Mean–variance approximations to expected utility

doi:10.1016/j.ejor.2012.08.023

European Journal of Operational Research

Volume 234, Issue 2, 16 April 2014, Pages 346-355

https://doi.org/10.1016/j.ejor.2012.08.023 Get rights and content

Abstract

It is often asserted that the application of mean–variance analysis assumes normal (Gaussian) return distributions or quadratic utility functions. This common mistake confuses sufficient versus necessary conditions for the applicability of modern portfolio theory. If one believes (as does the author) that choice should be guided by the expected utility maxim, then the necessary and sufficient condition for the practical use of mean–variance analysis is that a careful choice from a mean–variance efficient frontier will approximately maximize expected utility for a wide variety of concave (risk-averse) utility functions. This paper reviews a half-century of research on mean–variance approximations to expected utility. The many studies in this field have been generally supportive of mean–variance analysis, subject to certain (initially unanticipated) caveats.

Highlights

► Critically evaluates attacks on assumptions of modern portfolio theory. ► Argues attacks confuse sufficient versus necessary conditions for applying theory. ► Reviews a half-century of research on mean–variance approximations to expected utility.

Section snippets

Why not just maximize expected utility?

If one believes that action should be in accord with the max EU rule, why seek to approximately maximize it via a mean–variance analysis? Why not just maximize expected utility? In addressing this question I distinguish three types of expected utility maximization:

•
explicit,
•
MV-approximate,
•
implicit.

I refer to it as “explicit” EU maximization when a utility function is given and analytical or numerical methods are used to find the portfolio that maximizes the expected value of this function. In

Utility of return vs. utility of wealth

Eqs. (3), (4) assume that an investor seeks to maximize the expected value of Ln(1 + R), where R is the return on the portfolio during the forthcoming period.

Loistl’s erroneous analysis

In a paper titled “The Erroneous Approximation of Expected Utility by Means of a Taylor’s Series Expansion: Analytical and Computation Results,” Loistl (1976) concludes that “the mean–variance approximation is not a good approximation of the expected value of utility at all.” [p. 909]. The reason Loistl reaches such a negative conclusion is as follows: note that in Table 1 a 6% return is represented as R = 0.06, not as R = 6.0. In particular, in Eq. (9), if starting wealth is W_t = $1,000,000 and the

Levy and Markowitz (1979)

The Levy–Markowitz study had two principal objectives:

(1)
to see how good mean–variance approximations are for various utility functions and portfolio return distributions; and
(2)
to test an alternate way of estimating expected utility from a distribution’s mean and variance.

The Levy–Markowitz “alternate way” was to fit a quadratic approximation to U at three values of R: $(E - k σ), (E), (E + k σ)$ They tried their approach for $k = 0.01, 0.1, 0.6, 1.0 and 2.0$ Of these, k = 0.01 did best in almost every case. This is

Highly risk-averse investors

The Levy–Markowitz results for the exponential with b = 10 differ markedly from those of the other utility functions reported in Table 2. In this section we explore the reason for this.

For E = 0.1 and σ = 0.15, Table 3 compares the exponential utility function with the quadratic approximation, Q_E in Eq. (2). The utility function is rescaled as follows $U = - 1000 e^{- 10 (1 + R)}$ With this scaling, the difference between U (0.5) and U (−0.3) is of the same order of magnitude as that for Ln(1 + R) in Table 1, namely,

Highly risk-averse investors and a risk-free asset.

Simaan (1993) explores the efficacy of MV-approximate maximization for investors with the exponential utility function when a risk-free asset is available versus when such a risk-free asset is not available. He finds that, for investors with exponential utility functions with large values of b, MV-approximate EU maximization is highly efficacious when a risk-free asset is available, and much less so when it is not.

In deriving these results, Simaan assumes that security returns follow a factor

Portfolios of call options

It is often said that mean–variance analysis is not valid if applied to portfolios which contain securities with asymmetric distributions, or whose returns vary in a nonlinear manner as a function of some security or index. Hlawitschka (1994) examines the efficacy of mean–variance and higher order approximations to expected utility for portfolios of calls, and concludes that the afore mentioned view is wrong. For his individual stock return series, Hlawitschka uses the monthly returns for 10

Ederington’s quadratic and Gaussian approximations to expected utility.

