Prospect and Markowitz stochastic dominance

Levy and Wiener (J Risk Uncertain 16(2), 147–163, 1998), Levy and Levy (Manage Sci 48(10), 1334–1349, 2002; Rev Fin Stud 17(4), 1015–1041, 2004) develop the prospect and Markowitz stochastic dominance theory with S-shaped and reverse S-shaped utility functions for investors. In this paper, we extend their work on prospect stochastic dominance theory (PSD) and Markowitz stochastic dominance theory (MSD) to the first three orders and link the corresponding S-shaped and reverse S-shaped utility functions to the first three orders. We also provide experiments to illustrate each case of the MSD and PSD to the first three orders and demonstrate that the higher order MSD and PSD cannot be replaced by the lower order MSD and PSD. Furthermore, we formulate the following PSD and MSD properties: hierarchy exists in both PSD and MSD relationships; arbitrage opportunities exist in the first orders of both PSD and MSD; and for any two prospects under certain conditions, their third order MSD preference will be ‘the opposite of’ or ‘the same as’ their counterpart third order PSD preference. By extending the work of Levy and Wiener and Levy and Levy, we provide investors with more tools to identify the first and third order PSD and MSD prospects and thus they could make wiser choices on their investment decision.     

A Generalisation of the Mean-Variance Analysis

In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above this reference point. We also suppose that the decision maker generally distorts the objective probabilities. First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This 'mean-partial moments' utility generalises not only mean-variance utility of Tobin and Markowitz, but also mean-semivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present a solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure which generalises the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker's skewness preferences have first-order impact on risk measurement even when the risk is small.     

A Benchmark Approach to Portfolio Optimization under Partial Information

This paper proposes a filtering methodology for portfolio optimization when some factors of the underlying model are only partially observed. The level of information is given by the observed quantities that are here supposed to be the primary securities and empirical log-price covariations. For a given level of information we determine the growth optimal portfolio, identify locally optimal portfolios that are located on a corresponding Markowitz efficient frontier and present an approach for expected utility maximization. We also present an expected utility indifference pricing approach under partial information for the pricing of nonreplicable contracts. This results in a real world pricing formula under partial information that turns out to be independent of the subjective utility of the investor and for which an equivalent risk neutral probability measure need not exist.      

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods

The fallacy that a many-period expected-utility maximizer should maximize (a) the expected logarithm of portfolio outcomes or (b) the expected average compound return of his portfolio is now understood to rest upon a fallacious use of the Law of Large Numbers. This paper exposes a more subtle fallacy based upon a fallacious use of the Central-Limit Theorem. While the properly normalized product of independent random variables does asymptotically approach a log-normal distribution under proper assumptions, it involves a fallacious manipulation of double limits to infer from this that a maximizer of expected utility after many periods will get a useful approximation to his optimal policy by calculating an efficiency frontier based upon (a) the expected log of wealth outcomes and its variance or (b) the expected average compound return and its variance. Expected utilities calculated from the surrogate log-normal function differ systematically from the correct expected utilities calculated from the true probability distribution. A new concept of ‘initial wealth equivalent’ provides a transitive ordering of portfolios that illuminates commonly held confusions. A non-fallacious application of the log-normal limit and its associated mean-variance efficiency frontier is established for a limit where any fixed horizon period is subdivided into ever more independent sub-intervals. Strong mutual-fund Separation Theorems are then shown to be asymptotically valid.     

A note on the existence of the power investor’s optimizer

Karatzas et al. (SIAM J. Control Optim. 29:707–730, 1991) ensure the existence of the expected utility maximizer for investors with constant relative risk aversion coefficients less than one. In this note, we explain a simple trick that allows us to use this result to provide the existence of utility maximizers for arbitrary coefficients of relative risk aversion. The simplicity of our approach is to be contrasted with the general existence result provided in Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999).     

Reward–risk portfolio selection and stochastic dominance

The portfolio selection problem is traditionally modelled by two different approaches. The first one is based on an axiomatic model of risk-averse preferences, where decision makers are assumed to possess a utility function and the portfolio choice consists in maximizing the expected utility over the set of feasible portfolios. The second approach, first proposed by Markowitz is very intuitive and reduces the portfolio choice to a set of two criteria, reward and risk, with possible tradeoff analysis. Usually the reward–risk model is not consistent with the first approach, even when the decision is independent from the specific form of the risk-averse expected utility function, i.e. when one investment dominates another one by second-order stochastic dominance. In this paper we generalize the reward–risk model for portfolio selection. We define reward measures and risk measures by giving a set of properties these measures should satisfy. One of these properties will be the consistency with second-order stochastic dominance, to obtain a link with the expected utility portfolio selection. We characterize reward and risk measures and we discuss the implication for portfolio selection.     

