Moral-hazard premium
Takashi Misumi
Hitotsubashi University

Hisashi Nakamura∗
Hitotsubashi University

Koichiro Takaoka
Hitotsubashi University

September 26, 2014

Abstract
We provide an equilibrium asset-pricing formula under moral hazard, which is deﬁned as a ﬁrm’s strategic change ofmeasure that is incontractible, on the assumption of power utility and an endogenous riskless rate. Moral hazard distorts
equilibrium asset prices: the market price of diﬀusive risk is twisted in the opposite direction of an investor’s marginalutility; the market price of jump risk is ampliﬁed; a positive premium, called a moral-hazard premium here, is stipulated
on a riskless rate. Thus the risk-free rate puzzle is more serious under moral hazard. Also, ﬁnancial markets alleviate theallocation conﬂict caused by moral hazard.
Keywords: moral hazard, asset prices, diﬀusive risk, jump risk, riskless rate.
JEL Classiﬁcation Codes: D51, D82, G12.

1Introduction

It is well known that moral hazard is one of the most serious problems in ﬁnance. There exists a huge literature on moral hazard
in corporate ﬁnance (e.g. Tirole (2006, Section 3.2)). In a typical moral-hazard model, the probability of success of a ﬁrm’s project
depends on its manager’s hidden eﬀort, which is incontractible. An investor then needs to give the manager an incentive to avoid his
opportunistic misbehavior. Therefore, the moral hazard distorts optimal risk sharing and allocation. Still, such micro eﬀects are not
all the distortion causes in the ﬁnancial world. The moral hazard twists the investor’s marginal utility (i.e., pricing kernel) as well andthus aﬀects the valuation of not only the ﬁrm but also all other ﬁnancial assets in macro markets. How much return on investment
does the investor demand to compensate for a loss caused by moral hazard?
Surprisingly, however, there have been not so many studies of such valuation of moral hazard in asset pricing and ﬁnancialengineering (e.g. ﬁxed-income investment, the term structure of interest rates, corporate risk management, and actuarial insurance).
A notable exception is Ou-Yang (2005), who studies equilibrium asset pricing in the presence of moral hazard on the assumption of theexponential utility function and an exogenous constant riskless rate. Still, Ou-Yang (2005, p.1283) himself discusses the importance ofincorporating more general utility functions such as the power utility function. In addition, the assumption of the exogenous risklessrate seems to have limited its applicability to ﬁnancial practices.
The purpose of this paper is to provide an explicit asset-pricing formula in the presence of moral hazard on the assumption of
the power utility function and an endogenous riskless rate. Speciﬁcally, we incorporate moral hazard into a continuous-time generalequilibrium exchange economy. We solve the problem of the investor’s optimal consumption/wealth allocation in ﬁnancial markets
under two types of risk, i.e., regular (diﬀusive) risk and rare-event (negative jump) risk, when the process of production is subject to aﬁrm manager’s moral hazard. We then obtain explicitly an equilibrium state price, which can be used for pricing any kinds of securities
and loans in the presence of moral hazard.
The more speciﬁc model setup is as follows. There exist two players: the representative investor and a representative ﬁrm (i.e.,the ﬁrm manager) over [0, T ]. Both players rank their consumption based on time-separable utility with common time preference.
The investor’s instantaneous utility function is of a constant relative risk aversion (CRRA) type while the ﬁrm’s is of a log type. The
ﬁrm produces a single non-storable consumption good over time and shares it with the investor. No productive resources are utilized:
the production is an endowment, which stands for real macro outcomes (GDP). The ﬁrm can control ex-post the probability measure
by incurring eﬀort costs: moral hazard is deﬁned here as the ﬁrm’s ex-post, costly, strategic change of measure that is incontractible.
The endowment process is thus subject to moral hazard. The investor controls ex-post optimally her consumption/wealth with accessto ﬁnancial markets while designing ex-ante optimally an incentive contract for the ﬁrm so that the manager does not expect better
rewards from his opportunistic misbehavior. We look at the macroeconomic eﬀect of moral hazard.
This paper consequently provides an explicit asset-pricing formula under the moral-hazard problem. We obtain equilibrium state
prices (i.e., riskless rate and the market prices of diﬀusive risk and jump risk) and make clear the structural eﬀect of moral hazard on∗ This is a short version of our paper, namely Misumi et al. (2014), modiﬁed for the 2014 JARIP annual meeting proceedings. For thedetails of the paper, see the full version. Corresponding author: Hisashi Nakamura, Graduate School of Commerce and Management,
Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan. Phone: +81-42-580-8826. Fax: +81-42-580-8747. Email:hisashi.nakamura@r.hit-u.ac.jp. We would like to thank Jakša Cvitanić for his very helpful advice at an early stage of this research.
We are also thankful to Masaaki Fukasawa, Chiaki Hara, Tokuo Iwaisako, Hong Joo Jung, Masaaki Kijima, Yukinobu Kitamura,
Katsumasa Nishide, Kazuhiko Ohashi, Hideyuki Takamizawa, and participants at APRIA Moscow 2014 for their valuable comments.

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