Ba r t i c l e i n f o

Article history:Received July 2009Received in revised formNovember 2009Accepted 3 November 2009

JEL classification:C61G11

MSC:IE13IE43IE53IB10IB81

Keywords:Assetliability managementPortfolio optimizationBenchmarking

a b s t r a c t

We solve the optimal asset allocation problem for an insurer or pension fund by using a benchmarkingapproach. Under this approach the objective is an increasing function of the relative performance of theasset portfolio compared to a benchmark. The benchmark can be, for example, a function of an insurersliability payments, or the (either contractual or target) payments of a pension fund. The benchmarkingapproach tolerates but progressively penalizes shortfalls, while at the same time progressively rewardsoutperformance. Working in a general, possibly non-Markovian setting, a solution to the optimizationproblem is presented, providing insights into the impact of benchmarking on the resulting optimalportfolio. We further illustrate the results with a detailed example involving an option based benchmarkof particular interest to insurers and pension funds, and present closed form solutions.

2009 Elsevier B.V. All rights reserved.

1. Introduction

Traditional optimal asset allocation problems in the invest-ment management literature typically involves maximizing theexpected utility of a terminal portfolio value, and where the util-ity is a concave function possibly satisfying additional properties.However it is well known (see, for example, Panjer et al. (1998)and Yang and Zhang (2005)) that it is also important to considerthe effects of the liability payments when determining the optimalasset allocation for insurers and pension funds. Moreover, adapta-tion of standard financial economic methods (for example, max-imization of expected power utility of terminal surplus) in aninsurance/pensions setting is often not straightforward due to thenature of insurance and pension problems. In particular, typical

Corresponding address: School of Actuarial Studies, Australian School ofBusiness, University of New South Wales, NSW 2052, Australia. Tel.: +61 2 93852827; fax: +61 2 9385 1883.E-mail addresses: lim@ieor.berkeley.edu (A.E.B. Lim),

bernard.wong@unsw.edu.au (B. Wong).

utility functions favored in the financial economics literature (e.g.power utility) often involve a strict floor (typically at zero sur-plus) in the terminal wealth which cannot be violated (see alsoTepl (2001) for a related approach where the solvency guaranteeis placed in the constraints). While such absolute solvency guaran-teesmay be a desirable property in some applications, it iswell rec-ognized in the insurance literature that such guarantees may notbe financially desirable, or, indeed, feasible. For example, in under-funded defined benefit pension plans the assets are not sufficientto cover the liabilities with certainty by definition (see, for exam-ple, the discussion in Detemple and Rindisbacher (2008)), and inmany insurance models the probabilistic structures of many claimprocesses often mean that liabilities cannot be funded with cer-tainty under a realistic initial asset wealth, even if a guarantee isachievable at all. Such situations lead to an ill-posed optimizationproblem if the floor cannot be met with certainty, and no optimalsolutions are available.Hence in insurance and pension optimal asset allocation prob-

lems it is vital to consider approaches that can tolerate shortfalls.In this paper we consider the asset allocation problem by using abenchmarking approach. Under this approach the objective is anInsurance: Mathematics and E

Contents lists availa

Insurance: Mathema

journal homepage: www

A benchmarking approach to optimal asspension fundsAndrew E.B. Lim a, Bernard Wong b,a Department of Industrial Engineering and Operations Research, University of California,b School of Actuarial Studies, University of New South Wales, Australia0167-6687/$ see front matter 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2009.11.005conomics 46 (2010) 317327

ble at ScienceDirect

tics and Economics

.elsevier.com/locate/ime

et allocation for insurers and

erkeley, United States

318 A.E.B. Lim, B. Wong / Insurance: Mathem

increasing and concave function of the relative performance of theasset portfolio versus a benchmark, possibly with an additionalfloor if required. In insurance and pension applications the bench-mark will be a function of either target, or contractual liabilitypayments. This approach tolerates but progressively penalizes un-derperformance, while at the same time progressively rewardsoutperformance. Uncertainty in the financial market will be drivenby underlying Brownian motions, while the assets and the bench-mark are assumed to be general stochastic processes adapted to thefiltration of the Brownian motions. In particular, it is not requiredthat the assets or the benchmark follow geometric Brownian mo-tion (a common assumption in the literature), nor do they in factneed to be Markov, and can depend on other underlying processesin a similarmanner to traditional actuarial investment returnmod-els. It could also be from a large class of probability distributions.The benchmarks that we consider are very general random vari-ables measurable with respect to the underlying filtration, are notrequired to be attainable with the investors current wealth. Wepresent a general solution to the asset allocation problem, and il-lustrate our results with closed form solutions for an option basedbenchmark of particular relevance to insurers and pension funds.Several methods have been proposed in the insurance lit-

