Abstract:
We introduce an equilibrium asset pricing model, which we build on the relationship between a novel risk measure, the Expected Downside Risk (EDR) and the expected return. On the one hand, our proposed risk measure uses a nonparametric approach that allows us to get rid of any assumption on the distribution of returns. On the other hand, our asset pricing model is based on loss-averse investors of Prospect Theory, through which we implement the risk-seeking behaviour of investors in a dynamic setting. By including EDR in our proposed model unrealistic assumptions of commonly used equilibrium models - such as the exclusion of risk-seeking or price-maker investors and the assumption of unlimited leverage opportunity for a unique interest rate - can be omitted. Therefore, we argue that based on more realistic assumptions our model is able to describe equilibrium expected returns with higher accuracy, which we support by empirical evidence as well.

Abstract:
Traditionally, risk aversion (both absolute and relative) has been expressed as a function of wealth alone. The charac- teristics of risk aversion as wealth changes have been extensively studied. However, prices, as well as wealth, enter the indirect utility function, from which the typical risk aversion measures are calculated. Given that, changes in prices will affect risk aversion, although exactly how has not been considered in the literature. This paper provides such an analysis. In particular, we firstly remind the reader that both absolute and relative risk aversion are homogeneous functions, and as such independently of their particular slopes in wealth, there is a natural effect that holds relative risk aversion constant and decreases absolute risk aversion when prices and wealth are increased by a common factor. We also show that the size of relative risk aversion as compared to the number 1, which is of much importance to the comparative statics of the economics of risk and uncertainty, depends on how changes in prices affect marginal utility. Under plausible (and standard) theoretical assumptions we find that relative risk aversion is likely to be increasing, and that increases in prices will have a tempering effect on risk aversion.

Abstract:
The standard asset pricing models (the CCAPM and the Epstein-Zin non-expected utility model) counterintuitively predict that equilibrium asset prices can rise if the representative agent's risk aversion increases. If the income effect, which implies enhanced saving as a result of an increase in risk aversion, dominates the substitution effect, which causes the representative agent to reallocate his portfolio in favour of riskless assets, the demand for securities increases. Thus, asset prices are forced to rise when the representative agent is more risk adverse. By disentangling risk aversion and intertemporal substituability, we demonstrate that the risky asset price is an increasing function of the coefficient of risk aversion only if the elasticity of intertemporal substitution (EIS) exceeds unity. This result, which was first proved par Epstein (1988) in a stationary economy setting with a constant risk aversion, is shown to hold true for non-stationary economies with a variable or constant risk aversion coefficient. The conclusion is that the EIS probably exceeds unity.

Abstract:
This paper determines a new sufficient condition of the (von Neumann-Morgenstern) utility function that preserves comparative risk aversion under background risk. It is the single crossing condition of risk aversion. Because this condition requires monotonicity in the local sense, it may satisfy the U-shaped risk aversion observed in the recent empirical literature.

Abstract:
This paper determines a new sufficient condition of the (von Neumann-Morgenstern) utility function that preserves comparative risk aversion under background risk. It is the single crossing condition of risk aversion. Because this condition requires monotonicity in the local sense, it may satisfy the U-shaped risk aversion observed in the recent empirical literature. 1. Introduction Pratt [1] and Arrow [2] introduced the notion of risk aversion and its associated order in the expected utility framework. These are represented by concavity and the degree of concavity of the (von Neumann-Morgenstern) utility function. Comparative risk aversion, which is the order of risk aversion, has intuitive and reasonable properties in many decision-making problems that have appeared in economics and finance. For example, in typical portfolio problems, more risk averse investors hold less risky and more risk-free assets. In this example, investors face single risk. However, it is natural that investors face other risk which cannot traded in asset markets, for example, human capital risk cannot be traded in asset markets. In other situations, we face risk which investors cannot control and trade. This risk is called background risk. Optimal decision problems are rather complex by the presence of background risk. Over the past three decades, many researchers have examined how background risk influences optimal decisions in economics and finance. (Gollier [3] provided an excellent survey of this topic.) Since comparative risk aversion may be different with and without background risk, comparative risk aversion too weal to compare optimal decisions in the presence of background risk. This leads to the following question on this topic: “what conditions guarantee that comparative risk aversion is preserved in the presence of background risk?" This paper provides a new answer to this question. Important contributions to this question are from Kihlstrom et al. [4], Nachman [5], and Pratt [6]. The first two studies obtained a sufficient condition for the preservation of comparative risk aversion in the presence of background risk. As in the case of additive and multiplicative background risk, these sufficient conditions are decreasing absolute risk aversion (DARA) and decreasing relative risk aversion (DRRA), respectively. Pratt [6] established a necessary and sufficient condition for the preservation of comparative risk aversion in the presence of background risk. This paper proposes a new sufficient condition, which is the single crossing condition of risk aversion. The

