Comparative statics under κ‐ambiguity for log‐Brownian asset prices

• DOI: http://doi.org/10.1111/ijet.12100
• Author: Dejian Tian,Weidong Tian
• Published date: 01 December 2016
• Date: 01 December 2016

doi: 10.1111/ijet.12100

Comparative statics under κ-ambiguity for log-Brownian

asset prices

Dejian Tian∗and Weidong Tian†

This paper examines the comparative statics of optimal risky demand when economic agents

are both risk averse and ambiguity averse. In a setting with log-Brownian asset prices and

κ-ambiguity but virtually for all utility functions, we show that the greater the Arrow–Pratt

coefficient of absolute risk aversion under ambiguity, the less the optimal demand for risky

assets. This monotonic property is demonstrated with varying risk-free interest rate and allows

for distinct estimated model parameters across agents.

Key wor ds comparative statics, risk aversion, κ-ambiguity

JEL classification D81, G11, G12

Accepted 26 February 2015

1 Introduction

Arrow(1963) and Pratt (1964) show that, in a static model, risk aversion affects the amount invested in

risky assets in a monotone manner.1Borell (2007) and Xia (2011) demonstrate similar monotonicity

properties for a log-Brownian asset price process.2This paper extends the analysis in Borell (2007)

and Xia (2011) on the optimal demand for a risky asset when an agent has κ-ambiguity in the sense

of Chen and Epstein (2002), a special type of Knightian uncertainty.

In the setting presented, the risky asset follows a log-Brownian price process, and an economic

agent’s risk preference is represented by a general von Neumann–Morgenstern utility function.

The agent also has Knightian uncertainty (or ambiguity) on the true probability distribution for

the excess return on the risky asset. The ambiguity is represented by κ-ignorance ambiguity,

where the constant κdenotes an ambiguity level. We demonstrate that both risk preference and

κ-ambiguity jointly affect the optimal demand for the risky asset. More precisely, the greater the

Arrow–Pratt coefficient of absolute risk aversion under κ-ambiguity, the less the optimal demand

∗School of Science, China University ofMining and Technology, Xuzhou, China. Email: djtian@cumt.edu.cn

†Belk College of Business, University of NorthCarolina at Charlotte, Charlotte, North Carolina. USA.

Weare very grateful to the editor and an anonymous referee for several constructive and insightful comments on how to

improve the paper.Dr Tian Dejian acknowledges financial support from the NSFC grants (11601509 and 11371362), and

NSF of Jiangsu Province grant (BK20150167).

1For an instance of its extensions in a one-period setting, see Athey (2002).

2More specifically, Borell (2007) considers a multi-asset setting and proves a relatively different monotonicity property,

that is, preservation over time of spatial properties of the risk tolerance. Xia (2011) examines the monotonicity properties

for a standard log-Brownian asset price process. This paper focuses on the comparativeanalysis on the optimal demand

or strategy when both the risk tolerance and ambiguity vary, by following an economic setting similar to Xia (2011).

International Journal of Economic Theory 12 (2016) 361–378 © IAET 361

International Journal of Economic Theory

Comparative statics under κ-ambiguity Dejian Tian and WeidongTian

for the risky asset.3We further extend this monotonic property of the optimal demand for the risky

asset when the risk-free interest rate changes.

Others have analyzed questions related to optimal demand under Knightian uncertainty in a

continuous-time setting. For instance, Schroder and Skiadas (1999) investigate the problem for a

large class of stochastic differential utilities. Hansen and Sargent (2001) study the effect of ambiguity

aversion on investment in a robust optimization approach. Uppal and Wang (2003) and Maenhout

(2004) examine the portfolio choice problem under uncertainty by using a robust approach. Ju

and Miao (2012) study a dynamic continuous-time infinite-horizon portfolio choice problem for

an agent with smooth ambiguity aversion. Despite a rich structure on the asset price process and

Knightian ambiguity in these papers, the economic agent’s utility functions usually have limited

specifications to ensure analytical expressions for the optimal demand.

In contrast, this paper provides an alternative approach in a continuous-time framework to

the comparative analysis on optimal demand under ambiguity. Our approach does not rely on the

specification of the agent’s risk preference and we allow parameter estimation uncertainty for each

agent. However, we impose a log-Brownian asset price process and a special version of ambiguity:

κ-ambiguity. In this setting, we obtain robust comparative statics for virtually all risk preferences,

because an analytical expression for the optimal demand is not necessary for comparative statics.

Therefore, this paper contributes to the comparative statics analysis under uncertainty by comple-

menting previous studies of that question.

Wepresent two main results in this paper. Theorem 1 considers two risk-averseand κ-ambiguit y-

averse economic agents; and both the expected return rate and the volatility parameter might be

different for these two agents, even though they invest in the same market.4We show that the product

of the volatility parameter and the optimal demand negatively depends on the corresponding Arrow–

Pratt coefficient of absolute risk aversion under ambiguity.In particular, a higher volatility prediction

ensures a smaller risky position. Moreover,if both agents have the same estimation of the risky asset’s

future price movement, Theorem 1 claims the monotonicity of the optimal demands: the higher the

Arrow–Pratt coefficient of absolute risk aversion under ambiguity for one agent, the less optimal the

demand for the risky asset by this agent. Both the risk preference and κ-ambiguity jointly affect the

optimal demand through the Arrow–Pratt coefficient of absolute risk aversion under ambiguity.

Theorem 2 extends Theorem 1 in which two agents have differentr isk-freeinterest rate environ-

ments. Wedemonstrate that the r isk-freeinterest rate for each agent interrelates with the agent’s risk

preference in this monotonic theory. When the risk-free interest rates are the same for both agents,

Theorem 2 presents the same monotonicity as in Theorem 1. In a different risk-free interest rate

environment, the comparative statics in Theorem 1 still holds when one agent has either increasing

or decreasing relative aversion, depending on the levels of the risk-free interest rates. Theorem 2

explicitly characterizes the condition on the volatility, the risk-free interest rate, the risk preference

and the excess return per unit (Sharpe ratio) under which one agent demands a higher risky posi-

tion than the other. In particular, when both agents have the same risk preference and κ-ambiguity,

Theorem 2 offers a qualitative analysis between changes in the risk-free interest rate.

While several remarkable approaches have been developed toinvestigate the economic situation

with risk and uncertainty,5we follow the multi-prior framework of Gilboa and Schmeidler (1989) in

3The Arrow–Pratt coefficient of absolute risk aversion under κ-ambiguity is defined precisely in Section 2.

4See Garlappi et al. (2007) and Easley and O’Hara (2009) for model parameter uncertainty and its applications to the

portfolio choice problem. In particular, Theorem 1 implies the comparativestatics result when the risky asset’s expected

return and volatility parameter vary.

5See, for instance, Hansen and Sargent (2001), Klibanoff et al. (2005) and Maccheroniet al. (2006) .

362 International Journal of Economic Theory 12 (2016) 361–378 © IAET