More on Properness: The Case of Mean-Variance Preferences

This paper focuses on the situations where individuals with mean-variance preferences add independent risks to an already risky situation. Pratt and Zeckhauser (Econometrica, 55, 143–154, 1987) define a concept called proper risk aversion in the expected utility framework to describe the situation where an undesirable risk can never be made desirable by the presence of an independent undesirable risk. The assumption of mean-variance preferences allows us to study proper risk aversion in an intuitive manner. The paper presents an economic interpretation for the quasi-concavity of a utility function derived over mean and variance. The main result of the paper says that quasi-concavity plus decreasing risk aversion is equivalent to proper risk aversion.     

Risk aversion in the expected and the nonexpected utility functions

If asset returns are i.i.d. over time, the preference parameter in the time additive von Neumann-Morgenstern expected utility is the risk aversion coefficient in the Epstein-Zin nonexpected utility. By distinguishing between risk aversion and intertemporal substitution, this article provides an explanation about the observed discrepancy in the empirical estimates of the risk aversion coefficient.     

Machine learning and asset allocation

Investors have access to a large array of structured and unstructured data. We consider how these data can be incorporated into financial decisions through the lens of the canonical asset allocation decision. We characterize investor preference for simplicity in models of the data used in the asset allocation decision. The simplicity parameters then guide asset allocation along with the usual risk aversion parameter. We use three distinct and diverse macroeconomic data sets to implement the model to forecast equity returns (the equity risk premium). The data sets we use are (a) price‐dividend ratios, (b) an array of macroeconomic series, and (c) text data from the Federal Reserve's Federal Open Market Committee (FOMC) meetings.     

Long‐Run Stockholder Consumption Risk and Asset Returns

We provide new evidence on the success of long‐run risks in asset pricing by focusing on the risks borne by stockholders. Exploiting microlevel household consumption data, we show that long‐run stockholder consumption risk better captures cross‐sectional variation in average asset returns than aggregate or nonstockholder consumption risk, and implies more plausible risk aversion estimates. We find that risk aversion around 10 can match observed risk premia for the wealthiest stockholders across sets of test assets that include the 25 Fama and French portfolios, the market portfolio, bond portfolios, and the entire cross‐section of stocks.     

Habit formation and the equity–premium puzzle: a skeptical view

We argue that, ceteris paribus, introducing a habit that resolves the equity–premium puzzle is equivalent to increasing the Arrow-Pratt coefficient of relative risk aversion, AP-RRA. If we constrain the AP-RRA to a constant ‘acceptable’ level, the effect on the equity premium is quantitatively insignificant. In a dynamic setting, the fluctuations of the habit increase the equity premium, slightly, though generates unrealistic fluctuations in the risk-free interest rate. We conclude a habit is observationally equivalent, up to a first-order approximation, to a higher AP-RRA and to a preference shock. These effects cannot resolve the equity–premium puzzle.      

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.     

Asymmetric information in insurance: general testable implications

Several recent articles on empirical contract theory and insurance have tested for a positive correlation between coverage and ex post risk, as predicted by standard models of pure adverse selection or pure moral hazard. We show here that the positive correlationproperty can be extended to general setups: competitive insurance markets and cases where risk aversion is public. We test our results on a French dataset. Our tests confirm that the estimated correlation is positive; they also suggest the presence of market power.     

Consumption-based CAPM models: International evidence

We examine the performance of several types of the consumption-based CAPM (C-CAPM) models to explore if consumption factors matter for determining excess returns across 17 MSCI country indexes. While the classic world C-CAPM does exhibit some power in explaining cross-sectional variations of expected excess returns, the model seems to require an implausibly large coefficient of risk aversion. The more sophisticated models including the heterogeneous C-CAPM, the world surplus consumption and the habit-formation models provide more reasonable estimates and add substantial explanatory power for the variation in the cross section of excess stock returns. Our results suggest that country-specific consumption risk is not fully diversified thus implying that stock returns are related to idiosyncratic consumption risk.     

