Expected Utility Regret Rule: Minimax and Bayes Optimal Portfolio Choice

Masahiro Kato

econ.EMOct 1, 2026

Expected Utility Regret Rule: Minimax and Bayes Optimal Portfolio Choice

Authors: Masahiro Kato

Organizations: The University of Tokyo

Abstract

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.

Figures & tables

Figure 1: The EUR rule with at least two portfolio classes. Each arrow points from a sample or quantity to a step that uses it. The preliminary sample

D_{0}

determines the candidate set

C

and, when

\left|C\right|\geq 3

, the covariance

\widehat{\Omega}_{C}

of the finite Gaussian comparison, whereas every other estimate is computed from the second sample

D_{1}

alone. The index

j

is the class selected in the branch that applies.

$n$	$\sqrt{n}\,\mathcal{R}$	$\sqrt{n}\,\mathcal{R}_{\rm sel}$	$n\mathcal{R}_{\rm alloc}$	$n\mathcal{R}_{H_{1}}$	$n\mathcal{R}_{H_{2}}$
250	0.02348	0.01684	0.10491	0.21002	0.25757
1,000	0.02095	0.01447	0.20511	0.27138	0.34676
4,000	0.01752	0.01392	0.22801	0.29507	0.37310
16,000	0.01540	0.01369	0.21598	0.29591	0.37166
64,000	0.01409	0.01327	0.20752	0.28768	0.36089
256,000	0.01351	0.01312	0.20155	0.27946	0.35065

Table 1: Expected regret of the two-class EUR rule, normalized by

\sqrt{n}

(total, selection) or

n

(allocation, Bayes).

$(\Omega_{11},\Omega_{12},\Omega_{22})$	Grid $0.5$	Grid $0.25$	Largest observation	LP objective
$(2,1,2)$	0.37748	0.37327	0.37215	0.37175
$(1,0.8,1)$	0.23691	0.23207	0.23069	0.23024
$(1,0,4)$	0.45896	0.45324	0.46145	0.45202
$(1,-0.5,2)$	0.38741	0.37940	0.39103	0.37743

Table 2: Expected regret in the three-class Gaussian selection problem.

Figure 2: Regular allocation (left: normalized regret against the exact constant) and correlated class selection (right: normalized selection regret against the complexity of the utility differences at

n=50,100,250,500,1{,}000

). Error bars are Monte Carlo standard errors.

Figure 3: Regret at a decision boundary, divided by the theoretical constant: integrated expected regret under three evaluation priors (left) and worst-case expected regret (right).

Figure 4: Moment separation: worst-case expected utility regret over the two orderings of return distributions with identical means and second moments.

$n$	EUR rule, $v=-1$	MV, $v=-1$	EUR rule, $v=1$	MV, $v=1$
4096	0.50027	0.79945	0.50021	0.45595
16384	0.50006	0.70719	0.50005	0.47903
65536	0.50002	0.65570	0.50001	0.49902
262144	0.50000	0.62444	0.50000	0.51521
1048576	0.50000	0.60449	0.50000	0.52779
Limit	0.50000	0.56250	0.50000	0.56250

Table 3: Expected regret multiplied by

n

in the two-point return model.

$n$	Log, $\theta=.075$	Log, prior	CRRA, $\theta=.075$	CRRA, prior
250	0.978 (0.008)	0.911 (0.007)	0.995 (0.008)	0.927 (0.007)
1000	0.992 (0.008)	0.990 (0.008)	0.996 (0.008)	0.992 (0.008)
4000	0.994 (0.008)	0.988 (0.008)	1.013 (0.008)	1.003 (0.008)
16000	1.001 (0.008)	0.987 (0.008)	1.014 (0.008)	0.999 (0.008)
64000	1.009 (0.008)	1.010 (0.008)	0.994 (0.008)	1.004 (0.008)
256000	0.996 (0.008)	0.996 (0.008)	1.009 (0.008)	0.999 (0.008)

Table 4: Expected regret of the risk-parity form of the EUR rule, multiplied by

n

and divided by the constant in ( 96 ) or its prior integral.

$n$	EUR rule	Known scales	Difference	Contribution RMSE
250	0.10119 (0.00076)	0.08822 (0.00073)	0.01297	0.01583
1000	0.10175 (0.00075)	0.08912 (0.00072)	0.01263	0.00790
4000	0.09936 (0.00073)	0.08742 (0.00071)	0.01194	0.00393
16000	0.10060 (0.00075)	0.08808 (0.00072)	0.01252	0.00198
64000	0.10037 (0.00074)	0.08796 (0.00071)	0.01241	0.00099
Limit	0.10005	0.08781	0.01224	0

Table 5: The EUR rule with estimated relative volatilities.

$n$	$h$	EUR rule	Imposed composition
4096	0	0.24760 (0.00142)	0.12422 (0.00102)
4096	0.5	0.24879 (0.00141)	0.17266 (0.00103)
4096	1	0.24344 (0.00136)	0.30786 (0.00103)
4096	2	0.20501 (0.00126)	0.85311 (0.00102)
262144	0	0.24935 (0.00143)	0.12504 (0.00102)
262144	0.5	0.25020 (0.00144)	0.16984 (0.00101)

Table 6: Expected regret multiplied by

n

near the symmetric model.

$\Delta$	Exact loss	Budget expansion	Marginal-utility approximation
0.0200000	16.352575	16.343947	16.351785
0.0100000	4.086526	4.085987	4.086477
0.0050000	1.021530	1.021497	1.021527
0.0025000	0.255376	0.255374	0.255376
0.0012500	0.063844	0.063844	0.063844
0.0006250	0.015961	0.015961	0.015961

Table 7: The cost of imposing equal standardized holdings under a tighter money budget.

Rule	Mean return	Volatility	Certainty equivalent	Maximum drawdown	Annual turnover	CE difference	$p$ -value
Full expected utility	8.88	16.34	1.90	-50.10	0.87	-2.06	0.41
Second order	8.84	16.34	1.85	-49.61	0.87	-2.11	0.40
Full simplex	6.16	14.22	0.83	-49.86	0.58	-3.13	0.10
Minimum variance	9.06	12.69	4.98	-35.91	0.24	1.02	0.54
Equal weight	9.85	15.16	3.96	-47.61	0.10	0.00	–

Table 8: Out-of-sample performance of the industry portfolios over the test period, net of transaction costs.

Figure 5: Industry portfolios: cumulative net wealth in the primary specification (left) and annualized certainty equivalents compared with equal weighting (right).

Procedure	CE (%)	Difference (pp)	95% interval	Turnover
Full expected utility	1.903	0.000	–	0.874
Second-order objective	1.851	-0.052	[-0.613, 0.320]	0.868
Minimum variance	0.475	-1.427	[-5.401, 1.954]	0.920
Equal weights	1.808	-0.094	[-3.175, 3.842]	0.702

Table 9: Portfolio comparisons among classes holding at most three industries.

Component	CE contribution	Lower limit	Upper limit	$t$ -statistic	$p$ -value
Within-class allocation	-0.122	-0.283	0.038	-1.302	0.193
Class selection	0.174	-0.164	0.704	0.599	0.549
Total difference	0.052	-0.320	0.613	0.185	0.854

Table 10: Contributions from class selection and from allocation within a class, in certainty-equivalent terms.

Figure 6: Contributions of class selection and of allocation within a class across specifications, in annualized percentage points of certainty equivalent, with paired percentile intervals from resampling annual blocks.

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