Vijay S. Bawa

Optimal rules for ordering uncertain prospects

Abstract 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.

Capital market equilibrium in a mean-lower partial moment framework

In this paper, we develop a Capital Asset Pricing Model (CAPM) using a mean-lower partial moment framework. We explicitly derive formulae for the equilibrium values of risky assets that hold for arbitrary probability distributions. We show that when the probability distributions and portfolio returns are either normal, stable (with the same characteristic exponent between 1 and 2 and the same skewness parameter, not necessarily zero), or Student-t distributions, our CAPM reduces to the traditional mean-scale CAPMs. Consequently, since the traditional equilibrium models are special cases of our model, the mean-lower partial moment framework is guaranteed to do at least as well in explaining market data. As an application of our theory, we derive an acceptance criterion for capital investment projects and note that corporate finance theory results developed, for example, in the well-known mean- variance framework carry over to the mean-lower partial moment framework.

The effect of estimation risk on optimal portfolio choice

Abstract 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.

Safety-First, Stochastic Dominance, and Optimal Portfolio Choice

Stochastic Dominance rules are playing an increasingly prominent role in the literature on choice under uncertainty. Their foundation is the mainstream VonNeumann-Morgenstern expected utility paradigm. Their essence is to provide an admissible set of choices under restrictions on the utility functions that follow from prevalent and appealing modes of economic behavior: The admissible sets generated are useful for a large group of individual decision makers and the optimal choice for an individual can then be obtained from among the smaller set of admissible choices.

Portfolio choice and equilibrium in capital markets with safety-first investors

Abstract This paper develops optimal portfolio choice and market equilibrium when investors behave according to a generalized lexicographic safety-first rule. We show that the mutual fund separation property holds for the optimal portfolio choice of a risk-averse safety-first investor. We also derive an explicit valuation formula for the equilibrium value of assets. The valuation formula reduces to the well-known two-parameter capital asset pricing model (CAPM) when investors approximate the tail of the portfolio distribution using Tchebychevs inequality or when the assets have normal or stable Paretian distributions. This shows the robustness of the CAPM to safety-first investors under traditional distributional assumptions. In addition, we indicate how additional information about the portfolio distribution can be incorporated to the safety-first valuation formula to obtain alternative empirically testable models.

The effect of limited information and estimation risk on optimal portfolio diversification

Abstract This paper analyzes the optimal portfolio choice problem when security returns have a joint multivariate normal distribution with unknown parameters. For the case of limited, but sufficient (sample plus prior) information, we show that for a general family of conjugate priors, the optimal portfolio choice is obtained by the use of a mean-variance analysis that differs from traditional mean-variance analysis due to estimation risk. We also consider two illustrative cases of insufficient sample information and minimal prior information and show that in these cases it is asymptotically optimal for an investor to limit diversification to a subset of the securities. These theoretical results corroborate observed investor behavior in capital markets.

Abstract: Capital Market Equilibrium in a Mean-Lower Partial Moment Framework

In this paper, we develop a Capital Asset Pricing Model (CAPM) using a mean-lower partial moment framework. We explicitly derive the valuation formulas for the equilibrium value of risky assets and provide a distribution-free testable hypothesis for empirical validation of the new CAPM. We show the invariance of our results to the problem of estimation risk. We also show that when the probability distribution of security rer turns is the normal distribution, the stable Paretian distribution (with the same characteristic exponent between 1 and 2 and the same skewness parameter (not necessarily zero)), or the multivariate t-distribution, our CAPM reduces to the traditional two-parameter CAPM.

Abstract: The Effect of Limited Information and Estimation Risk on Optimal Portfolio Diversification

This paper analyzes the effect of limited information and estimation risk on optimal portfolio choice when the joint probability distribution of security returns is multivariate normal and the underlying parameters (means and variance-covariance matrix) are unknown. We first consider the case of limited, but sufficient information (the number of observations per security exceeds the number of securities or the prior distribution of the underlying parameters is “sufficiently†informative). We show that for a general family of conjugate priors, the admissible set of portfolios, taking estimation risk into account, may be obtained by the traditional mean-variance analysis. As a result of estimation risk the optimal portfolio choice differs from that obtained by traditional analysis.

Abstract–The Effect of Estimation Risk on Optimal Portfolio Choice under Uncertainty

Choice under uncertainty may be viewed as choice between alternative probability distributions of returns. Under Von Neumann and Morgensterns assumptions, an individuals optimal choice is a distribution that maximizes the expected utility of returns. In the theoretical analysis, the distribution functions are assumed to be known However, in most realistic cases, the distributions of returns are unknown and are estimated using available economic data. The traditional mode of analysis is to neglect the estimation risk and use the estimated distribution (in lieu of the true distribution) in determining the optimal choice under uncertainty. In this paper, we consider the portfolio choice problem and determine the effect of estimation risk on an individuals optimal choice under uncertainty.