On Determination of Stochastic Dominance Optimal Sets

Applying Fishburn's [4] conditions for convex stochastic dominance, exact linear programming algorithms are proposed and implemented for assigning discrete return distributions into the first- and second-order stochastic dominance optimal sets. For third-order stochastic dominance, a superconvex stochastic dominance approach is defined which allows classification of choice elements into superdominated, mixed, and superoptimal sets. For a choice set of 896 security returns treated previously in the literature, 454, 25, and 13 distributions are in the first-, second-, and third-order convex stochastic dominance optimal sets, respectively. These optimal sets compare with admissible first-, second-, and third-order stochastic dominance sets of 682, 35, and 19 distributions, respectively. The applicability of superconvex stochastic dominance for continuous distributions defined over a bounded interval is then shown. The difficulties in identifying the elements of the superdominated set for distributions defined over the entire real line are demonstrated in the determination of the dominated choices for a set of normally distributed mutual fund returns previously examined by Meyer [9]. Specifically, we find that the dominated set determined by Meyer is too large.     

THE MEAN‐GINI EFFICIENT PORTFOLIO FRONTIER

A main advantage of the mean‐variance (MV) portfolio frontier is its simplicity and ease of derivation. A major shortcoming, however, lies in its familiar restrictions, such as the quadraticity of preferences or the normality of distributions. As a workable alternative to MV, we present the mean‐Gini (MG) efficient portfolio frontier. Using an optimization algorithm, we compute MG and mean‐extended Gini (MEG) efficient frontiers and compare the results with the MV frontier. MEG allows for the explicit introduction of risk aversion in building the efficient frontier. For U.S. classes of assets, MG and MEG efficient portfolios constructed using Ibbotson (2000) monthly returns appear to be more diversified than MV portfolios. When short sales are allowed, distinct investor risk aversions lead to different patterns of portfolio diversification, a result that is less obvious when short sales are foreclosed. Furthermore, we derive analytically the MG efficient portfolio frontier by restricting asset distributions. The MG frontier derivation is identical in structure to that of the MV efficient frontier derivation. The penalty paid for simplifying the search for the MG efficient frontier is the loss of some information about the distribution of assets.     

Mean-variance vs. stochastic dominance some empirical findings on efficient sets

The study attempts to shed additional light on the issue of the costs and benefits of using the mean-variance criterion as opposed to stochastic dominance criteria for investment decisions. Relevant probabilities which facilitate measurement of these costs and benefits are identified. The mean-variance criterion is shown to be useful to some extent in identifying potentially optimal portfolios. However, it is shown that the informationally less demanding mean-variance criterion admits two types of errors: (i) including portfolios that no expected utility maximizing risk averters would choose, and (ii) excluding portfolios which some risk averters would find optimal. The empirical investigation also indicates that although the composition of the efficient sets appears to be unstable over time, the relationships between the efficient sets are persistent over time.     

Ordering Uncertain Options under Inflation: A Note

Stochastic dominance rules (SD) have been extended to the case where investors are allowed to borrow and lend at the riskless interest rate. Stochastic dominance rules with a riskless asset (SDR) are much more effective than SD rules. However, it seems that this benefit is eliminated by an uncertain inflation, since riskless assets become risky once uncertain inflation is considered. We prove in this paper that SDR criteria are valid also in the face of uncertain (and independent) inflation. Moreover, while the mean-variance (MV) efficient set increases with uncertain inflation, the stochastic dominance efficient sets decrease.     

Econometrical analysis of the sample efficient frontier

The efficient frontier is a parabola in the mean-variance space which is uniquely determined by three characteristics. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we derive tests and confidence sets for all possible arrangements of these characteristics. Note that all of our results are based on the exact distributions for a finite sample size. Moreover, we determine a confidence region of the whole efficient frontier in the mean-variance space. It is shown that this set is bordered by five parabolas.     

Optimal rules for ordering uncertain prospects

In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of alternatives, those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm to obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in finance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule to others previously advocated in the literature, including the more popular mean-variance rule as well as the semi-variance rule.     

Reduction of estimation risk in optimal portfolio choice using redundant constraints

It is well known that when the moments of the distribution governing returns are estimated from sample data, the out-of-sample performance of the optimal solution of a mean–variance (MV) portfolio problem deteriorates as a consequence of the so-called “estimation risk”. In this document we provide a theoretical analysis of the effects caused by redundant constraints on the out-of-sample performance of optimal MV portfolios. In particular, we show that the out-of-sample performance of the plug-in estimator of the optimal MV portfolio can be improved by adding any set of redundant linear constraints. We also illustrate our findings when risky assets are equally correlated and identically distributed. In this specific case, we report an emerging trade-off between diversification and estimation risk and that the allocation of estimation risk across portfolios forming the optimal solution changes dramatically in terms of number of assets and correlations.     

We don't need no fancy hedges! Or do we?