erature to consider the optimal asset allocation problem withtolerance for possible shortfalls. We note, however, that thesemethods differ from ours in either the objective function that isused and/or the stochastic model for the assets and liabilities. Theobjective functions that have been studied in the literature canbe broadly classified into two alternative groups. The first groupconsiders as an objective the probability of reaching some desiredoutcome (for example, solvency). Examples include Browne (1995,1999, 2000), Josa-Fombellida and Rincn-Zapatero (2006). From abehavioral perspective however this is often undesirable as it im-plies that the decision maker will solely focus on achieving the de-sired outcome (for example, to stay solvent), and in particular thelevel of final outperformance or underperformance is ignored. Thesecond approach is tomaximize the expected utility of the surplus.In order to tolerate shortfalls such an approach requires a utilityfunction that can be defined over negative values. The two primeexamples of such utility functions are those of exponential andquadratic type. The application of an exponential utility functionin dynamic assetliabilitymodellingwas explored recently in KornandWiese (2008), Yang and Zhang (2005), Wang (2007) andWanget al. (2007). Quadratic utility/loss functions have been explored inthe insurance literature by Cairns (2000), Chiu and Li (2006), Xieet al. (2008), Chen et al. (2008), and is also related to the problemof mean-variance hedging (see Schweizer (2001), Lim and Zhou(2001), Lim (2004, 2005) and the references therein). While theuse of the exponential and quadratic approaches provide a numberof benefits including tractability, both have undesirable propertieswhich are well known. Exponential utility, for example, impliesconstant absolute risk aversion, while quadratic utility possessesa saturation level beyond which utility decreases with increasingwealth. (See Luenberger (1998) for additional discussion).The use of a benchmarked return in a utility function has also

been used in the literature for problems related to real returnsand also with target pension funding ratios. In particular, Brennanand Xia (2002) and De Jong (2008) considered a dynamic assetallocation problem for long term investors where the objectivefunction is a power transformation of the real wealth, whileCairns et al. (2006) investigate stochastic lifestyling strategies forpension plans by considering as an objective function a powertransformation of the terminal wealth divided by terminal salary.Davis and Lleo (2008) considered a risk sensitive asset allocationproblem where the objective is related to the power utility of

the benchmarked return, and where the benchmark is a variant ofGeometric Brownian Motion. We note however that the previousatics and Economics 46 (2010) 317327

papers assume that prices processes and benchmarks are variantsof Geometric Brownian motion, and utility functions of powertype. In contrast, we consider very general dynamics for asset pricedynamics and the benchmark, and general increasing concavefunctions of the relative performance. In particular, benchmarkssuch as the maximum of a random quantity (such as a stock index)and a minimum return, which are not naturally formulated asGeometric BrownianMotion but are of interest in asset and liabilitymanagement, can be handled in our model.In related areas, although not of direct relevance to the problem

weconsider, utility on benchmarkedperformancehave also beenused in van Binsbergen et al. (2008) for themodelling of decentral-ized investment management, and in Lim et al. (forthcoming) forrobust asset allocation problems.Finally, note that the benchmarking approach is also applicable

to general investment management problems without a liability.Indeed, it is also particularly suited to problems where the per-formance of an investment fund is measured in a relative sense,with the benchmark being, for example, the performance of amar-ket index, or a quantile of peer performance, or any combinationsthereof.An outline of our paper follows. In Section 2we setup the finan-

cial market we consider. Section 3 defines a benchmark, and pro-vides discussion regarding the concept of a benchmarking functionand its relationship to standard utility functions. A general solutionto the optimization problem using martingale techniques is pre-sented in Section 4. A detailed example involving an option basedbenchmark of particular interest to insurers and pension funds isstudied in Section 5. Section 6 concludes.

2. Financial market

We use a standard financial market model setup (cf. Karatzasand Shreve (1998)). Consider the probability space (,F , P)and the time interval [0, T ], the filtration being generated by k-dimensional Brownian MotionWi(), i = 1, 2, . . . , k, augmentedto satisfy the usual conditions.

