Wealth-path dependent utility maximization in incomplete markets

Motivated by an optimal investment problem under time horizon uncertainty and when default may occur, we study a general structure for an incomplete semimartingale model extending the classical terminal wealth utility maximization problem. This modelling leads to the formulation of a wealth-path dependent utility maximization problem. Our main result is an extension of the well-known dual formulation to this context. In contrast with the usual duality approach, we work directly on the primal problem. Sufficient conditions for characterizing the optimal solution are also provided in the case of complete markets, and are illustrated by examples.Received: December 2003, Mathematics Subject Classification (2000): 91B28, 91B16, 49N15, 49N30JEL Classification: G11The authors would like to thank the anonymous referees for their remarks and suggestions which greatly improved this paper. We also thank participants at the Oberwolfach workshop in 2003 for comments and discussions.     

Robust utility maximization for complete and incomplete market models

We investigate the problem of maximizing the robust utility functional . We give the dual characterization for its solution for both a complete and an incomplete market model. To this end, we introduce the new notion of reverse f-projections and use techniques developed for f-divergences. This is a suitable tool to reduce the robust problem to the classical problem of utility maximization under a certain measure: the reverse f-projection. Furthermore, we give the dual characterization for a closely related problem, the minimization of expenditures given a minimum level of expected utility in a robust setting and for an incomplete market.Received: September 2004, Mathematics Subject Classification (2000): 62C20, 62O05, 91B16, 91B28JEL Classification: D81, G11I thank Hans Föllmer for his help when writing this paper. Furthermore, I thank Alexander Schied for discussing the topic with me and Michael Kupper and the referees for their helpful remarks.     

Derived factors in event studies

We examine the utility of the statistical factor model of the process generating stock returns in the context of event studies. For a variety of estimation procedures and experimental designs we find limited value added relative to the use of a simple market model. We would attribute this finding to misspecification of the statistical factor analysis model, and suspect that there exist more robust procedures for estimating the factor structure of stock returns.     

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.     

CAPM and Option Pricing With Elliptically Contoured Distributions

This article offers an alternative proof of the capital asset pricing model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modeling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modeling the thickness of tails associated with the possibility that asset returns take extreme values with nonnegligible probabilities. As summarized in this article, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Stein's lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is log‐elliptically distributed. The Black–Scholes call option price is a special case of this general result when the underlying is log‐normally distributed.     

Index tracking with utility enhanced weighting

Passive index investing involves investing in a fund that replicates a market index. Enhanced indexation uses the returns of an index as a reference point and aims at outperforming this index. The motivation behind enhanced indexing is that the indices and portfolios available to academics and practitioners for asset pricing and benchmarking are generally inefficient and, thus, susceptible to enhancement. In this paper we propose a novel technique based on the concept of cumulative utility area ratios and the Analytic Hierarchy Process (AHP) to construct enhanced indices from the DJIA and S&P500. Four main conclusions are forthcoming. First, the technique, called the utility enhanced tracking technique (UETT), is computationally parsimonious and applicable for all return distributions. Second, if desired, cardinality constraints are simple and computationally parsimonious. Third, the technique requires only infrequent rebalancing, monthly at the most. Finally, the UETT portfolios generate consistently higher out-of-sample utility profiles and after-cost returns for the fully enhanced portfolios as well as for the enhanced portfolios adjusted for cardinality constraints. These results are robust to varying market conditions and a range of utility functions.     

Investor attention in cryptocurrency markets

We examine the relationship between investor attention, and measures of uncertainty, with the market dynamics of Bitcoin and other cryptocurrencies. We find that increases in investor attention are associated with higher returns, more volatility, and greater illiquidity in cryptocurrency markets. In contrast, cryptocurrency uncertainty (UCRY) and financial market uncertainty (VIX) are also positively related to volatility and illiquidity but have a negative contemporaneous relationship with returns. The identified relationships are accentuated during the COVID-pandemic, and are robust to different measures of investor attention, volatility, and illiquidity. Our results suggest that monitoring investor attention could assist both investors and policymakers.     

