Short-Term Investments and Indices of Risk

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2007

A decision-maker observes sequentially a given permutation of n uniquely rankable options. He has to invest capital into these opportunities at the moment when they appear. At each step only relative ranks are known. At the end the true rank of the option, at which the investment has been made, is known. [Bruss and Ferguson(2002)] have considered such problems under the assumption that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return. Uninvested capital keeps its risk-free value. Wrong investments lose their value. In this paper we partially extend results by [Bruss and Ferguson(2002)]. We confine our study to linear utility but a wider range of payoffs is taken into account. Two cases are considered. The first-type payoff gives a positive rate of return if the investment is made on the best or the second best option. The second-type payoff pays when the investment is at the second best option. We motivate these payoff choices. A few examples are explicitly solved.

Oxford Bulletin of Economics and Statistics

This paper studies the long-term asset allocation problem of an investor with different risk aversion attitudes to the short and the long term. We characterize investor's preferences with a utility function exhibiting a regime shift in risk aversion at some point of the multiperiod investment horizon that is estimated using threshold nonlinearity methods. Our empirical results for a portfolio of cash, bonds and stocks suggest that long-term risk aversion is higher than short-term risk aversion and increases with the investment horizon. The exposure of the investment portfolio from stocks to bonds and cash increases with the degree of risk aversion.

SSRN Electronic Journal, 2015

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Finance Research Letters, 2012

We show that if an agent is uncertain about the precise form of his utility function, his actual relative risk aversion may depend on wealth even if he knows his utility function lies in the class of constant relative risk aversion (CRRA) utility functions. We illustrate the consequences of this result for optimal asset allocation: poor agents that are uncertain about their risk aversion parameter invest less in risky assets than wealthy investors with identical risk aversion uncertainty.

Journal of Industrial Engineering International

Loss-averse behavior makes the newsvendors avoid the losses more than seeking the probable gains as the losses have more psychological impact on the newsvendor than the gains. In economics and decision theory, the classical newsvendor models treat losses and gains equally likely, by disregarding the expected utility when the newsvendor is loss-averse. Moreover, the use of unbounded utility to model risk attitudes fails to explain some decision-making paradoxes. In contrast, this paper deals with the utility maximization of the newsvendor using a class of bounded utility functions to study the effect of loss aversion on the newsvendor certainty equivalents and risk premiums. New formulas are introduced to find the utility-optimal order quantity of the normal distribution. The results show that when an exponential loss aversion exists, the classical newsvendor optimal quantity serves as a lower bound when the overage costs are high and as an upper bound when the underage costs are high. In addition, we show that high loss aversion entails higher risk premiums. Similar conclusion holds when the overage/underage costs increase. Higher standard deviations, on the other hand, mean lower utility-optimal quantities and higher risk premiums. The presented formulas are advantageous in finding the optimal order quantities and risk premiums of a stochastic short-shelf life inventory when the loss is a key factor in the decision-making process.

This paper studies and describes stochastic orderings of risk/reward positions in order to define in a natural way risk/reward measures consistent/isotonic to investors' preferences. We begin by discussing the connection among the theory of probability metrics, risk measures, distributional moments, and stochastic orderings. Then, we demonstrate how further orderings could better specify the investor's attitude toward risk. Finally, we extend these concepts in a dynamic context by defining and describing new risk measures and orderings among stochastic processes with and without considering the available information in the market.

Journal of Economic Theory, 1996

We analyze the class of increasing utility functions exhibiting all derivatives of alternating sign. This property, that we call mixed risk aversion, is satisfied by the utility functions most commonly used in financial economics. The utility functions displaying mixed risk aversion can be expressed as mixtures of exponential functions. We characterize stochastic dominance and we find conditions for both mutual aggravation and mutual amelioration of risks when agents are mixed risk averse. Finally, the analysis of the distribution function describing a mixed utility allows one to characterize the behaviour of its indexes of risk aversion and to discuss its implications for portfolio selection.

Economic and Financial Decisions under Risk, 2011

Journal of Banking & Finance, 1988

This study extends the theoretical analysis and empirical research of risk aversion in securities markets. The analysis of the determinants of the market price of risk, part of an equilibrium modei of asset pricing, involves relative risk aversion and is carried out for the continuous time case. Micro relationships which are equilibrium demand functions of individual investors are derived; on the macro level the determinants of the market price of risk are determined. The analysis is carried out first assuming that a!! assets are marketable; then this assumption is relaxed and non-marketable assets (human-capital) are considered. Finally, we consider explicitly the effects of uncertain inflation on risk aversion. The major empirical results are: the assumption of constant relative risk seems to be a reasonable approximation of the market; secondly, the coefficient of relative risk aversion seems to be greater than unity; thirdly, for the first time trends in risk aversion were estimated. where r. is the return on an asset uncorrelated with the market return (a zero-beta asset), and Yk is the ratio of the kth individual wealth (wk) to total wealth W A specific form of the assumption of an infinitesimal planning horizon and no finite changes in value in an infinitesimal period, would imply for a finite interval a log normal distribution of returns.* If we assume that all wealth is invested in risky assets, i.e. W= K we get here an identical expression to the discrete case, except that y. is substituted for rf. It should be noted that the measure of absolute risk aversioa in the discrete case is replaced here by the relative measure, i.e. wkak =ck. Parallel to Friend and Blume (1975) we apply the continuous time solution to the situation in which not all wealth is invested in risky assets, but rather as in the discrete case, part of the wealth is invested in risk-free asset with a certain rate of return. *See Merton (19?3), I! Landskroner, Risk aversion in securities markets 133 We assume in common to all studies cited above homogenous expectations for all individuals. In the literature we find models focusing on differences in expectations or on differences in attitudes towards risk. The reason that different models focus on one or the other is that to understand how each of them works they are best studied in isolation-in our study, to understand the effects of differences in attitudes towards risk on portfolio allocation we assume that individuals have the same expectations.* We can write the wealth dynamics for individual k in stochastic difference equation form and then, by taking limits, in stochastic differential equation form. Thus for individual k, f&t +dr) = Wkr[ 1 + (1-ak)rf dt + a&, dt], (4) where LI indicates a random variable; t a point in time; ak the proportion invested in risky assets (the existence of 'many risky assets does not pose a problem where the separation theorem holds). First assume that the market rate of return ?,,,, il generated by a continuous Gaussian (Wiener) process:3

arXiv: Mathematical Finance, 2015

Under expected utility the local index of absolute risk aversion has played a central role in many applications. Besides, its link with the "global" concepts of the risk and probability premia has reinforced its attractiveness. This paper shows that, with an appropriate approach, similar developments can be achieved in the framework of Yaari's dual theory and, more generally, under rank-dependent utility.