Return Ambiguity, Portfolio Choice, and Asset Pricing∗

Yu Liu Hao Wang Lihong Zhang†

January 25, 2018

Abstract

We formulate a continuous-time rational expectations model that considers

return ambiguity and derives closed-form portfolio allocation and asset pricing rules.

We propose an economically motivated ambiguity penalty function through which

ambiguity aversion plays an “independent” role in affecting an agent’s utility and

identify the respective, interactive effects of return ambiguity and risk on decision

rules. Our Lucas-style power utility model simultaneously reconciles annual and

quarterly U.S. equity premia, risk-free rates, and consumption features with a

reasonable risk aversion coefficient of five. The calibration shows that about 23%,

41%, and 36% of the U.S. equity premium is attributable to risk, return ambiguity,

and their interaction, respectively.

JEL Classification: G11, G12.

Keywords: Return ambiguity, portfolio choice, asset pricing, equity premium puzzle,

risk-free rate puzzle.

∗We would like to thank Lars Peter Hansen, Zhongfei Li, Tan Wang, Yan Zeng, Hao Zhou, DongmingZhu, and seminar participants at the Chinese Academy of Sciences, Peking University, Sun Yat-senUniversity, and Tsinghua University for helpful discussions. All remaining errors are ours. LihongZhang acknowledges funding support from the National Natural Science Foundation of China (Grant No.71471099).†All authors are from Tsinghua University, School of Economics and Management. Liu can be reached

at liuy5.13@sem.tsinghua.edu.cn. Wang can be reached at wanghao@sem.tsinghua.edu.cn or +86 1062797482. Zhang can be reached at zhanglh2@sem.tsinghua.edu.cn or +86 10 62789963.

Return Ambiguity, Portfolio Choice, and Asset Pricing

Abstract

We formulate a continuous-time rational expectations model that considers return

ambiguity and derives closed-form portfolio allocation and asset pricing rules. We propose

an economically motivated ambiguity penalty function through which ambiguity aversion

plays an “independent” role in affecting an agent’s utility and identify the respective,

interactive effects of return ambiguity and risk on decision rules. Our Lucas-style power

utility model simultaneously reconciles annual and quarterly U.S. equity premia, risk-free

rates, and consumption features with a reasonable risk aversion coefficient of five. The

calibration shows that about 23%, 41%, and 36% of the U.S. equity premium is attributable

to risk, return ambiguity, and their interaction, respectively.

JEL Classification: G11; G12.

Keywords: Return ambiguity, portfolio choice, asset pricing, equity premium puzzle,

risk-free rate puzzle.

1 Introduction

We formulate a continuous-time rational expectations model that considers ambiguity

in the drift function of the asset return process and derive analytical portfolio and asset

pricing rules. We find that, on the one hand, return ambiguity independently affects

consumption, asset allocation, and asset pricing. On the other hand, it interacts with

risk in shaping these decision rules. Return ambiguity helps the Lucas (1978)-style power

utility model simultaneously reconcile annual and quarterly U.S. equity premia, risk-free

rates, and consumption features with a reasonable relative risk aversion (RRA) coefficient

of five, which presents a sensible channel to resolve the asset pricing puzzles. In addition,

the calibration shows that 23%, 41%, and 36% of the U.S. equity premium is attributable

to risk, return ambiguity, and their interaction, respectively.

Continuous-time rational expectations models have been extensively used to study

portfolio choice and asset pricing. A fundamental assumption underlying the models is that

agents have perfect knowledge about the probability law governing the stochastic process of

asset returns. However, the true model is often unknown and any specified probability laws

would be subject to potential misspecification. Model ambiguity (Knightian uncertainty)

then arises because there is no probability distribution available to describe the potential

misspecification due to lack of information and knowledge. Return ambiguity constitutes

one prominent source of ambiguity, since the first moment of asset returns is extremely

difficult to estimate (Merton, 1980; Blanchard, 1993; Cochrane, 2001; Maenhout, 2004).

In our model, we quantitatively describe return ambiguity as an ambiguity-induced

adjustment to the drift function of the asset return process. Compared to the notion of

relative entropy that is used by prior works to describe model uncertainty,1 our ambiguity

1In statistics, relative entropy (also called the Kullback–Leibler divergence) is a measure of how oneprobability distribution diverges from an expected probability distribution. Relative entropy plays afundamental role in information theory, continuous-time stochastic process models, and the examinationof statistical models of inference. See Hansen and Sargent (2001), Anderson et al. (2003), Uppal and Wang(2003), and Maenhout (2004) for applications of relative entropy as the primary quantity of (Knightian)model uncertainty for robust control. In their models, relative entropy, as a Radon–Nikodym derivativebetween a reference “approximating” probability measure and a statistically indistinguishable alternativeprobability measure, is projected via a penalty function to the drift function of the asset return process.

1

quantity is straightforward and more general in terms of not requiring the “ambiguity-

neutral” and “ambiguity-averse” probability measures to be equivalent and, hence, it can

be applied to general stochastic asset return processes.2

Although we focus on examining return ambiguity in this paper, we also develop a

general framework to study ambiguity and ambiguity aversion in economic and financial

decision making. This framework is applicable to general stochastic processes for general

utilities in exploring the economic implications of various sources of ambiguities, such

as volatility ambiguity, and allows for analytical solutions. In particular, we propose an

economically motivated quadratic ambiguity penalty that presents an explicit channel for

ambiguity and ambiguity aversion to play an “independent” role in affecting the agent’s

expected utility,3 and follow Hansen and Sargent (2001) to impose the ambiguity penalty

directly to the objective function in reflecting the agent’s skeptical and conservative

perspective. Thus, our model can identify the respective effects of return ambiguity,

risk, and their interaction on optimal portfolio choice and asset pricing. For example,

we decompose equity premium into the risk premium, ambiguity premium, and joint

risk–ambiguity premium, which offers a quantifiable solution to the equity premium puzzle

from the perspective of ambiguity.

The negative impact of return ambiguity on portfolio allocation to the risky asset is

economically significant. Numerical analysis with reasonable parameter values shows that,

for the case of constant RRA (CRRA) utility, the optimal fraction of wealth allocated

to the risky asset falls from 67% to 10% as the degree of ambiguity aversion increases

from zero to a reasonable level of 90 (see Section 3.1 for details), which helps explain why

The difference between our direct adjustment and relative entropy as the primary model ambiguityquantity is subtle but substantial. More discussions on the difference are presented in Section 2.2.

2Two probability distributions are equivalent if they are absolutely continuous with each other; inother words, they completely agree on the events with zero probability. This technical requirement limitsapplications of relative entropy to a subclass of stochastic processes. See Maenhout (2004) and Epsteinand Ji (2013) for more discussions.

3Independence between risk and ambiguity can be illustrated using an Ellsburg paradox type of ballgame. There are two urns containing white or black balls. The player is (truthfully) informed that allballs in each of the urns are of the same color and asked to pick one white ball from one of the two urns.In this situation, there is ambiguity about the colors of the balls in the urns, while there is no risk becausethe probability of white balls in one urn is either zero or one.

2

people do not try to hold more stocks that generate a substantially higher average return

than government bonds over a long horizon (Munk, 2013; Cochrane, 2017). In addition,

return ambiguity adversely affects consumption. Reduction in investment in the risky

asset leads to slower expected wealth accumulation, so the agent is less willing to borrow

from future wealth to boost current consumption, which, in turn, implies a lower risk-free

rate in equilibrium.