2.1. Primary securities

Assume that there are k+1primary securities traded in the timeinterval [0, T ]. One of the securities represents a savings accountprocess S0(), with dynamics

S0(t) = exp{ t0r(u)du

}, (1)

where r() is a progressively measurable process representing theshort rate, with T0|r(t)| dt 0, i = 1, . . . ,k}, and dynamics

dSi(t) = i(t)Si(t)dt + Si(t)kj=1

ij(t)dWj(t) i = 1, 2, . . . , k, (3)

for some progressively measurable, k-dimensional vector valued(), and k k-dimensional matrix valued () processes,satisfying T0

ki=1|i(t)| dt

A.E.B. Lim, B. Wong / Insurance: Mathem

almost surely. Assume that () has full rank for every t . Define themarket price of risk process () by(t) = ( (t))1((t) r(t)1k), (5)with 1k being the k 1 vector (1, 1, . . . , 1).An implication of the above assumptions is that the market

is complete. As discussed in Detemple and Rindisbacher (2008),this assumption, while strong, may be reasonable in view of theexpanding number of assets available in the financial markets. Inparticular, incomplete markets may be rendered complete by theaddition of new assets such as derivatives.It is worth noting that the above setup is very general in that the

coefficients can be stochastic. For example, the stock prices couldexhibit stochastic volatility (cf. Heston (1993) and Fouque et al.(2000)), or follow constant elasticity of variance processes (Coxand Ross, 1976, see also Xiao et al. (2007)), or be driven by ergodicdiffusions (Wong, 2009). Furthermore, while for convenience wehave called the assets stocks, the models are sufficiently generalthat they include other asset classes including inflation, currency,or interest rate linked securities.The interest rates process is also very general, and includes for

example thewell knownVasicek (1977) and Cox et al. (1985)mod-els. It is worth noting that stochastic interest rates are particularlyimportant for insurance and pension applications due to the longtermnature of the problems. See for example Deelstra et al. (2000),Boulier et al. (2001) and Gao (2008) for optimal portfolio problemswith stochastic interest rates.It is also worth noting that, subject to the assumption of

completeness of markets (for example, via the introduction ofderivatives to complete an incomplete market), our setup includesmodels where stock returns are driven by other underlyingeconomic variables such as inflation and unemployment. Thesemodels are in the spirit of traditional actuarial investment returnmodels (cf. Panjer et al. (1998)). See also Detemple and Rindis-bacher (2008). Blake et al. (2001) provides additional discussionon some implications of different asset return models for pensionasset allocation strategies.We will make the additional assumption that, almost surely, T0(t)2 dt 0 and initial

value x, we have the following budget constraint as a consequenceof the supermartingale property of H0()X().Definition 3. A budget constraint for a portfolio with terminalvalue X(T ) is

E[H0(T )X(T )] x. (12)Consider a non-negative, FT measurable random variable ,

and constant x > 0. In our market we have the following wellknown result regarding the attainability of through hedgingstrategies (cf. Karatzas and Shreve (1998, Theorem 2.3.5)).

Lemma 4. Let x > 0 be given, and let be a non-negative, FTmeasurable random variable such that

E[H0(T ) ] = x. (13)Then there exist a portfolio process pi() A(x) with associatedterminal value = X(T ).The exact formofpi() is closely related to themartingale represen-tation theorem (cf. Karatzas and Shreve (1991, Theorem 3.4.15)).In practice pi() can be calculated with the help of the Malliavincalculus (cf. Fourni et al. (1999), Elworthy and Li (1994), Detem-ple et al. (2005)) and related results (cf. Broadie and Glasserman(1996) and Glasserman (2004)). In general, for complicated mod-els numerical methods such as simulation can be used. Boyle et al.(2008) provides additional discussion on the application of MonteCarlo techniques to the calculation to optimal portfolios. In somespecial cases closed form solutions are available. A detailed exam-ple with closed form solutions is provided in Section 5.

3. Benchmark and benchmarking function

3.1. Benchmark

A benchmark is a strictly positive, FT measurable random vari-able satisfying amild technical condition below. In applications thebenchmark could be any deterministic or stochastic outcome. Forexample, it could be a function of the insurers liability payments,or a reference investment index of some sort, or the outcome of atraditional conservative (or otherwise) hedging strategy. In partic-ular, there is no need for the index to be a simple function of theterminal value of the primary assets.

