Variance Vulnerability, Background Risks, and Mean-Variance Preferences

An agent with two-parameter, mean-variance preferences is called variance vulnerable if an increase in the variance of an exogenous, independent background risk induces the agent to choose a lower level of risky activities. Variance vulnerability resembles the notion of risk vulnerability in the expected utility (EU) framework. First, we characterize variance vulnerability in terms of two-parameter utility functions. Second, we identify the multivariate normal as the only distribution such that EU- and two-parameter approach are compatible when independent background risks prevail. Third, presupposing normality, we show that—analogously to risk vulnerability—temperance is a necessary, and standardness and convex risk aversion are sufficient conditions for variance vulnerability.     

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.     

The Spillover effect of compulsory insurance

The assumption usually made in the insurance literature that risks are always insurable at the desired level does not hold in the real world: some risks are not—or are only partially—insurable, while others, such as civil liability or health and workers' injuries, must be fully insured or at least covered for a specific amount. We examine in this paper conditions under which a reduction in the constrained level of insurance for one risk increases the demand of insurance for another independent risk. We show that it is necessary to sign the fourth derivative of the utility function to obtain an unambiguous spillover effect. Three different sufficient conditions are derived if the expected value of the exogenous risk is zero. The first condition is that risk aversion be standard—that is, that absolute risk aversion and absolute prudence be decreasing. The second condition is that absolute risk aversion be decreasing and convex. The third condition is that both the third and the fourth derivatives of the utility function be negative. If the expected value of the exogenous risk is positive, a wealth effect is added to the picture, which goes in the opposite direction if absolute risk aversion is decreasing.     

Risk aversion,prudence, and compensation

In a standard principal-agent setting, we use a comparative approach to study the incentives provided by different types of compensation contracts, and their valuation by managers with utility function u who are risk averse (u″<0) and prudent (u″′>0). We show that concave contracts tend to provide more incentives to risk averse managers, while convex contracts tend to be more valued by prudent managers. This is because concave contracts concentrate incentives where the marginal utility of risk averse managers is highest, while convex contracts protect against downside risk. Thus, managerial prudence can contribute to explain the prevalence of stock-options in executive compensation. However, convex contracts are not optimal when the principal is sufficiently prudent relative to the manager.     

The independence axiom and asset returns

This paper integrates models of atemporal risk preference that relax the independence axiom into a recursive intertemporal asset-pricing framework. The resulting models are amenable to empirical analysis using market data and standard Euler equation methods. We are thereby able to provide the first nonlaboratory-based evidence regarding the usefulness of several new theories of risk preference for addressing standard problems in dynamic economics. Using both stock and bond returns data, we find that a model incorporating risk preferences that exhibit first-order risk aversion accounts for significantly more of the mean and autocorrelation properties of the data than models that exhibit only second-order risk aversion. Unlike the latter class of models which require parameter estimates that are outside of the admissible parameter space, e.g., negative rates of time preference, the model with first-order risk aversion generates point estimates that are economically meaningful. We also examine the relationship between first-order risk aversion and models that employ exogenous stochastic switching processes for consumption growth.     

Diffidence Theorem, State-Dependent Preferences, and DARA

C. Gollier (The Economics of Risk and Time. Cambridge: MIT Press, 2001) has developed a standard technique based on the diffidence theorem. This theorem provides a very simple instrument to solve relatively sophisticated problems when preferences are state-independent. The object of this article is to show that the theorem is also very useful to derive significant results with state-dependent preferences. Using the reference set notion and an extension of the diffidence theorem, we establish formally necessary and sufficient conditions on the reference set, in order to obtain prudence and decreasing absolute risk aversion. Examples of DARA utility functions compatible with non-linear reference sets are presented in the Appendix.     

Commodity futures hedging,risk aversion and the hedging horizon

This paper examines the impact of management preferences on optimal futures hedging strategy and associated performance. Applying an expected utility hedging objective, the optimal futures hedge ratio is determined for a range of preferences on risk aversion, hedging horizon and expected returns. Empirical results reveal substantial hedge ratio variation across distinct management preferences and are supportive of the hedging policies of real firms. Hedging performance is further shown to be strongly dependent on underlying preferences. In particular, hedgers with high risk aversion and short horizon reduce hedge portfolio risk but achieve inferior utility in comparison to those with low aversion.     

CVP under Uncertainty and the Manager's Utility Function Revisited

This paper re-examines the model of Kim, Abdolmohammada, and Klein (KAK, 1996) in which owners of a firm delegate the production decision to a risk-averse manager. Conflict of interest between the owners and the manager emerges as the latter maximizes the expected utility of his/her own wealth rather than that of the firm's profits. This paper shows that the results of KAK on the expected contribution margin and the excess return on the risky asset are flawed. Furthermore, while KAK study the effects of delegating the production decision to the manager on the firm's optimal output based on the mean-variance analysis, this paper derives parallel results within utility functions exhibiting constant absolute risk aversion and any arbitrary probability distribution functions.     