The evidence shows that return ambiguity fundamentally affects asset pricing. Tradi-

tional power utility models, such as that of Lucas (1978), have problems in reconciling

historical asset returns and stylized consumption features with plausible levels of risk

aversion, leading to the equity premium puzzle and the risk-free rate puzzle (Mehra and

Prescott, 1985; Weil, 1989). We calibrate our Lucas-style power utility model to the

same century-long annual and post-war quarterly samples of U.S. consumption, equity

premia, and risk-free rates used by Campbell (1999) and Maenhout (2004). A reasonable

RRA coefficient of five is required to simultaneously reconcile both samples. Thus, the

evidence suggests that return ambiguity gives rise to a sensible channel to resolve the

equity premium and risk-free rate puzzles. The calibration implies a pure equity risk

premium (worst-case equity premium) of 1.44%, which is comparable to the pessimistic

equity premium of 2% (the fifth percentile) estimated by financial economists (Welch, 2000).

In addition, calibrating the model to the pessimistic equity premium with both annual

and quarterly samples generates highly robust estimates of the risk aversion coefficient,

discount rate, and consumption rate.

In the setting of our model, the asset premium can be decomposed into the risk premium,

ambiguity premium, and joint risk–ambiguity premium. Campbell and Cochrane (1999)

question why people do not exploit the asymptotic arbitrage opportunities emerging due to

trading inertia when asset return volatility is extremely low while a positive asset premium

persists. The ambiguity premium helps answer their question: when trading inertia arises,

ambiguity remains and is likely to be high, since the information is extremely chaotic.

Investors would demand substantial ambiguity premia for holding risky assets, ruling

3

out arbitrage opportunities, even in the absence of risks. Generally, return ambiguity

accounts for 41% of the equity premium, 2.56% of 6.26% in our annual sample. The

figure is even higher for the quarterly sample, since ambiguity could be more prominent

over a short horizon. In addition, 36% of the equity premium is attributable to risk and

ambiguity jointly, suggesting that return ambiguity and risk tend to interact and reinforce

each other in determining asset prices. A further implication is that return ambiguity

perhaps plays a very influential but overlooked role in shaping financial markets. One

could seriously underestimate the consequences of financial shocks, such as a financial

crisis, on investors’ behaviors and asset prices if one considering only the risk effects while

ignoring the ambiguity effects.

This paper belongs to the model ambiguity literature and follows the max-min expected

utility theory pioneered by Gilboa and Schmeidler (1989). The study is particularly related

to the work of Uppal and Wang (2003) and Maenhout (2004) in exploring the implications

of return ambiguity on portfolio allocation and asset pricing.4

Our paper has novel features and makes unique contributions. Uppal and Wang (2003)

and Maenhout (2004) describe model (Knightian) uncertainty with the notion of relative

entropy that is a Radon–Nikodym derivative between two equivalent probability measures,

while we quantitatively describe return ambiguity as an adjustment to the drift function

of the asset return process. Our ambiguity quantity, unlike relative entropy, does not

require equivalence between ambiguity-neutral and ambiguity-averse measures and, hence,

can be applied to any stochastic processes to study other sources of ambiguities, such as

volatility ambiguity. Our model straightforwardly imposes an ambiguity penalty on the

objective function that specifies how ambiguity aversion independently affects an agent’s

4There are two separate but related lines of research following the seminal work of Gilboa and Schmeidler(1989). Hansen and Sargent (2001), Anderson et al. (2003), Uppal and Wang (2003), and Maenhout (2004)focus on the robustness of state variable processes. Another branch of research focuses on multiplier utilityfunctions. Among them, Epstein and Wang (1994) analyze the asset pricing implications of max-minutility in discrete time. Chen and Epstein (2002) provide a continuous-time extension. Uppal and Wang(2003) examine the implications of ambiguity for underdiversification. Klibanoff et al. (2005) and Ju andMiao (2012) propose a recursive utility model involving ambiguity to resolve asset pricing puzzles. Miaoand Rivera (2016) study contract theory with regard to ambiguity.

4

expected utility. The approach not only substantially simplifies the techniques for solving

the model in closed form (without relying on the Girsanov theorem to change probability

measures), but also enables one to examine the respective effects of ambiguity, risk, and

their interaction on the decision rules. Challenging the conventional wisdom, we show

that a large portion of the equity premium is indeed attributable to return ambiguity. To

our knowledge, this is the first rational expectations model with return ambiguity that

can simultaneously reconcile annual and quarterly U.S. equity premia, risk-free rates, and

consumption features with levels of risk aversion within the reasonable range given by

Mehra and Prescott (1985) and Munk (2013).

This paper enriches the asset pricing literature, showing both theoretically and em-

pirically that return ambiguity constitutes a fundamental asset pricing factor. Previous

research tends to resolve asset pricing puzzles from the angle of risk and risk preferences.

For example, Bansal and Yaron (2004) and Barro (2006) investigate different risk processes,

while Epstein and Zin (1989), Abel (1990), Constantinides (1990) and Campbell and

Cochrane (1999) explore various specifications of risk preferences. However, the existence

of multiple risk models and lack of consensus imply that economists are uncertain about

the true model. As depicted by Cochrane (2017), “one observer’s fragile assumption is

another observer’s well-identified parameters (page 959).” Our paper examines the puzzles

from the perspective of asset return ambiguity, which is inherent in many economic and

financial models. The results indicate that return ambiguity could play a profound yet

to-be-discovered role in economics and finance.

The remainder of the paper is organized as follows: Section 2 presents the model with

return ambiguity. Section 3 examines how return ambiguity affects the portfolio and

consumption rules. Section 4 explores the implications of return ambiguity on equilibrium

asset pricing and the asset pricing puzzles. Section 5 concludes the paper.

5

2 Model Setup

This section formulates a continuous-time rational expectations model that considers return

ambiguity and ambiguity aversion. It first presents the ambiguity-neutral reference model

and then introduces return ambiguity and ambiguity aversion to the decision process.

2.1 Reference Model

Consider the simplest continuous-time model, as Merton (1971) does: a representative

agent consumes a single good and invests in a risk-free asset that generates a constant

risk-free rate of return, r, and a risky asset whose price, St, follows the following geometric

Brownian motion:

dStSt

= µdt+ σdZt,

where µ and σ are constants and denote the expected asset return and return volatility,

respectively, and Zt denotes standard Brownian motion. At time t, the agent maximizes

expected utility, expressed in the following objective function:

V (W, t) = sup{C,π}

E

(∫ ∞t

e−δsU(Cs)ds∣∣∣Wt = W

)(2.1)

s.t. dWt = [Wt (r + πt(µ− r))− Ct] dt+ πtWtσdZt,

where U(Ct) denotes the agent’s utility function; Ct and πt denote consumption and the

fraction of wealth, Wt, allocated to the risky asset at time t, respectively; and δ denotes a

strictly positive discount rate. An implicit assumption underlying the model is that the

agent has perfect knowledge about the probability law governing the asset return process.

The Hamilton–Jacobi–Bellman (HJB) equation for the agent’s utility maximization

problem is

sup{C,π}{U(C)− δV + VW [W (r + π(µ− r))− C] + Vt + 1

2VWWπ

2W 2σ2}

= 0, (2.2)

6

with boundary condition V (W,∞) = 0, where Vx and Vxx denote the first- and second-order

partial derivatives of V with respect to x, respectively.

