Short-Term Investments and Indices of Risk

Abstract

We study various decision problems regarding short-term investments in risky assets whose returns evolve continuously in time. We show that in each problem, all risk-averse decision makers have the same (problem-dependent) ranking over short-term risky assets. Moreover, in each problem, the ranking is represented by the same risk index as in the case of CARA utility agents and normally distributed risky assets. JEL Classification: D81, G32.We analyze four decision problems that are important in economic settings. In general, different risk-averse agents rank the desirability of gambles differently. However, our main result shows that in each of these problems, all risk-averse agents have the same (problem-dependent) ranking over short-term investments in risky assets whose returns evolve continuously. Moreover, in each problem, the ranking is represented by the same risk index obtained in the commonly used mean-variance preferences (e.g., Markowitz, 1952), which are induced by CARA utility agents and normally distributed gambles. Brief Description of the ModelWe consider an agent who has to make an investment decision related to a gamble. We think of a gamble as the additive return on a financial investment. We assume that the agent has (1) an initial wealth w, and (2) a von Neumann-Morgenstern utility u that is increasing and risk-averse (i.e, u ′ > 0 and u ′′ < 0). We assume that a gamble is represented by a random variable with (1) positive expectation, and (2) some negative values in its support. For each problem the agents' choices are modeled by a decision function that assigns a number to each agent and each gamble, where a higher number is interpreted as the agent finding the gamble to be more attractive (i.e., less risky) for the relevant decision problem.We study four decision problems in the paper: (1) acceptance/rejection, in which the agent faces a binary choice between accepting and rejecting the gamble (e.g., Hart, 2011);(2) capital allocation, in which the agent has a continuous choice of how much to invest in the gamble (e.g., Markowitz, 1952; Sharpe, 1964); (3) the optimal certainty equivalent, in which the agent evaluates how much an opportunity to invest in the gamble (according to the optimal investment level) is worth to the agent (e.g., Hellman & Schreiber, 2018); and (4) risk premium, in which the agent evaluates how much investing in the gamble is inferior to obtaining the gamble's expected payoff (Arrow, 1970). 1 A risk index is a function that assigns to each gamble a nonnegative number, which is interpreted as the gamble's riskiness. We say that a risk index is consistent with a decision function f over some set of agents and gambles, if each agent in the set ranks all gambles in the set according to that risk index; that is, f assigns for each agent a higher value for gamble g than for gamble g ′ iff the risk index assigns a lower value to g. A risk-aversion index is a function that assigns to each agent a non-negative number, which is interpreted as the agent's risk aversion. We sa...