Decision-making in incomplete markets with ambiguity—a case study of a gas field acquisition

We apply utility indifference pricing to solve a contingent claim problem, valuing a connected pair of gas fields where the underlying process is not standard Geometric Brownian Motion and the assumption of complete markets is not fulfilled. First, empirical data are often characterized by time-varying volatility and fat tails; therefore, we use Gaussian generalized autoregressive score (GAS) and GARCH models, extending them to Student’s t-GARCH and t-GAS. Second, an important risk (reservoir size) is not hedgeable. As a result, markets are incomplete which makes preference free pricing impossible and thus standard option pricing methodology inapplicable. Therefore, we parametrize the investor’s risk preference and use utility indifference pricing techniques. We use Least Squares Monte Carlo simulations as a dimension reduction technique in solving the resulting stochastic dynamic programming problems. Moreover, an investor often only has an approximate idea of the true probabilistic model underlying variables, making model ambiguity a relevant problem. We show empirically how model ambiguity affects project values, and importantly, how option values change as model ambiguity gets resolved in later phases of the projects. We show that traditional valuation approaches will consistently underestimate the value of project flexibility and in general lead to overly conservative investment decisions in the presence of time-dependent stochastic structures.     

Multivariate risk aversion and intertemporal substitution

Researchers often assume that preferences over uncertain consumption streams are representable by $$E\left[ {\left( {{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)\sum\limits_{t = 0}^x {\delta ^t \tilde c_t^\gamma } } \right]$$ , where \(\tilde c_t \) , is (random) period t consumption. It is moreover often asserted that estimates of γ cannot be unambiguously interpreted, since the quantity 1 ? γ measures both relative risk aversion and the reciprocal of the elasticity of substitution. Clearly, this ambiguity arises only if 1 ? γ indeed measures risk aversion. Although changes in γ cannot reflect changes in risk aversion according to standard definitions of comparative multivariate risk aversion, we show that γ is rationalizable as a risk aversion measure provided that the “acceptance set” of sure prospects is restricted. We also show, however, that there is essentially no relationship between changes in γ and optimal consumption, even in a simple two period model; this finding casts doubt upon the interpretation of γ as a risk aversion measure.     

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.     

Modeling Asset Premiums and the Riskfree Rate in General Equilibrium CCAPM

We elaborate on the consumption capital asset pricing model (CCAPM) to reveal a set of underlying forces that determine asset returns. We use generalized preferences, allow for labor-leisure choice, a broad asset portfolio, and holding international claims. A calibration of the model with US data learns that excess stock and bond returns can be replicated. At the same time, however, the riskfree interest rate generally appears to be mispriced, consistent with Weil (1989). Additional results show that in general two optimal values of the intertemporal substitution parameter correspond with a specified coefficient of risk aversion. Tests that assess the dynamic properties of the model yield mixed results, but are most favorable when home bias is allowed.     

Multi-asset portfolio optimization and out-of-sample performance: an evaluation of Black–Litterman,mean-variance,and naïve diversification approaches

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems of mean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman model and implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices, covering the period from January 1993 to December 2011. We test its out-of-sample performance relative to other asset allocation models and find that Black–Litterman optimized portfolios significantly outperform naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better than mean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios, even after controlling for different levels of risk aversion, investment constraints, and transaction costs. The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversification across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixed return estimates that incorporate the reliability of return predictions, smaller estimation errors, and lower turnover.     

Age‐Dependent Increasing Risk Aversion and the Equity Premium Puzzle

We introduce a new preference structure—age‐dependent increasing risk aversion (IRA)—in a three‐period overlapping generations model with borrowing constraints, and examine the behavior of equity premium in this framework. We find that IRA preferences generate results that are more consistent with U.S. data for the equity premium, level of savings and portfolio shares, without assuming unreasonable levels of risk aversion. We find that the relative difference between the two risk aversions (how much more risk‐averse old agents are relative to the middle‐aged) matters more than the average risk aversion in the economy (how much more risk‐averse both cohorts are). Our findings are robust with respect to a number of model generalizations.     