2.2 Return Ambiguity

It is the case that the true asset return process is unknown and the agent worries that

the expected asset return, µ, is subject to model misspecification. Given the agent’s

information about the economy as represented by the asset return process, the agent’s

problem is to determine how to take into account possible misspecification in decision

making. To reflect the agent’s skeptical and conservative perspective, the agent imposes

an ambiguity penalty to the objective function, turning the expected utility problem in

Equation (2.1) into the following max-min expected utility problem:

sup{C,π}

inf{u}

E

[∫ ∞t

[U(Cs) + P (u)] e−δsds∣∣∣Wt = W

](2.3)

s.t. dWt = [Wt(r + (µ+ u− r)πt)− Ct] dt+ πtWtσdZt,

where P represents the ambiguity penalty function. It constitutes a channel through which

return ambiguity and ambiguity aversion play a central role in affecting the agent’s utility.

Without this explicit ambiguity penalty, return ambiguity can only affect the agent’s

utility through the preexisting risk channel and, then, its influence on the decision rules

would depend entirely on the specification of the agent’s utility function and the degree

of risk aversion. Relative to risk, ambiguity exists independently and should, intuitively,

play at least a partially independent role in determining the ambiguity-averse agent’s

objective. The penalty function facilitates such a role. In this sense, the ambiguity penalty

effectively transforms the objective function from an ambiguity-neutral setting to an

ambiguity-averse one. The functional form of the ambiguity penalty is presented below.

Parameter u is an endogenous ambiguity adjustment to the drift of the asset return process,

µ. The adjustment quantitatively describes the magnitude of the return ambiguity. The

infinimization over u reflects the agent’s aversion to ambiguity; that is, the agent chooses

the adjustment that leads to the lowest expected utility.

7

In a nutshell, the penalty function should specify with a clear economic rationale the

dynamics through which return ambiguity and ambiguity aversion affect the agent’s utility.

The penalty must be of the same unit as the objective function and the effects of ambiguity

preferences on the decision rules will not fade out as the scales of the state variables

increase (Uppal and Wang, 2003; Maenhout, 2004). Parsimony and analytical solutions

are always desirable model features. Taking the above-mentioned features into account,

we propose the following quadratic functional form:

P (u) =

(∂V0∂µu)2

2θ|V0|, (2.4)

where θ represents the degree of the agent’s ambiguity aversion based on the agent’s

information about the reference model. A higher θ implies that the agent is more ambiguity

averse or possesses less information about the reference model and has less confidence in

it, which, in turn, leads to a more significant ambiguity adjustment, u. For the extreme

case in which θ = 0, the agent is ambiguity neutral or has perfect information about

(full confidence in) the reference model, the max-min expected utility problem reduces to

the expected utility problem specified in Equation (2.1). Generally, the agent balances

knowledge about the economy as represented by the reference model and concerns about

model misspecification. The agent wants neither to completely forgo the information of the

reference model by setting θ ≈ ∞ nor to overlook the possibility of model misspecification

by setting θ = 0. We include θ in the denominator of the equation to ensure that a greater

ambiguity adjustment (in the numerator) is made to justify the penalty as the agent

becomes more ambiguity averse or is less confident in the reference model. For a given θ,

the agent chooses the level of u that yields the lowest expected utility.

The term V0 is the ambiguity-neutral value function in Equation (2.1). The derivative

∂V0∂µ

represents the sensitivity of the value function with respect to the drift of the asset

return process and ∂V0∂µu describes by how much the agent’s utility would change due

to return ambiguity. The more sensitive the value function with respect to the drift,

8

the stronger the effects return ambiguity would have on the objective function and,

subsequently, the decision rules. The derivative ∂V0∂µ

not only constitutes an economic

dynamic through which return ambiguity and ambiguity aversion independently affect the

agent’s utility via the ambiguity adjustment u, but also naturally takes the features of

the reference model into account via V0. We use the functional form of(∂V0∂µu)2

to ensure

non-negativity and for analytical convenience. We then divide it by |V0| to make the units

of the penalty function consistent with those of the objective function.

It is important to note that we use ∂V0∂µ

rather than the derivative of the ambiguity-averse

value function, ∂V∂µ

, in specifying the penalty, for the following considerations. Logically,

the purpose of the ambiguity penalty is to present by how much V deviates from its

ambiguity-neutral benchmark V0 due to return ambiguity. Therefore, the penalty should

be specified according to how the ambiguity-induced return adjustment u changes V0

rather than how it changes V . In other words, since V already carries the effects of u,

specifying the penalty as a function of ∂V∂µu would recursively count the effects of u on V ,

which could lead to biased quantitative implications for the decision rules. In addition, an

important advantage of our specification is that V0 is exogenous to the penalty function,

so it helps separate the effect of ambiguity attitude from that of the ambiguity impact

channel ∂V0∂µ

on the decision rules. Mathematically, V0 is well defined, so we do not need to

prove that it satisfies necessary conditions, such as having a valid solution, to specify the

penalty function.

The HJB equation for the ambiguity-averse agent’s max-min expected utility problem

specified in Equation (2.3) is

sup{C,π}

inf{u}

[U(C) +

(∂V0∂µu)2

2θ|V0|− δV + VW [W (r + π(µ+ u− r))− C] + Vt +

1

2VWWπ

2W 2σ2

]= 0,(2.5)

with boundary condition V (W,∞) = 0.

Let

A =W |V0|(∂V0∂µ

)2 .9

Solving the infinimization part of Equation (2.5) yields

u∗ = −πθAVW , (2.6)

where the sign of u∗ depends on the sign of π. In particular, π is positive if µ > r; that

is, the agent takes a long position in the risky asset if it generates an expected return

above the risk-free rate. In this reasonable case, the agent tends to adjust the expected

risky asset return downward if the agent is uncertain about it. The magnitude of the

ambiguity adjustment u increases with the degree of the agent’s ambiguity aversion or

as the agent has less information about the reference model. In addition, the magnitude

of u increases with the fraction of wealth allocated to the risky asset, π, suggesting that

the ambiguity-averse agent would be increasingly conservative and pessimistic about the

expected asset return as the agent demands more the risky asset. Substituting Equation

(2.6) back into Equation (2.5) turns the HJB equation into

sup{C,π}

[U(C)− δV + VW [W (r + π(µ− r))− C] + Vt +

1

2VWWπ

2W 2σ2 − 1

2AθV 2

Wπ2W

]= 0.(2.7)

The first-order conditions with respect to C and π are, respectively,

UC − VW = 0, (2.8)

VWW (µ− r) + VWWW2σ2π − AθV 2

WπW = 0. (2.9)

Solving for the optimal consumption and portfolio choice rules, we obtain the following

proposition:

Proposition 1. For an ambiguity-averse agent with a general utility function, the optimal

consumption satisfies the following envelope condition:

UC

∣∣∣C=C∗

= VW . (2.10)

10

The optimal fraction of wealth allocated to the risky asset is

π∗ =VW (µ− r)

−VWWWσ2 + AθV 2W

. (2.11)

Substituting Equation (2.6) into Equation (2.9) solves for

π∗ =VW (µa − r)−VWWWσ2

, (2.12)

where µa is the ambiguity-adjusted expected asset return,

µa = µ+ u.

