Constant Risk Aversion Research Articles

Portfolio Optimization and Martingale Measures

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

The Construction of a Simple Book

A book is made for a horse race, and punters place their bets. The problem considered here is how the bookmaker should construct his book. Before this can be solved, it has to be determined how the punters will react to any proposed book. Much of the detailed discussion is confined to a race with two horses, though some results apply in the general case. The punters' problem is solved using a utility function, special attention being paid to the case of constant risk-aversion. Two solutions are provided for the bookmaker's problem, dependent on whether it is desired to maximize expected gain, or achieve the same gain whatever horse wins.

Effective Return, Risk Aversion and Drawdowns

We derive two risk adjusted performance measures for investors with risk averse preferences. Maximizing these measures is equivalent to maximizing the expected utility of an investor. The first measure, X(eff), is derivedassuming a constant risk aversion while the second measure, R(eff),is based on a stronger risk aversion to clustering of losses than of gains. The clustering of returns is captured through a multi-horizon framework. The empirical properties of X(eff), R(eff) are studied within the context of real-time trading models for foreign exchange rates and their properties are compared to those of more traditional measures like the annualized return, the Sharpe Ratio and the maximum drawdown. Our measures are shown to be more robust against clustering of losses and has the ability to fully characterize the dynamic behavior of investment strategies.

Constant Risk Aversion

Constant risk aversion means that adding a constant to all outcomes of two distributions, or multiplying all their outcomes by the same positive number, will not change the preference relation between them. We prove several representation theorems, where constant risk aversion is combined with other axioms to imply specific functional forms. Among other things, we obtain a form of disappointment aversion theory without using the concept of reference point in the axioms, and a form of the rank dependent model without making references to the ranking of the outcomes. This axiomatization leads to a natural generalization of the Gini index.Journal of Economic LiteratureClassification Number: D81

Implied Risk Aversion Parameter from Option Prices

AbstractThis 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.

Risk Aversion and the Intertemporal Behavior of Asset Prices

In this article, we characterize economies in which both cash flows and forward prices follow random walks. We show in the case of geometric random walks that the preferences of the representative investor are of the constant proportional risk-aversion type. We also show the conditions under which spot prices follow random walks and under which the equivalent martingale measure is non-state-dep endent.

Risk attitudes for nonlinear measurable utility

This paper concerns SSB utility theory in the monetary context. It introduces a measure of risk aversion and establishes necessary and sufficient conditions for comparative risk aversion. Nonseparable decompositions of an SSB utility function that have the form Φ(x, y)=w(x)w(y)f(x−y) are examined. Risk properties such as constant risk aversion and a delta property are shown to give specific functional forms for an SSB utility function.

Dynamic Probabilistic Decision Processes

This paper presents a general model for the formulation and solution of the risk‐sensitive dynamic decision problem that maximizes the certain equivalent of the discounted rewards of a time‐varying Markov decision process. The problem is solved by applying the principle of optimality and stochastic dynamic programming to the immediate rewards and the certain equivalent associated with the remaining transitions of a time‐varying Markov process over a finite or infinite time horizon, under the assumptions of constant risk aversion and discounting of future cash flows. The solution provides transient and stationary optimal decision policies that depend on the presence or absence of discounting. The construction equipment replacement problem serves as an example application of the model to illustrate the solution methodology and the sensitivity of the optimal policy to the discount factor and the degree of risk aversion.

Preference Among Risky Prospects Under Constant Risk Aversion

AbstractRisk analyses often require a measure of individual risk aversion. Here a procedure is presented to calculate risk aversion parameter ranges wherein individuals would exhibit preference among a set of risky prospects.

Constant vs. Variable Risk‐Aversion in Foraging Bananaquits

Recent foraging theories predict that risk—averse foragers will trade off increases in mean reward against increases in reward variability. Different formulations of this trade—off have been proposed, with some models assuming constant risk—aversion and others assuming decreasing risk—aversion. A detailed analysis of indifference curves (when a forager shows equal preference for certain mean—variance combinations) can theoretically be used to discriminate between models. We follow the analysis of previous workers by equating decreasing risk—aversion with the predictions of the z—score model because those predictions suggest indifference curves that can be discriminated from the constant risk—aversion of the variance discounting model. Experiments conducted with captive Bananaquits (Coereba flaveola) foraging on nectar in an artificial floral patch indicate that they show a simple mean—variance trade—off. Bananaquit foraging behavior was consistent with the constant risk—aversion of the variance—discounting model, but was significantly different from decreasing risk—aversion represented by the z—score model.

On the Consistency of the Black-Scholes Model with a General Equilibrium Framework

We construct a simple economy with consumption only at the final date in which we “endogenize” the stochastic behavior of prices assumed in the Black-Scholes model. Certain preferences (constant proportional risk aversion) and beliefs are shown to be sufficient and necessary, in certain respects, for the existence of such an equilibrium. The analysis is then generalized to a continuous-consumption framework, in which we embed the Merton proportional dividend model.

How constant is the constant of risk-aversion?

We describe an experiment designed to distinguish between two models of risk-sensitive feeding behaviour: the variance discounting model and the z-score model. The variance discounting model assumes that mean reward levels do not affect preferences over reward variability, but the z-score model assumes that mean reward levels do affect preferences over variability. We presented two choices to feeding rufous hummingibrds, Selaphoruous rufus. One alternative had a higher mean and a higher variance than the other. After measuring preference, we increased the mean of both alternatives by adding the same amount to all possible outcomes. The variance discounting model predicts that such a general shift should not change preferences, but the z-score model predicts that preferences will change. Our results support the z-score model. The variance discounting model's assumption of constant risk-aversion fails.

