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Mathematical Finance, Vol. 17, No. 3 (July 2007), 319–343

PORTFOLIO MANAGEMENT WITH CONSTRAINTS

PHELIM BOYLE AND WEIDONG TIAN

University of Waterloo, Ontario

The traditional portfolio selection problem concerns an agent whose objective is tomaximize the expected utility of terminal wealth over some horizon. This basic prob-lem can be modified by adding constraints. In this paper we investigate the portfolioselection problem for an investor who desires to outperform some benchmark indexwith a certain confidence level. The benchmark is chosen to reflect some particular in-vestment objective and it can be either deterministic or stochastic. The optimal strategyfor this class of problems can lead to nonconvex constraints raising issues of existenceand uniqueness. We solve this optimal portfolio selection problem and investigate theprocedure for both deterministic and stochastic benchmarks.

KEY WORDS: portfolio selection, nonconvex constraints, beating benchmark

1. INTRODUCTION

In his seminal paper on optimal portfolio selection Merton (1971) assumed that an agentmaximized expected utility of terminal wealth or consumption over a finite investmenthorizon. While this is a very natural objective, investors can have other objectives thathave both theoretical and practical appeal. We first discuss a simple example that willgive a preview of the constraints we study in this paper.

Consider a portfolio selection problem where the available assets are a risky asset andthe T period zero coupon bond. An agent with initial wealth x0 wishes to maximizeexpected utility of terminal wealth at the end of the horizon, T . The agent can modifythis problem by adding a constraint so that a certain guaranteed return is obtained. Onepossible constraint is that the investor’s terminal wealth has to be at least x0. In this casethe agent can devise an optimal strategy to maximize expected utility of terminal wealthunder this constraint. However if the constraint is changed so that the investor’s terminalwealth is to be at least

x0

P(0, T)+ ε,

where P(0, T) is the current price of the zero coupon bond and ε is strictly positive,the agent cannot meet this constraint with probability one since this would violate theno-arbitrage condition. But, it is still possible for the agent to design a strategy to meet

The authors thank two anonymous referees for their numerous constructive and insightful suggestionson previous versions and they acknowledge research support from the Natural Sciences and EngineeringResearch Council of Canada. We are also grateful to the Editor Robert Jarrow for his invaluable assistance.

Manuscript received December 2004; final revision received February 2006.Address correspondence to Weidong Tian, Department of Statistics and Actuarial Science, University of

Waterloo, Waterloo, Ontario Canada N2L 3G1; e-mail: [email protected].

C© 2007 The Authors. Journal compilation C© 2007 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148,USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.

319

320 P. BOYLE AND W. TIAN

this revised benchmark with a certain confidence level. This is the type of problem wewill be considering.

We derive the optimal investment strategy for an investor who desires to obtain areturn that exceeds some benchmark with a given confidence level. The confidence levelis fixed as an objective parameter and we denote this strategy as the Desired BenchmarkStrategy (DBS). At one extreme, a conservative investor might want a minimal guaranteedreturn. This corresponds to portfolio insurance, which has been studied by several authorsincluding Brennan and Solanki (1981), Brennan and Schwartz (1989), Basak (1995),Grossman and Zhou (1996), Sorensen (1999), and Jensen and Sorensen (2001). Thereare stringent restrictions on the benchmark in the portfolio insurance case: its returnmust be less than the risk-free rate return. However, the DBS strategy admits a richerset of constraints. First, the benchmark index can be either deterministic or stochastic.Second, the benchmark return can be greater than risk-free rate. To obtain higher returnsthe investor must assume additional risk.1 This risk is reflected in the confidence levelparameter. The desired benchmark provides the constraint that has to be satisfied with acertain probability.

Our aim is to solve this portfolio selection problem, and derive the optimal strategy forsuch an investor. Of course, the trade-off between the target return and the probability hasto permit feasible solutions. For instance, given a probability α, there exists a maximumpossible return among all possible investments in this strategy. Equivalently, given adesired target return, there exists a maximum probability (strictly less than 1 if the targetreturn is strictly greater than risk-free rate). This nature of this trade-off has been analyzedby Browne (1999), Cvitanic and Karatzas (1999), Follmer and Leukert (1999), and Spivakand Cvitanic (1999) under their quantile hedging framework. If the trade-off betweenreturn and risk is satisfied, our main theorem (Theorem 2.1) states that, under somefairly general assumptions, we can always construct the optimal strategy explicitly. Thestructure of the optimal portfolio selection problem is quite different from the usual casebecause of the nature of the portfolio constraints (see Cox and Huang [1989], Pliska [1986],Karatzas, Lehoczky, and Shreve [1989] for the standard treatment with the usual typesof constraints, Cvitanic and Karatzas [1992], Detemple and Murthy [1997] for convexconstraints). This is because the constraints involved in the DBS are typically nonconvexwhere questions of existence and uniqueness are delicate. Our paper contributes newexistence results to this literature.

Interestingly, the same analysis can be used to deal with portfolio managers who wantto make investment decisions under the VaR constraint. As will be explained2 the VaRconstraint is just a special case of our DBS. Therefore, the results of this paper apply tothese risk managers as well. A similar analysis is also useful for credit risk modeling as inBasak and Shapiro (2005).3 Basak and Shapiro (2001) examined the VaR constraint in thecase of deterministic interest rates and a single risky asset. Within a deterministic interestrate framework, Basak, Shapiro, and Tepla (2006) studied a stochastic index for one-dimensional Brownian motion. Bielecki, Pliska, and Sherris (2000), Stutzer (2003), andPham (2003) included a stochastic benchmark index. These papers focused on the infinitehorizon problem rather than the finite horizon case. In general they do not consider theexistence issue.

1 We might characterize aggressive investors as those who want to beat the market. Wachter (2003) showedthat the optimal strategy for the most conservative investors is to hold the T-maturity bond. Therefore theinvestors who follow the DBS might be described as aggressive investors.

2 See in our Example 2.3.3 Indeed the papers by Basak and Shapiro (2001, 2005) helped to motivate our general analysis.

PORTFOLIO MANAGEMENT WITH CONSTRAINTS 321

The current paper is organized as follows. Section 2 presents the main result of thepaper. We derive the optimal solution for an investor who maximizes expected utility ofterminal wealth subject to the constraint of beating a benchmark with a certain probabil-ity. To illustrate the result more detailed analyses are provided in the subsequent sections.Section 3 deals with the case where the benchmark is deterministic. Section 4 considers astochastic benchmark. Section 5 concludes. To improve readability some of the technicaldetails are covered in the Appendix.

2. THE PORTFOLIO SELECTION PROBLEM

In this section we describe the model and present the main result of the paper. We derive thesolution to the optimal portfolio selection problem under the DBS. First we describe thefinancial market. We assume that the market is complete and that there is no arbitrage.Then in our main theorem we derive the optimal terminal wealth for the problem in thetradition of Cox and Huang (1989). From the martingale representation theorem we canderive the corresponding trading strategies that will replicate this wealth.

2.1. The Model Setup

We assume a financial market M, which can be described as follows. Let (�,F, P)be a probability space, T < ∞ be the fixed time horizon, and let F = {Ft : 0 ≤ t ≤ T}be the natural filtration generated by the random noise W (t) = (W1(t), . . . , Wn(t)), ann-dimensional Brownian motion. Suppose there are n risky assets that are traded con-tinuously. The evolution of the price of the ith asset at time t is modeled by the linearstochastic differential equation

dSi = Si

[bi (t) dt +

n∑j=1

σij(t) dWj (t)

],(2.1)

where bi(t), σij(t) are adapted processes. We assume there is a locally risk-free asset B (thebank account), whose price B(t) at time t evolves according to the equation

dB(t) = r (t)B(t) dt, B(0) = 1,(2.2)

where r(t) denotes the spot interest rate at time t. In this paper we assume that r(t) isdeterministic.4

Throughout this paper we make use of an admissible trading strategy. This is repre-sented by an F-predictable stochastic process π = {(π1(t), . . . , πn(t))′, 0 ≤ t ≤ T} with∫ T

0||π ′(t)σ (t)||2 dt +

∫ T

0|π ′(t)(b(t) − r (t)1n)| dt < ∞, a.s.

and which satisfies the following condition:

X x0,π (t)B(t)

:= x0 +∫ t

0B(s)−1π (s)′σ (s) dW (s), 0 ≤ t ≤ T

is nonnegative a.s. The π (t) denote the dollar amounts the fund manager invests in therisky assets at time t, given initial wealth x0. The time t wealth of the portfolio (obtained

4 The extension to the stochastic interest rate case is straightforward and available from the authors uponrequest.


by following the trading strategy π ) is denoted by X x 0,πt or Xπ

t . Thus X x 0,π0 = Xπ

0 = x0.We assume standard technical assumptions on {r(t), b(t), σ (t)}, which imply that themarket M is frictionless, arbitrage free, and complete. Hence, there exists a unique risk-neutral measure Q, with state-price process {ξt}. (For details see Karatzas [1996].) We alsoassume that for all t, ξt has finite moments, (both positive or negative). This follows if{σ (s)−1(b(s) − r(s)1n)} is uniformly bounded.