Abstract:
Risk aversion is a common behavior universal to humans and animals alike. Economists have traditionally defined risk preferences by the curvature of the utility function. Psychologists and behavioral economists also make use of concepts such as loss aversion and probability weighting to model risk aversion. Neurophysiological evidence suggests that loss aversion has its origins in relatively ancient neural circuitries (e.g., ventral striatum). Could there thus be an evolutionary origin to risk avoidance? We study this question by evolving strategies that adapt to play the equivalent mean payoff gamble. We hypothesize that risk aversion in the equivalent mean payoff gamble is beneficial as an adaptation to living in small groups, and find that a preference for risk averse strategies only evolves in small populations of less than 1,000 individuals, while agents exhibit no such strategy preference in larger populations. Further, we discover that risk aversion can also evolve in larger populations, but only when the population is segmented into small groups of around 150 individuals. Finally, we observe that risk aversion only evolves when the gamble is a rare event that has a large impact on the individual's fitness. These findings align with earlier reports that humans lived in small groups for a large portion of their evolutionary history. As such, we suggest that rare, high-risk, high-payoff events such as mating and mate competition could have driven the evolution of risk averse behavior in humans living in small groups.

Abstract:
Spectral risk measures are attractive risk measures as they allow the user to obtain risk measures that reflect their risk-aversion functions. To date there has been very little guidance on the choice of risk-aversion functions underlying spectral risk measures. This paper addresses this issue by examining two popular risk aversion functions, based on exponential and power utility functions respectively. We find that the former yields spectral risk measures with nice intuitive properties, but the latter yields spectral risk measures that can have perverse properties. More work therefore needs to be done before we can be sure that arbitrary but respectable utility functions will always yield 'well-behaved' spectral risk measures.

Abstract:
This paper explores the dynamics of risk aversion of a representative agent
with an iso-elastic utility function. In contrast to most of the existing literature,
we estimate the coefficient of relative risk aversion from option prices.
To do this, we transform the risk-neutral density function obtained from a
cross-section of option prices to an objective distribution function that reflects
individuals’ risk aversion through a CRRA utility function. The dynamics
of the relative risk-aversion coefficient are obtained by repeating the same
estimation procedure over rolling windows. This procedure uncovers strong
variation in risk aversion over time. We also propose a simulation procedure
to construct confidence intervals for the risk-aversion coefficient in each period.
We assess the robustness of these confidence intervals under different
assumptions on the data generating process of stock prices. The results imply
a strong influence of volatility on the variation of risk aversion. In an empirical
application, we compare the forecasting performance of our approach
based on our risk-aversion estimates against the method proposed in [1].
Overall, we find that our simulation based approach obtains better forecasting
results than bootstrap methods.

Abstract:
We study the feasibility and noise sensitivity of portfolio optimization under some downside risk measures (Value-at-Risk, Expected Shortfall, and semivariance) when they are estimated by fitting a parametric distribution on a finite sample of asset returns. We find that the existence of the optimum is a probabilistic issue, depending on the particular random sample, in all three cases. At a critical combination of the parameters of these problems we find an algorithmic phase transition, separating the phase where the optimization is feasible from the one where it is not. This transition is similar to the one discovered earlier for Expected Shortfall based on historical time series. We employ the replica method to compute the phase diagram, as well as to obtain the critical exponent of the estimation error that diverges at the critical point. The analytical results are corroborated by Monte Carlo simulations.

Abstract:
We develop efficient algorithms to construct utility maximizing mechanisms in the presence of risk averse players (buyers and sellers) in Bayesian settings. We model risk aversion by a concave utility function, and players play strategically to maximize their expected utility. Bayesian mechanism design has usually focused on maximizing expected revenue in a {\em risk neutral} environment, and no succinct characterization of expected utility maximizing mechanisms is known even for single-parameter multi-unit auctions. We first consider the problem of designing optimal DSIC mechanism for a risk averse seller in the case of multi-unit auctions, and we give a poly-time computable SPM that is $(1-1/e-\eps)$-approximation to the expected utility of the seller in an optimal DSIC mechanism. Our result is based on a novel application of a correlation gap bound, along with {\em splitting} and {\em merging} of random variables to redistribute probability mass across buyers. This allows us to reduce our problem to that of checking feasibility of a small number of distinct configurations, each of which corresponds to a covering LP. A feasible solution to the LP gives us the distribution on prices for each buyer to use in a randomized SPM. We next consider the setting when buyers as well as the seller are risk averse, and the objective is to maximize the seller's expected utility. We design a truthful-in-expectation mechanism whose utility is a $(1-1/e -\eps)^3$-approximation to the optimal BIC mechanism under two mild assumptions. Our mechanism consists of multiple rounds that processes each buyer in a round with small probability. Lastly, we consider the problem of revenue maximization for a risk neutral seller in presence of risk averse buyers, and give a poly-time algorithm to design an optimal mechanism for the seller.