Risk aversion, intertemporal substitution, and the aggregate investment-uncertainty relationship

We analyze the role of risk aversion and intertemporal substitution in a simple dynamic general equilibrium model of investment and savings. Our main finding is that risk aversion cannot by itself explain a negative relationship between aggregate investment and aggregate uncertainty, as the effect of increased uncertainty on investment also depends on the intertemporal elasticity of substitution. In particular, the relationship between aggregate investment and aggregate uncertainty is positive even if agents are very risk averse, as long as the elasticity of intertemporal substitution is low. A negative investment-uncertainty relationship requires that the relative risk aversion and the elasticity of intertemporal substitution are both relatively high or both relatively low. We also show that the implications of our model are consistent with the available empirical evidence.     

Interest rate rules and price determinacy: The role of transactions services of bonds

Interest rate rules have been associated with price indeterminacy when they do not respond aggressively enough to inflation. Price indeterminacy is typically associated with indeterminacy of real bond balances, suggesting that the missing element is a meaningful role for government bonds. We assume that government bonds provide arbitrarily small transactions services and show that this can dramatically change the local and global determinacy conditions. In particular, the specification of fiscal policy affects the aggressiveness with which monetary policy must respond to inflation to deliver local determinacy—a range of passive monetary policies, even an interest rate peg, may yield determinacy.     

TIME VARIABILITY IN MARKET RISK AVERSION

We adopt realized covariances to estimate the coefficient of risk aversion across portfolios and through time. Our approach yields second moments that are free from measurement error and not influenced by a specified model for expected returns. Supporting the permanent income hypothesis, we find risk aversion responds to consumption-smoothing behavior. As income increases, or as the consumption-to-income ratio falls, relative risk aversion decreases. We also document variation in risk aversion across portfolios: risk aversion is highest for small and value portfolios.     

A Utility-Based Model of Common Stock Price Movements

This paper develops and tests a nonlinear utility-based econometric model of the temporal behavior of aggregate stock price movements based on a constant relative risk aversion utility function and an observable information set consisting of aggregate consumption, aggregate dividends, and past stock prices. The stochastic process derived from time-series analyses of consumption and dividends measured over annual intervals is used to derive and empirically test a closed-form solution for stock-price movements. The endogenization of discount rate changes in the utility-based model is shown to be more consistent with aggregate stock price movements over a twenty-year holdout period than constant discount rate models. The model is also used to estimate the representative investor's relative risk aversion. The estimate of 4.22 is consistent with that used by Grossman and Shiller in their perfect foresight model and is significantly higher than the relative risk aversion of 1.0 implied by logarithmic utility.     

Risk preferences in multi-period consumption models, the equity premium puzzle, and habit formation utility

The relationship between the relative risk aversion measure for the utility function for consumption and that for the value function for wealth is a derived relationship whose properties depend on how consumption and wealth are defined and measured. This fact together with information concerning estimates for these two relative risk aversion measures is used to give another perspective on the equity premium puzzle, and to explain why it is that the habit formation utility function is effective in eliminating that puzzle. A time separable utility function that can serve as an alternative to the assumption of habit formation is also presented.     

Endogenous Price Stickiness,Trend Inflation,and Macroeconomic Stability

Previous studies show that higher trend inflation is more likely to induce indeterminacy of equilibrium in sticky‐price models based on micro evidence that each period a fraction of prices is kept unchanged. This paper demonstrates that when the degree of price stickiness is endogenously determined in a Calvo model, indeterminacy caused by higher trend inflation is less likely. A key factor for determinacy is the long‐run inflation elasticity of output implied by the New Keynesian Phillips curve. This elasticity declines substantially with higher trend inflation in the case of exogenously given price stickiness, whereas in the case of endogenous price stickiness the decline in the elasticity is mitigated because higher trend inflation leads to a higher probability of price adjustment.     

Heterogeneity and Option Pricing

An economy with agents having constant yetheterogeneous degrees of relative risk aversion prices assetsas though there were a single decreasing relative risk aversion``pricing representative' agent. The pricing kernel has fattails, and option prices do not conform to the Black-Scholesformula. Implied volatility exhibits a ``smile.' Heterogeneityas the source of non-stationary pricing fits Rubenstein's (1994)interpretation of the ``over-pricing' as an indication of ``crash-o-phobia'.Rubinstein's term suggests that those who hold out-of-the moneyput options have relatively high risk aversion (or relativelyhigh subjective probability assessments of low market outcomes).The essence of this explanation is investor heterogeneity.     