The utility function u(·) is a continuously differentiable, strictly increasing, and strictlyconcave function defined over (0, ∞), which satisfies the following uniform continuitycondition: For all (an), (bn) > 0,

bn

an→ 1 ⇐⇒ u′(bn)

u′(an)→ 1.(2.3)

u(0) is defined and equal to the right-hand limit at 0 if the right-hand limit limC↓0u(C)exists. limC↓0u′(C) denotes the right-hand limit of u′(·) at 0. This limit can be +∞.

This uniform continuity condition implies the Inada condition (see Dybvig, Rogers,and Back [1999]):

limx→∞ u′(x) = 0.(2.4)

Define the inverse function of u′(x) as follows:

I(x) =

(u′)−1(x), x < limC↓0

u′(C),

0, otherwise.

Therefore I(x) is defined everywhere, and

limx↓0

I(x) = ∞, limx→+∞ I(x) = 0.(2.5)

We can now formulate the central task of this paper.

(Desired Benchmark Strategy [DBS] Problem). Given an investor with a utility functionu(·) and initial wealth x0, a FT-measurable, positive a.s. random variable � with E[�ξT ] <

∞, and a probability α ∈ (0, 1). The problem is to choose an admissible process {πt} tomaximize E[u(X x 0,π (T))] subject to the constraint P(X x0,π (T) ≥ �) ≥ α.

This represents the formal statement of the problem we want to solve. Without theconstraint, it reduces to the standard optimal portfolio selection problem. The randomvariable � represents the benchmark. This benchmark can be either deterministic orstochastic. If α = 1, this problem becomes the familiar portfolio insurance problem.In the portfolio insurance case, the benchmark � must satisfy E[ξT�] ≤ x0. When thebenchmark � has a higher return than the risk-free rate it is impossible5 to beat thebenchmark in all scenarios. In this case α < 1 represents the confidence level for thisstrategy. We now give some special cases.

EXAMPLE 2.1. Assume α = 0. In this well-known6 Merton portfolio selection problemthere exists an admissible process π∗

t with terminal wealth X x 0,π∗(T) such that

E[u(X x0,π∗(T))] ≥ E[u(X x0,π(T))]

for any admissible process {πt}.Xu(T) := X x 0,π∗(T) is the optimal terminal wealth.

Xu(T) = I(λuξT ) for some positive real number λu. We select λu so that the budget con-straint is binding; that is E[ξT Xu(T)] = x0. Moreover, the admissible process is unique

5 By the no-arbitrage principle.6 See Pliska (1986), Cox and Huang (1989), Karatzas, Lehoczky, and Shreve (1989), Dybvig, Rogers,

and Back (1999).


in the sense that if there is another optimal terminal wealth X x 0,π (T), then X x 0,π (T) =Xu(T), a.s.

EXAMPLE 2.2. Assume α = 1. The solution can be obtained by standard arguments(as in El Karoui, Jeanblanc-Picque, and Lacoste [2005]). Special cases have been de-rived by Basak (1995), Brennan and Schwartz (1989), Dybvig, Rogers, and Back (1999),Grossman and Vila (1989). We know that: If x0 > E[ξT�], then there exists an admissibleprocess π∗

t with terminal wealth Xg(T) := Xx0,π∗(T) such that

Xg(T) ≥ �, a.s., E[u(X g(T))] ≥ E[u(Xx0,π (T))],

for any admissible process {πt} subject to Xx0,π (T) ≥ �, a.s. Furthermore, the optimalterminal wealth Xg(T) can be written explicitly as

X g(T) = max(I(λgξT

), �

)for a positive real number λg. In addition, the optimal portfolio strategy is unique in thesense that if there is another optimal terminal wealth Xg2 (T) such that Xg2 (T) ≥ �, a.s.,then X g(T) = Xg2 (T), a.s.

The benchmark � can be an equity index or a bond index, or any portfolio of assets.When the benchmark is deterministic, the strategy corresponds to classical portfolioinsurance. Under portfolio insurance, the optimal strategy is equivalent to an optimalstrategy without any constraints plus a long position in a put option. The put optionguarantees the basic floor of protection. Pension plans are often interested in achievingsome guaranteed return because they have fixed liabilities. See Jensen and Sorensen (2001).Tepla (2001) analyzed a problem with an equity index benchmark in a model with constantinterest rates.

EXAMPLE 2.3. The VaR Constraint. The DBS problem can also be motivated froma risk management perspective. Assume L is the VaR limit for time horizon T . Thetypical VaR constraint on a portfolio is P(Loss > L) ≤ α where Loss = x − Xπ

T , and α

is the confidence level. This VaR constraint is equivalent to P(XπT ≥ x − L) ≥ 1 − α.

(See Basak and Shapiro [2001] in a continuous time setting, and Alexander and Baptista[2004], Danielsson Jorgensen, de Vries, and Yang [2001] in a discrete-time framework.)

In Examples 2.1 and 2.2, the constraints on the terminal wealth XπT are convex. There-

fore uniqueness of the optimal terminal wealth follows from standard arguments. How-ever, the constraint in Example 2.3 is nonconvex. The portfolio selection problem undernonconvex constraints is very different from the corresponding problem with convexconstraints. The implication of the difference will emerge as we proceed.

2.2. The Main Theorem

We will present the solution to the DBS problem in this section. We employ a con-structive proof. We postulate the functional form of a class of random variables whereeach member of the class is indexed by a positive real number λ. Then we show thatthere exists one optimal terminal wealth under the DBS in this specially constructedfamily.

At this stage some comments on the constraints to be satisfied by the terminal wealthmay be helpful. There are two constraints on the candidate terminal wealth XT at timeT . The first one is the budget constraint that E[ξT XT ] ≤ x0. This type of constraint isstandard from the original Cox–Huang approach. The second constraint arises fromrequirement that P(XT ≥ �) ≥ α. Following Cox and Huang (1989) and Pliska (1986),


if these two constraints are binding simultaneously for one candidate terminal wealthXT , then XT should solve the optimal portfolio selection problem under both these twoconstraints. We now state an important lemma7 that puts this intuition on a more formalfooting and serves as a starting point for the subsequent discussion.

LEMMA 2.1. Given λ > 0, α ∈ (0, 1), and a measurable random variable �. Assumethere exists a nonnegative measurable random variable Xλ,α satisfying the following threeconditions:

(1) P(Xλ,α ≥ �) = α.(2) E[ξT Xλ,α] = x0.(3) There exists a real number C ≥ 0 such that for each ω ∈ �, Xλ,α(ω) solves the

following optimization problem A:

Max{u(x) − λξT(ω)x + C1{x≥�(ω)}

}.

Then for any random variable XT with constraints

E[ξT XT] ≤ x0, P(XT ≥ �) ≥ α

we have

E[u(Xλ,α)] ≥ E[u(XT)].

Proof of Lemma 2.1. Note that

E[u(Xλ,α) − u(XT)] ≥ E[u(Xλ,α) − λξT Xλ,α + C1{Xλ,α≥�}

]− E

[u(XT) − λξT XT + C1{XT≥�}

]≥ 0,

where the first inequality comes from the condition that x0 = E[ξT Xλ,α] ≥ E[ξT XT ], andP(XT ≥ �) ≥ P(Xλ,α ≥ �) = α, the second inequality comes from condition (3). Thiscompletes the proof of Lemma 2.1. �

Note that both λ and C correspond to the Lagrange multipliers of the constraints.We will use this lemma in the proof of our main result. However, note that the random

variable Xλ,α has to satisfy three conditions for the lemma to be true. In the next few pagesour task will be to assemble the tools to enable us to prove these three conditions for theterminal wealth. We do so by constructing the terminal wealth in a specific manner.

