Optimal portfolios under worst-case scenarios

In standard portfolio theories such as Mean–Variance optimization, expected utility theory, rank dependent utility heory, Yaari’s dual theory and cumulative prospect theory, the worst outcomes for optimal strategies occur when the market declines (e.g. during crises), which is at odds with the needs of many investors. Hence, we depart from the traditional settings and study optimal strategies for investors who impose additional constraints on their final wealth in the states corresponding to a stressed financial market. We provide a framework that maintains the stylized features of the SP/A theory while dealing with the goal of security in a more flexible way. Preferences become state-dependent, and we assess the impact of these preferences on trading decisions. We construct optimal strategies explicitly and show how they outperform traditional diversified strategies under worst-case scenarios.     

The implied Sharpe ratio

In an incomplete market, including liquidly traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky assets, there is no such measure available to help investors choose among the different European options. We introduce a new concept—the implied Sharpe ratio—which allows investors to make such a comparison in an incomplete financial market. Specifically, when comparing various European options, it is the option with the highest implied Sharpe ratio that, if included in an investor's portfolio, will improve his expected utility the most. Through the method of Taylor series expansion of the state-dependent coefficients in a nonlinear partial differential equation, we also establish the behaviour of the implied Sharpe ratio with respect to an investor's risk-aversion parameter. In a series of numerical studies, we compare the investment attractiveness of different European options by studying their implied Sharpe ratio.     

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.     

Risk aversion with state-dependent preferences in the rank-dependent expected utility theory

We extend the analysis of risk aversion with state-dependent preferences to the rank-dependent expected utility theory. We find that in this extended theory, for two preference relations to be comparable in risk aversion not only do their reference sets need to coincide (a condition first introduced by Karni [1983, 1985] in the original expected utility framework), but they must also rank the prospective state-dependent outcomes in the same manner. We formalize this additional condition by introducing the concept of certainty sets. Under our condition of comparability, various results and characterizations of interpersonal comparison of risk aversion are obtained. The implications for a specific insurance problem are also discussed.     

The Optimal Log-Utility Asset Management under Incomplete Information

In this paper, we propose a theory for deriving the optimal portfolio that assures the log-utility investors of maximizing their expected utility. Restricting investors' information at defined levels, we propose the sample path-wise optimal portfolio (SPOP), which is consistent with the back-test framework used in actualinvestment. It is proven that, at any finite terminal time, this SPOP is asymptotically optimal among all the portfolios which are predictable under investors' incompleteinformation. The optimality is guaranteed by the continuous Bayesian updating formula. Finally, we discuss an algorithm for searching the SPOP, based on asset prices at discrete time intervals.     

Microstructure and Ambiguity

A goal for stock exchanges is to increase participation by firms and investors. We show how specific features of the microstructure can reduce perceived ambiguity, and induce participation by both investors and issuers. We develop a model with sophisticated traders, who we view as expected utility maximizers with rational expectations, and unsophisticated traders, who we view as rational traders facing ambiguity about the payoffs to participating in the market. We show how designing markets to reduce ambiguity can benefit investors through greater liquidity, exchanges through greater volume, and issuing firms through a lower cost of capital.     

Noise Trading in Small Markets

Considering noise traders as agents with unpredictable beliefs, we show that in an imperfectly competitive market with risk averse investors, noise traders may earn higher expected utility than rational investors. This happens when, by deviating from the Nash equilibrium strategy, noise traders hurt rational investors more than themselves. It follows that the willingness of arbitrageurs to exploit noise traders' misperceptions is lower relative to a perfectly competitive economy. This result reinforces the theory that noise trading may explain closed-end fund discounts and small firms' returns, since these markets are less competitive than the market for large firms' stock.     

Disagreement,tastes, and asset prices

Standard asset pricing models assume that: (i) there is complete agreement among investors about probability distributions of future payoffs on assets; and (ii) investors choose asset holdings based solely on anticipated payoffs; that is, investment assets are not also consumption goods. Both assumptions are unrealistic. We provide a simple framework for studying how disagreement and tastes for assets as consumption goods can affect asset prices.     

The curious case of Canadian corporate emissions valuation

This study examines the relevance to investors of the greenhouse gas (GHG) emissions of publicly-traded Canadian firms over 2006–2018. Based on two independent datasets, we document that firm value varies positively in the level of emissions. This result suggests that the Canadian setting differs from those studied previously, notably because of low climate litigation risk and national and subnational expenditure policies to offset climate impacts on the economy. While national and subnational expenditures to mitigate emissions affect firms' on-balance-sheet costs and profits, investors price the future payoffs to these expenditures into firm value. Supporting this view, we find that the positive relation between emissions and firm value in Canada is amplified for high GHG-intensity firms (mainly energy firms in Alberta), whose future payoffs to environmental policies and spending exceed those of low GHG-intensity firms. Our results are consistent with investors’ recognition of the benefits to firm value of national and subnational policies to decarbonize the Canadian economy.     

Infrequent Extreme Risks

The main tools and concepts of financial and actuarial theory are designed to handle standard, or even small risks. The aim of this paper is to reconsider some selected financial problems, in a setup including infrequent extreme risks. We first consider investors maximizing the expected utility function of their future wealth, and we establish the necessary and sufficient conditions on the utility function to ensure the existence of a non degenerate demand for assets with extreme risks. This new class of utility functions, called LIRA, does not contain the classical HARA and CARA utility functions, which are not adequate in this framework. Then we discuss the corresponding asset supply-demand equilibrium model.     

