Generalized Multiplicative Risk Apportionment

Hongxia Wang

doi:10.3390/risks7020065

Abstract

This work examines apportionment of multiplicative risks by considering three dominance orderings: first-degree stochastic dominance, Rothschild and Stiglitz’s increase in risk and downside risk increase. We use the relative nth-degree risk aversion measure and decreasing relative nth-degree risk aversion to provide conditions guaranteeing the preference for “harm disaggregation” of multiplicative risks. Further, we relate our conclusions to the preference toward bivariate lotteries, which interpret correlation-aversion, cross-prudence and cross-temperance.

Highlights

Since Eeckhoudt and Schlesinger (2006) showed the simple lottery pair characterization for such concepts of risk apportionment as risk prudence and temperance by the idea of “harm disaggregation”, plenty of both theoretical and empirical/experimental research has explored preferences over various lottery pairs.Consider the statistically independent random variables Xi, Yi, i = M, N. Eeckhoudt et al (2009a) looked at a preference toward the additive 50–50 lottery [ X N + YM, YN + X M ] and 50–50 lottery [ X N +X M, YN + YM ] to extend Eeckhoudt and Schlesinger’s (Eeckhoudt and Schlesinger 2006) work by defining relatively “good” and “bad” via stochastic dominance
Among others, developed previous results to the bivariate framework, most researchers focus on additive lottery pairs
Ekern (1980) generalized a F ( y ) dy and G ( z ) = G ( z ), G. These dominance orderings to nth-degree risk increase, which corresponds to first-degree stochastic dominance (FSD), Rothschild and Stiglitz’s increase in risk (RSIR), and downside risk increase (DRI) when n = 1, 2, 3, respectively

Summary

Introduction

Since Eeckhoudt and Schlesinger (2006) showed the simple lottery pair characterization for such concepts of risk apportionment as risk prudence and temperance by the idea of “harm disaggregation”, plenty of both theoretical and empirical/experimental research has explored preferences over various lottery pairs. Let u( x ) be the utility function of decision-maker, denote as RRA(n) the coefficient of relative nth-degree risk aversion ( n +1). We present our model and explain multiplicative risk apportionment in terms of both “harms disaggregation” (Eeckhoudt et al 2009b) and “mutual aggravation of risk changes” (Ebert et al 2018). Ekern (1980) generalized a F ( y ) dy and G ( z ) = G ( z ), G these dominance orderings to nth-degree risk increase, which corresponds to FSD, RSIR, and DRI when n = 1, 2, 3, respectively. F dominates G via nth-degree risk increase if and only if risk aversion decision-maker u( x ). That is, following Ebert et al (2018), we can explain A B by borrowing terminology “mutual aggravation” Kimball (1993) of multiplicative risk changes rather than “harm disaggregation”

Main Result

Relate to Multiplicative Risk Apportionment

Relate to Preferences over Bivariate Lottery Pairs

Conclusions