We start by defining a sequence of random variables f (λ, ξT ), for each λ > 0, by

f (λ, ξT) := u(I(λξT)) − λξT I(λξT) + λξT� − u(�).

These random variables f (λ, ξT ) involve both the utility function and the benchmark andthus connect the two dimensions of the problem that relate to the two constraints. Theyplay an important role in the construction of the solution. Because u′ is decreasing, it iseasy to see that f (λ, ξT ) ≥ 0 for all positive λ.8

Define the bivariate function G(λ, x) and the univariate function H(λ), for λ > 0, x ∈[0, ∞) such that

G(λ, x) = P(I(λξT) < �, f (λ, ξT) ≥ x)(2.6)

7 This lemma is essentially similar to a result in Basak and Shapiro (2001).8 This follows from the concavity of u(·).


and

H(λ) = P(I(λξT) < �)(:= G(λ, 0)).(2.7)

The purpose of the bivariate function G(·) is again related to the presence of two con-straints for a general benchmark. As we will see later, if the benchmark is deterministic,we do not need to introduce the bivariate function G(λ, x) because it can be essentiallyreduced to a univariate function. Define

λα = Sup{λ > 0 : H(λ) < 1 − α}.(2.8)

If there is no λ > 0 such that H(λ) < 1 − α, we set λα = 0. If H(λ) < 1 − α for any λ, thenwe set λα = +∞. We will show shortly that λα ∈ (0, +∞). Let G−1(λ, a) := {x : G(λ, x) =a}, and H−1(a) = {λ : H(λ) = a}.

Note that

G(λ, x) = H(λ) − K(λ, x)

with

K(λ, x) = P(I(λξT) < �, f (λ, ξT) < x).

It is evident that H(λ) is an increasing function of λ, and G(λ, x) is a decreasing functionwith respect to x. Hence G−1(λ, a) is an interval. Similarly, H−1(a) is an interval aswell.

We next introduce two technical assumptions on the function G(λ, x) and then discussthem.

ASSUMPTION 2.1. G(λ, x) is jointly continuous with respect to both λ and x.

ASSUMPTION 2.2. For any λ ≥ λα, G−1(λ, 1 − α) has Lebesgue measure zero. MoreoverH−1(1 − α) has Lebesgue measure zero.

Assumption 2.1 ensures the existence of the construction that underpins our solution.In Section 4.2, we examine cases where Assumption 2.1 does not hold and show that thesolution to the portfolio selection problem does not exist. Since G−1(λ, a) and H−1(a) areintervals, Assumption 2.2 means essentially that there exists at most one element in eachG−1(λ, 1 − α) and H−1(1 − α). Assumption 2.2 is used to derive the uniqueness of theconstruction procedure.

We now show that 0 < λα < ∞. For every λn ↓ 0, using the Lebesgue dominated con-vergence theorem,9 we have

limλn↓0

E[1{I(λnξT )<�}

] = 0.

Equivalently,

limλn↓0

P(I(λnξT) < �) = 0.(2.9)

Similarly, we see that limλn↑+∞ P(I(λnξT) < �) = 1. Therefore 0 < λα < +∞.According to Assumption 2.1, the function H(λ) is continuous. Then from for-

mula (2.8),

H(λα) = 1 − α.

9 Even though the indicator function is not continuous everywhere, it is easy to see that 1{I(λnξT )<�} →0, a.s.


Moreover, by the previous discussion, Assumption 2.2 implies that H(λ) > 1 − α forevery λ > λα.

Let λu denotes the Lagrange multiplier in Example 2.1. If λα ≥ λu, then I(λuξT ) ≥I(λαξT ), and

P(I(λuξT

) ≥ �) ≥ P(I(λαξT) ≥ �) = α.

Thus I(λuξT ) presents the unique optimal terminal wealth for our portfolio selectionproblem. In the sequel, we assume that

λα < λu .(2.10)

To prove our main theorem (Theorem 2.1), we need the next three lemmas that dealmainly with the convergence of certain sequences that are used in the construction of oursolution.

LEMMA 2.2. Assume Assumption 2.1 and 2.2.

1. For any λ > λα, there exists a unique positive real number d(λ, α) such that

G(λ, d(λ, α)) = 1 − α.(2.11)

2. d(λ, α) is continuous with respect to λ for λ > λα.3. The function d(λ, α) can be extended continuously over the region {λ ≥ λα} with

d(λα, α) = 0.

Proof of Lemma 2.2. See Appendix.

By using Lemma 2.2, we define a family of random variables Xλ,α(T) for each λ >

λα,

Xλ,α(T) ={

�, if I(λξT) < �, f (λ, ξT) < d(λ, α)

I(λξT), otherwise.

Moreover, we can define Xλα,α(T) := I(λαξT), since d(λα, α) = 0 and f (λ, ξT ) ≥ 0 for allλ > 0.

Since

P(Xλ,α(T) < �) = P(I(λξT) < �, f (λ, ξT) ≥ d(λ, α)) = G(λ, d(λ, α)) = 1 − α,

the family {Xλ,α(T) : λ > λα} is an obvious candidates for the terminal wealth under theDBS strategy.

We now define three set-valued functions of λ, which are necessary for the formulationof our next lemma.

For every λ ≥ λα, define

A(λ) := {I(λξT) < �, f (λ, ξT) < d(λ, α)},(2.12)

B(λ) := {I(λξT) < �, f (λ, ξT) ≥ d(λ, α)},(2.13)

and

C(λ) := {I(λξT) ≥ �}.(2.14)


LEMMA 2.3. Assume Assumption 2.1 holds, then for every λn → λ in the region λ > λα,or λn ↓ λα, we have

1A(λn ) → 1A(λ), a.s., 1B(λn ) → 1B(λ), a.s., 1C(λn ) → 1C(λ), a.s.


The next lemma is used to study the convergence when λ → ∞.

LEMMA 2.4.

limλ→∞

1A(λ) = limλ→∞

1{ f (λ,ξT )≤d(λ,α)}, a.s.(2.15)


We are now in a position to present our solution to the portfolio selection problem.The solution is given by the family {Xλ,α(T) : λ > λα}, which we have just constructed.

THEOREM 2.1. Assume that Assumptions 2.1 and 2.2 hold, E[ξT�] < +∞, and assumethat

limλ→∞

E[ξT�1{ f (λ,ξT )≤d(λ,α)}

]< x0,(2.16)

then there exists a positive λ such that Xλ,α(T) is the optimal feasible terminal wealth in theDBS problem.

Proof of Theorem 2.1. By Lemma 2.1, we need to show that all three conditions ofLemma 2.1 are satisfied for at least one Xλ,α(T). The first condition follows, for everyXλ,α(T), by the construction of d(λ, α).

To show that the second condition is satisfied for at least one Xλ,α(T), we needto prove that E[ξT Xλ,α(T)] = x0 for at least one λ. Note that I(x) is bounded by aconstant plus a power times a constant (See Dybvig, Rogers, and Back [1999], for-mula (48)). By assumption, ξT possesses all finite moments, and E[ξT�] < ∞. Then,E[ξT Xλ,α(T)] < ∞ for λ > λα. Moreover, each Xλ,α(T) is dominated by � + I(λαξT ), andE[ξT {� + I(λαξT )}] < ∞. Let g(λ) := E[ξT Xλ,α(T)] for λ > λα. For every λn → λ > λα,or λn ↓ λα, by using Lemma 2.3, we have Xλn ,α(T) → Xλ,α(T), a.s. Then by the Lebesguedominated convergence theorem, g(λ) is continuous for λ > λα, and g(·) is right continuousat λα. Moreover, for λ ↓ λα

g(λ) → E[ξT Xλα,α(T )] ≥ E[ξT I(λαξT)] ≥ E[ξT I

(λuξT

)] = x0,(2.17)

because λα < λu. Furthermore, E[ξT I(λαξT )] > x0 because of the uniqueness result fromExample 2.1.10 Hence limλ↓λα

g(λ) > x0.We now prove that

limλ→∞

g(λ) < x0.(2.18)

To see this, from the Inada condition, we have

limλ→∞

g(λ) ≤ limλ→∞

E[ξT�1A(λ)].(2.19)

10 Otherwise, E[I(λαξT )] = E[I(λuξT )]. Therefore, λα = λu by the uniqueness result in Example 2.1.


By Lemma 2.4 and the condition (2.16), we then obtain

limλ→∞

g(λ) ≤ limλ→∞

E[ξT�1{ f (λ,ξT )≤d(λ,α)}

]< x0.(2.20)

By formulas (2.17) and (2.18), then there exists a positive λ satisfying g(λ) = x0.Let C = d(λ, α) > 0. We want to prove Lemma 2.1 (3) holds for each Xλ,α(T), λ >

λα. Then by previous arguments, there exists one Xλ,α(T) satisfying all conditions inLemma 2.1.