Foraging juncos: interaction of reward mean and variability

Recent stochastic theories of feeding strategy propose that a risk-averse forager will trade off increases in reward variability against increases in average reward. We show that two models of risk-aversion (variance discounting and the z-score model) imply different forms for this interaction of mean and variability. Variance discounting suggests constant risk-aversion, since the effect of reward variance on expected fitness is assumed to be independent of the mean. The z-score model suggests decreasing risk-aversion, since the effect of a given level of reward variance on expected fitness is assumed to decrease as the mean increases. We report two series of experiments with dark-eyed juncos ( Junco hyemalis). Each series investigated the mean-variability trade-off under physiological conditions promoting risk-aversion; reward sizes were generally greater in the second series. The results clearly indicate a trade-off, but within a series the data do not discriminate between the two models' predictions. Comparing the two series suggests that the juncos' level of aversion to reward variability decreases after an overall increment in environmental profitability.

Insurance and saving: some further results

The purpose of this paper is to provide a joint treatment of the saving and insurance decisions which have very often been dealt with separately in the economic literature. In the present model any decrease in the current consumption level can be used to finance either insurance purchase or an increase in the stock of safe assets held in financial institutions (called ‘deposits’ or ‘contingency reserves fund’ for brevity).The most important result is that, under decreasing temporal risk aversion, deposits and insurance are pure substitutes in the Hicksian sense. Besides, it is shown among other comparative statics results that insurance is not necessarily an inferior good, contrarily to the prevailing view in the literature. Finally, we indicate under which conditions a separation theorem between consumption, insurance and deposits holds. These conditions are either a fair insurance premium or a constant temporal risk aversion. Finally our results are compared to related ones in the literature.

The Valuation of Multivariate Contingent Claims in Discrete Time Models

ABSTRACTThere are several examples in the literature of contingent claims whose payoffs depend on the outcomes of two or more stochastic variables. Familiar cases of such claims include options on a portfolio of options, options whose exercise price is stochastic, and options to exchange one asset for another. This paper derives risk neutral valuation relationships (RNVRs) in a discrete time setting that facilitate the pricing of such complex contingent claims in two specific cases: joint lognormally distributed underlying variables and constant proportional risk aversion on the part of investors, and joint normally distributed underlying variables and constant absolute risk aversion preferences, respectively. This methodology is then applied to the valuation of several interesting complex contingent claims such as multiperiod bonds, multicurrency option bonds, and investment options.

Technical Note—On the Evaluation of Intertemporal Lotteries

We consider the evaluation of intertemporal lotteries using two tools: a single univariate utility function and a set of discount coefficients. If each of the lotteries consists of a single stochastic return to be received in a future period, then these tools are sufficient only if the utility function exhibits constant proportional risk aversion. Further, if the lotteries involve returns in more than one time period, we demonstrate that the tools are sufficient if, and only if, the utility function is linear.

Dynamic investment, risk aversion, and foresight sensitivity

Since optimal investment strategies generally cannot be obtained in closed form when utility functions exhibit non-constant risk aversion, most dynamic investment studies have focused on the constant risk aversion case. The present paper investigates a general class of dynamic investment models with final-period expected wealth objective for which the final-period utility of wealth function is not restricted to be constantly risk averse. Existence, monotonicity, concavity, differentiability, and absolute risk aversion properties are established for the optimal feedback investment strategies and dynamic programming indirect utility functions. The loss in final-period expected utility resulting from the use of limited foresight investments is shown to be bounded above by terms dependent both on the variance of myopically achievable utility and on the relative size of myopic and global absolute risk aversion. Finally, simulation results are presented which indicate the optimality of a rolling 2-period foresight horizon for a class of exponential utility functions exhibiting decreasing absolute risk aversion.

Implications of constant risk aversion

The assumption of constant risk aversion often leads to a considerable simplification of decision theoretic analyses. It is shown that this restriction on constant risk aversion permits the description of a wide range of risk averse patterns between the extreme cases of risk neutrality and the exclusive orientation on the pessimistic maxmin criterion. In particular, a new axiomatic foundation of the constant risk aversion is given and a series of properties for the certainty equivalent are derived. Possible applications are shown by the treatment of risk premiums, values of information, portfolio selection, capital market theory and syndicates.

Necessary and Sufficient Conditions for the Mean-Variance Portfolio Model with Constant Risk Aversion

The familiar two-parameter model for portfolio decisions, attributed to Markowitz [11], has individuals maximizing an objective function, ϕ [E(Y), V(Y)], of mean and variance of end-of-period wealth, subject to a constraint imposed by initial wealth. In the usual version there is an arbitrary number, n, of risky assets with stochastic end-of-period values (price plus dividend) represented by the vector X with exogenously given mean vector μ and nonsingular variance matrix σ. There is also one riskless asset, whose certain end-of-period value per dollar invested is p. Final wealth, as constrained by initial wealth, W, is given by Y = WP + a' (X – OP), where a and P are vectors of risky asset quantities and prices. Assuming ϕE > 0 (wealth preference), ϕV < 0 (risk aversion), and that the Hessian of $ is negative and semidefinite, portfolio optimum calls for ϕE(μ − OP) + 2ϕVσa = 0, or

Conditions on Risk Attitude for a Single Attribute

For a decision problem having consequences described by a single attribute, the task of determining a utility function can be facilitated by verifying that the decision maker's risk attitude satisfies a condition such as constant risk aversion. We investigate a general class of conditions on risk attitude, and show that a utility function for such a condition may exist only when the condition is of a special type. Next, we discuss and interpret conditions of this special type. Then, we define two conditions which imply that the decision maker's risk attitude satisfies a condition of this type and is represented by a generalized logarithmic utility function or a linear fractional utility function.