Definition 5. A benchmark is a strictly positive, FT measurablerandom variable satisfying

E[H0(T ) ]

320 A.E.B. Lim, B. Wong / Insurance: Mathem

distribution is general. In particular, can be constructed to havea gamma distribution or any other distribution that is commonlyused in insurance modelling. Specific distributions can be con-structed by appropriate choice of the drift and volatility coeffi-cients for the diffusion Y (). See for example the monograph ofKarlin and Taylor (1981), and alsoMadan and Yor (2002) andWong(2009) for additional discussion on the construction of diffusionswith specified distributions.Let y > 0 be the value of the expectation in (14). Define a

process Y () by

Y (t) = 1H0(t)

E[H0(T ) |Ft ]. (15)The starting and terminal values of Y () satisfiesY (0) = y (16)Y (T ) = ,and as H0() is strictly positive we see that Y ()will always be welldefined.Eq. (14) can be interpreted, via Lemma 4, as a terminal wealth

that can be attained with finite initial wealth y. In particular thatthere exist a portfolio, with associated value process Y (). Hencewe can view Y () as themarked-to-market value of the benchmark . Note that it is not necessary that the value of the benchmark yequals the investors initial wealth x. (That is, we do not assumethat Y (T ) is attainable from the initial wealth x).For simplicity of notation, in the remainder of this paperwewill

refer to Y (T ) as the benchmark.

3.2. Benchmarking function

Consider F(X(T )Y (T )

), where X() it the dollar value of our portfolio

strategy and Y (T ) is the (strictly positive) outcome of the bench-mark. We call F() a benchmarking function. As we wish to rewardoutperformance F() should have a positive gradient. We also as-sume (just like a utility function) that F() is concave to representdecreasing marginal gains as the benchmarked performance in-creases (see Gerber and Pafumi, 1998).Specifically, define a benchmarking function F() on the ben-

chmarked terminal wealth as a concave, non-decreasing, uppersemicontinuous function F : R [,). By analogy withthe analytics of standard utility functions (cf. Karatzas and Shreve(1998), Section 3.3.4) we assume that F() satisfies the followingproperties:

1. the half-line dom(F) , { R; F() > } is a non-emptysubset of [0,]

2. F is continuous positive and strictly decreasing on the interiorof dom(F), and F () = lim F () = 0.

3. We also set , inf { R; F() > }, with [0,).4. The strictly decreasing, continuous function F : (,) onto(0,F (+)) has a strictly decreasing, continuous inverse :(0, F (+)) onto(,). We further set () = for F (+) . This implies that () is well defined, finite andcontinuous on (0,], with

F ( ()) ={; 0 < < F (+)F (+) F (+). (17)

The above definition is very general. Note also that we do notassume = 0, nor do we assume the Inada condition. This for ex-ample allows for the use of benchmarking functions of exponentialtypeF() = e; > 0, (18)atics and Economics 46 (2010) 317327

with constant > 0. In this paper we are particularly interested inbenchmarking functions of power type

F() = p

p; > 0, (19)

for p = (, 1) \ {0}. In addition, in cases where we wish to im-pose a minimal relative outcome , with a strictly positive con-stant, we can consider benchmarking functions

F (q)() =

( )qq; >

0; = ; < ,

(20)

for q = (, 1) \ {0}.Note that whilst themathematical description of the properties

of F() mimic that of standard utility functions, it is importantto note that F() is not a traditional utility function due to itsdependence on Y (T ). An alternative interpretation of our objectivefunction could be as a stochastic, state dependent utility functionU(X(T ), ), with the benchmark acting as a numeraire.

4. Optimization problem and solution: General case

For a given benchmarking function F(), benchmark Y (), andinitial wealth x, the optimization problem is as follows: Find anoptimal portfolio pi() A(x) for the problem

J(x) = suppi()A(x)

E[F(X(T )Y (T )

)](21)

of maximizing the expected benchmarked utility from terminalwealth, with

A(x) ={pi() A(x); E

[F(X(T )Y (T )

)]

A.E.B. Lim, B. Wong / Insurance: Mathem

The problem (23) is analyzed using Lagrangian duality. Specifi-cally, introducing a Lagrange multiplier > 0, we have

E[F(

Y (T )

)]+

(x E

[H0(T )Y (T )

Y (T )

])= x+ E

[F(

Y (T )

) H0(T )Y (T ) Y (T )