Market pricing of executive stock options and implied risk preferences

When managers get to trade in options received as compensation, their trading prices reveal several aspects of subjective option pricing and risk preferences. Two subjective pricing models are fitted to show that executive stock option prices incorporate a subjective discount. It depends positively on implied volatility and negatively on option moneyness. Further, risk preferences are estimated using the semiparametric model of Aït-Sahalia and Lo (2000). The results suggest that relative risk aversion is just above 1 for a certain stock price range. This level of risk aversion is low but reasonable, and it may be explained by the typical manager being wealthy and having low marginal utility. Related to risk aversion, it is found that marginal rate of substitution increases considerably in states with low stock prices.     

Extracting Information from Options Markets: Smiles, State–Price Densities and Risk Aversion

In this paper, recent techniques of estimating implied information from derivatives markets are presented and applied empirically to the French derivatives market. We determine nonparametric implied volatility functions, state–price densities and historical densities from a high–frequency CAC 40 stock index option dataset. Moreover, we construct an estimator of the risk aversion function implied by the joint observation of the cross–section of option prices and time–series of underlying asset value. We report a decreasing implied volatility curve with the moneyness of the option. The estimated relative risk aversion functions are positive and globally consistent with the decreasing relative risk aversion assumption.     

What Asset Prices Should Be Targeted by a Central Bank?

This paper investigates the monetary policy design for restoring equilibrium determinacy. Our interests are whether a central bank should respond to asset price fluctuations, and if so, what asset prices should be targeted. We show that a monetary policy response to the price of a productive tangible asset (capital price) is helpful for equilibrium determinacy, while that to the price of an intangible asset that reflects a firm's profit (share prices) is a source of equilibrium indeterminacy. This result comes from the two assets' prices moving in opposite directions in response to a permanent increase in inflation.     

Markets versus governments

We provide a simple framework for comparing market allocations with government-regulated allocations. Governments can collect information about individuals’ types and enforce transfers across individuals. Markets (without significant government intervention) have to rely on transactions that are ex post beneficial for individuals. Consequently, governments achieve better risk sharing and consumption smoothing than markets. However, politicians in charge of collective decisions can use the centralized information and the enforcement power of government for their own benefits. This leads to political economy distortions and rents for politicians, making government-operated allocation mechanisms potentially worse than markets. We provide conditions under which it is ex ante beneficial for the society to tolerate the political economy distortions in exchange for the improvement in risk sharing. For example, more effective controls on politicians or higher discount factors of politicians make governments more attractive relative to markets. Moreover, when markets cannot engage in self-enforcing risk-sharing arrangements and income effects are limited, greater risk aversion and greater uncertainty make governments more attractive relative to markets. Nevertheless, we also show theoretically and numerically that the effect of risk aversion on the desirability of markets may be non-monotonic. In particular, when markets can support self-enforcing risk-sharing arrangements, a high degree of risk aversion improves the extent of risk sharing in markets and makes governments less necessary. The same pattern may also arise because of “income effects” on labor supply. Consequently, the welfare gains of governments relative to markets may have an inverse U-shape as a function of the degree of risk aversion of individuals.     

Equilibrium Indeterminacy and Asset Price Fluctuation in Japan: A Bayesian Investigation

This paper investigates sources of asset price fluctuation in Japan using an estimated financial accelerator model. For explicit treatment of expectational beliefs characterized by sunspots, the model is analyzed over the parameter space where the equilibrium can be indeterminate. We show that indeterminacy arises if the financial accelerator effect is sufficiently large. According to our Bayesian estimation results, Japan's economy was affected by sunspot shocks; however, the contribution of the sunspots to asset price volatility was low. Rather, net worth and cost shocks drove the asset price fluctuation. We find, however, that the sunspots substantially affected capital investment.     

Indeterminacy and Forecastability

Recent studies document the deteriorating performance of forecasting models during the Great Moderation, which conversely implies that forecastability was higher in the preceding era when the economy was unexpectedly volatile. We explain this phenomenon in the context of equilibrium indeterminacy in dynamic stochastic general equilibrium (DSGE) models. We first analytically show that a model under indeterminacy exhibits richer dynamics that can improve forecastability. Then, using a sticky‐price DSGE model, we numerically demonstrate that indeterminacy arising from passive monetary policy generates persistent dynamics that lead to superior forecastability. We also point out the possibility that forecastability under indeterminacy deteriorates when the degree of uncertainty about sunspot fluctuations is large.     