There are two complementary ways of illustrating that the optimal asset allocation π∗

has exactly the same functional form as its counterpart in the ambiguity-neutral model but

is different due to return ambiguity. First, as shown in Equation (2.11), the difference is that

−VWWWσ2 +AθV 2W replaces −VWWWσ2 in the denominator. The terms −VWWWσ2 and

AθV 2W represent the respective effects of risk and ambiguity. Both are positive, suggesting

that, like risk, ambiguity adversely affects the optimal fraction of wealth allocated to the

risky asset. Second, as depicted by Equation (2.12), the ambiguity-adjusted drift of µa

replaces µ in the asset allocation rule. The effects of return ambiguity and ambiguity

aversion are entirely captured by the ambiguity-adjusted expected asset return. Overall,

an ambiguity-averse agent would appear to be more risk averse compared to an otherwise

identical but ambiguity-neutral agent in making asset allocation decisions. The dynamic is,

however, not that return ambiguity simply amplifies volatility. Ambiguity and ambiguity

aversion influence the decision rule not only by altering the wealth function, but also via

the additional ambiguity penalty.

11

2.3 Special Case of Power Utility

When the agent has the power (CRRA) utility function, U(C) = C1−γ

1−γ , and γ 6= 1, according

to Merton (1971), the HJB equation specified in Equation (2.2) can be solved to obtain

V0 =a−γ

1− γW 1−γ, (2.13)

where

a =1

γ

[δ − (1− γ)r − (1− γ)(µ− r)2

2γσ2

]. (2.14)

The optimal controls are

C∗0 = aW,

π∗0 =µ− rγσ2

and the sensitivity of V0 with respect to the parameter µ is

∂V0∂µ

=(µ− r)γσ2

a−γ−1W 1−γ.

In the presence of return ambiguity, let λ = W |V0|

( ∂V0∂µ )2W−γ = AW−γ, then

λ =a2+γγ2σ4

|1− γ|(µ− r)2. (2.15)

Note that λ is a strictly positive function, which does not involve the state variable W

and the control variables π and C. Substituting Equation (2.14) into Equation (2.15), we

rewrite

λ =aγ{

[δ − (1− γ)r]σ2 − (1−γ)(µ−r)22γ

}2

|1− γ|(µ− r)2

=aγ{

[δ − (1− γ)r]2 σ2 − (1−γ)(µ−r)2[δ−(1−γ)r]γ

}|1− γ|(µ− r)2

σ2 +aγ|1− γ|(µ− r)2

4γ2(2.16)

, λ1σ2 + λ2,

12

where

λ1 =aγ{

[δ − (1− γ)r]2 σ2 − (1−γ)(µ−r)2[δ−(1−γ)r]γ

}|1− γ|(µ− r)2

; (2.17)

λ2 =aγ|1− γ|(µ− r)2

4γ2. (2.18)

In addition, λ1 6= 0 and λ2 6= 0 if σ = 0.

With the help of the notation and results, we obtain the following proposition:

Proposition 2. For an agent with power utility U(C) = C1−γ

1−γ and γ 6= 1, Equations (2.10)

and (2.11) can be solved, conjecturing that V = κ−γ

1−γW1−γ. The optimal controls are

C∗ = κW, (2.19)

π∗ =µ− r

κ−γλθ + γσ2, (2.20)

respectively, where λ is a strictly positive parameter given by Equation (2.16) and κ is a

strictly positive solution to the equation

κ =1

γ

[δ − (1− γ)r − 1

2(1− γ)(µ− r)π∗

]. (2.21)

We can rewrite Equation (2.20) as

π∗ =µ− r

[γ + κ−γλ1θ]σ2 + κ−γλ2θ, (2.22)

where λ1 and λ2 are given by (2.17) and (2.18), respectively.

In the setting of our model, return ambiguity and volatility drive the asset allocation

rule with very different dynamics. Their effects are both independent and interactive.

Proposition 2 and Equation (2.16) show that return ambiguity affects the asset allocation

rule through two channels. One channel involves its interaction with volatility, captured

by κ−γλ1θσ2. Another channel is more independent, represented by κ−γλ2θ. Overall,

its effects do not entirely depend on the level of risk or the degree of risk aversion. For

a reasonable level of σ and µ > r, π∗ approaches zero as θ increases to high but not

13

implausible levels. When the agent is extremely ambiguity averse or, given the agent’s

information, has very little confidence in the expected asset return, the agent will optimally

demand no risky asset, resulting in no participation in risky asset markets and offering

an ambiguity-based explanation to the phenomena, such as trading inertia and liquidity

shortage, observed during financial crises.

We obtain the following proposition on the existence and uniqueness of the solutions

presented in Proposition 2:

Proposition 3. For an agent with power utility U(C) = C1−γ

1−γ and γ 6= 1, Equations (2.20)

and (2.21) have unique solutions if one of the following conditions is satisfied:

(1) 0 < γ ≤ 1 and 1− 1γδ + 1−γ

γr > 0 and

(2) γ > 1 and the consumption rate κ < γ−12γ .

See the Appendix for a proof of Proposition 3. Numerically, for a historical U.S. annual

risk-free rate of 1.96%, an equity premium of 6.26%, equity return volatility of 18.53%,

and δ = 0.14, condition (1) is satisfied if γ ≥ 0.125. Condition (2) is satisfied for the

reasonable range of 2 < γ < 10 according to Mehra and Prescott (1985) and Munk (2013)

because reasonable levels of κ are substantially below the implied constraints of 84–89%.

Therefore, conditions (1) and (2) are satisfied for reasonable asset returns, consumption

features, and preference parameter values and Equations (2.20) and (2.21) have unique

solutions.

3 Implications for Consumption and Asset Alloca-

tion Rules

This section analyzes the implications of return ambiguity for optimal portfolio and

consumption rules for the special case of CRRA utility. It presents a decomposition of

the asset premium into the risk premium, the joint risk–ambiguity premium, and the

ambiguity premium. A comparison with Maenhout’s (2004) model concludes this section.

14

3.1 Portfolio Choice and Consumption

We start this section with the following proposition:

Proposition 4. For an agent with power utility U(C) = C1−γ

1−γ and γ 6= 1, if Equations

(2.20) and (2.21) have unique solutions as shown in Proposition 3, the following relations

are obtained:

(1) The optimal allocation π∗ is a decreasing function of the degree of ambiguity aversion

θ, that is, dπ∗

dθ< 0 when µ > r. The optimal allocation π∗ is an increasing function of the

degree of ambiguity aversion θ, that is, dπ∗

dθ> 0 when µ < r.

(2) The optimal consumption rate κ is a decreasing function of the degree of ambiguity

aversion θ, that is, dκdθ< 0 when γ > 1. The optimal consumption rate κ is an increasing

function of the degree of ambiguity aversion θ, that is, dκdθ> 0 when γ < 1.

See the Appendix for a proof.

Numerical analysis shows that the negative effects of return ambiguity on consumption

and asset allocation are economically significant. Table 1 shows that, for µ > r, π∗

monotonically decreases as θ increases. For the risk aversion coefficient γ = 4 and other

reasonable parameter values, the value of π∗ decreases from 66.67% to 10.02% as θ increases

from zero to 90.5

[Insert Table 1 here.]

The worst-case asset premium, denoted EPA, equals the ambiguity-adjusted risky asset

return minus the risk-free rate, that is, EPA = µ+ u∗ − r, which reflects the ambiguity-

averse agent’s pessimistic perception of the expected asset return. It can also be regarded

5The values of θ used in Table 1 are substantially lower than those estimated in the calibrationsreported in Table 2. The reason is that, in the calibrations, we follow the literature and the implicationsof the Lucas (1978) model to use the covariance between consumption growth rates and equity returns tosubstitute for the equity return variance. As shown in Table 2, the covariances are substantially lowerthan the variances, which requires high levels of θ to reconcile with the empirical data. Untabulatedresults of calibrations with the equity return variances estimate the values of θ at around 90 (and almostthe same levels of the risk aversion coefficient γ as reported in Table 2). Therefore, we use θ = 90 to beconsistent with the use of asset return volatility in the numerical analysis.