Ederington (1995) principal results are summarized in our Table 7, extracted from Ederington’s Table 5. Ederington refers to the series represented in the table as “semi-ex ante simulated returns.” He argues that historical return series such as those used by Levy and Markowitz are an “ex post” sample drawn from some unknown population. Other possible samples from this same population might have randomly drawn more extreme returns than the one actual historical series, and therefore would have

Mean–variance approximations to the geometric mean

Markowitz (2012a) presents historical comparisons between the geometric mean G and six mean–variance approximations to it, for two data sets: one consisting of the historical returns on asset classes widely used in asset allocation decisions, the other containing the real returns during the 20th Century of the equity markets of sixteen countries. I will refer to the first of these as the AC (Asset Class) database, and the second as the DMS (Dimson et al., 2002) database. Both were obtained

Other measures of risk

As described in the preceding section, Markowitz (2012a) compared six different MV methods for approximation ELn(1 + R). Among these the Markowitz (1959) approximation in Eq. (6) here did as well as any. Markowitz (forthcoming) asks could another of the commonly proposed risk-measures have done better. The measures compared in Markowitz (forthcoming) are:

Variance (V),
Mean Absolute Deviation (MAD),
Semivariance (SV),
Value at Risk (VaR),
Conditional Value at Risk (CVaR).

From Eq. (6) we see that $Ln (1 + E)$

Other pioneers

In previous sections I have tried to review what I consider to be the most critical steps towards our present understanding of MV approximations of EU, these include: the Levy, Markowitz observation that MV approximations to EU are quite good for many U(R)—except those with extremely high risk-aversion, i.e., implausibly low VBC; Simaan’s observation that such investors will nevertheless be well served if the MV analysis includes a risk-free asset; Hlawitschka’s observation that MV

Conclusion

It is now over a half-century since Markowitz (1959) first defended MV analysis as a practical way to approximately maximize EU. In light of repeated confirmation since then of the efficacy of MV approximations to EU, the persistence of the Great Confusion—that MV analysis is applicable in practice only when return distributions are Gaussian or utility functions quadratic—is as if geography textbooks of 1550 still described the Earth as flat.

References (33)

G. Chamberlain
A characterization of the distributions that imply mean–variance utility functions
Journal of Economic Theory
(1983)
A.S. Dexter et al.
Portfolio selection in a lognormal market when the investor has a power utility function: computational results
Dimson, E., Marsh P., Staunton M., 2002. Triumph of the Optimists. Princeton Univercity...
Ederington, L.H., 1995. Mean–variance as an approximation to expected utility maximization: semi ex-ante results. In:...
R.R. Grauer
Normality, solvency, and portfolio choice
Journal of Financial and Quantitative Analysis
(1986)
N.H. Hakansson
Capital growth and the mean–variance approach to portfolio selection
Journal of Financial and Quantitative Analysis
(1971)
Walter Hlawitschka
The empirical nature of Taylor-series approximations to expected utility
The American Economic Review
(1994)
W.H. Jean et al.
Geometric mean approximations
Journal of Financial and Quantitative Analysis
(1983)
Y. Kroll et al.
Mean variance versus direct utility maximization
Journal of Finance
(1984)
H.A. Latané
Criteria for Choice Among Risky Ventures
Journal of Political Economy
(1959)

H.A. Latané et al.

Criteria for Portfolio Building

The Journal of Finance

(1967)

H. Levy et al.

Approximating expected utility by a function of mean and variance

American Economic Review

(1979)

Otto. Loistl

The erroneous approximation of expected utility by means of a Taylor’s series expansion: analytic and computational results

The American Economic Review

(1976)

H.M. Markowitz

Portfolio selection

The Journal of Finance

(1952)

H.M. Markowitz

Portfolio Selection: Efficient Diversification of Investments

(1970)

H.M. Markowitz

Foundations of portfolio theory

Journal of Finance

(1991)

Cited by (220)