Let

φω(x) := u(x) − λξT(ω)x + C1{x≥�(ω)}.(2.21)

To solve the optimization problem A in Lemma 2.1 for each ω, we consider two differentcases.

Case 1. Assume that �(ω) ≤ I(λξT (ω)).

In this case, the function φω(x) is increasing when x < �(ω), and I(λξT (ω)) solvesthe optimization problem Max{x≥�(ω)}φω(x). Since �(ω) ≤ I(λξT (ω)), and C > 0, thenI(λξT (ω)) solves the optimization problem A of Lemma 2.1.

Case 2. Assume that I(λξT (ω)) < �(ω).

In this case, I(λξT (ω)) solves the problem Max{x<�(ω)}φω(x), and �(ω) solves the sameoptimization problem for the region {x ≥ �(ω)}. Therefore, we need to compare two localmaximums. One is φω(I(λξT )), which equals to u(I(λξT (ω)) − λξT (ω)I(λξT (ω)). The otherone is φω(�(ω)), which equals to u(�(ω)) − λξT (ω)�(ω) + C. I(λξT (ω)) solves Problem Aif and only if

u(I(λξT(ω)) − λξT(ω)I(λξT(ω)) ≥ u(�(ω)) − λξT(ω)�(ω) + C,

which is equivalent to

f (λ, ξT(ω)) ≥ C.

Otherwise, �(ω) solves the problem A, which is equivalent to f (λ, ξT (ω)) < C.Therefore, Xλ,α(T) solves the optimization Problem A. This completes the proof ofTheorem 2.1. �

It may be helpful to discuss the interpretation of Xλ,α(T). Given λ, Xλ,α(T) is a time Tcash flow that is constructed to equal either I(λξT ) or the benchmark�, which correspondsto the optimal terminal wealth that satisfies the constraints. The properties of this cashflow essentially depend on the variable f (λ, ξT ), which in turn depends on both the utilityfunction u(·) and benchmark �. Therefore, the future cash flow Xλ,α(T) is explicitlydetermined by both u(·) and the benchmark �. The fair value of this future cash flowdepends on λ. The key idea of this theorem is to construct a suitable financial derivativewith payoff Xλ,α(T) whose market value is the same as the initial wealth x0. In the nexttwo sections, we give more examples of how to construct Xλ,α(T) in specific cases.


We now give some technical remarks on the theorem.

REMARK 2.1. As we will show in the next section,11 the condition (2.16) is necessaryfor the existence of feasible wealth. This condition cannot be improved. In Section 4, weprove an nonexistence result if Assumptions 2.1 and 2.2 do not hold.

REMARK 2.2. Assumptions 2.1 and 2.2 are used to derive the existence of d(λ, α) withcertain properties. We do not have to check these assumptions if d(λ, α) is computeddirectly.

REMARK 2.3. In Example 3.1 below and other examples, g(λ) = x0 has only one positivesolution, since g(·) is strictly monotone. However, this does not imply the uniqueness ofthe optimal terminal wealth. The uniqueness issue of the problem is more subtle becauseof the nonconvex constraints, and is beyond the scope of this paper.

3. DETERMINISTIC BENCHMARK INDEX

In this section, we illustrate the application of the theorem using a specific example with adeterministic benchmark. We assume that the distribution function of ξT is continuous, forall T > 0. This continuity assumption plays essentially the same role as Assumptions 2.1and 2.2 for a general benchmark. Then, we derive an explicit solution for the case when thestock returns are lognormal. This explicit solution is used to show that the condition (2.16)cannot be improved. We assume that the benchmark is

� = x0

P(0, T)eεT,

where x0 is the initial wealth, P(0, T) is the current price of the T period zero coupon bond,and �0 := x0

P(0,T) is the risk-free total proceeds of the initial investment of x0. Therefore,

ε, which equals to 1T log(�/�0), is the excess return over the risk-free return. We assume

that ε > 0. We note that it is impossible to improve upon the risk-free rate without takingon some risk; α denotes the probability of achieving a positive excess return ε over therisk-free rate.

For the deterministic benchmark index �, we define a function f (λ, x) as follows:

f (λ, x) = u(I(λx)) − λxI(λx) + λx� − u(�)

satisfying

∂ f (λ, x)∂x

= λ(� − I(λx)).

Hence f (λ, x) is strictly increasing with respect to x when I(λx) ≤ �. In this case the func-tion G(λ, x) of the last section can be simplified; G(λ, x) = P(I(λξT ) < �, ξT ≥ f −1(λ, x)).Moreover, as we will see later (from Theorem 3.1) we can choose d(λ, α) such thatf −1(λ, d(λ, α)) does not depend on λ.

Define ξα, for any α ∈ (0, 1), satisfying

ξα := inf {ξ ∈ R : P(ξT > ξ ) ≤ 1 − α}.(3.1)

11 See Remarks 3.2 and 3.3.


Then, P(ξT > ξα) = P(ξT ≥ ξα) = 1 − α. Moreover, for every λ > 0

{I(λξT) < �} ={ξT >

u′(�)λ

}.

Then by the definitions of λα and ξα, we have

λα = u′(�)

ξα

,(3.2)

where we have used the deterministic property of �.We now present a construction12 of the terminal wealth. For every λ > λα, define a

measurable random variable Yλ,α as follows:

Yλ,α(T)(ω) ={

�, if I(λξT(ω)) < �, ξT(ω) < ξα

I(λξT(ω)), otherwise,

for each ω ∈ �.

THEOREM 3.1. Assume that E[ξT�1{ξT<ξα}] < x0. Then there exists one positive λ suchthat Yλ,α(T) provides the optimal terminal wealth of the DBS problem.

Proof. The proof is similar to the proof of Theorem 2.1. We rely once more onLemma 2.1. The first condition in Lemma 2.1 is satisfied by construction. As for thesecond, for every λ > λα, the Lebesgue dominated convergence theorem implies thatE[ξT Yλ,α(T)] < ∞. Write h(λ) = E[ξT Yλ,α(T)]; then h(λ) is continuous when λ > λα.Moreover, by formula (3.2), if λ ↓ λα, then Yλ,α(T) → I(λαξT), a.s. Hence by the Lebesguedominated convergence theorem once more

h(λα) := limλ↓λα

h(λ) = E[I(λαξT)] ≥ E[I(λuξT

)] = x0.

Because of Remark 2.1, we have h(λα) > x0. On the other hand, when λ → ∞, by usingthe Inada condition, we have Yλ,α(T) → �1{ξT<ξα}, a.s. Then, limλ→∞ h(λ) < x0. Hencethere exists at least one λ such that E[ξT Yλ,α(T)] = x0.

For the last condition, choose C = f (λ, ξα) ≥ 0. Note that f (λ, ·) is strictly increasingin the region ( u′(�)

λ, ∞). The proof is exactly the same as in Theorem 2.1 with d(λ, α) =

f (λ, ξα). �

REMARK 3.1. The terminal wealth Yλ,α(T) is essentially the terminal wealth Xλ,α(T)constructed in the last section. In fact, for the deterministic benchmark index, d(λ, α)has a simple expression: d(λ, α) = f (λ, ξα).13

REMARK 3.2. The condition E[ξT�1{ξT<ξα}] < x0, in Theorem 3.1, cannot be improved.To see this, assume that ξT is lognormally distributed: log(ξT ) ∼ N(µξ , σ

2ξ ). Then ξα =

exp(µξ + σξ�−1(α)) and

E[ξT�1{ξT<ξα}

] = �(�−1(α) − µξ

),(3.3)

where �(·) is the standard normal distribution function. Let

12 See also Basak and Shapiro (2001) for a similar construction.13 By using formula (3.2), for every λ > λα , we see that {I(λξT) < �, f (λ, ξT) ≥ f (λ, ξα)} = {I(λξT) <

�, ξT ≥ ξα} = {ξT ≥ ξα}. Thus, G(λ, f (λ, ξα)) = P(ξT ≥ ξα) = P(ξT > ξα) = 1 − α, by the definition of ξα

and the continuity of the distribution function of ξT .