Optimal health insurance

I formulate expected-utility-maximizing models for health insurance with a single optimal coinsurance (C*) and (separately) a single optimal deductible (D*). While so-doing, I formalize Nyman's challenge to standard welfare-loss models, clarifying when and by how much this alters unadjusted models. Using MEPS-calibrated lognormal distributions and incorporating skewness and kurtosis measures of financial risk, I show how C* shifts as various economic parameters change. For reasonable parameter values, C* < 0.1, much lower than variance-only estimates would conclude. Omitting higher-order risk parameters importantly understates risk and hence understates optimal insurance coverage. I separately develop methods to determine D*, showing that it is approximately a fixed percentage of income that falls as the distribution of financial risks rise. This finding contrasts with existing US public policy regarding high-deductible health plans, which employ fixed deductibles, independent of income.     

Loss aversion,asymmetric market comovements,and the home bias

Loss aversion has been used to explain why a high equity premium might be consistent with plausible levels of risk aversion. The intuition is that the first-order-different utility impact of wealth gains and losses leads loss-averse investors to behave similarly to investors with high risk aversion. But if so, should those agents not perceive larger gains from international diversification than standard expected-utility investors with plausible levels of risk aversion? They might not, because comovements in international stock markets are asymmetric: correlations are higher in market downturns than in upturns. This asymmetry dampens the gains from diversification relatively more for loss-averse investors. We analyze the portfolio problem of such an investor who has to choose between home and foreign equities in the presence of asymmetric comovement in returns. Perhaps surprisingly, in the context of the home bias puzzle we find that loss-averse investors behave similarly to those with standard expected-utility preferences and plausible levels of risk aversion. We argue that preference specifications that appear to perform well with respect to the equity premium puzzle should be subjected to this “test”.     

Explaining the equity premium in Hong Kong with C-CAPM: The use of emigration growth as an instrument

We test the C-CAPM with CRRA utility using Hong Kong data. In 2SLS regressions, we obtain rather high estimates of the coefficient of relative risk aversion, which could explain the high equity premium in Hong Kong. Because we use lagged emigration growth as an instrument in the first-stage regression, which has significant negative impact on future stock market return in Hong Kong, the first-stage R² and F-statistics are rather high and the weak instrument critique of the validity of 2SLS regressions is potentially resolved. Weak-instrument-robust tests also confirm that the degree of risk aversion is indeed high for Hong Kong.     

Option pricing under time-varying risk-aversion with applications to risk forecasting

We present a two-factor option-pricing model, which parsimoniously captures the difference in volatility persistences under the historical and risk-neutral probabilities. The model generates an S-shaped pricing kernel that exhibits time-varying risk aversion. We apply our model for two purposes. First, we analyze the risk preference implied by S&P500 index options during 2001–2009 and find that risk-aversion level strongly increases during stressed market conditions. Second, we apply our model for Value-at-Risk (VaR) forecasts during the subprime crisis period and find that it outperforms several leading VaR models.     

Why Does Implied Risk Aversion Smile?

Implied risk aversion estimates reported in the literature arestrongly U-shaped. This article explores different potentialexplanations for these "smile" patterns: (i) preference aggregation,both with and without stochastic volatility and jumps in returns,(ii) misestimation of investors’ beliefs caused by stochasticvolatility, jumps, or a Peso problem, and (iii) heterogeneousbeliefs. The results reveal that preference aggregation andmisestimation of investors’ beliefs caused by stochasticvolatility and jumps are unlikely to be the explanation forthe smile. Although a Peso problem can account for the smile,the required probability of a market crash is unrealisticallylarge. Heterogeneous beliefs cause sizable distortions in impliedrisk aversion, but the degree of heterogeneity required to explainthe smile is implausibly large. (JEL: G12, G13)     

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.