Insurance demand in the presence of loss-dependent background risk

We analyze insurance demand when insurable losses come with an uninsurable zero-mean background risk that increases in the loss size. If the individual is risk vulnerable, loss-dependent background risk triggers a precautionary insurance motive and increases optimal insurance demand. Prudence alone is sufficient for insurance demand to increase in two cases: the case of fair insurance and the case where the smallest possible loss exceeds a certain threshold value (referred to as the large loss case). We derive conditions under which insurance demand increases or decreases in initial wealth. In the large loss case, prudence determines whether changes in the background risk lead to more insurance demand. We generalize this result to arbitrary loss distributions and find conditions based on decreasing third-degree Ross risk aversion, Arrow–Pratt risk aversion, and Arrow–Pratt temperance.     

Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting

A general class of fair valuations which are both market-consistent (mark-to-market for any hedgeable part of a claim) and actuarial (mark-to-model for any claim that is independent of financial market evolutions) was introduced in Dhaene et al. [Insurance: Mathematics & Economics, 76, 14–27 (2017)] in a single period framework. In particular, the authors considered mean-variance hedge-based (MVHB) valuations where fair valuations of insurance liabilities are expressed in terms of mean-variance hedges and actuarial valuations. In this paper, we generalize this MVHB approach to a multi-period dynamic investment setting. We show that the classes of fair valuations and MVHB valuations are equivalent in this generalized setting. We derive tractable formulas for the fair valuation of equity-linked contracts and show how the actuarial part of their MVHB valuation decomposes into a diversifiable and a non-diversifiable component.     

International exchange risk and asset substitutability

This paper estimates subsitutability/complementarity relations among financial assets denominated in foreign currencies. Utilizing a representative investor and a flexible functional form methodology, a mean-variance utility function was estimated and used to determine expected return and variance elasticities between assets in the world portfolio. The hypothesis that international assets are perfect substitutes was rejected. It was also found that relative changes in variance tended to have a bigger impact on asset demand than did relative changes in expected returns. Substituability/complementarity relationships were not strong except in specific cases where strong relationships were expected a priori.     

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.     

Multivariate risk aversion and uninsurable risks: Theory and applications

The objective of this paper is to develop conditions for global multivariate comparative risk aversion in the presence of uninsurable, or background, risks, and thus generalize Kihlstrom and Mirman [1974] and Karni [1979,1989]. We analyze von Neumann-Morgenstern (VNM) utility functionsas well as smooth preference functionals which are nonlinear in distribution but locally linear in probabilities. In each case we provide an economic application which illustrates how our theorems can be used. We analyze a risk sharing, a portfolio choice, and a labor supply problem for VNM utility functions, and the optimal allocation of effort to risky technologies in the presence of a random supply (or quality) of a public good for nonlinear preference functionals. We consider thecase where the random variables are mean-independent as well as the case where they are independent. In the labor supply application for VNM utility functions, we show that if the two risks are independent, the comparative statics effect of greater risk aversion on labor supply in the presence of a background non-wage income risk is determined by a monotonic relationship between labor supply and the wage rate under certainty. That is, we extend the applicability of the Diamond-Stiglitz [1974]-Kihlstrom-Mirman [1974]single-crossing property to the case where an independent background risk is present.     

Loss aversion and the price of risk

This paper derives a simple theoretical relationship between the degree of loss aversion, the concavity/convexity of the value function, and the equilibrium market price of risk. We show that while the degree of loss aversion is key in determining the market price of risk, the convexity/concavity of the value function is much less important in this respect. The theoretical relationship obtained is tested empirically by using international data from 16 different countries during over 100 years, as documented by Dimson et al. [Triumph of the Optimists: 101 Years of Global Investment Returns, 2002 (Princeton University Press)]. The empirical data yield an estimate of λ=2.3 for the loss aversion index. This value is in striking agreement with estimates obtained in the very different methodology of laboratory experiments of individual decision-making.     

Mean-variance vs. stochastic dominance some empirical findings on efficient sets

The study attempts to shed additional light on the issue of the costs and benefits of using the mean-variance criterion as opposed to stochastic dominance criteria for investment decisions. Relevant probabilities which facilitate measurement of these costs and benefits are identified. The mean-variance criterion is shown to be useful to some extent in identifying potentially optimal portfolios. However, it is shown that the informationally less demanding mean-variance criterion admits two types of errors: (i) including portfolios that no expected utility maximizing risk averters would choose, and (ii) excluding portfolios which some risk averters would find optimal. The empirical investigation also indicates that although the composition of the efficient sets appears to be unstable over time, the relationships between the efficient sets are persistent over time.     