Oil market uncertainty and excess returns on currency carry trade

This paper examines the role of oil market uncertainty on currency carry trade payoffs. We find that oil market uncertainty can impact currency carry trade excess returns. When oil market uncertainty rises, expected currency excess returns will increase. Our findings are robust to alternative measures of oil market uncertainty and, after controlling for traditional uncertainties, different types of oil shocks. The results also hold well in both developed and emerging markets, as well as for oil-related currencies, non-oil currencies, commodity currencies, and non-commodity currencies. Additionally, oil market uncertainty can be priced in the cross section of currency carry trade excess returns. This effect can be explained by investors becoming more risk averse under high oil market uncertainty and requiring greater compensation for bearing such risk. Moreover, our measure of oil market uncertainty, the downside risk from the oil market, is quite different from that of traditional aggregate measures.     

Portfolio diversification and value at risk under thick-tailedness†

This paper focuses on the study of portfolio diversification and value at risk analysis under heavy-tailedness. We use a notion of diversification based on majorization theory that will be explained in the text. The paper shows that the stylized fact that portfolio diversification is preferable is reversed for extremely heavy-tailed risks or returns. However, the stylized facts on diversification are robust to heavy-tailedness of risks or returns as long as their distributions are moderately heavy-tailed. Extensions of the results to the case of dependence, including convolutions of α-symmetric distributions and models with common shocks are provided.     

How do investors react under uncertainty?

It has long been accepted that risk plays an important role in determining valuation where risk reflects that investors are unsure of future returns but are able to express their prior expectations by a probability distribution of these returns. Knight (1921) introduced the concept of uncertainty where investors possess incomplete knowledge about this distribution and so are unable to formulate priors over all possible outcomes. One common approach for making uncertainty tractable is to assume that investors faced with uncertainty will base their decisions on the worst case scenario (i.e. follow maxmin expected utility). As a consequence it is postulated that investors will become more pessimistic as uncertainty increases, upgrading bad news and downgrading good news. Using Australian data, we find evidence that investors react to bad news at times of high market uncertainty but largely ignore good news which is consistent with them taking on a pessimistic bias. However, we also find evidence of the reverse when market uncertainty is low with investors taking on an optimistic stance by ignoring bad news but reacting to good news. We also find that the impact that market uncertainty has on the reaction of investors to new information is modified by the prevailing market sentiment at the time of the announcement. Besides throwing light on the question of how uncertainty impacts on investor behaviour, our findings seriously challenge the common assumption made that investors consistently deal with uncertainty by applying maxmin expected utility.     

Information uncertainty,investor sentiment,and analyst reports

This study investigates the relationships among information uncertainty, investor sentiment, analyst reports, and stock returns in a unified framework. The effects of analyst reports on stock returns depend on the degrees of information uncertainty, indicating that recommendation upgrades (downgrades) convey more valuable positive (negative) information under higher information uncertainty. Such stock market reactions are significantly explained by investor sentiment when information uncertainty is high. Our empirical findings are robust to changes in abnormal return measures and information uncertainty proxies.     

Bounds for Ratios of Posterior Expectations: Applications in the Collective Risk Model

This paper considers the collective risk model for the insurance claims process. We will adopt a Bayesian point of view, where uncertainty concerning the specification of the prior distribution is a common question. The robust Bayesian approach uses a class of prior distributions which model uncertainty about the prior, instead of a single distribution. Relatively little research has dealt with robustness with respect to ratios of posterior expectations as occurs with the Esscher and Variance premium principles. Appropriate techniques are developed in this paper to solve this problem using the k -contamination class in the collective risk model.     

Duality theory for robust utility maximisation

In this paper, we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real line. Our results are inspired by – and can be seen as the robust analogues of – the seminal work of Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999). Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.