15

as the pure risk premium of the risky asset. The worst-case asset premium decreases as

the degree of ambiguity aversion increases. For reasonable parameter values, EPA declines

from 6.00% to 0.92% as θ increases from zero to 90. The agent effectively expects a much

lower expected asset return when making consumption and portfolio allocation decisions.

The consumption rate declines as the degree of ambiguity aversion rises. For γ = 4

and other reasonable parameter values, the value of κ decreases from 3.63% to 2.35% as θ

increases from zero to 90. Ambiguity aversion reduces investment in the risky asset and,

subsequently, the speed of future wealth accumulation, because the risk-free asset generates

lower expected returns than the risky asset does (µ > r). The agent intends to maximize

utility by smoothing out consumption over time, since consumption shortfalls deliver more

pain than surpluses give pleasure. However, the agent tends to borrow less from future

wealth to boost current consumption in this situation. Asset allocation decisions appear

to be more sensitive to changes in ambiguity (aversion) than consumption decisions are,

which is coherent with the notion that consumption growth rates are less volatile than

stock market returns.

3.2 Asset Premium Decomposition

Substituting Equation (2.20) into Equation (2.6) leads to

u∗ = −π∗λθκ−γ = −π∗λ1σ2θκ−γ − π∗λ2θκ−γ. (3.23)

The ambiguity-induced adjustment is determined by the degree of ambiguity aversion.

Risk and risk aversion affect the adjustment through λ and κ, respectively. Ambiguity

and risk tend to interact with each other in determining asset prices. The asset premium

can be expressed as

µ− r = −u∗ + γσ2π∗ = π∗{[γ + κ−γλ1θ

]σ2 + κ−γλ2θ

}. (3.24)

16

Equation (3.24) demonstrates that, given optimal consumption and asset allocation, the

risky asset premium can indeed be decomposed into three parts: (1) the pure risk premium,

that is, γσ2π∗; (2) the pure ambiguity premium, that is, π∗κ−γλ2θ; and (3) the joint

risk–ambiguity premium, that is, π∗κ−γλ1θσ2. The impact of return ambiguity does not

entirely depend on the level of risk, which is represented by σ. This finding provides

an explanation to the question of Campbell and Cochrane (1999) on why investors do

not exploit asymptotic arbitrage opportunities that emerge due to trading inertia when

the level of volatility is low while a positive asset premium persists. When inertia arises,

ambiguity persists and could be high, since information is ambiguous. Investors would

demand high ambiguity premia for holding risky assets, which justifies the positive risky

asset premia and rules out arbitrage, even in the absence of risks. Generally, the finding

suggests that the ambiguity premium could play a fundamental but largely overlooked role

in justifying empirical equity premia and offer an ambiguity-based solution to the equity

premium puzzle documented in prior literature.

3.3 Relation between Consumption and Asset Allocation

Both risk and return ambiguity simultaneously affect consumption and asset allocation.

Do they affect the relation between consumption and asset allocation differently? For the

case of CRRA utility, the reference model implies the following consumption–investment

relation:

da

dπ∗0=

dadσdπ∗

0

dσ

= −(1− γ)(µ− r)2γ

. (3.25)

This shows that the consumption rate increases with the optimal fraction of wealth

allocated to the risky asset for the reasonable case of µ > r and γ > 1. The sensitivity of

consumption with respect to portfolio allocation is entirely determined by the risky asset

premium µ− r and the degree of risk aversion γ. Asset risk (volatility) has no impact on

17

sensitivity. Considering return ambiguity, we obtain the following expression:

dκ

dπ∗=

dκdθdπ∗

dθ

= −(1− γ)(µ− r)2γ

,

which is identical to Equation (3.25). The result suggests that return ambiguity, like asset

risk, does not alter the sensitivity of the consumption rate with respect to asset allocation.

3.4 Remarks on Maenhout (2004)

This work is inspired by that of Maenhout (2004). We conclude this section with a

comparison between Maenhout’s model and ours, which generalizes Maenhout’s.

Maenhout carries out the investigation in the setting of CRRA utility, while we design

a model for general utilities. In the special case of CRRA utility, the HJB equation,

Equation (2.7), differs from that of Maenhout (2004) in the last term. In Maenhout (2004),

the last term is

1

2

θσ2

(1− γ)VV 2Wπ

2W 2.

The last term of Equation (2.7) is

1

2AθV 2

Wπ2W =

1

2

θ|V0|(∂V0∂µ

)2V 2Wπ

2W 2.

The above equations show that, in Maenhout’s model, ambiguity aversion, θ, is attached

to σ. The effects of ambiguity preferences crucially depend on the level of risk and would

be significantly undermined if σ were small. In our model, return ambiguity and ambiguity

aversion play an independent role through the new ambiguity penalty function. Here,

∂V0∂µ6= 0 if σ = 0, suggesting that ambiguity would affect the decision rules even in the

absence of risk. This finding is intuitive, since ambiguity exists independently of risk and

should play a role independent of risk. For example, the independence of ambiguity premia

helps rule out asymptotic arbitrage opportunities emerging from trading inertia when

asset return volatility tends to zero and a positive asset premium persists.

18

Let us further zoom in on the difference between the above equations, that is, σ2

(1−γ)V

versus |V0|

( ∂V0∂µ )2 , which describes how ambiguity affects the objective functions in the respective

models. For the case of CRRA utility, in Maenhout (2004),

σ2

(1− γ)V= σ2aγMW

γ−1,

where aM = 1γ

[δ − (1− γ)r − (1−γ)(µ−r)2

2(γ+θ)σ2

]. In our model,

|V0|(∂V0∂µ

)2 =γ2σ4aγ+2

(1− γ)(µ− r)2W γ−1.

To facilitate the analysis, we combine the two equations in the form of their ratio:[δ − (1− γ)r − (1−γ)(µ−r)2

2(γ+θ)σ2

]γ[δ − (1− γ)r − (1−γ)(µ−r)2

2γσ2

]γ+2

(1− γ)(µ− r)2

σ2.

For Maenhout (2004), the complete effects of ambiguity aversion are realized by amplifying

risk aversion, whereas, in our model, ambiguity aversion affects the agent’s utility through

different dynamics from those of volatility and risk aversion. Ambiguity preferences

are both independent and interactive with risk preferences. Parameter θ does not show

up in the denominator of the equation, suggesting that the ambiguity aversion θ and

the ambiguity impact channel |V0|

( ∂V0∂µ )2 are explicitly separate in our model. This setup

helps single out the unique effects of ambiguity aversion, given that ∂V0∂µ6= 0 for any

value of σ. In contrast, the ambiguity aversion θ is mixed with the ambiguity impact

channel σ2

(1−γ)V in Maenhout’s model. The difference is partly due to our use of ∂V0∂µ

rather

than ∂V∂µ

in specifying the ambiguity penalty, which intends to capture by how much the

ambiguity-averse value function V deviates from its ambiguity-neutral benchmark V0 due

to return ambiguity. Specifying the penalty as a function of V0 instead of V not only

avoids recursively counting the effect of V on itself, but also helps to disentangle the

unique effects of ambiguity aversion on the decision rules.

19

4 Implications for Asset Pricing

This section examines the implications of return ambiguity for equilibrium asset pricing.