Mean-variance versus utility maximization revisited: The case of constant relative risk aversion

We examine if mean-variance (M-V) is a good proxy for portfolios based on the Constant Relative Risk Aversion (CRRA) utility function. M-V portfolios are considered good proxies for portfolios from several utility functions which is why they are routinely used in the portfolio theory literature as the benchmark. Our results clearly show this is not the case. Low risk CRRA portfolios are in many cases very different to M-V portfolios, especially with respect to downside risk. If a risk-free asset is available, in many cases, no M-V portfolio is an adequate proxy for CRRA portfolios. The implications of our findings are that: i) M-V portfolios are a poor proxy for investors with CRRA preferences, ii) CRRA portfolios are more suited to investors who care about downside risk than M-V portfolios, and iii) the efficacy of M-V to proxy for utility maximization should be examined more thoroughly.     

Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria

In this paper, an ambiguity-averse insurer (AAI) whose surplus process is approximated by a Brownian motion with drift, hopes to manage risk by both investing in a Black–Scholes financial market and transferring some risk to a reinsurer, but worries about uncertainty in model parameters. She chooses to find investment and reinsurance strategies that are robust with respect to this uncertainty, and to optimize her decisions in a mean-variance framework. By the stochastic dynamic programming approach, we derive closed-form expressions for a robust optimal benchmark strategy and its corresponding value function, in the sense of viscosity solutions, which allows us to find a mean-variance efficient strategy and the efficient frontier. Furthermore, economic implications are analyzed via numerical examples. In particular, our conclusion in the mean-variance framework differs qualitatively, for certain parameter ranges, with model-uncertainty robustness conclusions in the framework of utility functions: model uncertainty does not always result in an agent deciding to reduce risk exposure under mean-variance criteria, opposite to the conclusions for utility functions in Maenhout and Liu. Our conclusion can be interpreted as saying that the mean-variance problem for the AAI explains certain counter-intuitive investor behaviors, by which the attitude to risk exposure, for an AAI facing model uncertainty, depends on positive past experience.     

Disappointment aversion and the equity premium puzzle: new international evidence

Drawing upon the seminal study of Ang, Bekaert, and Liu [2005. “Why Stock May Disappoint?” Journal of Financial Economics 76 (3): 471–508], we incorporate disappointment aversion (DA, that is, aversion to outcomes that are worse than prior expectations) within a simple theoretical portfolio-choice model. Based on the results of this model, we then empirically address the portfolio allocation problem of an investor who chooses between a risky and a risk-free asset using international data from 19 countries. Our findings strongly support the view that DA leads investors to reduce their exposure to the stock market (i.e. DA significantly depresses the portfolio weights on equities in all cases considered). Overall, our study shows that in addition to risk aversion, DA plays an important role in explaining the equity premium puzzle around the world.     

Increasing return response to changes in risk

Risk aversion theory is based on an individual's choice among risky assets with expected utility in its foundation. It is about investor behavior (i.e., investor choice), under normal circumstances, toward assets with various levels of risk. A positive and marginally diminishing relationship between risk and return exists. This study is about investor behavior related to their response (not choice) to risk. We present an argument and supporting evidence that investors’ return response to risk is increasing with the level of risk. Thus, investor behavior is subject to change and the level of risk is a determinant of such change. We also explain the negative time‐series correlation between risk and return.     

The effects of risk‐taking tendency on risk choice and pre‐ and post‐decisional information selection

Ideally, people seek and select information about unfamiliar risks with which they are confronted, before they make a risk choice. This study investigated what happens when people do not have this opportunity. The main question was how risk‐taking tendency influences intuitive risk decisions and how this impacts subsequent information search and subsequent choices. In the present study, participants had to make a choice about an unfamiliar risk, either before or after they had had the opportunity to search for (risk‐promoting or risk‐averse) information. In the condition where they could only seek for information after they had made a choice, they had to reconsider their first choice and make a second risk choice. We expected that (1) risk‐taking tendency would impact people's risk choices, but only in the situation where they have little information. On the basis of cognitive dissonance theory, it was furthermore predicted that (2) risk‐taking tendency and (3) initial risk choice would affect risk information selection. Furthermore, we predicted that (4) the first risk choice and (5) the risk information selected would influence the subsequent risk choice. The results suggest that if people make a first, intuitive decision about an unknown risk, risk‐taking tendency has an effect on the choice, but that this does not happen when people can first select information. Risk‐taking tendency did not influence information selection, but initial choice did (although in another way than we expected). Furthermore, both the first risk choice and the risk information selected affected subsequent risk choices. These findings suggest that people often make initial intuitive decisions that are influenced by personality characteristics, and that are subsequently difficult to change.