Exponential utility

Abstract: OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.

Abstract: We consider the problem of providing incentives over time for an agent with constant absolute risk aversion. The optimal compensation scheme is found to be a linear function of a vector of N accounts which count the number of times that each of the N kinds of observable events occurs. The number N is independent of the number of time periods, so the accounts may entail substantial aggregation. In a continuous time version of the problem, the agent controls the drift rate of a vector of accounts that is subject to frequent, small random fluctuations. The solution is as if the problem were the static one in which the agent controls only the mean of a multivariate normal distribution and the principal is constrained to use a linear compensation rule. If the principal can observe only coarser linear aggregates, such as revenues, costs, or profits, the optimal compensation scheme is then a linear function of those aggregates. The combination of exponential utility, normal distributions, and linear compensation schemes makes computations and comparative statics easy to do, as we illustrate. We interpret our linearity results as deriving in part from the richness of the agent's strategy space, which makes it possible for the agent to undermine and exploit complicated, nonlinear functions of the accounting aggregates.

Abstract: This paper tests implications of full consumption insurance. The object is to determine how much mileage can be obtained from a model with complete markets, with such features as private information or liquidity constraints omitted. The implication exploited is that individual consumption responds to aggregate risk but not to idiosyncratic risk. The test involves regressing the change in household consumption onto the change in aggregate consumption and other right-hand-side variables such as the change in household income and change in employment status. All variables other than the change in aggregate consumption are predicted to be insignificant in explaining the change in household consumption. With observations on consumption and income for 10,695 households from the Consumer Expenditure Survey, the results are mixed. The results for one specification (exponential utility) are mostly consistent with full consumption insurance; the results for the other specification (power utility) are not.

Abstract: We consider a firm that is faced with an uncontrollable stochastic cash flow, or random risk process. There is one investment opportunity, a risky stock, and we study the optimal investment decision for such firms. There is a fundamental incompleteness in the market, in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy, since the random risk process ensures that there is always a positive probability of ruin bankruptcy. We therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin. In particular, we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level. This policy also minimizes the probability of ruin. We prove that when there is no risk-free interest rate, this policy is equivalent to the policy that maximizes utility from terminal wealth, for a fixed terminal time, when the firm has an exponential utility function. This validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility. When there is a positive risk-free interest rate, the conjecture is shown to be false. We also solve for the optimal policy for the related objective of minimizing the expected discounted penalty paid upon ruin.