] x+ E

[sup

{F(

Y (T )

) H0(T )Y (T ) Y (T )

}]= x+ E

[sup

{F() H0(T )Y (T )

}]= x+ E [F(H0(T )Y (T ))], (24)

where

F() = sup

(F() ) (25)is the LegendreFenchel transform of F(). The basic idea is that theinequality in (24) holds with equality if the Lagrange multiplier is chosen so that the budget constraint is satisfied with equality,and that themaximizer over in such a case should be the optimalterminal wealth for the problem (23).Our conditions on F() imply that the supremum in (25) is

achieved by some = ()whenever > 0, and thatF() = F( ()) (). (26)Since H0(T )Y (T ) is strictly positive a.s. themaximizer in (24) sat-isfies

= Y (T )

= (H0(T )Y (T )) (27)or equivalently

= Y (T ) (H0(T )Y (T )). (28)The next step is to find the constant > 0 so that the terminalwealth satisfies the budget constraint, namely

E[H0(T ) ] = E[H0(T )Y (T ) (H0(T )Y (T ))] = x. (29)With this in mind, let

X() := E[H0(T )Y (T ) (H0(T )Y (T ))], (30)define a function of on the interval (0,), and let G() denotethe inverse ofX() defined byX(G()) = .It follows that the budget constraint (12) (or (24)) is satisfied withequality if and only if we choose = G(x). A candidate for theoptimal terminal wealth is thus given by

= Y (T ) (G(x)H0(T )Y (T )), (31)with associated portfolio process X() determined by Lemma 4.The following result formalizes this heuristic derivation. Ac-

companying technical results can be found in the Appendix.

Theorem 7. Assume

X() := E[H0(T )Y (T ) (H0(T )Y (T ))]

322 A.E.B. Lim, B. Wong / Insurance: Mathem

and hence

X(g) = E[H0(T )Y (T )(gH0(T )Y (T ))

1p1]

= g 1p1 E[(H0(T ))

pp1 (Y (T ))

pp1]. (40)

Now we havepp 1 < 1 (41)and furthermore by assumption on the financial market and thedefinition of a benchmark,

P(H0(T )Y (T ) > 0) = 1. (42)This implies, with (14), that

E[(H0(T ))

pp1 (Y (T ))

pp1] 1+ E[H0(T )Y (T )]< , (43)

and hence we see that (32) is satisfied for general Y (), as claimedin Remark 8.It follows that

G(x) = xp1(

E[(H0(T ))

pp1 (Y (T ))

pp1])p1 ,

and, by applying Theorem 7, the optimal terminal wealth is

X(T ) = Y (T ) (G(x)H0(T )Y (T ))

= Y (T ) xp1H0(T )Y (T )(E[(H0(T ))

pp1 (Y (T ))

pp1])p1

1p1

= x(H0(T ))1p1 (Y (T ))

pp1

E[(H0(T ))

pp1 (Y (T ))

pp1] . (44)

The optimal terminal wealth in (44) can be compared with the op-timal terminal wealth in a standard (non-benchmarked) optimalportfolio problem with power utility function. The standard opti-mization problem has optimal terminal wealth

x(H0(T ))1p1

E[(H0(T ))

pp1] (45)

and hence we see on comparison that the introduction of thebenchmark has effectively tilted the optimal terminal wealth by afactor equal to Y (T )

pp1 . Observe also that, in the limit as p,

we havepp 1 11p 1 0hence

X(T ) xY (T )E[H0(T )Y (T )] =

xyY (T ). (46)

Hence in the limit when p the optimal wealth is simply thebenchmark Y (T ) scaled by x/y.Note that in the limit when p 0 the benchmarking function

becomes that of a log type. Consider

F() = ln, (47)

with inverseatics and Economics 46 (2010) 317327

() = 1, (48)

hence

X(g) = E[H0(T )Y (T )gH0(T )Y (T )

]= 1g,

and in particular notice that assumption (32) is satisfied. We thenhave

G(x) = 1x. (49)

In aggregate, the optimal terminal wealth is

X(T ) = Y (T ) (G(x)H0(T )Y (T ))= Y (T )1

xH0(T )Y (T )

= xH0(T )

, (50)

implying that the benchmark is totally ignored. The optimal port-folio is the growth optimal portfolio (cf. Luenberger (1998), Platen(2005) and references therein).The reason for this result can be seen as follows. The objective

function is

suppi()A(x)