ERα =−log�

(logξα − µξ

σξ

− σξ

)T

,(3.4)

where �(·) is the standard normal cumulative distribution function. By using E[ξT] =exp(µξ + 1

2σ 2ξ ) = P(0, T), it is straightforward to see that the aforementioned condition

is equivalent to ε < ERα. Then by Theorem 3.1, given a probability α, an investor canalways find an optimal strategy to have an excess return that is less than but arbitrarilyclose to ERα.

In fact, as we will see in the theorem 3.2, ERα is the maximum possible excess re-turn rate with probability α. Hence the excess return cannot be larger than ERα. There-fore, to find an admissible strategy such that P(X x0,π

T ≥ �) ≥ α, the benchmark � mustsatisfy E[ξT�1{ξT<ξα}] < x0. By Remark 3.1, condition (2.16) of Theorem 2.1 cannot beimproved.

Given any ε > 0, Let

C(ε) := MaxP(X x 0,π (T) ≥ �),(3.5)

where π runs through admissible trading strategies. C(ε) is the maximum possible proba-bility and the underlying strategy is the well-known maximum probability strategy, solvedby Spivak and Cvitanic (1999), and Follmer and Leukert (1999) in a fairly general setting.The next result is an easy application of Follmer and Leukert’s general theorem.

THEOREM 3.2. Assume the Black–Scholes world with r = 0. Then ξT is lognormallydistributed. Write log(ξT ) ∼ N(µξ , σ

2ξ ), we have

C(ε) = �(�−1(e−εT) + σξ ).(3.6)

REMARK 3.3. Solving the equation C(ε) = α, given a probability α, we have ε =ERα. Therefore ERα is the maximum possible excess return rate with probabilityα.

Proof of Theorem 3.2. Assume that the price of the risky asset follows:

dSt

St= µ dt + σ dW t.

Let

θ = µ

σ.(3.7)

Then

dPdQ

= exp{−θWT − 1

2θ2T

}.

Define a set

A(λ) :={

dPdQ

> λ�

}.

If there exists a λ such that E[ξT 1{A(λ)}] = x0, then by the Follmer and Leukert (1999)theorem 2.22, the maximum probability C(ε) = P(A(λ)). It is straightforward to checkthat λ has to satisfy


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Probability $α$

Exc

ess

Ret

urn

Excess Return Rate

Available Region

Efficient Frontier

FIGURE 3.1. µ = 0.05, r = 0, σ = 30%, and T = 5

e−εT = �

−

log(λ�) + 12θ2T

θ√

T

.(3.8)

On the other hand

P(A(λ)) = �

−

log(λ�) − 12θ2T

θ√

T

.(3.9)

Hence

C(ε) = P(A(λ)) = �(�−1(e−εT) + θ√

T). �(3.10)

The relationship between ε → C(ε) and α → ERα is the risk-return trade-off for theinvestor. The higher α is, then the higher is the excess return ERα. Conversely, to obtaina higher excess return ε, the investor has to take on higher risk C(ε).

Figure 3.1 plots the efficient frontier between the probability α and excess return ε.The parameters are µ = 0.05, r = 0, σ = 30%, and T = 5. The graph plots the excessreturn with respect to the probability α. Given a probability α, the possible excess returnmust lie below the frontier, or inside the available region. This relationship has beendocumented by Browne (1999, 2000). According to Theorem 3.1, for any (α, ε) inside theavailable region, there always exists an optimal strategy.

Because we know the specific expression for the optimal wealth in this example, theimplementation of the optimal portfolio strategy is straightforward. In the rest of thissection, we show how to construct the explicit solution in Theorem 3.1 using a simpleexample. For simplicity we again assume the Black–Scholes world with r = 0. We usesame notation as in Theorem 3.2 with input parameters θ, �, α, and x0. The constraintson {�, α, x0} are satisfied. We assume u(x) = log(x). Then ξα = exp[θ

√T�−1(α) − 1

2θ2T]


0 0.5 1 1.5 2 2.59

9.5

10

10.5

11

11.5

12

12.5

lambda

pric

e

Initial Price of terminal wealth X

h(0.19)=9.5

Excess Return: 5%

FIGURE 3.2. µ = 0.06, r = 0, σ = 30%, and T = 1

and λα = 1�ξα

. For each λ > λα, there is an explicit expression h(λ) = E[ξT Yλ,α(T)] asfollows:

h(λ) = 1λ

�

(−log(�λ)

θ√

T+ 1

2θ√

T)

+ 1λ

[1 − �

(log(ξα)

θ√

T+ 1

2θ√

T)]

+ �

[�

(log(ξα)

θ√

T− 1

2θ√

T)

− �

(−log(�λ)

θ√

T− 1

2θ√

T)]

.

According to the proof of Theorem 3.1, any Yλ,α(T) such that h(λ) = x0 is an optimalterminal wealth. Figure 3.2 plots the graph of the function h(λ) with respect to λ. Theassumed model parameters are:

µ = 0.06, σ = 0.2, T = 1, α = 0.95, x0 = 9.5, and ε = 5%.

Then, � = 9.987, ξα = 1.5659, and λ = 0.1541 solve the equation h(λ) = x0. The functionX0.1541,0.95 denotes the optimal wealth for the DBS such that the terminal wealth in oneyear has an excess return of at least 5% with probability 95% or more. The terminal wealthX0.1541,0.95 can be viewed as a payoff of a contingent claim written on the underlying S.It is easy to check that

1ξT

=(

ST

S0

) θσ

e[ θ (σ−θ)T2

].

Therefore X0.1541,0.95 has the following payoff in terms of the risky asset:

X0.1541,0.95(T)(ω) =

9.987, if − 0.2890 ≤ log(

ST

S0

)≤ 0.2976

6.3914 ∗(

ST

S0

)3/2

, otherwise.

The replication strategy of X0.1541,0.95 is the admissible strategy to replicate this payoff.


It is interesting to compare the optimal wealth Xt(0 < t < T) in this case with thecorresponding optimal wealth under portfolio insurance. For every t < T ,

Xt = 9.987{�(B + 0.3√

T − t) − �(A+ 0.3√

T − t)}

+ 6.3914(

St

S0

)3/2

exp[0.015(T − t)]{1 + �(A) − �(B)},

where

A =−0.2890 − log

(St

S0

)+ 0.04(T − t)

0.2√

T − t, B =

0.2976 − log(

St

S0

)+ 0.04(T − t)

0.2√

T − t.

It is easy to see that Xt is not a convex function of the risky asset St. This contrasts withthe portfolio insurance case of Leland (1980) where the wealth is a convex function ofthe risky asset. Moreover, the investor following the DBS does not necessarily buy lowand sell high, since ∂ Xt

∂Stis not a monotonic function of St.

4. STOCHASTIC BENCHMARK INDEX

We now discuss the stochastic benchmark case. This case is of practical interest sinceinstitutional investors often are evaluated14 against some reference portfolio such as themarket index. One interesting choice of a stochastic benchmark is the portfolio of anotherinvestor. We consider two cases. In the first case the benchmark is the optimal portfolioof a different investor. As an example we can assume it is the portfolio of the representa-tive15 investor. We can think of this as trying to beat the market. In the second case thebenchmark is the portfolio of an investor who is identical in every way with the original.We call this second case outperforming your identical twin.

We analyze both cases assuming that both the individual investor and the benchmarkinvestor have utility functions that belong to the CRRA class, and assume that the dis-tribution function of ξT is continuous for every T > 0. There is a remarkable and veryinteresting difference between these two cases. We can obtain the solution in the first casebut in the second case a solution does not exist. In the first case Assumptions 2.1 and 2.2will normally hold, and there exists an optimal terminal wealth. However in the secondcase where the investor tries to beat his identical twin, Assumptions 2.1 and 2.2 do nothold. We prove that, there exists no optimal strategy in this case (see Theorem 4.2 below).

Similar to the deterministic benchmark case of the last section, if � is a function of thestate-price density, say � = β(ξT ), we introduce a function f (λ, x) as follows:16

f (λ, x) = u(I(λx)) − λxI(λx) + λxβ(x) − u(β(x)).

4.1. Outperforming Another Investor with Different Preferences

We consider two investors who construct their portfolios in an optimal way. We assumethat one investor wishes to outperform the other with a certain probability. In other words,

14 Roll (1992) has noted professional money managers are often judged by total return performance relativeto a benchmark.

15 Rubinstein (1974) has shown the aggregate investor’s utility function, can be constructed from the indi-vidual investor’s utility functions under some assumptions.

