Effects of background risks on cautiousness with an application to a portfolio choice problem

We provide necessary and sufficient conditions on an individual's expected utility function under which any zero-mean idiosyncratic risk increases cautiousness (the derivative of the reciprocal of the absolute risk aversion), which is the key determinant for this individual's demand for options and portfolio insurance.     

HETEROGENEOUS RISK ATTITUDES IN A CONTINUOUS‐TIME MODEL*

We prove that every continuous‐time model in which all consumers have time‐homogeneous and time‐additive utility functions and share a common probabilistic belief and a common discount rate can be reduced to a static model. This result allows us to extend some of the existing results of the representative consumer and risk‐sharing rules in static models to continuous‐time models. We show that the equilibrium interest rate is lower and more volatile than in the standard representative consumer economy, and that the individual consumption growth rates are more dispersed than in the absence of uncertainty.     

Risk aversion and allocation to long-term bonds

As risk aversion approaches infinity, the portfolio of an investor with utility over consumption at time T is shown to converge to the portfolio consisting entirely of a bond maturing at time T. Previous work on bond allocation requires a specific model for equities, the term structure, and the investor's utility function. In contrast, the only substantive assumption required for the analysis in this paper is that markets are complete. The result, which holds regardless of the underlying investment opportunities and the utility function, formalizes the “preferred habitat” intuition of Modigliani and Sutch (Amer. Econom. Rev. 56 (1966) 178).     

Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium

I analyze the optimal intertemporal portfolio problem of an investor who worries about model misspecification and insists on robust decision rules when facing a mean-reverting risk premium. The desire for robustness lowers the total equity share, but increases the proportion of the intertemporal hedging demand. I present a methodology for calculation of detection-error probabilities, which is based on Fourier inversion of the conditional characteristic functions of the Radon-Nikodym derivatives. The quantitative effect of robustness is more modest than in i.i.d. settings, because model discrimination between the benchmark and the worst-case alternative model is easier, as indicated by the detection-error probabilities.     

State-price densities under heterogeneous beliefs, the smile effect, and implied risk aversion

It has been widely noted in the empirical literature that state-price densities implicit in financial asset prices are not log-normal. This paper shows that this phenomenon can be caused by heterogeneity in investors’ beliefs. It derives the state-price density under heterogeneous beliefs in closed form and demonstrates that heterogeneous beliefs can give rise to multimodal state-price densities. Consequences for the “smile effect” in implied option volatility and for measures of risk aversion inferred from empirical state-price densities are discussed.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Modeling non-monotone risk aversion using SAHARA utility functions

We develop a new class of utility functions, SAHARA utility, with the distinguishing feature that it allows absolute risk aversion to be non-monotone and implements the assumption that agents may become less risk averse for very low values of wealth. The class contains the well-known exponential and power utility functions as limiting cases. We investigate the optimal investment problem under SAHARA utility and derive the optimal strategies in an explicit form using dual optimization methods. We also show how SAHARA utility functions extend the class of contingent claims that can be valued using indifference pricing in incomplete markets.     

Convex and decreasing absolute risk aversion is proper

Proper risk aversion, a pivotal concept in the study of behavioral conditions on utility functions, states that an undesirable risk can never be made desirable by the presence of an independent risk. It is well known that standard risk aversion is sufficient for this concept. We show in this short article that convex and decreasing absolute risk aversion is an alternative sufficient condition.     

An optimal auction with correlated values and risk aversion

We consider an auction setting where the buyers are risk averse with correlated private valuations (CARA preferences, binary types), and characterize the optimal mechanism for a risk-neutral seller. We show that the optimal auction extracts all buyer surplus whenever the correlation is sufficiently strong (greater than 1/3 in absolute value), no matter how risk averse the buyers are. In contrast, we note that a sufficiently risk-averse seller would not use a full rent extracting mechanism for any positive correlation of the valuations even if the buyers were risk neutral.     

Production and insurance under regret aversion

This paper examines the behavior of a regret-averse producer facing revenue risk. To insure against the revenue risk, the producer can purchase a coinsurance contract with an endogenously chosen coinsurance rate. Regret-averse preferences are characterized by a utility function that includes disutility from having chosen ex-post suboptimal alternatives. We show that the regret-averse producer never fully insures against the revenue risk even though the coinsurance contract is actuarially fair. When the producer is sufficiently regret averse and the loss probability is high, we further show that the regret-averse producer chooses not to purchase the actuarially fair coinsurance contract. Even when purchasing the actuarially fair coinsurance contract is optimal, we derive sufficient conditions under which the regret-averse producer reduces the optimal output level as compared to that without the coinsurance contract. These results are distinct from those under pure risk aversion, thereby making the consideration of regret aversion crucial.     

Housing and relative risk aversion

This paper derives closed-form and numerical solutions for relative risk aversion in a standard consumption-based model enriched with housing. The presence of housing enables the household to hedge against unexpected shocks and may decrease relative risk aversion. In addition, housing may generate state-dependent, time-varying risk aversion.     