     

Optimal portfolio positioning within generalized Johnson distributions

Many empirical studies have shown that financial asset returns do not always exhibit Gaussian distributions, for example hedge fund returns. The introduction of the family of Johnson distributions allows a better fit to empirical financial data. Additionally, this class can be extended to a quite general family of distributions by considering all possible regular transformations of the standard Gaussian distribution. In this framework, we consider the portfolio optimal positioning problem, which has been first addressed by Brennan and Solanki [J. Financial Quant. Anal., 1981, 16, 279–300], Leland [J. Finance, 1980, 35, 581–594] and further developed by Carr and Madan [Quant. Finance, 2001, 1, 9–37] and Prigent [Generalized option based portfolio insurance. Working Paper, THEMA, University of Cergy-Pontoise, 2006]. As a by-product, we introduce the notion of Johnson stochastic processes. We determine and analyse the optimal portfolio for log return having Johnson distributions. The solution is characterized for arbitrary utility functions and illustrated in particular for a CRRA utility. Our findings show how the profiles of financial structured products must be selected when taking account of non Gaussian log-returns.     

Estimating Private Equity Returns from Limited Partner Cash Flows

We introduce a methodology to estimate the historical time series of returns to investment in private equity funds. The approach requires only an unbalanced panel of cash contributions and distributions accruing to limited partners and is robust to sparse data. We decompose private equity returns from 1994 to 2015 into a component due to traded factors and a time‐varying private equity premium not spanned by publicly traded factors. We find cyclicality in private equity returns that differs according to fund type and is consistent with the conjecture that capital market segmentation contributes to private equity returns.     

Optimal investment of an insurer with regime-switching and risk constraint

We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly.     

Speculative trading and stock returns: A stochastic dominance analysis of the Chinese A-share market

The pricing of A-shares in China has long puzzled financial economists. This paper applies recent tests of stochastic dominance (SD) to examine whether differences in the return distributions of A- and B-shares in China are consistent with market efficiency. As SD is nonparametric, market efficiency can be examined without the joint test problem arising from misspecifications in the asset pricing benchmark. Our results show A-shares have second-order dominated B-shares from 1996 to 2005. This dominance was most significant during the market segmentation period, but has continued, albeit to a lesser extent even after the B-share market was opened to local investors in 2001. Our results are robust to using residual returns from an international asset pricing model instead of raw returns. We conclude that the superior performance of A-shares cannot be attributed to risk. The results are more likely due to a return bias caused by intense speculation among retail individuals under limited arbitrage.     

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.     

Pricing assets with higher moments: Evidence from the Australian and us stock markets

This paper investigates the importance of higher moments of return distributions in capturing the variation of average stock returns for companies listed in the leading S&P US and Australian indices. We find that Australian stocks are more negatively skewed but less leptokurtic than US stocks. As a result, we find that co-skewness plays a more important role in explaining Australian returns while co-kurtosis is consistently influential for US stock returns. We postulate that the differences in results are related to the underlying firm characteristics of the companies in the two indices, where principally the Australian firms are noticeably smaller than their US counterparts and concentrated in a smaller number industry sectors. This implies that for many smaller exchanges around the world higher moment characteristics displayed by the US market may not be applicable. We also show our results are robust to partly explaining average stock returns in the presence of size, value, and momentum effects.     

Policy uncertainty and seasoned equity offerings methods

Based on a sample of U.S. seasoned equity offering (SEO) during the period 2002–2017, we examine how the choice of equity issuance method changes in response to policy uncertainty. We find that firms subject to high policy uncertainty are less likely to use accelerated offerings rather than other types of traditional seasoned equity offerings. Our results are robust to alternative variable specifications, propensity score matching method, IV approach, and the inclusion of additional controls. Also, the effect of policy uncertainty on accelerated offering decision is weaker for firms with better information environment, earnings quality, and governance structures. Further, policy uncertainty increases the cost of funds and lowers long-run abnormal returns after SEOs for firms subject to high levels of policy uncertainty.