Infrequent Extreme Risks

The main tools and concepts of financial and actuarial theory are designed to handle standard, or even small risks. The aim of this paper is to reconsider some selected financial problems, in a setup including infrequent extreme risks. We first consider investors maximizing the expected utility function of their future wealth, and we establish the necessary and sufficient conditions on the utility function to ensure the existence of a non degenerate demand for assets with extreme risks. This new class of utility functions, called LIRA, does not contain the classical HARA and CARA utility functions, which are not adequate in this framework. Then we discuss the corresponding asset supply-demand equilibrium model.     

Variable Selection for Portfolio Choice

We study asset allocation when the conditional moments of returns are partly predictable. Rather than first model the return distribution and subsequently characterize the portfolio choice, we determine directly the dependence of the optimal portfolio weights on the predictive variables. We combine the predictors into a single index that best captures time variations in investment opportunities. This index helps investors determine which economic variables they should track and, more importantly, in what combination. We consider investors with both expected utility (mean variance and CRRA) and nonexpected utility (ambiguity aversion and prospect theory) objectives and characterize their market timing, horizon effects, and hedging demands.     

Two Paradigms and Nobel Prizes in Economics: a Contradiction or Coexistence?

Abstract Markowitz and Sharpe won the Nobel Prize in Economics for the development of Mean‐Variance (M‐V) analysis and the Capital Asset Pricing Model (CAPM). Kahneman won the Nobel Prize in Economics for the development of Prospect Theory. In deriving the CAPM, Sharpe, Lintner and Mossin assume expected utility (EU) maximisation in the face of risk aversion. Kahneman and Tversky suggest Prospect Theory (PT) as an alternative paradigm to EU theory. They show that investors distort probabilities, make decisions based on change of wealth, exhibit loss aversion and maximise the expectation of an S‐shaped value function, which contains a risk‐seeking segment. Can these two apparently contradictory paradigms coexist? We show in this paper that although CPT (and PT) is in conflict to EUT, and violates some of the CAPM's underlying assumptions, the Security Market Line Theorem (SMLT) of the CAPM is intact in the CPT framework. Therefore, the CAPM is intact also in CPT framework.     

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.     

Equilibrium with investors using a diversity of deviation measures

It has been argued that investors who optimize their portfolios with attention paid only to mean and standard deviation will all end up choosing some multiple of a certain master fund portfolio. Justification for the capital asset pricing model of classical portfolio theory, which relates individual assets to such a master fund, has come from this direction in particular. Attempts have been made to provide solid mathematical support by showing that the imputed behavior of investors is a consequence of price equilibrium in a market in which assets are traded subject to budget constraints, and optimization is carried out with respect to utility functions that depend only on mean and standard deviation.     

A decision theoretic formulation of insurance risk theory

Abstract

In a number of papers Borch has shown how certain insurance problems can be formulated using the concept of utility. (See Borch [3], [4], [5], [6], [7] and [8].) Borch's work is used as a building block in Part I of this report, which presents a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory. Part I makes use of the concepts of utility and subjective probability. It is admitted that these concepts are more commonly associated with individuals rather than groups of individuals such as insurance companies. However, in this report, we will refer to an insurance company as an individual (albeit a neuter one) and assume that it can quantify its preferences for consequences and its opinions about the occurrence of events. Further, we assume that a company “behaves” according to certain rules of consistent behavior which imply that when presented with several risky courses of action, the company will take the action which has the greatest expected utility. Formal treatments of assumptions that lead to this mode of behavior can be found in Savage [17] and Pratt, Raiffa, and Schlaifer [15].     

Utilitarianism and fairness in portfolio positioning

The paper introduces the theory of optimal positioning of financial products. It is illustrated in the context of long-term intertemporal portfolio allocation and can be applied for example to asset allocation funds. We embed this problem in location theory: the portfolio is optimized within the investors’risk aversion dimension. For the CRRA utility functions, we compute explicitly the distance functions. For the first (utilitarian criterion), the average utility of the investors is maximized. For the second one (fairness criterion), the choice of portfolio is optimized so that the average monetary loss due to the lack of customization is minimized. Given the distribution of investors’ risk aversion, we provide a solution method and an algorithm to optimally position standardized portfolio along one of these two criteria.     

Speculative Betas

The risk and return trade‐off, the cornerstone of modern asset pricing theory, is often of the wrong sign. Our explanation is that high‐beta assets are prone to speculative overpricing. When investors disagree about the stock market's prospects, high‐beta assets are more sensitive to this aggregate disagreement, experience greater divergence of opinion about their payoffs, and are overpriced due to short‐sales constraints. When aggregate disagreement is low, the Security Market Line is upward‐sloping due to risk‐sharing. When it is high, expected returns can actually decrease with beta. We confirm our theory using a measure of disagreement about stock market earnings.     

Discount Rates,Debt Maturity,and the Fiscal Theory

This paper examines how the transmission of government portfolio risk arising from maturity operations depends on the stance of monetary/fiscal policy. Accounting for risk premia in the fiscal theory allows the government portfolio to affect expected inflation, even in a frictionless economy. The effects of maturity rebalancing on expected inflation in the fiscal theory depend directly on the conditional nominal term premium, giving rise to an optimal debt-maturity policy that is state-dependent. In a calibrated macrofinance model, we demonstrate that maturity operations have sizable effects on expected inflation and output through our novel risk transmission mechanism.