16 Clearly, this definition is consistent with the previous definition of the random variable f (λ, ξT ).


the benchmark index is chosen as another investor’s optimal terminal wealth. By the no-arbitrage principle, to obtain a higher return than this benchmark, the investor has toaccept some risk.

The results presented in this section are of interest both from a theoretical and practicalperspective. Individual investors or institutional investors (such as hedge funds) desire tobeat the market, and we can let the benchmark correspond to the market portfolio. We canmodel the market portfolio in terms of a representative agent. Typically the parametersof the representative agent’s utility function are different from the individual investor’sutility function (Rubinstein 1974). Leland (1980) examined portfolio insurance when themarket utility function differs from (or is equal to) that of the individual investor. Thenext theorem describes the solutions. We use the term benchmark agent for the investorwhose portfolio is the benchmark.

THEOREM 4.1. Assume both the individual investor and the benchmark agent utilityfunctions belong to the CRRA class with risk-aversion parameters γind, γben, respectively.Moreover, one of the risk-aversion parameters is equal to one, corresponding to the log utilityfunction. Then, there exists an optimal strategy to solve the corresponding DBS problem.The optimal terminal wealth Xλ,α(T) is constructed from Theorem 2.1, such that

(a) If γind < γben, then

Xλ,α(T) ={

�, if I(λξT) < �, ξT < ξα

I(λξT(ω)), otherwise.

(b) If γind > γben, then

Xλ,α(T) ={

�, if I(λξT) < �, ξT > ξ1−α

I(λξT(ω)), otherwise.

Proof. We first consider the case where the individual agent’s preferences are repre-sented by u(x) = log(x). The benchmark agent’s preferences are given by a CRRA utilityfunction with absolute risk aversion γ so that

uγ (x) = 11 − γ

x1−γ .

In this case the notation becomes γind = 1, γben = γ . For the CRRA investor the inverseof the marginal utility function is given by Iγ (x) = x− 1

γ . Hence the stochastic bench-mark index is � = Iγ (λγ ξT ). Furthermore, β(x) = Iγ (λγ x), where λγ is determined byE[ξT Iγ (λγ ξT )] = x0. We have

f (λ, x) =(

1γ

− 1)

log(x) + λx(λγ x)−1γ + log(λγ )

γ− log(λ) − 1,(4.1)

so that

∂ f (λ, x)∂x

=(

1γ

− 1)

1x

+ λλ− 1

γ

γ

(1 − 1

γ

)x− 1

γ .(4.2)

Let xλ := (λγ λ−1γ )

11−γ . We note that

f (λ, x) = H(

1λ

λ−1/γγ x(1−γ )/γ

),


where H(x) = u − 1 − ln(u). Since u = 1 is the unique positive root of the function H(u),then xλ is the unique positive root of the function f (λ, .). It is easy to check that

∂f (λ, x)∂x

|x=xλ= 0,(4.3)

where · · · |x=xλis evaluated at x = xλ.

Case I (a) γ > 1.

In this case, note that

{I(λξT) < �} = {ξT > xλ}.

Then λα = λ1γ

γ ξ1γ−1

α . For each λ > λα, then

x1− 1

γ

λ = λ−1λ1γ

γ < λ−1α λ

1γ

γ = ξ1− 1

γ

α .(4.4)

Hence xλ < ξα. Choose d(λ, α) = f (λ, ξα), then

{I(λξT) < �, f (λ, ξT) ≥ x} = {ξT > xλ, f (λ, ξT) ≥ f (λ, ξα)}= {ξT > xλ, ξT ≥ ξα} = {ξT ≥ ξα},

where the first equality comes from the nature of the condition that I(λξT ) < �, the secondone comes from the increasing property of the function f (λ, x) when x ≥ xλ, and the lastone is implied by xλ < ξα. With this choice of d(λ, α), the terminal wealth constructedas in Theorem 3.1 is the Xλ,α(T) presented in this theorem. To prove the existence of theoptimal terminal wealth, it is enough to check that E[ξT�1{ξT<ξα}] < x0. This comes fromthe condition that x0 = E[ξT�] and the positive probability of {ξT ≥ ξα}.

Case I (b) γ < 1.

In this case, {I(λξT) < �} = {ξ1γ−1

T < λλ− 1

γ

γ }. Note that P(ξT < ξ1−α) = 1 − α, and 1γ

>

1, then λα = λ1γ

γ ξ1γ−1

1−α . Therefore, for each λ > λα, we have

ξ1−α < xλ.(4.5)

Hence, by choosing d(λ, α) = f (λ, ξ1−α), and by the same reasons as in Case I (a), wehave

{I(λξT) < �, f (λ, ξT) ≥ x} = {ξT < xλ, ξT ≤ ξ1−α}= {ξT ≤ ξ1−α}.

The existence of optimal terminal wealth among Xλ,α(T) is shown in the same way asbefore.

We now prove the theorem when the benchmark agent has log utility. In this casethe notation becomes γind = γ, γben = 1. That is u(x) = x1−γ

1−γbut the benchmark is � =

I1(λ1ξT) = 1λ1ξT

. In this case, I1(x) = 1x , λ1 = 1

x0, and β(x) = 1

λ1x . Note that

f (λ, x) = γ

1 − γλ

1− 1γ x1− 1

γ + λ

λ1− 1

1 − γ

(1λ1

)1−γ

xγ−1.(4.6)

Then

∂ f (λ, x)∂x

= λγ−11 xγ−2 − λ

1− 1γ x− 1

γ .(4.7)


Define yλ > 0 such that yγ−2+1/γ

λ = (λ1λ−1/γ )1−γ . Then

y1− 1

γ

λ = λ−11 λ

1γ .

Then it is easy to check that

f (λ, yλ) = 0,∂f (λ, x)

∂x|x=yλ

= 0.(4.8)

The following proof is then similar to the previous case and we omit the details.This completes the proof of the theorem. �

REMARK 4.1. This theorem shows how the optimal strategy varies as we vary the relativerisk-aversion parameters between the individual agent and the benchmark. In Case I(a), there exists three different states for the optimal terminal wealth. The “good state”corresponding to lower state prices (when ξT < xλ) and the “bad state” correspondingto higher state prices, that is ξT > ξα, and other states are termed as the “middle state”(see Basak and Shapiro [2001] for a similar discussion on VaR constraints). Therefore,the bad state is fixed (above the ξα) in Case I (a). In Case I (b), even though there are stillthree different states, the good state is fixed (that is, below than ξ1−α). The ratio γind

γbenof

the risk-aversion parameters plays a key role in determining the optimal strategy.

REMARK 4.2. If neither of the risk-aversion parameters is equal to 1, the function f (λ, x)has a much more complicated shape. Suppose that the individual agent is denoted by agent1 and the benchmark by agent 2. We have u1(x) = x1−γ1

1−γ1, u2(x) = x1−γ2

1−γ2, � = Iγ2 (λγ2ξT),

where

Iγ 2 (x) = x− 1γ2 .

In this case, β(x) = Iγ 2 (λγ 2 x). Then

f (λ, x) = γ1

1 − γ1(λx)1− 1

γ1 − 11 − γ1

(λγ 2 x)γ1−1γ2 + λλ

− 1γ2

γ 2 x1− 1γ2 .(4.9)

If γ1 > γ2, then λα = λ

γ1γ2γ2 ξ

γ1γ2

−1α . Otherwise, λα = λ

γ1γ2γ2 ξ

γ1γ2

−1

1−α . For every λ > λα, f (λ, .) mighthave finitely many local extreme points. The region {I(λξT ) ≤ �, f (λ, ξT ) ≥ x} is compli-cated for each ω ∈ � and x > 0. But it is still possible to check both Assumptions 2.1and 2.2. Then Theorem 2.1 is used to find the optimal feasible strategy explicitly for thegeneral situation.

4.2. Benchmark Agent Has Identical Preferences to the Investor

We discuss whether an investor can strictly17 outperform the benchmark with someconfidence level when the benchmark utility function is the same as the individual in-vestor’s utility function. This could correspond to the case where an investor wants tooutperform his identical twin with some confidence level. In this case, the benchmarkis � = I(λuξT ). We now investigate the corresponding optimal portfolio selection prob-lem. Note that P(I(λξT ) ≥ I(λuξT )) equals to either 0 or 1 depending on λ > λu or not.Therefore,

λ < λu ⇐⇒ P(I(λξT) < I

(λuξT

))< 1 − α,

∀α ∈ (0, 1) ⇐⇒ P(I(λξT) < I

(λuξT

)) = 0.