Hedging decisions in the real world often contradict the literature. We reverse-engineer the optimal hedging problem by identifying patterns of price behavior that warrant using strategies more sophisticated than variance minimization (MV). Historical time series of spot and futures prices for the crack spread components (crude oil, gasoline, and heating oil) are used to generate different patterns of price dependency. A copula approach is used to model the joint dependence between spot and futures price shocks of the three commodities. We find that minimizing a downside risk criterion (what actual hedgers do) leads to consistently better outcomes than MV, as measured by Expected Utility. This is especially true in scenarios corresponding to strong upward or downward price movements. We provide a simple decision heuristic for hedgers by identifying price patterns whereby using sophisticated strategies for multi-commodity hedging is optimal in practice.     

The 4% strategy revisited: a pre-commitment mean-variance optimal approach to wealth management

In contrast to single-period mean-variance (MV) portfolio allocation, multi-period MV optimal portfolio allocation can be modified slightly to be effectively a down-side risk measure. With this in mind, we consider multi-period MV optimal portfolio allocation in the presence of periodic withdrawals. The investment portfolio can be allocated between a risk-free investment and a risky asset, the price of which is assumed to follow a jump diffusion process. We consider two wealth management applications: optimal de-accumulation rates for a defined contribution pension plan and sustainable withdrawal rates for an endowment. Several numerical illustrations are provided, with some interesting implications. In the pension de-accumulation context, Bengen (1994)’s [J. Financial Planning, 1994, 7, 171–180], historical analysis indicated that a retiree could safely withdraw 4% of her initial retirement savings annually (in real terms), provided that her portfolio maintained an even balance between diversified equities and U.S. Treasury bonds. Our analysis does support 4% as a sustainable withdrawal rate in the pension de-accumulation context (and a somewhat lower rate for an endowment), but only if the investor follows an MV optimal portfolio allocation, not a fixed proportion strategy. Compared with a constant proportion strategy, the MV optimal policy achieves the same expected wealth at the end of the investment horizon, while significantly reducing the standard deviation of wealth and the probability of shortfall. We also explore the effects of suppressing jumps so as to have a pure diffusion process, but assuming a correspondingly larger volatility for the latter process. Surprisingly, it turns out that the MV optimal strategy is more effective when there are large downward jumps compared to having a high volatility diffusion process. Finally, tests based on historical data demonstrate that the MV optimal policy is quite robust to uncertainty about parameter estimates.     

Investment Policy Implications of the Capital Asset Pricing Model

The results of previous generalized Security Market Line (SML) tests of the Mean Variance (MV) and Linear Risk Tolerance (LRT) Capital Asset Pricing Models indicate that the models are empirically identical. A very widely accepted, but technically incorrect, explanation for the results is that with normal return distributions all expected utility maximizing riskaverse investors will pick MV portfolios. The paper shows that the generalized SML tests cannot distinguish between the MV model and a much wider variety of power utility LRT models than has previously been entertained. On the other hand, with approximately normal, or real world, return distributions the investment policies of the various models are shown to be different from each other, and from the MV policy in particular. To the extent the results of the portfolio selection calculations are robust, the results of, and implications drawn from, the tests of the macro pricing relations are not based on firm micro foundations.     

Capital Asset Pricing Compatible with Observed Market Value Weights

We show that the set of expected return vectors, for which an observed portfolio is mean variance (MV) efficient, is a two-parameter family. We identify ten ways to specify the time series behavior of the two parameters; the result highlights a number of inconsistencies involved in MV modelling. For each of the cases, it permits the inference of the time series of expected return vectors, as well as all the other Capital Asset Pricing Model (CAPM) variables, compatible with a known covariance matrix and the observed time series of market value weights. The empirical work shows that there are substantial case-to-case differences in the time series of mean vectors and many of them are quite different from the constant mean vector envisioned in tests of the CAPM.     

Potential performance and tests of portfolio efficiency

The potential performance of an asset set may be obtained by choosing the portfolio proportions to maximize the Sharpe (1966) performance measure. If a portfolio has a Sharpe measure equivalent to the potential performance of the underlying set of assets, then it is efficient. Multivariate statistical procedures for comparing potential performance and testing portfolio efficiency are developed and then evaluated using simulations. Two likelihood ratio statistics are then used to compare stock and bond indices against sets of 20 and 40 portfolios. The procedures are also compared to the Gibbons (1982) methodology for testing financial models.     

Confidence regions for the mean-variance efficient set: An alternative approach to estimation risk

The mean-variance efficient set is used extensively in portfolio analysis and in addition underlies many of the models and tests of asset pricing. Despite this vital role, little direct attention has been paid to efficient frontier estimation. This article illustrates that an estimator for the efficient set hyperbola is composed of three mutually independent statistics whose distributions are known. This result is used to develop a confidence region for the efficient set hyperbola in (σ, μ) space. Two alternative approaches are used to define a confidence region. The first approach can be used to obtain an expression for a confidence region for σ given μ or for μ given σ. The second approach defines a confidence region in (σ, μ) space that contains the true hyperbola with a specified probability. In addition, anF test for mean-variance efficiency is used to generate a sample acceptance region. The sample acceptance region and the two confidence regions are compared graphically. A simulation experiment is used to examine the properties of the various procedures.     

