Can cooperative game theory solve the low‐risk puzzle?

• DOI: http://doi.org/10.1002/ijfe.1696
• Author: Tobias Hiller,Benjamin R. Auer
• Published date: 01 April 2019
• Date: 01 April 2019

Received: 21 February 2018 Revised: 25 July 2018 Accepted: 10 September 2018

DOI: 10.1002/ijfe.1696

RESEARCH ARTICLE

Can cooperative game theory solve the low-risk puzzle?

Benjamin R. Auer1,2,3 Tobias Hiller3

1Brandenburg University of Technology

Cottbus-Senftenberg, Chair of Finance,

Cottbus, Germany

2Research Network Area Macro, Money

and International Finance, CESifo

Munich, Munich, Germany

3Departments of Finance and

Microeconomics, University of Leipzig,

Leipzig, Germany

Correspondence

Benjamin R. Auer,Chair of Finance,

Brandenburg University of Technology

Cottbus-Senftenberg, Erich-Weinert-Str. 1,

03046 Cottbus, Germany.

Email: auer@b-tu.de

JEL Classification: G10; G11; C71

Abstract

In this article, we illustrate that cooperative game theory may havethe potential

to solve the low-risk puzzle, which has become one of the most important in

modern finance because its implication of a negative risk-return trade-off poses a

challenge to traditional models of asset prices. Using several simulation settings,

we highlight that quantifying risk by means of assets' Shapley values, that is,

assets' contributions to overall portfolio risk, instead of classic measures supplies

a (more) positive risk-return relationship in manypractically relevant cases. This

is partially attributable to the fact that the game-theoretic risk measure captures

more investment-relevant information than commonly used alternatives.

KEYWORDS

cooperative game theory, low-risk puzzle, Shapley value

1INTRODUCTION

Empirical research has revealed an effect in the cross

section of asset returns, which is now known as the

low-risk puzzle. According to this phenomenon, invest-

ment opportunities with low risk consistently tend to out-

perform their high-risk counterparts (see Ang et al., 2006,

2009; Dutt and Humphery-Jenner,2013; Frazzini and Ped-

ersen, 2014). Because it holds across different markets and

asset classes and seriously challenges asset pricing theo-

ry's traditional notion of a positive risk-return trade-off,

thelow-riskpuzzleisconsideredtobeoneofthemost

important capital market anomalies discovered so far.

Although many authors have tried to explain the puz-

zle in the context of restricted borrowing, preferences for

assets with lottery-like payoffs, or fund managers' man-

dates to beat fixed benchmarks (see Blitz et al., 2014), a

more straightforward rationale might be that researchers

are simply using “the wrong measure of risk” (see Baker

et al., 2011, p. 43). Previous studies have relied on the stan-

dard deviation of returns (see Dutt and Humphery-Jenner,

2013), idiosyncratic volatility (see Ang et al., 2006), or beta

(see Frazzini and Pedersen, 2014) to document the puzzle

and to point out its relevance for investors.1Inother words,

they have shown that these measures produce risk rank-

ings of individual assets that are inconsistent with the idea

that investors should be rewarded for bearing risk. How-

ever, all these measures have serious limitations because

they do not fully capture the risk that is actually relevant

for typical investment decisions and/or because they are

bound to restrictive theoretical assumptions. For example,

measures of total risk (such as the standard deviation) treat

risk in an isolated fashion and thus do not consider poten-

tial diversification benefits in a portfolio context. Further-

more, popular measures of systematic risk (such as beta)

can cover diversification issues (see Fama and French,

2004) but are bound to an unobservable market portfolio

(see Roll, 1977), which is not held by most investors (see

De Bondt, 1998). In other words, they capture how an asset

contributes to the risk of a portfolio that is not relevant

from a practical perspective.

We argue that a measure of risk, which can be derived

from cooperative game theory, has the potential to solve

the low-risk puzzle. We suggest ranking assets based on

their Shapley (1953) contribution to the risk of an invest-

ment portfolio. This is because the process of portfolio

884 © 2018 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/ijfe IntJ Fin Econ. 2019;24:884–889.