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.     

Implied Risk Aversion Parameter from Option Prices

This paper estimates the representative investor's coefficient of relative risk aversion using option price data. Estimation is carried out using the method of simulated moments. Employing the following assumptions: a) agents have constant proportional risk averse preferences, b) complete markets exist, and c) asset returns are distributed lognormally, an objective function is constructed within the equivalent martingale measure framework. Unlike the case of equity markets, the implied risk aversion parameter from option prices is quite low and stays between zero and one.     

Precautionary Saving and Consumption Smoothing across Time and Possibilities

This paper examines how aversion to risk and aversion to intertemporal substitution determine the strength of the precautionary saving motive in a two-period model with Selden/Kreps–Porteus preferences. For small risks, we derive a measure of the strength of the precautionary saving motive that generalizes the concept of "prudence" introduced by Kimball (1990b) . For large risks, we show that decreasing absolute risk aversion guarantees that the precautionary saving motive is stronger than risk aversion, regardless of the elasticity of intertemporal substitution. Holding risk preferences fixed, the extent to which the precautionary saving motive is stronger than risk aversion increases with the elasticity of intertemporal substitution. We derive sufficient conditions for a change in risk preferences alone to increase the strength of the precautionary saving motive and for the strength of the precautionary saving motive to decline with wealth. Within the class of constant elasticity of intertemporal substitution, constant-relative risk aversion utility functions, these conditions are also necessary.     

Risk,uncertainty, and asset prices

We identify the relative importance of changes in the conditional variance of fundamentals (which we call “uncertainty”) and changes in risk aversion in the determination of the term structure, equity prices, and risk premiums. Theoretically, we introduce persistent time-varying uncertainty about the fundamentals in an external habit model. The model matches the dynamics of dividend and consumption growth, including their volatility dynamics and many salient asset market phenomena. While the variation in price–dividend ratios and the equity risk premium is primarily driven by risk aversion, uncertainty plays a large role in the term structure and is the driver of countercyclical volatility of asset returns.     

Stock performance by utility indifference pricing and the Sharpe ratio

We compare stock performance based on utility indifference pricing and the Sharpe ratio assuming that stock returns follow the class of discrete normal mixture distributions. The utility indifference price with an exponential utility function satisfies several desirable properties that a suitable value measure should satisfy. For utility indifference pricing, we employ the inner rate of risk aversion proposed by Miyahara [Evaluation of the scale risk. RIMS Kokyuroku, No. 1886, Financial Modeling and Analysis (2013/11/20-2013/11/22), 181–188, 2014], which is the degree of risk aversion that makes the utility indifference price with the exponential utility function zero in order to evaluate stock performance. Using a selection of U.S. stocks, the results show that the evaluation of stock performance based on the inner rate of risk aversion is more relevant for risk-averse investors than that based on the Sharpe ratio, which represents performance by the first two moments.     

Government Spending and the Taylor Principle

This paper explores how government size affects the scope for equilibrium indeterminacy in a New Keynesian economy, where part of the population live hand-to-mouth. The main result is that a higher level of public consumption is likely to generate indeterminacy and render the Taylor principle insufficient as criterion for equilibrium uniqueness. This holds even though fiscal policy serves to reduce swings in current income. Only if government consumption is a substitute for private consumption, will it narrow the scope for indeterminacy. Hence monetary policy should be conducted with an eye to the amount and composition of government consumption.     

Time-varying risk aversion and asset prices

This paper uses a variant of the consumption-based representative agent model in Campbell and Cochrane [Campbell, J.Y., Cochrane, J.H., 1999. By force of habit: Consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107, 205–251] to study how investors’ time-varying risk aversion affects asset prices. First, we show that a countercyclical variation of risk aversion drives a procyclical conditional risk premium. Second, we show that with a small value for the volatility of the log surplus consumption ratio, a large value of risk aversion may not determine whether the equity premium and the risk-free rate puzzles can be resolved or not. Third, we show that countercyclical risk aversion may not help explain the predictability of long-horizon stock returns, the univariate mean-reversion of stock prices and the “leverage effect” in return volatility.