A Shrinkage Approach to Model Uncertainty and Asset Allocation

This article takes a shrinkage approach to examine the empiricalimplications of aversion to model uncertainty. The shrinkageapproach explicitly shows how predictive distributions incorporatedata and prior beliefs. It enables us to solve the optimal portfoliosfor uncertainty-averse investors. Aversion to uncertainty aboutthe capital asset pricing model leads investors to hold a portfoliothat is not mean-variance efficient for any predictive distribution.However, mean-variance efficient portfolios corresponding toextremely strong beliefs in the Fama–French model areapproximately optimal for uncertainty-averse investors. Theempirical Bayes approach does not result in optimal portfoliosfor investors who are averse to model uncertainty.     

A NOTE ON THE EFFECTS OF EFFICIENCY CRITERIA AND PORTFOLIO SIZE ON CHARACTERISTICS OF EFFICIENT PORTFOLIOS*

Comparisons of the results of using alternative efficiency criteria for portfolio selection, while plentiful overseas, are seemingly scarce in Australia. Here, mean-variance efficient sets are compared with and found to be not too different from sets efficient according to second and third degree stochastic dominance. In examining the effect of increasing the number of securities allowed in each portfolio, it is found that the size of efficient sets of portfolios diminishes, as does the mean return and, most significantly, the variance of return for efficient portfolios.     

Collusion Proof Transfer Payment Schemes with Multiple Agents

Most of the current studies on transfer pricing under asymmetric information focus on a single principal and a single agent. Under a separating management and ownership assumption, transfer pricing is at minimum a three-person problem involving one principal and two agents. This paper considers a transfer pricing problem with two agents who possess private information and seek to maximize their net cash flows, instead of divisional accounting profits. The objectives of this paper are: (1) to derive a direct-revelation mechanism that induces truth telling and efficient allocation; (2) to study the agents' collusion behaviors under the direct-revelation mechanism. The findings indicate that when agents have the option to quit after contracting, it is optimal for the center to produce less than the first-best output level unless the costs for both divisions are at their lowest levels. The optimal amount of underproduction varies according to the demand condition. In addition, two sets of transfer functions, named as identical and nonidentical functions, are derived to induce truth-telling and yield optimal equilibrium output. The two sets of transfer functions are subject to collusion. However, the functions induce different collusion behaviors among agents, that is, the collusion sets for both functions are not common sets. This property enables us to eliminate any collusion between agents, particularly prior to their observation of private information.     

Downside risk and asset pricing

We analyze if the value-weighted stock market portfolio is stochastic dominance (SD) efficient relative to benchmark portfolios formed on size, value, and momentum. In the process, we also develop several methodological improvements to the existing tests for SD efficiency. Interestingly, the market portfolio seems third-order SD (TSD) efficient relative to all benchmark sets. By contrast, the market portfolio is inefficient if we replace the TSD criterion with the traditional mean–variance criterion. Combined these results suggest that the mean–variance inefficiency of the market portfolio is caused by the omission of return moments other than variance. Especially downside risk seems to be important for explaining the high average returns of small/value/winner stocks.     

Optimal asset allocation under linear loss aversion

We study the asset allocation of a linear loss-averse (LA) investor and compare it to the more traditional mean-variance (MV) and conditional value-at-risk (CVaR) investors. First we derive conditions under which the LA problem is equivalent to the MV and CVaR problems and solve analytically the two-asset problem of the LA investor for a risk-free and a risky asset. Then we run simulation experiments to study properties of the optimal LA and MV portfolios under more realistic assumptions. We find that under asymmetric dependence LA portfolios outperform MV portfolios, provided investors are sufficiently loss-averse and dependence is large. Finally, using 13 EU and US assets, we implement the trading strategy of a linear LA investor who reallocates his/her portfolio on a monthly basis. We find that LA portfolios clearly outperform MV and CVaR portfolios and that incorporating a dynamic update of the LA parameters significantly improves the performance of LA portfolios.     

On the Theory of Rational Insurance Purchasing: A Note

The authors consider the optimal amount of insurance purchased by an individual who behaves according to the Hurwicz criterion of choice under uncertainty. Their results are compared with earlier results obtained in alternative frameworks (expected utility maximization and Savage's regret criterion). It is shown that a positive amount deductible is often suboptimal.     

The effect of estimation risk on optimal portfolio choice

This paper determines the effect of estimation risk on optimal portfolio choice under uncertainty. In most realistic problems, the parameters of return distributions are unknown and are estimated using available economic data. Traditional analysis neglects estimation risk by treating the estimated parameters as if they were the true parameters to determine the optimal choice under uncertainty. We show that for normally distributed returns and ‘non-informative’ or ‘invariant’ priors, the admissible set of portfolios taking the estimation uncertainty into account is identical to that given by traditional analysis. However, as a result of estimation risk, the optimal portfolio choice differs from that obtained by traditional analysis. For other plausible priors, the admissible set, and consequently the optimal choice, is shown to differ from that in traditional analysis.