(4.10)

17 The significance of the word “strictly” will emerge as we proceed.


Hence λα = λu, and the function λ → P(I(λξT) < I(λuξT)) is discontinuous at λu. Thenthe Assumption 2.1 does not hold.

In this framework we want see if one can devise a strategy that strictly beats thebenchmark (based on the same utility function) with some probability level. In otherwords, for a given α ∈ (0, 1), we seek the optimal feasible terminal wealth X x 0,π (T) forwhich P(X x 0,π (T) > �) ≥ α. This last constraint is different from the probabilistic con-straint in Theorem (2.1), which is P(X x 0,π (T) ≥ �) ≥ α. With some abuse of terminologylet us denote the constraint in Theorem (2.1) as the closed constraint and the current con-straint, where the terminal wealth strictly exceeds the benchmark, as the open constraint.The existence of the open constraint is the main18 reason why we cannot derive an optimalsolution in this case. However from an economic viewpoint it is natural19 to consider theopen constraint when the benchmark is constructed based on the same utility functionas the investor.

We assume both investors have the same CRRA utility with γ1 = γ2 = γ so that

u(x) = x1−γ

1 − γ, γ ∈ (0, 1).

We prove the following nonexistence result for this case.

THEOREM 4.2. There are feasible trading strategies for which the terminal wealthX x 0,π (T) satisfies P(X x 0,π (T) > �) ≥ α. However, there is no optimal strategy that solvesthe problem MaxE[u(X x 0,π (T))] subject to the constraint P(X x 0,π (T) > �) ≥ α.

Proof. In this case � = I(λuξT ). Recall that ξα is defined by P(ξT ≤ ξα) = α. Forany λ1 < λu , and λ1 > λu�(−d)γ , where

d =(

1 − 1γ

)σξ + µξ − logξα

σξ

let

λ2 = (λu)−

1γ − λ1

− 1γ �(d)

�(−d)

γ

then λ1 < λu < λ2. Define a random variable Xλ1(ω) = I(λ1ξT(ω)), if ξT ≤ ξα;

otherwise, Xλ1(ω) = I(λ2ξT(ω)). Then Xλ1

is a feasible terminal wealth satisfying theconstraints that P(Xλ1

> I(λuξT)) ≥ α, and E[ξT Xλ1] = x0.

If λ1 → λu , we have λ2 → λu as well. Define a function θ (λ1) := E[u(Xλ1)]. Then

by Lebesgue’s dominated convergence theorem (by using similar arguments as inLemma 2.3), if λ1 → λu ,

θ (λ1) → E[u(Xu

T

)].(4.11)

Now suppose there is one optimal feasible wealth XT for the current portfolio se-lection problem. Then E[u(XT)] ≥ θ (λ1) for all λ1 ∈ (λu�(−d)γ , λu). Let λ1 → λu , we

18 We are grateful to one of the referees for clarifying this point.19 It is not reasonable to require that

XT ≥ I(λξT)

because I(λξT ) is optimal already.


have E[u(XT )] ≥ E[u(XuT )]. Note that Xu

T is a feasible terminal wealth for Merton’s op-timization problem. Then by Remark 2.1, E[u(XT )] = E[u(Xu

T )] and XT = XuT = �, a.s.

Hence P(XT > �) = 0, which contradicts the constraint condition that P(XT > �) ≥α > 0. �

5. CONCLUSIONS

The constraints discussed in this paper are related to important risk-reward character-istics of actual investment practice. These include performance evaluation relative to abenchmark, guaranteed returns in the insurance and pension fund industry, aggressiveinvestment strategies and several aspects of portfolio risk management. These featurescan be modeled in an expected utility framework with (nonconvex) portfolio constraints.In this paper we provided an analytical approach to discuss these nonstandard portfolioselection problems and proved the existence of an explicit solution.

The results developed in the current paper provide both theoretical and practical in-sights. Using a probability measure to beat benchmark opens many ways to managewealth more effectively. However there are several extensions that we do not addresshere. These include the extension of the model to handle problems where there are shortselling constraints. Another possible extension is to extend the recent analysis of Bank andEl Karoui (2004), Bank and Follmer (2003), El Karoui, Jeanblanc-Picque, and Lacoste(2005), and El Karoui and Meziou (2006) to the case where the guarantee incorporatesan American feature. These extensions are left for future work.

APPENDIX

Proof of Lemma 2.2.

(1) Since limx↓0 G(λ, x) = H(λ) > 1 − α, for every λ > λα, and limx→∞ G(λ, x) = 0,then, by Assumption 2.1, there exists a d(λ, α) such that G(λ, d(λ, α)) = 1 − α.The uniqueness of d(λ, α) comes from Assumption 2.2 because G−1(λ, 1 − α) isan interval.

(2) For any sequence λn → λ > λα, we first prove that {d(λn, α) : n =1, 2, . . .} is bounded. For otherwise, if {d(λn, α) : n = 1, 2, . . .} is notbounded, then there exists a subsequence λnk such that d(λnk, α)goes to infinity. Hence limk→∞G(λnk, d(λnk, α)) = 0, which contradictsto limk→∞ G(λnk, d(λnk, α)) = 1 − α > 0 by the definition of d(λ, α). Then{d(λn, α) : n = 1, 2, . . .} is bounded.

For any convergent subsequence, say d(λnk, α), of d(λn, α), choose thelimit when nk → ∞. By Assumption 2.1, we have G(λ, limnk→∞ d(λnk, α)) =G(λ, d(λ, α)) = 1 − α. Then by Assumption 2.2 again, limnk→∞ d(λnk, α) =d(λ, α). This proves that d(λ, α) is the only one limit point of{d(λn, α) : n = 1, 2, . . .}. Thus, the required continuity property of the functiond(λ, α) with respect to λ is proved.

(3) We prove that

limλ↓λα

d(λ, α) = 0.(A.1)

If this were not the case, since {d(λ, α) : λ ↓ λα} is bounded by the same proof ofthe first part in Lemma 2.2 (2), there exists a sequenceλn ↓ λα such that d(λn, α) →x > 0. Thus, by Assumption 2.1, G(λα, x) = 1 − α. Note that H(λα) = 1 − α.


Then, G(λα, y) = 1 − α for every 0 < y < x. But this is impossible by Assump-tion 2.2. Then, d(λ, α) can be extended continuously over {λ ≥ λα} with d(λα,

α) = 0. �

To prove Lemma 2.3 we need the following preliminary lemma first.

LEMMA A.1. Assume 2.1 holds, then for every λ ≥ λα,x ≥ 0 we have

(1) P(I(λξT ) = �) = 0(2) P(I(λξT ) ≤ �, f (λ, ξT ) = x) = 0.

Proof of Lemma A.1. Choose a sequence λn ↓ λ, say λn = λ + 1n , since I(·) is de-

creasing, then

{I(λξT) ≤ �} =⋂

n

{I(λnξT) < �}.(A.2)

Then by using the continuity of H(·), we see that

P

(⋂n

{I(λnξT) < �})

= limn→+∞ P(I(λnξT) < �) = P(I(λξT) < �).(A.3)

Thus P(I(λξT ) = �) = 0. To prove (2), by using (1) of this lemma, it suffices to prove that,for every x ≥ 0,

P(I(λξT) < �, f (λ, ξT) = x) = 0.(A.4)

Choose a sequence xn ↓ x, say xn = x + 1n , since G(λ, x) is decreasing with respect to x,

{I(λξT) < �, f (λ, ξT) > x} =⋃

{I(λξT) < �, f (λ, ξT) ≥ xn}.(A.5)

Then by using the continuity of G(·), we see that

P(I(λξT) < �, f (λ, ξT) ≥ x) = limn→∞ G(λ, xn) = G(λ, x).(A.6)

Thus P(I(λξT ) < �, f (λ, ξT ) = x) = 0. The proof of (2) is then completed. �

Proof of Lemma 2.3. In the set of ω such that 1A(λn )(ω) does not converge to 1A(λ)(ω),there exists either infinitely many ones in the set {1A(λn )(ω) : n ≥ 1} but ω does not belongto A(λ), or there exists infinitely many zeros in the set {1A(λn )(ω) : n ≥ 1} but ω ∈ A(λ).More precisely

{1A(λn ) → 1A(λ)} ⊆(

lim supn→∞

A(λn)⋂

A(λ)c) ⋃ (

lim supn→∞

A(λn)c⋂

A(λ))

.(A.7)

Because I(·), f (., .), and d(λ, α) are continuous functions, we see that

lim supn→∞

A(λn) ⊆ {I(λξT) ≤ �, f (λ, ξT) ≤ d(λ, α)}.(A.8)

On the other hand, A(λ)c = B(λ)⋃

C(λ), then

lim supn→∞

A(λn)⋂

A(λ)c ⊆(

lim supn→∞

A(λn)⋂

B(λ)) ⋃ (

lim supn→∞

A(λn)⋂

C(λ))

⊆ {I(λξT) ≤ �, f (λ, ξT) = d(λ, α)}⋃

{I(λξT) = �}.


