Asset pricing under quantile utility maximization

• Published date: 01 November 2013
• Author: Bruno C. Giovannetti
• Date: 01 November 2013
• DOI: http://doi.org/10.1016/j.rfe.2013.05.008

Asset pricing under quantile utility maximization☆

Bruno C. Giovannetti ⁎

Department of Economics, University of Sao Paulo, Brazil

abstractarticle info

Article history:

Received 14 September 2012

Received in revised form 29 March 2013

Accepted 28 May 2013

Available online 11 June 2013

JEL classification:

G11

G12

Keywords:

Asset prices

Downside risk

Quantile utility

“Focus on the downside, and the upside will take care of itself”is a famous quote among professional inves-

tors. By considering an agent who follows this advice, we reproduce the first and second moments of stock

returns, risk-free rate and consumption growth. The agent's behavior toward risk is analogous to a relative

risk aversion of about 3 under expected utility, the elasticity of intertemporal substitution is about 0.5 and

the time discount factor is below 1. In particular, the proposed model separates time and risk preferences

in an innovative way.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

A famous quote among professional investors is “Focus on the

downside, and the upside will take care of itself”. In this paper, we

consider a consumer–investor who follows this advice. Surprisingly,

the consumption-based asset pricing model that emerges from this

idea explains the main existing puzzles found within the asset pricing

literature. These include the equity premium and the risk-free

rate puzzles, the countercyclicality of the equity premium and the

procyclicality of the risk-free rate.

In the proposed model, the consumer–investor is concerned with

the so-called downside risk. This is done by replacing the standard

setting of expected utility optimizing agents with the concept of

quantile utility. Under this framework, the agent summarizes a risky

situation using a worst-case scenario which is a function of his down-

side risk aversion. The more downside risk averse the agent, the

worse the worst-case scenario he considers. The τquantile of a

continuous random variable can be interpreted as the worst possible

outcome that can occur with probability 1 −τ. Hence, instead of

maximizing the expected value of his utility function, the agent max-

imizes a given τquantile of it. As we will see, τdefines his downside risk

aversion:the lower τ, the higher the down side risk aversion.

1

The crucial difference between quantile and expected utility is

straightforward. Under expected utility, an agent, when facing a

situation where he has to choose among uncertain alternatives,

picks the one that maximizes the expected value of his utility func-

tion. However, under quantile utility, the agent picks the one that

maximizes some given quantile of the utility distribution, instead of

its mean. For instance, the given quantile can be the median of the

utility distribution, or the 0.25 quantile. In the case of the 0.25

quantile for example, when evaluating an uncertain situation, he

looks at the worst outcome that can occur with 75% probability (i.e.,

the chance of the realized scenario being better than the scenario he

considers is 75%).

The choice rule of quantile utility was axiomatized by Rostek

(2010). It nests the famous maxmin and maxmax decision criteria. In-

deed, decision makers who select an alternative that offers the

highest minimal or maximal payoff can be viewed as maximizing

the lowest or the highest quantile, respectively. Maxmax and maxmin

have been applied in broad literature, as game theory, robust control,

individual and social choice, bargaining, and voting. However, these

criteria have been commonly criticized for basing choice on what

may be extreme and unlikely outcomes (maxmin agents would not

invest, would not drive, and so on). The quantile utility captures

more moderate preferences while preserving the qualitative properties

Review of Financial Economics 22 (2013) 169–179

☆I would like to thank Dennis Kristensen for invaluable support on this work. I also

thank Andrew Ang, Pierre-André Chiappori and Marcelo Moreira for constant and cru-

cial advice, and Ricardo Brito, John Donaldson, Guilherme Martins, Marcos Nakaguma,

Walker Hanlon, Ricardo Reis, Bernard Salanié, and the participants at the Columbia

Econometrics Colloquium, Applied Micro Colloquium and Finance Colloquium, and

the Economics Seminars at Fundação Getúlio Vargas (EESP), Insper and Universidade

de São Paulo for important comments and discussions. All errors left are only mine.

⁎Tel.: +55 11965253300.

E-mail address: bcg@usp.br.

1

One could say that the agent's objective function is given by the value at risk (VaR)

of his utility. However, since τhere is a free parameter defining preference toward risk,

it is not restricted to being close to zero (as in standard VaR applications).

1058-3300/$ –see front matter © 2013 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.rfe.2013.05.008

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Review of Financial Economics

journal homepage: www.elsevier.com/locate/rfe