Risk aversion

Abstract: This paper concerns utility functions for money. A measure of risk aversion in the small, the risk premium or insurance premium for an arbitrary risk, and a natural concept of decreasing risk aversion are discussed and related to one another. Risks are also considered as a proportion of total assets.

Abstract: A menu of paired lottery choices is structured so that the crossover point to the high-risk lottery can be used to infer the degree of risk aversion. With “normal” laboratory payoffs of several dollars, most subjects are risk averse and few are risk loving. Scaling up all payoffs by factors of twenty, fifty, and ninety makes little difference when the high payoffs are hypothetical. In contrast, subjects become sharply more risk averse when the high payoffs are actually paid in cash. A hybrid “power/expo” utility function with increasing relative and decreasing absolute risk aversion nicely replicates the data patterns over this range of payoffs from several dollars to several hundred dollars. Although risk aversion is a fundamental element in standard theories of lottery choice, asset valuation, contracts, and insurance (e.g. Daniel Bernoulli, 1738; John Pratt, 1964; Kenneth Arrow, 1965), experimental research has provided little guidance as to how risk aversion should be modeled. To date, there have been several approaches used to assess the importance and nature of risk aversion. Using lottery choice data from a field experiment, Hans Binswanger (1980) concluded that most farmers exhibit a significant amount of risk aversion that tends to increase as payoffs are increased. Alternatively, risk aversion can be inferred from bidding and pricing tasks. In auctions, overbidding relative to Nash predictions has been attributed to risk aversion by some and to noisy decision-making by others, since the payoff consequences of such overbidding tend to be small (Glenn Harrison, 1989). Vernon Smith and James Walker (1993) assess the effects of noise and decision cost by dramatically scaling up auction payoffs. They find little support for the noise hypothesis, reporting that there is an insignificant increase in overbidding in private value auctions as payoffs are scaled up by factors of 5, 10, and 20. Another way to infer risk aversion is to elicit buying and/or selling prices for simple lotteries. Steven Kachelmeier and Mohamed Shehata (1992) report a significant increase in risk aversion (or, more precisely, a decrease in risk seeking behavior) as the prize value is increased. However, they also obtain dramatically different results depending on whether the choice task involves buying or selling, since subjects tend to put a high selling price on something they “own” and a lower buying price on something they do not, which implies This is analogous to the well-known “willingness to pay/willingness to accept bias.” Asking for a high selling price 1 implies a preference for the risk inherent in the lottery, and offering a low purchase price implies an aversion to the risk in the lottery. Thus the way that the pricing task is framed can alter the implied risk attitudes in a dramatic manner. The issue is whether seemingly inconsistent estimates are due to a problem with the way risk aversion is conceptualized, or to a behavioral bias that is activated by the experimental design. We chose to avoid this possible complication by framing the decisions in terms of choices, not purchases and sales. 3 risk seeking behavior in one case and risk aversion in the other. Independent of the method used to elicit 1 a measure of risk aversion, there is widespread belief (with some theoretical support discussed below) that the degree of risk aversion needed to explain behavior in low-payoff settings would imply absurd levels of risk aversion in high-payoff settings. The upshot of this is that risk aversion effects are controversial and often ignored in the analysis of laboratory data. This general approach has not caused much concern because most theorists are used to bypassing risk aversion issues by assuming that the payoffs for a game are already measured as utilities. The nature of risk aversion (to what extent it exists, and how it depends on the size of the stake) is ultimately an empirical issue, and additional laboratory experiments can produce useful evidence that complements field observations by providing careful controls of probabilities and payoffs. However, even many of those economists who admit that risk aversion may be important have asserted that decision makers should be approximately risk neutral for the low-payoff decisions (involving several dollars) that are typically encountered in the laboratory. The implication, that low laboratory incentives may be somewhat unrealistic and therefore not useful in measuring attitudes toward “real-world” risks, is echoed by Daniel Kahneman and Amos Tversky (1979), who suggest an alternative: Experimental studies typically involve contrived gambles for small stakes, and a large number of repetitions of very similar problems. These features of laboratory gambling complicate the interpretation of the results and restrict their generality. By default, the method of hypothetical choices emerges as the simplest procedure by which a large number of theoretical questions can be investigated. The use of the method relies of the assumption that people often know how they would behave in actual situations of choice, and on the further assumption that the subjects have no special reason to disguise their true preferences. (Kahneman and Tversky, 1979, p. 265) In this paper, we directly address these issues by presenting subjects with simple choice tasks that

Abstract: A menu of paired lottery choices is structured so that the crossover point to the high-risk lottery can be used to infer the degree of risk aversion With normal laboratory payoffs of several dollars, most subjects are risk averse and few are risk loving Scaling up all payoffs by factors of twenty, fifty, and ninety makes little difference when the high payoffs are hypothetical In contrast, subjects become sharply more risk averse when the high payoffs are actually paid in cash A hybrid "power/expo" utility function with increasing relative and decreasing absolute risk aversion nicely replicates the data patterns over this range of payoffs from several dollars to several hundred dollars

Abstract: This paper develops a class of recursive, but not necessarily expected utility, preferences over intertemporal consumption lotteries An important feature of these general preferences is that they permit risk attitudes to be disentangled from the degree of intertemporal substitutability Moreover, in an infinite horizon, representative agent context these preference specifications lead to a model of asset returns in which appropriate versions of both the atemporal CAPM and the intertemporal consumption-CAPM are nested as special cases In our general model, systematic risk of an asset is determined by covariance with both the return to the market portfolio and consumption growth, while in each of the existing models only one of these factors plays a role This result is achieved despite the homotheticity of preferences and the separability of consumption and portfolio decisions Two other auxiliary analytical contributions which are of independent interest are the proofs of (i) the existence of recursive intertemporal utility functions, and (ii) the existence of optima to corresponding optimization problems In proving (i), it is necessary to define a suitable domain for utility functions This is achieved by extending the formulation of the space of temporal lotteries in Kreps and Porteus (1978) to an infinite horizon framework A final contribution is the integration into a temporal setting of a broad class of atemporal non-expected utility theories For homogeneous members of the class due to Chew (1985) and Dekel (1986), the corresponding intertemporal asset pricing model is derived