Risk-Adjusted Economic Value Analysis

Abstract

The ability of commonly used profitability measures to reflect risk exposure appropriately is evaluated and found lacking. As an alternative, a modern portfolio theory approach, based on utility theory, is recommended. Generalized formulas for calculating risk-adjusted economic values by deriving risk adjustments from certainty equivalents are developed by using the Markowitz expected utility maxim. Practical applications are described. Where appropriate, simplifying assumptions are shown to result in closed-form solutions, thereby reducing the need for extensive, stochastic cashflow simulations. The resulting formulas can be used to measure financial performance on a risk-adjusted basis consistently across different lines of business or to evaluate risk exposures in strategic alternatives.     

Keeping up with the Joneses and optimal diversification

Peer-effects have been shown to affect behavior, and can generally lead to investments choices that are mean–variance inefficient. This paper analyzes optimal diversification with peer-effects. We show that if individuals have keeping-up with the Joneses preferences and they take their peer-group reference as the market portfolio, Markowitz’s mean–variance efficiency analysis and the CAPM equilibrium are intact. This holds for any keeping-up preferences, as well as heterogeneous combinations of such preferences. These results also extend to the Merton–Levy segmented-market model.     

Portfolio selection with higher moments

We propose a method for optimal portfolio selection using a Bayesian decision theoretic framework that addresses two major shortcomings of the traditional Markowitz approach: the ability to handle higher moments and parameter uncertainty. We employ the skew normal distribution which has many attractive features for modeling multivariate returns. Our results suggest that it is important to incorporate higher order moments in portfolio selection. Further, our comparison to other methods where parameter uncertainty is either ignored or accommodated in an ad hoc way, shows that our approach leads to higher expected utility than competing methods, such as the resampling methods that are common in the practice of finance.     

The relationship between expected utility and higher moments for distributions captured by the Gram–Charlier class

In this letter we derive the closed form solution for expected utility in terms of higher moments of the distribution of wealth when expected utility takes the CARA form and the distribution of wealth is captured by the Gram–Charlier class of distributions. We derive the condition under which positive skewness can be associated with a decrease in expected utility.     

Imperfect Agency and the Regulation of Hospitals

This article addresses the combined problem of imperfect agency and asymmetric information in the regulation of hospitals by modeling the physician as a utility maximizer with both the utility of patients and profit of the hospital as arguments in his or her utility function. The article concludes that optimal regulation of hospitals is based on three important factors: the doctor's marginal rate of substitution between profit of the hospital and utility of the patients, moral hazard in the relationship between the regulator and the hospital, and adverse selection in the same relation.     

Efficient Hedge Fund Strategy Allocations – Systematic Framework for Investors that Incorporates Higher Moments

In this paper, we provide a realistic framework that investors can use to optimize hedge fund portfolio strategy allocations. Our approach includes important aspects that investors need to address in the real world, such as how limited resources can restrict the ability to conduct multiple due diligences. Additionally, our approach is not based on a utility function, but on an easily quantifiable preference parameter, lambda. We need to account for higher moments of the return distribution within our optimization and approximate a best‐fit distribution. Thus we replace the empirical return distributions, which are often skewed or exhibit excess kurtosis, with two normal distributions. We then use the estimated return distributions in the strategy optimization. Our dataset comes from the Lipper TASS Hedge Fund Database and covers the June 1996‐December 2008 time period. Our results show in‐ and out‐of‐sample that our framework yields superior results to the Markowitz framework. It is also better able to manage regime switches, which tend to occur frequently during crises. Lastly, to test our results for stability, we use robustness tests, which allow for the time‐varying correlation structures of the strategies.     

Possible explanations of no-synergy mergers and small firm effect by the Generalized Capital Asset Pricing Model

The Sharpe-Lintner Capital Asset Pricing Model (CAPM) and the General Capital Asset Pricing Model (GCAPM) suggested by Levy (1978), Merton (1987), and Markowitz (1989) are compared and analyzed. Under the GCAPM we obtain the following main results: 1) the value additivity principle breaks down, which explains mergers and acquisitions; 2) beyond a certain limit, the profit from additional merger is negative; and 3) in a GCAPM equilibrium, small firms earn an abnormal profit in comparison to what is predicted by the CAPM. These results, which are indeed observed in the market, are fully consistent with the GCAPM, but are in contradiction to the CAPM.     