Does Islamic investing modify portfolio performance? Time-varying optimization strategies for conventional and Shariah energy-ESG-utilities portfolio
2024, Quarterly Review of Economics and Finance
This paper aims to assess the performance of Islamic portfolios vis-à-vis their conventional counterparts across two distinct periods: the pre-COVID-19 era and the COVID-19 era. Departing from prior studies, this study makes a novel contribution by employing an extensive array of 18 portfolio optimization techniques to construct optimal portfolios for conventional stock indices encompassing energy, utilities, and environmental, social, and governance (ESG) dimensions, as well as their Islamic equivalents. Performance comparisons are made utilizing three risk-adjusted performance measures, namely the Sharpe ratio, the Omega ratio, and the Sortino ratio. Our findings reveal that Shariah portfolios outperform conventional portfolios across all performance measures and risk-aversion levels when the most effective optimization methods are employed, both during pre-crisis and crisis periods. Additionally, our analysis highlights certain methods, namely EWMA, GM, DCC, and SHC, which produce portfolios exhibiting superior performance relative to alternative methods, as assessed by risk-adjusted metrics. Furthermore, Islamic portfolios demonstrate higher average returns compared to their conventional counterparts. Notably, incorporating ESG-related stocks into energy and utilities assets significantly enhances average returns, underscoring the potential of ESG investments. Collectively, our findings have noteworthy implications for investors, as they emphasize the role of Islamic stocks as effective diversifiers, yielding favorable resource allocation opportunities during times of crisis as well as stability. However, investors should exercise caution in selecting the optimal portfolio optimization method, as substantial performance disparities exist among different approaches.
TVP-VAR based time and frequency domain food & energy commodities connectedness an analysis for financial/geopolitical turmoil episodes
2024, Applied Energy
Amidst the current global inflationary challenges, the concurrent rise of energy and agricultural commodity prices, which constitute the primary components of consumer prices, has emerged as a matter of significant interest among both scholars and policymakers. To this end, this study examines the dynamic interlinkages between food and energy commodity indexes from 2005:1 to 2023:3, covering turmoil episodes such as the Global Financial Crisis (GFC), the COVID-19 pandemic, and the Russia-Ukraine Conflict (RUC). Additionally, following Broadstock et al. (2022), we perform dynamic portfolio analyses to determine portfolio performances under 3 different portfolio construction approaches. The empirical results presented in this paper allow for a number of important findings. First, both the time and frequency-domain connectedness indexes associate with major financial/geopolitical stress events. Second, the fuel energy and the crude oil price indexes are the largest propagators and recipients of spillovers Third, the cumulative portfolio returns exhibit significant growth during the early phase of the COVID-19, declining during the RUC, and a notable upswing during the GFC. Finally, our findings for frequency-dependent connectedness networks indicate that the market is particularly susceptible to short-term shocks. This paper has significant ramifications for investors, market players, and policymakers.
First passage times in portfolio optimization: A novel nonparametric approach
2024, European Journal of Operational Research
This paper introduces a portfolio optimization procedure that aims to minimize the intra-horizon (IH) risk subject to a minimum expected time to achieve a target cumulative return. To estimate the first passage probabilities and the expected time a novel nonparametric method and a new Markov chain order determination approach are developed. The optimization framework proposed allows us to include novel path-dependent measures of risk and return in the asset allocation problem. An empirical application to S&P 100 companies, a risk-free asset and stock indices is provided. Our empirical results suggest that the proposed framework exhibits more consistency between in-sample and out-of-sample performance than the mean-variance model and an alternative optimization problem that minimizes the MaxVaR measure of Boudoukh et al. (2004). Overall, the portfolio optimization approach we introduce results in higher out-of-sample annualized returns for relatively low levels of IH risk.
Fifty years of portfolio optimization
2024, European Journal of Operational Research
The allocation of resources to alternative investment opportunities is one of the most important decisions organizations and individuals face. These decisions can be guided by building and solving portfolio optimization models that capture the salient aspects of the investment problem, including decision-makers’ preferences, multiple objectives, and decision opportunities over the planning horizon. In this paper, we give a historically grounded overview of portfolio optimization which, as a field within operational research with roots in finance, is vast thanks to many decades of research and the huge diversity of problems that have been tackled. In particular, we provide a unified and therefore unique treatment that covers the full breadth of portfolio optimization problems, including, for instance, the allocation of resources to financial assets and the selection of indivisible assets such as R&D projects. We also identify opportunities for future methodological and applied research, hoping to inspire researchers to contribute to the growing field of portfolio optimization.
A general framework for portfolio construction based on generative models of asset returns
2023, Journal of Finance and Data Science
In this paper, we present an integrated approach to portfolio construction and optimization, leveraging high-performance computing capabilities. We first explore diverse pairings of generative model forecasts and objective functions used for portfolio optimization, which are evaluated using performance-attribution models based on least absolute shrinkage and selection operator (LASSO). We illustrate our approach using extensive simulations of crypto-currency portfolios, and we show that the portfolios constructed using the vine-copula generative model and the Sharpe-ratio objective function consistently outperform. To accommodate a wide array of investment strategies, we further investigate portfolio blending and propose a general framework for evaluating and combining investment strategies. We employ an extension of the multi-armed bandit framework and use value models and policy models to construct eclectic blended portfolios based on past performance. We consider similarity and optimality measures for value models and employ probability-matching (“blending”) and a greedy algorithm (“switching”) for policy models. The eclectic portfolios are also evaluated using LASSO models. We show that the value model utilizing cosine similarity and logit optimality consistently delivers robust superior performances. The extent of outperformance by eclectic portfolios over their benchmarks significantly surpasses that achieved by individual generative model-based portfolios over their respective benchmarks.
Estimating ensemble weights for bagging regressors based on the mean–variance portfolio framework
2023, Expert Systems with Applications
This paper presents a novel ensemble learning framework inspired by modern portfolio optimization to address regression problems. This formulation in the ensemble learning field allows determining the ensemble’s weights considering the error predictions of the base learners, the variability in their predictions, and the covariance among their predictions’ errors, which is completely aligned with the bias–variance–covariance theory. Under the framework proposed, four potential instantiations have also been provided. The first two ensemble models impose the non-negativity constraints on the ensemble’s weights (along with the equality constraint that the ensemble’s weights sum up to one) and are solved with the active set algorithm. The second two ensemble models do not include the non-negativity constraints on the ensemble’s weights (which in the financial literature are called the non-shorting constraints), giving rise to a convex quadratic programming (QP) problem (as the matrix included in the quadratic term is symmetric and positive definite) that is solved by the Lagrangian procedure. Extensive experiments with regression datasets evaluate the proposed ensemble framework. Comparisons with other state-of-the-art ensemble methods confirm that the ensemble framework yields the best overall performance.

