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.     

Tractable hedging with additional hedge instruments

In an uncertain volatility model where only the stock and the money market account are traded, the upper price bound of a European claim is given by the solution of a Black-Scholes-Barenblatt equation. If an additional hedge instrument is available, the price bound can be tightened. This is also true if the set of admissible strategies is restricted to tractable strategies, which are defined as sums of Black-Scholes strategies. We study the structure of both strategies, the general strategies and the tractable strategies, when an additional convex instrument is available. For a call and a bullish vertical spread, we give closed-form solutions for the optimal tractable hedge when the additional instrument is a call option. We show that the position in the additional convex claim as well as the reduction in the price bounds allow to capture the amount of convexity risk a claim is exposed to.     

Robust portfolio optimization with a generalized expected utility model under ambiguity

This paper proposes a robust approach maximizing worst-case utility when both the distributions underlying the uncertain vector of returns are exactly unknown and the estimates of the structure of returns are unreliable. We introduce concave convex utility function measuring the utility of investors under model uncertainty and uncertainty structure describing the moments of returns and all possible distributions and show that the robust portfolio optimization problem corresponding to the uncertainty structure can be reformulated as a parametric quadratic programming problem, enabling to obtain explicit formula solutions, an efficient frontier and equilibrium price system. We would like to thank Prof. Zengjing Chen from School of Mathematics and System Sciences, Shandong University for helpful suggestions, and to thank the anonymous referee for valuable comments.     

The generalized value at risk admissible set: constraint consistency and portfolio outcomes

Generalized value at risk (GVaR) adds a conditional value at risk or censored mean lower bound to the standard value at risk and considers portfolio optimization problems in the presence of both constraints. For normal distributions the censored mean is synonymous with the statistical hazard function, but this is not true for fat-tailed distributions. The latter turn out to imply much tighter bounds for the admissible portfolio set and indeed for the logistic, an upper bound for the portfolio variance that yields a simple portfolio choice rule. The choice theory in GVaR is in general not consistent with classic Von Neumann–Morgenstern utility functions for money. A re-specification is suggested to make it so that gives a clearer picture of the economic role of the respective constraints. This can be used analytically to explore the choice of portfolio hedges.     

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.     

Inflation targeting and nominal-income-growth targeting: When and why are they suboptimal?

We compare optimal and simple interest-rate rules. Our model features optimizing agents, monopolistic competition in both product and labor markets, and one-period nominal contracts (for wages alone or for both wages and prices) signed before shocks are known. Exact solutions ensure that we obtain correct welfare rankings. Optimal rules maximize the unconditional expected utility of the representative agent with commitment subject to the information set of the policymaker. Even with monopolistic distortions, the optimal full-information rule makes the economy mimic the hypothetical full-flexibility equilibrium. Strict versions of inflation targeting, nominal-income-growth targeting, and other such simple rules are suboptimal under both full and partial information but flexible versions are optimal under certain partial-information assumptions. Nominal-income-growth targeting dominates inflation targeting for plausible parameter values.     

Portfolio Performance Measurement: a No Arbitrage Bounds Approach

This paper presents a new method to examine the performance evaluation of mutual funds in incomplete markets. Based on the no arbitrage condition, we develop bounds on admissible performance measures. We suggest new ways of ranking mutual funds and provide a diagnostic instrument for evaluating the admissibility of candidate performance measures. Using a monthly sample of 320 equity funds, we show that admissible performance values can vary widely, supporting the casual observation that investors disagree on the evaluation of mutual funds. In particular, we cannot rule out that more than 80% of the mutual funds are given positive values by some investors. Moreover, we empirically demonstrate that potential inference errors embedded in existing parametric performance measures can be of important magnitude.     

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.     

Risk aversion under preference uncertainty

We show that if an agent is uncertain about the precise form of his utility function, his actual relative risk aversion may depend on wealth even if he knows his utility function lies in the class of constant relative risk aversion (CRRA) utility functions. We illustrate the consequences of this result for optimal asset allocation: poor agents that are uncertain about their risk aversion parameter invest less in risky assets than wealthy investors with identical risk aversion uncertainty.     

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

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 Tchebychev's 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.     

Multivariate utility maximization with proportional transaction costs

We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor’s preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for this include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer.     

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.     

On optimal stratifications for multivariate distributions

Summary

In the present paper we study the problem of optimal stratifications for estimating the mean vector y of a given multivariate distribution F(x) with covariance matrix ζ both in cases of proportionate and of optimal (or generalized Neyman) allocations. It is noted that an “optimal stratification” is meant for one to make the covariance matrix of an unbiased estimator X for μ minimal, in the sense of semi-order defined below, in the symmetric matrix space. We show the existence of an optimal stratification and the necessary conditions for a stratification to be optimal. Besides we prove that an optimal stratification can be represented by a “hyperplane stratification” or a “quadratic hypersurface stratification” according to the proportionate or optimal (or generalized Neyman) allocation, and that the set of all optimal (or admissible) stratifications is a minimal complete class in the analogous sense of decision theory. Further we discuss the optimal stratification when a criterion based on a suitable real-valued function is adopted instead of the semi-order.     

