Downside Risk Aversion and the Downside Risk Premium

• DOI: http://doi.org/10.1111/jori.12241
• Date: 01 June 2018
• Author: Qi Zeng,Richard C. Stapleton
• Published date: 01 June 2018

©2018 The Journal of Risk and Insurance. Vol.85, No. 2, 379–395 (2018).

DOI: 10.1111/jori.12241

Downside Risk Aversion and the Downside Risk

Premium

Richard C. Stapleton

Qi Zeng

Abstract

We search for a definition of the downside risk premium analogous to the

Pratt–Arrow definition of the risk premium. However,even in the local anal-

ysis difficulties arise. To overcome these, we propose a definition based

on the difference between two gambles. Further, a global analysis reveals

that higher-order terms affect the downside risk premium and these can-

not be ignored. We show that all five measuresof the intensity of downside

risk aversion that have been suggested are invalid in the case of the global

analysis.

Introduction

Menezes, Geiss, and Tressler (1980, hereafter MGT) define increasing downside

risk, and show that a downside risk-averse agent is characterized by the condition

u(x)>0. As their initial contribution, numerous attempts have been made to charac-

terize the intensity of downside risk aversion. In the literature there are no fewer than

five different measures of the intensity of downside risk aversions, all with strong

economic motivations.1

In this article, we investigate how these measures of the intensity of downside risk

aversion relate to the downside risk premium. The downside risk premium is the

amount that a downside risk-averse agent would pay to avoid the downside risk. In

the risk aversion analysis by Pratt (1964) and Arrow (1965), the close link between

risk aversion and the risk premium plays a critical role in terms of understanding the

risk aversion measure. Similarly, we will define a downside risk premium that we

Richard C. Stapleton is at the University of Manchester. Stapleton can be contacted via e-mail:

richard.stapleton1@btopenworld.com, Tel: +44 1524 381 172. Qi Zeng is at the University of

Melbourne. Zeng can be contacted via e-mail: qzeng@unimelb.edu.au, Tel:+61 3 8344 5359. We

thank C. Gollier (the editor), three anonymous referees, N. Trech, H.Schlesinger, A. Snow, and

the participants in the 39th EGRIE annual seminar for helpful comments.

1Kimball (1990) shows that prudence −u(x)/u(x) explains precautionary saving, Modica and

Scarsini (2005) show that u(x)/u(x) is related to the local premium for skewness, Huang

(2000) shows that the cautiousness measure u(x)u(x)/u(x)2explains the demand for options,

Keenan and Snow (2002) propose the Schwarzian derivative, S(x)=D(x)−3

2u2(x)/u2(x), and

Liu and Meyer (2012) advocate the use of the rate of change of risk aversion: (−u(x)/u(x)).

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argue reflects the downside risk aversion of agents. Then, we will use this downside

risk premium concept to investigate the five measures of downside risk aversion that

have been proposed in the literature.

In the original MGT , “increasing” downside risk is defined using a mean–variance

preserving transformation (MVPT) between two risky bets. One risky bet is defined as

having more downside risk than the other. In this article, we compare the downside

risk of risky bets. This is in contrast to the definition of risk in Arrow (1965) and

Pratt (1964), where a risky bet is compared to a certain payoff. The definition of the

downside risk premium in this article is thus associated with the two risky bets.

The foremost task for any downside risk premium definition is to ensure that for a

downside risk-averse agent, namely, one with u(x)>0, the risk premium is posi-

tive. More importantly, by using local analysis, we may associate the downside risk

premium with one or more of the measures of downside risk aversion. Specifically,

we wish to show that, for two agents with different downside risk aversion, the more

downside risk-averse agent will pay a higher downside risk premium to avoid the

risk. Finally, we would like to show that, globally, the relationship still holds. This is

analogous to the process spelled out in the Pratt (1964) analysis of risk aversion and

the associated risk premium.

Having introduced our definition of the downside risk premium using the difference

between two risky bets, we show that, locally, the risk premium is closely associated

with the measure of downside risk aversion proposed in Modica and Scarsini (2005)

and Crainich and Eeckhoudt (2008): D≡u(x)/u(x). Using local analysis, we also

show why it is difficult to define the risk premium explicitly using only one risky bet.

In the following section, we use our definition to investigate the global case. Here,

we find that the above measure do not fit nicely with the risk premium. Furthermore,

using the example of the HARA utility function, we show that in the case of all

five suggested measures of downside risk aversion, the link between downside risk

aversion and the downside risk premium is violated for some numerical examples.

Of course, one possible reason is that our definition of the downside risk premium

is not the best definition. However, we argue that an alternative similar definition

do not work either.2More importantly, we argue in the later discussion section that

the source of the problem is more likely to be found in the higher-order derivatives

of the utility function, whose effect is not easy to disentangle from the effect of the

third-order derivative.

In the remainder of the article, we first define the downside risk premium and com-

pare it with the monetary compensation proposed in Crainich and Eeckhoudt (2008).

Then, using local analysis, we discuss the difficulty that arises when the downside

risk premium is linked to just one risky bet. Using the difference between the pre-

miums of two bets, we then link our definition of the downside risk premium to the

2Crainich and Eeckhoudt (2008) suggest a premium that they term “the monetary compensa-

tion.” In tests (not reported here) this definition also failed.