The traditional continuous-time power utility model encounters challenges in reconciling

the historical (high) equity premium, (low) risk-free rate, and stylized consumption features

with plausible levels of risk aversion, leading to the equity premium puzzle (Mehra and

Prescott, 1985) and the risk-free rate puzzle (Weil, 1989). We investigate the extent to

which return ambiguity helps to resolve these prominent asset pricing puzzles. We first

describe our Lucas (1978)-style power utility model that accounts for return ambiguity

and then calibrate the model to annual and quarterly U.S. equity premia, risk-free rates,

and consumption data.

4.1 Asset Pricing Model

As Lucas (1978), we consider a simple exchange economy that has two assets, one risky

and one risk free. The agent consumes dividends paid by the risky asset. The dividend

process follows geometric Brownian motion:

dDt

Dt

= µDdt+ σDdZt, (4.26)

where the constants µD and σD denote dividend growth rate and the volatility of dividend

growth, respectively; µD > 0; and σD > 0. The risky asset is a claim to the dividend

stream, so its value follows geometric Brownian motion:

dSt +Dtdt

St= µdt+ σdZt, (4.27)

where µ and σ denote the expected asset return and return volatility, respectively. Following

the literature, we impose the following equilibrium conditions: (1) all dividends are

consumed in continuous time, that is, C∗ = D, and (2) the share of wealth allocated to the

risky asset, π∗, equals one. The markets clear continuously if the equilibrium conditions

are satisfied.

20

The agent worries that the expected asset return, µ, is subject to potential misspecifi-

cation. For robust pricing rule, the agent considers a worst-case alternative model:

dSt +Dtdt

St= (µ+ u∗)dt+ σdZt, (4.28)

where u∗ denotes ambiguity-induced adjustment to the drift function as in Equation (3.23).

The worst-case risky asset premium, which reflects the agent’s skeptical and conservative

perspective, is expressed as

EPA = Eu∗

t

(dSt +Dtdt

St

)− r = µ+ u∗ − r, (4.29)

where Eu∗t [·] is the conditional expectation under the ambiguity-averse probability measure

for the given information at time t. Thus, the agent’s wealth is expressed as

dW = [W (r + π(µ+ u∗ − r))− C] dt+ σπWdZt. (4.30)

Substituting Equations (2.19) and (2.20) into Equation (4.30), we rewrite the wealth as

dW

W= (µ− κ)dt+ πσdZt. (4.31)

The instantaneous market clearing condition of π∗ = 1 implies that

µ− r = λθκ−γ + γσ2.

Given that u∗ = −λθκ−γ in this case, we obtain

EPA = µ+ u∗ − r = µ− λθκ−γ − r.

The instantaneous market clearing condition of C∗ = D implies that σ2D = σ2

C = σCS,

where σC and σCS denote consumption growth volatility and the covariance between risky

asset returns and consumption growth, respectively. In traditional consumption-based

asset pricing models, the equity premium is determined by σCS. Hence, we follow the

21

literature (e.g. Mehra and Prescott, 1985; Maenhout, 2004) and replace σ2 with σCS in

the equations and obtain the following proposition:

Proposition 5. An agent considers an ambiguity-neutral asset pricing model such as

that of Lucas (1978) and an ambiguity-averse model as specified in Equation (2.3). The

ambiguity-neutral model implies an equity premium

EP ∗R = µ− r = κ−γλθ + γσCS =[γ + κ−γλ1θ

]σCS + κ−γλ2θ (4.32)

and the ambiguity-averse model implies a worst-case equity premium

EP ∗A = µ− κ−γλθ − r = γσCS, (4.33)

where EP ∗A represents the risk premium. The difference between EP ∗R and EP ∗A represents

the equity premium that is related to ambiguity, which can be decomposed into the ambiguity

premium, κ−γλ2θ, and the joint risk–ambiguity premium, κ−γλ1θσCS.

4.2 Asset Pricing Puzzles

To shed light on how ambiguity about expected asset return affects equilibrium asset

pricing and subsequently the prominent asset pricing puzzles, we calibrate our Lucas

(1978)-style model introduced above to the same set of U.S. asset price and consumption

data used by Campbell (1999) and Maenhout (2004). The data consist of a century-long

annual sample from 1891 to 1994 and a post-war quarterly sample from 1947 to 1996.

Panel A of Table 2 reports the input parameter values.

[Insert Table 2 here.]

For comparison, we first calibrate the ambiguity-neutral model to estimate the RRA

coefficient, γ, and the discount rate, δ, by matching the model-implied risk-free rate

and equity return to their empirical counterparts for the annual and quarterly samples,

respectively. We follow the model to impose the following restrictions in the calibration:

γ > 0 and δ > 0.

22

For the annual sample, Column (1) of Panel B of Table 2 reports that the ambiguity-

neutral model requires a risk aversion coefficient γ = 21 and a discount rate δ = 0.34 to

reconcile the equity premium of 6.26% and risk-free rate of 1.96%. The value of γ is the

same as that of Maenhout (2004), confirming that the traditional CRRA utility model

needs implausibly high levels of risk aversion to reconcile with empirical asset prices and

consumption features. The RRA coefficient, γ, is the inverse of intertemporal substitution

elasticity in the setting of CRRA utility models. A high value of γ implies rigid elasticity

and that the agent wants desperately to smooth consumption over time. That would

drive up the risk-free rate, since the agent has a stronger incentive to borrow from his

richer future to boost current consumption. To keep the model-implied risk-free rate down

to match its empirical level, an excessively high discount rate δ is needed. The result

echoes the risk-free rate puzzle; that is, the model-implied risk-free rate would otherwise

be significantly higher than its empirical counterpart.

Calibration to the post-war quarterly sample yields even more puzzling results. Column

(2) shows that the estimated risk aversion coefficient γ = 247, the same as for Maenhout

(2004). Much stronger risk aversion is necessary for the ambiguity-neutral model to

generate an equity premium of 7.85%, which is higher than its annual counterpart, and

a risk-free rate of 0.79%, which is lower than its annual counterpart. In addition, the

correlation between the quarterly consumption growth rates and equity returns, ρ, and

the quarterly consumption growth rate volatility, σC , are much lower than their annual

counterparts. All these features demand a higher degree of risk aversion and a greater

discount rate for reconciliation. The large inconsistency between the estimated parameter

values for the annual and quarterly samples implies that the traditional model could be

nontrivially misspecified, calling for consideration of model ambiguity in robust asset

pricing rules.

We calibrate our ambiguity-averse model in two ways. First, we let the model simulta-

neously fit the annual and quarterly asset prices and consumption data. That means we

estimate a single set of parameter values γ and θ to match both samples. We estimate

23

the δ values for the annual and quarterly samples, respectively. In the calibration, the

parameters are estimated by simultaneously matching the model-implied annual and

quarterly equity premia, EPR, and the risk-free rates, r, to their respective empirical

values. We would like to give the model the maximum level of freedom and follow the

model properties to impose only the following restrictions: γ > 0, θ > 0, and δ > 0.

Second, we consider a target for the worst-case equity premium of 2% based on the

pessimistic equity premium estimated by financial economists (Welch, 2000) and the

worst-case equity premium estimated by Maenhout (2004).6 For the annual sample, the

calibration estimates the parameters γ, θ, and δ by matching the model-implied asset

returns EPR, r, and EPA, respectively, to their empirical values or targets. For calibration

to the quarterly data, we use the annual sample calibration-estimated θ for consistency of

ambiguity preferences. Calibration to the quarterly data estimates γ and δ by matching

the model-implied EPR and EPA, respectively, to their empirical values or targets. Again,

we only require γ > 0, θ > 0, and δ > 0 in the calibrations.