A NOTE ON INVESTMENT DECISION RULES BASED ON UTILITY FUNCTIONS

Many investment decision models are formulated on the basis of certain assumptions regarding investor's tastes combined with the assumed objective of expected utility maximization. The rules are often expressed in the form of a finite number of moments of the returns distributions, depending on the specific utility function restrictions imposed. For example, mean/variance decisions have been derived using the assumption of a quadratic preference function. Borch (1969) and many others have demonstrated the ambiguity of mean/variance decision rules based on this specific utility functional form. In this note, Borch's findings are extended t o all investment decision rules based on the assumption of any finite order utility function for a general class of risk averters.     

A dynamic balance sheet management model for a Canadian chartered bank

This paper presents a linear programming (LP) model based on Markowitz portfolio theory for solving the balance sheet management problem for the domestic assets and liabilities of a Canadian chartered bank. Given the bank's initial position, its economic forecasts, and the constraints under which it operates, the model will determine the current and expected future balance sheet adjustments which will meet the bank's expected profits goal with the minimum possible risk. By parametrically varying the expected profits goal, the model will generate the set of risk-return efficient decisions. Bankers need examine only the set of efficient decisions to choose their optimal solution.     

Optimal dynamic reinsurance with dependent risks: variance premium principle

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the variance premium principle, we adopt a nonstandard approach to examining the existence and uniqueness of the optimal reinsurance strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived for the compound Poisson risk model as well as for the Brownian motion risk model. From the numerical examples, we see that the optimal results for the compound Poisson risk model are very different from those for the diffusion model. The former depends not only on the safety loading, time, and the interest rate, but also on the claim size distributions and the claim number processes, while the latter depends only on the safety loading, time, and the interest rate.     

Portfolio selection using hierarchical Bayesian analysis and MCMC methods

This paper contributes to portfolio selection methodology using a Bayesian forecast of the distribution of returns by stochastic approximation. New hierarchical priors on the mean vector and covariance matrix of returns are derived and implemented. Comparison’s between this approach and other Bayesian methods are studied with simulations on 25 years of historical data on global stock indices. It is demonstrated that a fully hierarchical Bayes procedure produces promising results warranting more study. We carried out a numerical optimization procedure to maximize expected utility using the MCMC (Monte Carlo Markov Chain) samples from the posterior predictive distribution. This model resulted in an extra 1.5 percentage points per year in additional portfolio performance (on top of the Hierarchical Bayes model to estimate μ and Σ and use the Markowitz model), which is quite a significant empirical result. This approach applies to a large class of utility functions and models for market returns.     

Dynamic Portfolio Choice with Parameter Uncertainty and the Economic Value of Analysts' Recommendations

We derive a closed-form solution for the optimal portfolio ofa nonmyopic utility maximizer who has incomplete informationabout the alphas or abnormal returns of risky securities. Weshow that the hedging component induced by learning about theexpected return can be a substantial part of the demand. Usingour methodology, we perform an "ex ante" empirical exercise,which shows that the utility gains resulting from optimal allocationare substantial in general, especially for long horizons, andan "ex post" empirical exercise, which shows that analysts’recommendations are not very useful. (JEL C61, G11, G24)     

Accounting information,disclosure, and expected utility: Do investors really abhor uncertainty?

Investors are said to “abhor uncertainty,” but if there were no uncertainty they could earn only the risk‐free rate. A fundamental result in the analytical accounting literature shows that investors buying into a CARA‐normal CAPM market pay lower asset prices, gain higher ex‐ante expected returns, and obtain higher expected utility, when the market payoff has higher variance. New investors obtain similar “welfare” gains from risk under a log/power utility CAPM. These results do not imply that investors “abhor information.” To realize investors' ex‐ante expectations, the subjective probability distributions representing market expectations must be accurate. Greater payoff risk can add to investors' expected utility, but higher ex‐post(realized) utility comes from better information and more accurate ex‐ante expectations. An important implication for accounting is that greater disclosure can have the simultaneous effects of (i) exposing more fully or perceptibly firms' payoff uncertainty, thereby increasing new investors' expected utility, and (ii) improving market estimates of firms' payoff parameters (means, variances, covariances), thereby giving investors a better chance of realizing their expectations. Paradoxically, better information can be valuable to new investors by exposing more fully and more accurately the risk in firms' business operations and results–new investors maximizing expected utility want both more risk and better information.