E[ln(X(T )Y (T )

)]

=(sup

pi()A(x)E[ln(X(T ))]

) E[ln(Y (T ))], (51)

which is equivalent to a standard log-optimal problem withoutbenchmarking as Y () is not affected by the portfolio pi()

4.2. Power benchmarking function with a minimum threshold ratio

In some applications a lower bound of 0 relative wealth may beinappropriate. In the following we will consider a benchmarkingfunction of power type which moreover enforces a non-zerolower bound on the relative levels of the terminal wealth and thebenchmark. Let > 0 be the lower bound. For q (, 1) \ {0}consider a class of benchmark functions of the form

F (q)() =

( )qq; >

0; = ; <

(52)

with inverse

() = + 1q1 . (53)Hence we have

X() = E[H0(T )Y (T ) (H0(T )Y (T ))]= E[H0(T )Y (T )] +

1q1 E

[(H0(T )Y (T ))

qq1]

= y+ 1q1 E[(H0(T )Y (T ))

qq1], (54)

and, by noting (43), it follows that (32) is satisfied.The inverse ofX() is

yq1G() = E[(H0(T )Y (T ))

qq1] , (55)

A.E.B. Lim, B. Wong / Insurance: Mathem

and hence, for initial wealth x > y, the optimal wealth is

X(T ) = Y (T ) (G(x)H0(T )Y (T ))

= Y (T )+ x yE[(H0(T )Y (T ))

qq1] (H0(T )) 1q1 Y (T ) qq1 . (56)

Hence the optimal portfolio allocation for the power benchmarkingfunction with required minimum ratio satisfies a fund separationproperty. Namely, y of the initial wealth is used to replicateY (T ), and the remainder of the initial wealth is invested in aportfolio corresponding (after scaling) to the power benchmarkingfunction case presented in Section 4.1. On the other hand if x < ythen theminimumthreshold cannot be guaranteed andno solutionexists.

4.3. Exponential benchmarking function

Consider a benchmarking function of exponential type

F() = e; > 0, (57)with constant > 0. The inverse is

() = 1ln(

), (58)

hence

X(g) = E[H0(T )Y (T )

ln(

gH0(T )Y (T )

)].

Assume that Y (T ) is chosen such that assumption (32) is satisfied.The optimal terminal wealth is

X(T ) = Y (T ) (G(x)H0(T )Y (T ))= cY (T ) 1

Y (T ) ln(H0(T )Y (T )) (59)

with c a constant determined to ensure that the starting wealthis x.

5. A portfolio insurance benchmark

In this section we focus on optimization problems where anoption based portfolio insurance strategy is used as a benchmark.Consider as a benchmark a portfolio strategy with payoff

Y (T ) = max(S(T ), K), (60)with a strictly positive constant. This strategy provides a floorguarantee of K . Let y be the initial cost of this portfolio. Clearly

y = E[H0(T )max(S(T ), K)] (61)(recall that H0() is the state price density). In this section we willconsider an asset allocation problem with Y (T ) as the benchmark(possibly with constrained minimum ). Note that it is notassumed that y = x. When y > x the benchmark Y (T ) cannot bereplicated with initial wealth x, corresponding to an underfundedsituation.The use of benchmarks of the form (60) can be interpreted as

that of a problem where the relative performance of a portfoliorelative to a stochastic outcome S(T ) is important when S(T )performs well (i.e. when it is greater than K ), but on the otherhand when S(T ) performs badly the measure of performance of aportfolio is of an absolute sense. This is particularly of interest forthe asset allocation problems for DC pension plans where S()willrepresent a market index.

The benchmark (60) is also of interest for the optimal asset al-

location of insurers selling equity linked guarantees of the point toatics and Economics 46 (2010) 317327 323

point type (cf. Hardy, 2003), provided that (possibly limited) short-falls can be tolerated by the insurer in exchange for the possibilityof outperformance. In this case this benchmark will represent di-rectly the liability payments.Consider now the power benchmark function in Section 4.1

(the corresponding results for power benchmarking functionswithminimum threshold ratios follows automatically by the discus-sion in Section 4.2). From (44) it follows that the optimal terminalwealth is

X(T ) = x(H0(T ))1p1 (max(S(T ), K))

pp1

E[(H0(T ))

pp1 (max(S(T ), K))

pp1] . (62)

with associated value process

X(t) =x 1H0(t)E

[(H0(T ))

pp1 (max(S(T ), K))

pp1Ft]