The results show that return ambiguity fundamentally affects asset pricing and generates

a sensible channel to resolve the equity premium puzzle and the risk-free rate puzzle.

Columns (3) and (4) of Table 2 report the results of the first calibration. The estimated

risk aversion coefficient γ is five, which is below the reasonable upper boundary of 10

(Mehra and Prescott, 1985) and within the reasonable range of two to five (Munk, 2013).

The estimated degree of ambiguity aversion θ is 585 and the discount rate δ is 11.7%. The

model-implied worst-case equity premium is 1.44%, slightly below the pessimistic equity

premium of 2% (the fifth percentile) estimated by financial economists (Welch, 2000). The

implied consumption rates are 6.5% and 6.7%, respectively, for the annual and quarterly

samples.

For the second calibration, Column (5) of Table 2 reports that, for the annual sample,

the estimated risk aversion coefficient γ equals seven and the estimated degree of ambiguity

6Welch (2000) reports 2% as the average pessimistic estimation (lower bound of the 95% confidenceinterval) of the equity premium estimated by financial economists. Maenhout (2004) reports the worst-caseequity premium of 2.1% in a calibration using the same data as ours.

24

aversion θ is 1708 when calibrating the model to EPA equal to 2%. Comparing these

results to those reported in Columns (3) and (4), we find the degree of risk aversion γ and

the degree of ambiguity aversion θ appear to be sensitive to the level of EPA. In particular,

parameter θ, which also represents the agent’s information about or confidence in the

expected equity return, changes substantially as the worse-case equity premium changes

from 1.44% to 2%. For the quarterly sample, we set θ = 1708 to preserve consistency of

ambiguity preferences. Column (6) reports that the estimated risk aversion coefficient γ

equals six and the implied discount rate δ is 0.144. It is noteworthy that the different

calibrations with annual and quarterly data generate quantitatively consistent values of

the discount rate δ around 14%.

Return ambiguity appears to play a fundamental role in asset pricing. Challenging

the conventional wisdom, it accounts for 41% of the equity premium: 2.56% of 6.26%

in our annual sample. In addition, 36% of the equity premium is jointly attributable to

risk and ambiguity. An important implication is that ambiguity could play an extremely

influential role in affecting financial markets, especially during financial crises. Besides its

own influence, it could dramatically amplify the effects of risk shocks. One could seriously

underestimate the impact of financial crises or other economic shocks on investment

behaviors and asset prices if taking only risk effects into consideration while ignoring

ambiguity effects.

5 Conclusion

Asset return ambiguity (Knightian uncertainty) is inherent in the rational expectations

models. This paper develops a new framework to consider return ambiguity in portfolio and

asset pricing decision rules. There are two innovative features: The first involves imposing

directly to the objective function an economically motivated ambiguity penalty function

through which ambiguity preferences independently affect the decision rules. In the second,

we use an ambiguity-induced adjustment to the drift function of the asset return process

25

as a quantity of direct return ambiguity. Compared to the notion of relative entropy that

is used by prior works to describe model uncertainty, this ambiguity quantity is more

direct and general in terms of not requiring the ambiguity-neutral and ambiguity-averse

probability measures to be equivalent, so it can be applied to any stochastic asset return

processes.

We find that return ambiguity plays a fundamental role in consumption, portfolio choice,

and asset pricing decisions. One important contribution of the paper is its decomposition

of the asset premium into the risk premium, the ambiguity premium, and the joint risk–

ambiguity premium. The ambiguity premium helps to address the well-known question

of Campbell and Cochrane (1999) of why investors do not exploit asymptotic arbitrage

opportunities emerging from trading inertia when asset return volatilities tend to zero but

positive asset premia persist. The ambiguity premium helps justify the asset premium and

rule out arbitrage, even in the absence of risk. Generally, return ambiguity enables the

Lucas (1978)-style power utility model to simultaneously reconcile annual and quarterly

U.S. asset returns and consumption features with plausible levels of risk aversion.

Perhaps more importantly, return ambiguity could potentially have profound yet to-

be-discovered implications for economics and finance. To facilitate the exploration, we

formulate a general framework that is applicable to any stochastic processes for general

utilities and can be utilized to investigate the economic implications of various sources of

ambiguities with analytical solutions.

26

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28

A Appendix

Proof of Proposition 3.

This section proves that Equations (2.20) and (2.21) have unique solutions.

Substituting Equation (2.20) into Equation (2.21) yields

κ− δ

γ+

1− γγ

r +(1− γ)(µ− r)2

2γ(λθκ−γ + γσ2)= 0. (A-1)

Let φ = κ−γ and define

f(φ) = φ−1γ − δ

γ+

1− γγ

r +(1− γ)(µ− r)2

2γ(λθφ+ γσ2).

Then

f ′(φ) =

(−1

γ

)φ−

1γ−1 +

λθ(γ − 1)(µ− r)2

2γ(λθφ+ γσ2)2.

Recall that κ ∈ (0, 1) and γ > 0; therefore, φ ∈ (1,+∞). Then we know that f(φ) is

continuous with respect to φ ∈ (1,+∞) and, further,

f(1) = 1− 1

γδ +

1− γγ

r +(1− γ)(µ− r)2

2γ(λθ + γσ2),

and

f(+∞) = −1

γδ +

1− γγ

r.

Recall the ambiguity-neutral (Merton) model. We know the rational consumption rate

a ∈ (0, 1). According to Equation (2.14), we have

γ > δ − (1− γ)r − (1− γ)(µ− r)2

2γσ2> 0. (A-2)

This is basic assumption in an ambiguity-neutral model.

In the following, we investigate the results in two different cases:

1. 0 < γ ≤ 1.

The inequality (A-2) yields

δ − (1− γ)r >(1− γ)(µ− r)2

2γσ2> 0,

29

which means that

f(+∞) = −1

γ[δ − (1− γ)r] < 0.

With the assumption 1 − 1γδ + 1−γ

γr > 0, we obtain f(1) > 0. By the continuous

property of function f(φ), where φ ∈ (1,+∞), we know that there must exist

φ∗ ∈ (1,+∞) such that f(φ∗) = 0. Combining the equation for f ′(φ) yields

f ′(φ) < 0 in this case. Therefore, the solution φ∗ is unique.

2. γ > 1.

In this case, f(+∞) = − 1γ(δ + (γ − 1)r) < 0 and

f(1) = 1− 1

γδ +

1− γγ

r +(1− γ)(µ− r)2

2γ(λθ + γσ2)

≥ 1− 1

γδ +

1− γγ

r +(1− γ)(µ− r)2

2γ2σ2

= 1− 1

γ

[δ − (1− γ)r − (1− γ)(µ− r)2

2γσ2

]> 0.

The last inequality holds because of Equation (A-2). For the same reason, we know

that there must exist φ∗ ∈ (1,+∞) such that f(φ∗) = 0.

Suppose that φ∗ is the solution. We further investigate the sign of f ′(φ∗):

f ′(φ∗) =

(−1

γ

)(φ∗)−

1γ−1 +

λθ(γ − 1)(µ− r)2

2γ(λθφ∗ + γσ2)2

= −(φ∗)−1

γ2· 1

γ

[δ + (γ − 1)r +

(γ − 1)(µ− r)2

2(λθφ∗ + γσ2)

]+λθ(γ − 1)(µ− r)2

2γ(λθφ∗ + γσ2)2

= −(φ∗)−1

γ2[δ + (γ − 1)r] +

(γ − 1)(µ− r)2

2γ(λθφ∗ + γσ2)

[λθ

(λθφ∗ + γσ2)− φ∗

γ

]= −(φ∗)−1

γ2[δ + (γ − 1)r] +

(γ − 1)(µ− r)2

2γ(λθφ∗ + γσ2)· −λθ(φ

∗)2 − γσ2φ∗ + λθγ

γ(λθφ∗ + γσ2)

<(γ − 1)(µ− r)2[−λθ(φ∗)2 − γσ2φ∗ + λθγ]

2γ2(λθφ∗ + γσ2)2.