The dynamics of efficient asset trading with heterogeneous beliefs

This paper analyzes the dynamic properties of portfolios that sustain dynamically complete markets equilibrium when agents have heterogeneous priors. We argue that the conventional wisdom that belief heterogeneity generates continuous trade and significant fluctuations in individual portfolios may be correct but it needs some qualifications. We consider an infinite horizon stochastic endowment economy populated by many Bayesian agents with heterogeneous priors over the stochastic process of the states of nature. Our approach hinges on studying the portfolios that decentralize Pareto optimal allocations. Since these allocations are typically history dependent, we propose a methodology to provide a complete recursive characterization when agents believe that the process of states of nature is i.i.d. but disagree about the probability of the states. We show that even though heterogeneous priors within that class can indeed generate genuine changes in the portfolios of any dynamically complete markets equilibrium, these changes vanish with probability one if the true process consists of i.i.d. draws from a common distribution and the support of some agent's prior belief contains the true distribution. Finally, we provide examples in which asset trading does not vanish because either (i) no agent learns the true conditional probability of the states or (ii) some agent does not know the true process generating the data is i.i.d.     

Risk taking with additive and multiplicative background risks

We examine the effects of background risks on optimal portfolio choice. Examples of background risks include uncertain labor income, uncertainty about the terminal value of fixed assets such as housing and uncertainty about future tax liabilities. While some of these risks are additive and have been amply studied, others are multiplicative in nature and have received far less attention. The simultaneous effect of both additive and multiplicative risks has hitherto not received attention and can explain some paradoxical choice behavior. We rationalize such behavior and show how background risks might lead to seemingly U-shaped relative risk aversion for a representative investor.     

Loss aversion and market crashes

This study proposes a rational expectation equilibrium model of stock market crashes with information asymmetry and loss averse speculators. We obtain a state-dependent linear optimal trading strategy, which makes the equilibrium price tractable. The model predicts nonlinear market depth and the result that small shocks to fundamentals (e.g., supply or informational shocks) can cause abrupt price movements. We demonstrate that short-sale constraints intensify asset price collapses relative to upward movements. The model also generates contagion between uncorrelated assets. These results are consistent with the main puzzling features observed during market crashes, namely abrupt and asymmetric price movements that are not driven by major news events but coupled with a spillover effect between unrelated markets.     

Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach

We analyze comparative risk aversion in a new way, through a comparative statics problem in which, for a cost, agents can shift from an initial probability distribution toward a preferred distribution. The Ross characterization arises when the original distribution is riskier than the preferred distribution and the cost is monetary, and the Arrow-Pratt characterization arises when the original distribution differs from the preferred distribution by a simple mean-preserving spread and the cost is a utility cost. Higher-order increases in risk lead to higher-order generalizations, and the comparative statics method yields a unified approach to the problem of comparative risk attitudes.     

Optimal annuitization, uncertain survival probabilities, and maxmin preferences

We consider a life-cycle model with bequest motives, and assume that the individual does not know his/her survival probability and has maxmin utility preferences; we show that it is optimal not to annuitize but to purchase pure life insurance policies instead.     

The tradeoff between risk sharing and information production in financial markets

The production of information in financial markets is limited by the extent of risk sharing. The wider a stock's investor base, the smaller the risk borne by each shareholder and the less valuable information. A firm which expands its investor base without raising capital affects its information environment through three channels: (i) it induces incumbent shareholders to reduce their research effort as a result of improved risk sharing, (ii) it attracts potentially informed investors, and (iii) it may modify the composition of the base in terms of risk tolerance or liquidity trading. Implications for individual firms and the market as a whole are derived.     

Direct evidence of risk aversion as a source of advantageous selection in health insurance

Using the self-stated degree of risk aversion regarding health from the GSOEP we find some evidence for risk aversion being a source of advantageous selection. Risk averse men more often procure supplementary insurance for hospital visits despite needing the additional coverage less.     

Observational equivalence and nonequivalence of subjective and robust mean–variance preferences

We identify the conditions where robust mean–variance preferences, which capture ambiguity aversion, are observationally nonequivalent to subjective mean–variance preferences. Conversely, we also provide an example showing that observational equivalence holds regardless of the degree of ambiguity aversion.     

A lattice approach for pricing convertible bond asset swaps with market risk and counterparty risk

This paper proposes a new lattice framework for valuing convertible bonds (CBs) and asset swaps on CBs (CBASs) with market risk and counterparty risk, where interest rate is assumed to follow a mean-reverting square root process. The reduced-form approach is generalized to include a constant elasticity of variance (CEV) process for equity price prior to default. In order to approximate the CEV process while taking into account stochastic interest rate and the correlation between stock price and interest rate, I first propose a transform that is uncorrelated with interest rate, and then construct a new lattice method which can ensure the validity of branching probabilities for all nodes. The lattice framework performs properly when it is used to value European call options. Based on the empirical results in Duffie et al. (J. Fin. Econ. 83(3): 635-665, 2007) and Jankowitsch et al. (J. Bank. Fin. 32(7): 1269-1285, 2008), a novel default intensity process is constructed which is specified as a function of time, stock price, and interest rate. When valuing the asset swaps, the counterparty risk is taken into consideration. Based on the results of the numerical experiments, the impacts of different parameters on the prices of CBs and CBASs are explained.