Stochastic dominance and portfolio analysis

The principle of stochastic dominance is used to characterize the optimal efficient sets when the distributions of the random prospects belong to a family. Most of the well-known distributions are considered. In each case, the optimal efficient sets are characterized by easily verifiable conditions on the parameters of the distributions. These optimal efficient sets are then compared with the corresponding mean-variance (MV) efficient set. It is often found that the optimal efficient sets are proper subsets of the MV efficient set. Thus, the MV criterion is a proper efficiency criterion, but the MV efficient set can be excessively large compared to the optimal efficient set.     

Optimal fiscal policy under multiple equilibria

We study optimal fiscal policy in an economy where (i) search frictions create a coordination problem and generate multiple, Pareto-ranked equilibria and (ii) the government finances the provision of a public good by taxing market activity. The government must choose the tax rate before it knows which equilibrium will obtain, and therefore an important part of the problem is determining how the policy will affect the equilibrium selection process. We show that when the equilibrium selection rule is based on the concept of risk dominance, higher tax rates make coordination on the Pareto-superior outcome less likely. As a result, taking equilibrium-selection effects into account leads to a lower optimal tax rate.     

Technical analysis: An asset allocation perspective on the use of moving averages

In this paper, we analyze the usefulness of technical analysis, specifically the widely employed moving average trading rule from an asset allocation perspective. We show that, when stock returns are predictable, technical analysis adds value to commonly used allocation rules that invest fixed proportions of wealth in stocks. When uncertainty exists about predictability, which is likely in practice, the fixed allocation rules combined with technical analysis can outperform the prior-dependent optimal learning rule when the prior is not too informative. Moreover, the technical trading rules are robust to model specification, and they tend to substantially outperform the model-based optimal trading strategies when the model governing the stock price is uncertain.     

Optimal design of derivatives in illiquid markets*

Abstract

The aim of this paper is to determine the optimal structure of derivatives written on an illiquid asset, such as a catastrophic or a weather event. This transaction involves two agents: a bank which wants to hedge its initial exposure towards this illiquid asset and an investor which may buy the contract. Both agents also have the opportunity to invest their residual wealth on a financial market.

Based on a utility maximization point of view, we determine an optimal profile (and its value) such that it maximizes the bank's utility given that the investor decides to make the deal only if it increases its utility. In the case of exponential utility, we show that the pricing rule is a non-linear function of the structure and that the bank always transfers the same proportion of its initial exposure. In the general case, an additional term appears, depending only on the relative log-likelihood of the two agents' views of the distribution of the illiquid asset.     

Portfolio optimization with a defaultable security

In this paper we derive a closed-form solution for a representative investor who optimally allocates her wealth among the following securities: a credit-risky asset, a default-free bank account, and a stock. Although the inclusion of a credit-related financial product in the portfolio selection is more realistic, no closed-form solutions to date are given in the literature when a recovery value is considered in the event of a default. While most authors have assumed some recovery scheme in their initial model set up, they do not address the portfolio problem with a recovery when a default actually occurs. Given the tractability of the recovery of market value, we solved the optimal portfolio problem for the representative investor whose utility function is a Constant Relative Risk Aversion utility function. We find that the investor will allocate larger fraction of wealth to the defaultable security as long as the default-event risk is priced. These results are very intuitive and reasonable since it indicates that if the default risk premium is not priced properly the investor purchases less defaultable securities.     

Dynamic preferences for popular investment strategies in pension funds

In this paper, we characterize dynamic investment strategies that are consistent with the expected utility setting and more generally with the forward utility setting. Two popular dynamic strategies in the pension funds industry are used to illustrate our results: a constant proportion portfolio insurance (CPPI) strategy and a life-cycle strategy. For the CPPI strategy, we are able to infer preferences of the pension fund’s manager from her investment strategy, and to exhibit the specific expected utility maximization that makes this strategy optimal at any given time horizon. In the Black–Scholes market with deterministic parameters, we are able to show that traditional life-cycle funds are not optimal to any expected utility maximizers. We also prove that a CPPI strategy is optimal for a fund manager with HARA utility function, while an investor with a SAHARA utility function will choose a time-decreasing allocation to risky assets in the same spirit as the life-cycle funds strategy. Finally, we suggest how to modify these strategies if the financial market follows a more general diffusion process than in the Black–Scholes market.