View all citing articles on Scopus

^☆: This paper is based on Chapter 2 of a book which the author is writing under the sponsorship of 1st Global of Dallas, TX. The author is delighted to thank 1st Global in general and, more specifically, its CEO Stephen A (Tony) Batman, its President David Knoch, and Kenneth Blay, my principal day-to-day contact at 1st Global.

View full text

Mean–variance approximations to expected utility☆

Abstract

Highlights

Section snippets

Why not just maximize expected utility?

Utility of return vs. utility of wealth

Loistl’s erroneous analysis

Levy and Markowitz (1979)

Highly risk-averse investors

Highly risk-averse investors and a risk-free asset.

Portfolios of call options

Ederington’s quadratic and Gaussian approximations to expected utility.

Mean–variance approximations to the geometric mean

Other measures of risk

Other pioneers

Conclusion

A characterization of the distributions that imply mean–variance utility functions

Journal of Economic Theory

Portfolio selection in a lognormal market when the investor has a power utility function: computational results

Normality, solvency, and portfolio choice

Journal of Financial and Quantitative Analysis

Capital growth and the mean–variance approach to portfolio selection

Journal of Financial and Quantitative Analysis

The empirical nature of Taylor-series approximations to expected utility

The American Economic Review

Geometric mean approximations

Journal of Financial and Quantitative Analysis

Mean variance versus direct utility maximization

Journal of Finance

Criteria for Choice Among Risky Ventures

Journal of Political Economy

Criteria for Portfolio Building

The Journal of Finance

Approximating expected utility by a function of mean and variance

American Economic Review

The erroneous approximation of expected utility by means of a Taylor’s series expansion: analytic and computational results

The American Economic Review

Portfolio selection

The Journal of Finance

Portfolio Selection: Efficient Diversification of Investments

Foundations of portfolio theory

Journal of Finance