in the… · p. 69
  • The Taylor remainder bound concerns objective approximation, whereas Theorem 7.2 concerns concentration of optimal and estimated portfolios and resulting utility regret.Source: G The Taylor Criteria Used in the… · p. 69

H Proofs for Allocations Based on an Observed State

This appendix develops proofs for conditional allocation-regret bounds, Bayes and minimax regret results, and portfolio-approximation results, using Gaussian comparisons and verifying likelihood and signed-Gamma conditions.

H Proofs for Allocations Based on an Observed State

  • For allocations based on an observed state, the proof bounds regret using integrated conditional-gradient estimation error and obtains a matching lower bound through sign perturbations.The upper and lower bounds have the same order after balancing the bandwidth terms.Source: H Proofs for Allocations Based on an… · pp. 70–71

I.1 Likelihood Concentration and Expected Allocation Regret

  • Likelihood concentration and local expansions yield the allocation formula, while a uniform normal approximation controls estimated utility differences near regular ties.The approximation also bounds the probability that estimated class values tie exactly.Source: §I.1 · p. 72

I.2 The Finite Gaussian Comparison

  • For two classes, selecting the class with the larger observation attains the Gaussian minimax value κσ, matching a two-point-prior lower bound.The prior places equal mass at means ±u⋆σ, where u⋆ attains κ.Source: §I.2 · p. 73

I.3 Sampling Approximation with an Estimated Covariance

  • With estimated covariance, the EUR rule’s normalized expected selection regret is bounded by the candidate-set Gaussian minimax constant plus o(1), and is uniformly O(n^-1/2).This applies when the candidate set contains an optimal class; with one candidate, selection loss is zero.Source: §I.3 · p. 74

I.4 Gaussian Comparison and the Worst-Case Bound

  • The Gaussian comparison and screening arguments establish worst-case upper bounds for the EUR rule, including the no-ties case.The bound is uniform over the evaluation set; with a unique optimum, the candidate set reduces to that class with exponentially high probability.Source: §I.4 · p. 75

I.5 A Minimax Lower Bound in the Original Return Model

  • A local asymptotic argument transfers the Gaussian minimax value at interior ties into a lower bound for normalized minimax regret in the original return model.Transversality extends the lower bound to ties on the boundary of the parameter set.Source: §I.5 · p. 76

I.6 Integration under the Original Prior

  • Integrating regret near pairwise optimal-class tie surfaces yields the Bayes selection upper bound, with each pair contributing σ^2/2.Regions with three or more nearly optimal classes are controlled separately, and inferior classes away from ties contribute exponentially small amounts.Source: §I.6 · p. 77

I.7 The Bayes Lower Bound and the No-Tie Minimax Value

  • The exact Bayes decision’s regret lower bounds add across unique-optimum regions and tie surfaces, matching the upper bound; a prior argument yields the minimax lower bound (60).The Bayes lower bound proves (74).Source: §I.7 · p. 78

J Portfolio Approximation

  • Strong concavity and Taylor expansion control the portfolio-weight approximation; taking expectations with Cauchy–Schwarz yields (48).The proof works in orthonormal coordinates of the portfolio class and uses the upper Hessian bound along feasible portfolio segments.Source: J Portfolio Approximation · p. 79

K Regular Likelihoods in One Portfolio Class

K.1 Expected Regret and the Lower Bound

  • In one portfolio class, full-sample estimation gives upper bounds under both evaluations, while van Trees yields Bayes and minimax lower bounds, proving the one-class conclusions.The minimax lower bound uses smooth priors concentrated near interior points whose allocation constants approach their supremum.Source: §K.1 · p. 80

K.2 Verification for the Signed-Gamma Family

  • For the signed-Gamma family, the proof verifies the estimation and curvature conditions and derives the fitted risk-parity allocation assertions from the general EUR results.The utility-derivative condition is γLs+ < 1, and the parameter bounds establish the interior condition.Source: §K.2 · p. 81

L Portfolio Restrictions

  • The final appendix part establishes the EUR rule’s risk-parity representation and compares it under prescribed-composition and money-budget restrictions.Source: L Portfolio Restrictions · p. 81

L.1 Proof of Proposition 7.6

The proofs establish a unique symmetric-coordinate optimum and its risk-contribution normalization, derive EUR regret expansions under local misspecification, and characterize the effect of tightening the budget.

L.1 Proof of Proposition 7.6

  • Strict concavity and an interior optimizer make αθ1 the unique standardized-coordinate optimum; normalizing risky holdings by their sum yields (95).The component risk contribution is σθ(w)/K.Source: §L.1 · p. 82

L.2 Proof of Proposition 7.7

  • The EUR regret expansion has trace constant c∥+c⊥, while a deterministic perturbation th changes the population portfolios by tD⊥h+O(t^2).Source: §L.2 · p. 82

L.3 Proof of Proposition 7.9

  • Under a tighter budget, the optimum scale falls by ∆/(c⊤1); the quadratic-loss gap is nonnegative and vanishes only when relative volatilities are equal.Source: §L.3 · p. 82

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