# Schedule - Parallel Session 1 - Risk and Time Preferences 1

IDL Auditorium - 11:00 - 12:30#### The Magnitude Paradox

Helga Fehr-Duda; Thomas Epper

#### Abstract

It is a common observation that risk aversion increases with stake size, implying that the elasticity of the utility function is decreasing. In the domain of time discounting, however, we often find patience increasing with stake size, which implies the opposite characteristic of the utility function. These contradictory findings have been interpreted as evidence of two different utility functions governing behavior in the domains of risk and time. We show that the magnitude effects in risk taking and time discounting can be reconciled if decision makers perceive the future as inherently uncertain and are prone to Allais-type common-ratio violations. In our model, decreasing elasticity of the utility function over risky outcomes predicts patience increasing with stake size. Therefore, the unity of utility can be preserved, which is desirable when dealing with prospects that are both risky and delayed.

**Helga Fehr-Duda**

#### Dynamic Portfolio Choice in a Non-Expected Utility Framework

Jinrui Pan

#### Abstract

This paper considers the dynamic optimal portfolio choice problem, in a discrete-time and non-expected utility setting. Standard portfolio choice models often assume that preferences are represented by a von Neuman-Morgenstern utility function and individuals choose prospects so as to maximise the expectation of the utility of possible outcomes. Although the expected utility model has long been the standard for choice under risk and uncertainty, questions have been raised concerning its validity. Behaviour patterns which are inconsistent with expected utility theory have often been observed as in Allais (1953) and Kahneman and Tversky (1979). Numerous alternatives to the expected utility model have been developed in static settings, which incorporate a non-linear treatment of probabilities. In dynamic settings, this raises the question of how such treatment of probabilities develops over time. In this paper, the optimal portfolio choice is derived for an investor who behaves according to Rank Dependent Utility Theory for all the time periods concerned. A model featuring a distinction between risk and time preferences is adopted, where intertemporal rate of substitution is captured by a general discounting function independent of probabilities and outcomes, utility of outcomes is captured by standard vNM utility independent of time, and a two-parameter probability weighting function, namely the constant relative sensitivity (CRS) probability weighting function proposed by Abdellaoui, L’Haridon and Zank (2010), captures intertemporal probabilistic risk attitudes, with one parameter being constant over time, the other being time-dependent. An index of optimism is derived that depends on both parameters, which allows to model the observed high risk tolerance for delayed prospects. The possibility of diversification depends on such index.

**Jinrui Pan**

#### Prudent Discounting for Risky Times

Sebastian Ebert

#### Abstract

How should we discount outcomes that occur at random (‘risky’) times? Through a symmetry in the discounted expected utility (DEU) model’s treatment between such time risks and outcome risks, well-known results for utility functions carry over to discount functions. I start this research agenda by defining the concepts of prudent and temperate discounting in analogy to the influential behavioral traits of prudent and temperate utility. I characterize time risk preferences, i.e. preferences towards delay risk, through preferences over simple and intuitive time risk apportionment lotteries, through behavior in optimal stopping problems (e.g., prudent optimal stopping), a precautionary patience motive, and by signing the discount function’s derivatives of all orders. Finally, noting that the derivatives of commonly used discount and utility functions are consistently opposite in sign, it follows that the standard parametrizations of the DEU model imply an anti-symmetric behavior towards outcome and time risk if and only if outcomes are desirable.

**Sebastian Ebert**

#### Intertemporal Consumption with Risk: A Revealed Preference Analysis

Songfa Zhong; Joshua Lanier; Bin Miao; John Quah; Songfa Zhong

#### Abstract

Motivation: Many important economic decisions involve agents choosing among risky consumption streams. The canonical way of representing preferences in this context is to combine the expected utility and discounted utility models into what is known as the discounted expected utility (DEU) model. DEU functions are additive across dates and across states. While this model has the advantage of simplicity, it has a strong implication: the coefficient of relative risk aversion is the reciprocal of that of intertemporal substitution. Partly because this relationship has been repeatedly confounded by data, alternative models have been proposed by various authors (for example, Selden (1978), Epstein and Zin (1989), Chew and Epstein (1990) and Halevy (2015)). These models depart from DEU either by dispensing with separability across states or separability across time. In this paper, we report a new experiment that was designed to elicit preferences of subjects over risky consumption streams. Our objective is to cast light on the precise ways in which agents depart from the DEU model and to adjudicate among various alternative hypotheses on the structure of individual preferences. Experimental Design: Every subject in our experiment is allocated 100 tokens with which to purchase four commodities, where each commodity pays out in state s at time t, where there are two equiprobable states and two time points corresponding to one week and nine weeks later. By varying the prices of these commodities, we obtain more than 40 different budget sets, and every subject makes an allocation decision for each budget set. Analysis: Eliciting preferences from budgetary decisions is not an uncommon experimental practice, but ours is the first experiment in which subjects choose among affordable alternatives where payoffs vary in both state and time. At the most general, we can ask whether the subject is maximizing some utility function defined over the four state-time commodities. The most stringent hypothesis is to require the utility function to have the DEU form. Between these two extremes, one could posit that the utility function is separable over time or over states and (maybe) impose structural assumptions on the sub-utility functions or on the aggregator function. Throughout the paper, we apply (non-parametric) revealed preference methods to test these alternative hypotheses. Afriat’s Theorem provides the basic test of consistency with utility-maximization while more recently developed methods are employed to test for further restrictions on the utility function, such as separability (Quah, 2014) or additive separability (Quah, Polisson, Renou, 2015). Our results broadly support the separation of preferences across states but not across time. Furthermore, restricting the sub-utility function (defined over intertemporal consumption) to be the same in the two states and to display positive time preference is also consistent with the data.

**Songfa Zhong**