Moreover,

lim supn→∞

A(λn)c ⊆ lim supn→∞

B(λn)⋃

lim supn→∞

C(λn).(A.9)

and

lim supn→∞

A(λn)c⋂

A(λ) ⊆(

lim supn→∞

B(λn)⋂

A(λ)) ⋃ (

lim supn→∞

C(λn)⋂

A(λ))

.(A.10)

It is easy to see that

lim supn→∞

B(λn)⋂

A(λ) ⊆ {I(λξT) ≤ �, f (λ, ξT) = d(λ, α)},(A.11)

and

lim supn→∞

C(λn)⋂

A(λ) = φ,(A.12)

where φ is the empty set. Thus 1A(λn ) → 1A(λ), a.s, follows from Lemma A.1. Similarly,we can prove that 1B(λn ) → 1B(λ), a.s, and 1C(λn ) → 1C(λ), a.s. �

Proof of Lemma 2.4. Since A(λ) ⊆ {f (λ, ξT ) ≤ d(λ, α)}, then limλ→∞1A(λ) ≤limλ→∞1{ f (λ,ξT )≤d(λ,α)}. Assume that limλ→∞1A(λ) = 0, it suffices to provethat limλ→∞1{ f (λ,ξT )≤d(λ,α)} = 0. Note that

{ f (λ, ξT) ≤ d(λ, α)} = A(λ)⋃

{I(λξT) ≥ �, f (λ, ξT) ≤ d(λ, α)}.(A.13)

By using Inada condition, limλ→∞ I(λξT) = 0, then

limλ→∞ P(I(λξT) ≥ �, f (λ, ξT) ≤ d(λ, α)) ≤ limλ→∞ P(I(λξT) ≥ �) = 0.(A.14)

Hence limλ→∞1{ f (λ,ξT )≤d(λ,α)} = 0. �

REFERENCES

ALEXANDER, G., and A. BAPTISTA (2004): Active Portfolio Management with Benchmarking:Adding a Value-at-Risk Constraint, J. Eco. Dyn. Control (forthcoming).

BASAK, S. (1995): A General Equilibrium of Portfolio Insurance, Rev. Finan. Stud. 8, 1059–1090.

BASAK, S., and A. SHAPIRO (2001): Value-at-Risk-Based Risk Management: Optimal Polices andAsset Prices, Rev. of Finan. Stud. 14, 371–405.

BASAK, S., and A. SHAPIRO (2005): A Model of Credit Risk, Optimal Policies, and Asset Prices,J. Business 78, 1215–1266.

BASAK, S., A. SHAPIRO, and L. TEPLA (2006): Risk Management with Benchmarking, Manage-ment Science 52, 542–557.

BANK, P., and N. EL KAROUI (2005): A Stochastic Representation Theorem with Applicationsto Optimization and Obstacle Problems, Ann. Prob. 32(1B), 1030–1067.

BANK, P., and H. FOLLMER (2003): American Options, Multi-armed Bandits and Optimal Con-sumption Plans: A Unifying View, Paris-Princeton Lectures on Mathematical Finance 2002,Lecture Notes in Mathematics 1814, 1–42, Berlin: Springer.

BIELECKI, T. R., S. R. PLISKA, and M. SHERRIS (2000): Risk Sensitive Asset Allocation, J. Econ.Dyna. Control 24, 1145–1177.

BRENNAN, M., and R. SOLANKI (1981): Optimal Portfolio Insurance, J. Financial Quant. Anal.16, 279–300.


BRENNAN, M., and E. S. SCHWARTZ (1989): Portfolio Insurance and Financial Market Equilib-rium, J. Business 62, 455–472.

BROWNE, S. (1999): Reaching Goals by a Deadline: Digital Options and Continuous-Time ActivePortfolio Management, Adv. Appl. Prob. 31, 551–577.

BROWNE, S. (2000): Risk Constrained Dynamic Active Portfolio Management, Manage. Sci. 46,1188–1199.

COX, J. C., and C. F. HUANG (1989): Optimal Consumption and Portfolio Policies When AssetPrices Follow a Diffusion Process, J. Econ. The. 49, 33–83.

CVITANIC, J., and I. KARATZAS (1992): Convex Duality in Constrained Portfolio Optimization,Ann. Appl. Prob. vol 2, 766–818.

CVITANIC, J., and I. KARATZAS (1999): On Dynamic Measure of Risk, Finance and Stochastics3, 451–482.

DANIELSSON, J., B. JORGENSEN, C. DE VRIES, and X. YANG (2001): Optimal Portfolio Allocationunder a Probability Risk Constraint and the Incentives for Financial Innovation, TinbergenInstitute Discussion Paper T1-2001-069/2.

DETEMPLE, J., and S. MURTHY (1997): Equilibrium Asset Prices and No Arbitrage with PortfolioConstraints, Rev. Finan. Stud. 10(4), 1133–1174.

DYBVIG, P. H., L. C. ROGERS, and K. BACK (1999): Portfolio Turnpikes, Rev. Finan. Stud. 12,165–195.

FOLLMER, H., and P. LEUKERT (1999): Quantile Hedging, Finance and Stochastics 3, 251–273.

GROSSMAN, S. J., and J. VILA (1989): Portfolio Insurance in Complete Market: A Note, J. Business62, 473–476.

GROSSMAN, S. J., and Z. ZHOU (1996): Equilibrium Analysis of Portfolio Insurance, J. Finance51, 1379–1403.

EL KAROUI, N., M. JEANBLANC-PICQUE, and V. LACOSTE (2005): Optimal Portfolio Managementwith American Capital Guarantee, J. Econ. Dyn. Control 29, 449–468.

EL KAROUI, N., and A. MEZIOU (2006): Constrained Optimization with Respect to StochasticDominance: Application to Portfolio Insurance, Math. Finance 16(1), 103–117.

JENSEN, B. A., and C. SORENSEN (2001): Paying for Minimal Interest Rate Guarantees: WhoShould Compensate Whom? Eur. Finan. Manage. 7, 183–211.

KARATZAS, I. (1996): Lecture on the Mathematics of Finance, CRM Monograph, Vol 8, Provi-dence, RI: American Mathematical Society.

KARATZAS, I., J. P. LEHOCZKY, and S. E. SHREVE (1987): Optimal Portfolio and ConsumptionDecisions for a “Small Investor” on a Finite Horizon, SIAM J. Control Optimization 25,1557–1586.

LELAND, H. E. (1980): Who Should Buy Portfolio Insurance? J. Finance 35(2), 581–594.

MERTON, R. C. (1971): Optimum Consumption and Portfolio Rules in a Continuous-TimeModel, J. Econ. Theor. 3, 373–413.

PHAM, H. (2003): A Large Deviation Approach to Optimal Long Term Investment, Finance andStochastics 7(2), 169–195.

PLISKA, S. (1986): A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios,Math. Operations Research 11, 371–384.

ROLL, R. (1992): A Mean/Variance Analysis for Tracking Errors, J. Portfolio Manage. 18(4),13–22.

RUBINSTEIN, M. (1974): An Aggregation Theorem for Securities Markets, J. Financ. Econ. 1,225–244.


SORENSEN, C. (1999): Dynamic Asset Allocation and Fixed Income Management, J. Financ.Quant. Anal. 34, 513–531.

SPIVAK, G., and J. CVITANIC (1999): Maximizing the Probability of a Perfect Hedge, Ann. Appl.Probab. 9, 1303–1328.

STUTZER, M. (2003): Portfolio Choice with Endogenous Utility: A Large Deviation Approach,J. Econo. 116, 365–386.

TEPLA, L. (2001): Optimal Investment with Minimum Performance Constraints, J. Econ. Dyn.Control 25, 1629–1645.

WACHTER, J. A. (2003): Risk Aversion and Allocation to Long-term Bonds, J. Econ. Theo. 112,325–333.

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