This inequality holds because

−(φ∗)−1

γ2[δ + (γ − 1)r] < 0.

30

So f ′(φ∗) < 0 if the second term component

−λθ(φ∗)2 − γσ2φ∗ + λθγ < 0.

We solve the quadratic equation

−λθx2 − γσ2x+ λθγ = 0.

Its larger solution xL =

√γ2σ4+4λ2θ2γ−γσ2

2λθ> 0. If φ∗ > xL, then −λθ(φ∗)2 − γσ2φ∗ +

λθγ < 0 and f ′(φ∗) < 0.

We solve for the conditions of φ∗ > xL:

φ∗ >

√γ2σ4 + 4λ2θ2γ − γσ2

2λθ,

2φ∗λθ + γσ2 >√γ2σ4 + 4λ2θ2γ,

4(φ∗)2λ2θ2 + γ2σ4 + 4φ∗λθγσ2 > γ2σ4 + 4λ2θ2γ,

(φ∗)2λθ + φ∗γσ2 > λθγ,

φ∗γσ2 > λθ(γ − (φ∗)2).

If γ − (φ∗)2 < 0, all the above inequalities are satisfied. Hence, φ∗ > γ12 is required,

that is, κ−γ > γ12 , which is rewritten as

κ < γ−12γ .

We know that, if φ∗ is the solution to f(φ) = 0, then f ′(φ∗) < 0. Suppose there

is more than one real number solutions for f(x) = 0. Without loss generality,

we consider the case in which there are two real number solutions denoted by

1 < x1 < x2 < +∞.

We have f ′(x1) < 0; hence, there exists x2 − x1 > ε1 > 0 such that, for all

x ∈ (x1 + 12ε1, x1 + ε1), f(x) < 0. Similar to the above discussions, since f(x2) = 0

and f ′(x2) < 0, there exists ε2 that satisfies x2 − x1 − ε1 > ε2 > 0 such that, for all

x ∈ (x2 − ε2, x2 − 12ε2), f(x) > 0.

Therefore, the value of function f(x) goes from strictly negative to strictly positive as

the variable x goes from x1 + 12ε1 to x2− 1

2ε2 and the continuity property of function

f(x) implies that there must exist x ∈ (x1 + 12ε1, x2 − 1

2ε2) such that f(x) = 0.

Continuing the process leads to an infinite number of solutions for f(x) that are all

in (x1, x2), which is a contradiction. Therefore, f(x) must have a unique solution.

�

31

Proof of Proposition 4.

Let φ∗ be the solution of f(φ) = 0, φ∗ ∈ (1,+∞). Then

κ = (φ∗)−1γ

and

π∗ =µ− r

κ−γλθ + γσ2

are solutions in Proposition 2.

Obviously, φ∗ is a function of θ and is denoted φ∗(θ). Therefore, κ and π∗ are also

functions of θ.

Taking the derivative of equation f(φ∗) = 0 yields

f ′(φ∗)dφ∗ − λφ∗(1− γ)(µ− r)2

2γ(λθφ∗ + γσ2)2dθ = 0, (A-3)

which can be rewritten as

f ′(φ∗)dφ∗

dθ=λφ∗(1− γ)(µ− r)2

2γ(λθφ∗ + γσ2)2. (A-4)

As proven in Proposition 3, f ′(φ∗) < 0; therefore, dφ∗

dθ< 0 when 0 < γ < 1 and dφ∗

dθ> 0

when γ > 1.

Recall that

κ = (φ∗)−1γ .

Then, we have dκdθ> 0 when 0 < γ < 1 and dκ

dθ< 0 when γ > 1.

Equation (2.21) gives

dκ

dπ∗=

(γ − 1)(µ− r)2γ

. (A-5)

Therefore,

dκ

dθ=

(γ − 1)(µ− r)2γ

· dπ∗

dθ.

We present the results for two different cases:

1. 0 < γ ≤ 1

• When µ > r, that is, π∗ > 0, then dπ∗

dθ< 0.

• When µ < r, that is, π∗ < 0, then dπ∗

dθ> 0.

32

2. γ > 1

• When µ > r, that is, π∗ > 0, then dπ∗

dθ< 0.

• When µ < r, that is, π∗ < 0, then dπ∗

dθ> 0.

�

33

Tab

le1:

Sim

ula

tion

This

table

rep

orts

the

num

eric

alre

sult

sof

the

opti

mal

frac

tion

ofw

ealt

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ted

toth

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mEP

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dth

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for

diff

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tle

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skav

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put

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rem

pir

ical

valu

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ticu

lar,µ

=8%

,σ

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=2%

.

γ=

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=4

γ=

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θπ∗

EPA

κπ∗

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κπ∗

EPA

κπ∗

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κ

013

3.33

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88%

4541

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1.86

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87%

25.7

1%2.

31%

2.70

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2.67

%2.

58%

16.4

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96%

2.49

%

9020

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0.90

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10.2

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92%

2.35

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180

9.66

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2.84

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450

3.76

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67%

0.15

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1.04

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2.11

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75%

0.13

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08%

34

Tab

le2:

Cali

bra

tion

Resu

lts

of

the

Ass

et

Pri

cing

Model

This

table

rep

orts

the

calibra

tion

resu

lts

ofth

eL

uca

s-st

yle

pow

eruti

lity

model

ofas

set

pri

cing

wit

hre

gard

tore

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ambig

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y.P

anel

Are

por

tsth

eva

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ofth

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put

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able

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(2)

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the

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Col

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put

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amet

ers

Sam

ple

Parameters

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(189

1-19

94)

Quar

terl

y(1

947.

2-19

96.1

)

Con

sum

pti

onG

row

thR

ate,µC

0.01

740.

0191

Con

sum

pti

onG

row

thV

olat

ilit

y,σC

0.03

260.

0108

Equit

yR

eturn

Vol

atilit

y,σS

0.18

530.

1522

Cor

rela

tion

bet

wee

nE

quit

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eturn

and

Con

sum

pti

onG

row

th,ρ

0.49

700.

1930

Equit

yP

rem

ium

,µS−r

6.25

8%7.

852%

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k-F

ree

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e,r

1.95

5%0.

785%

Pan

elB

:E

stim

ated

Par

amet

ers

Parameters

(1)

(2)

(3)

(4)

(5)

(6)

Ris

kA

vers

ion

Coeffi

cien

t,γ

2124

75

57

6

Deg

ree

ofA

mbig

uit

yA

vers

ion,θ

--

585

585

1708

1708

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count

Rat

e,δ

0.34

05.

024

0.11

70.

144

0.14

30.

173

Con

sum

pti

onR

ate,κ

--

0.06

50.

067

0.06

50.

067

Wor

st-C

ase

Equit

y(R

isk)

Pre

miu

m,EPA

--

1.44

2%0.

153%

2.00

0%0.

199%

Am

big

uit

yP

rem

ium

--

2.56

1%7.

361%

2.00

3%7.

265%

Am

big

uit

y–R

isk

Inte

ract

ion

Pre

miu

m-

-2.

255%

0.33

8%2.

255%

0.38

8%

35