E[(H0(T ))

pp1 (max(S(T ), K))

pp1] . (63)

5.1. Optimal portfolio in a market with constant coefficients

While the optimal wealth (62)(63) hold for general financialmarketmodels,we can obtain additional insights by the case of a fi-nancial market with constant coefficients. Specifically, we are ableto derive in closed form the optimalwealth and portfolio allocationusingmartingale techniques, in particular by using change of mea-sure techniques (cf. Geman et al. (1995)) and the BismutElworthyformula (cf. Elworthy and Li (1994) and Qin (2008)). The proofs ofthe following formulas can be found in the Appendix.We will first require the value of the denominator in (62),

E[(H0(T ))

pp1 (max(S(T ), K))

pp1]

= pp1(Spp1 (0)p(T )(N(c2,p(0, T )))

+ K pp1mp(T )N(c1,p(0, T )))

(64)

with

mp(T ) = exp{p1 p rT +

p2(1 p)2

2T}

(65)

p(T ) = exp{ p(1 p)2 ( r)T +

p2(1 p)2 (

2 + 2)T}, (66)

and where N() is the cumulative distribution function of a stan-dard normal random variable, and

c1,p(t, T ) =ln(KS(t)

)(+ p1p ( r) 12 2

)(T t)

T t , (67)

and

c2,p(t, T ) = c1,p(t, T )+ p1 pT t. (68)

The portfolio allocation follows by an application of the BismutElworthy formula andmeasure transformation techniques (detailsin the Appendix):

pi(t) = x(H0(t))1p1 e

p2(Tt)2(1p)2(

Spp1 (0)p(T )N(c2,p(0, T ))+ K

pp1mp(T )N(c1,p(0, T ))

)(Spp1 (t)e

p(r+ 12 2)(Tt)(1p)2

(

(1 p) p1 p

)N(c2,p(t, T ))

pr(Tt) ( ) )

+ K pp1 e (1p)2

(1 p) N(c1,p(t, T )) . (69)

324 A.E.B. Lim, B. Wong / Insurance: Mathem

Notice in particular that in the limit as pwe havepi(t) = x

(S(0)N(c2,(0, T ))+ KerTN(c1,(0, T ))) S(t)N(c1,(t, T )+

T t) (70)

where ci,(t, T ) = limp ci,p(t, T ) for i = 1, 2 is just the usualintegral limits as per the BlackScholes formula. Hencewe see thatin the limit as p the portfolio strategy (up to scaling by therelative initial wealth) is just the portfolio allocation strategy of thebenchmark. Similarly, in the limit as p 0 we have

pi(t) =(xH0(t)

)(

)(71)

which is just the log-optimal strategy.

5.2. Numerical illustrations

Example 1: Power benchmarking functionWe consider a single risky asset with parameters = 10%,

= 20%. The initial stock price is S(0) = 1 = K . There is a moneymarket account with interest rate r = 4%. The time horizon T is 1year.For the chosen parameter values, the price of the benchmark

y = 1.1255, and for convenience, we have chosen the initial valueof the portfolio x(0) = y = 1.1255 so that the optimal terminalwealth coincides with the benchmark max{S(T ), K}, when p =.Fig. 1 shows plots of the benchmark investors terminal wealth

for p = 0.3,2,4 and . Note that the terminal wealthcoincides with the benchmark when p = . Also shown isthe optimal wealth of a standard power utility maximizer (withrisk-aversion parameter p = 4) and the density function ofthe terminal stock price. Observe that the benchmark investorsunderperform relative to the standard power utility maximizerwhen the terminal stock price is moderate and low valued, butcompensates by outperforming when the terminal stock price ishigh. It can be seen from the plot that this outperformance is quitesubstantial with a reasonably high probability. Fig. 2 calculatesthe probability that the benchmark investors shown in Fig. 1outperform the maximizer of the standard power utility.

Example 2: Power benchmarking function with a minimum thresholdratioWe consider a single risky asset with parameters = 10%,

= 20%. The initial stock price is S(0) = 1 = K . There is a moneymarket account with interest rate r = 4%. The time horizon T is 1year.For the chosen parameter values, the price of the benchmark

y = 1.0776, and for convenience, we have chosen the initial valueof the portfolio x(0) = y = 1.0776 so that the optimal ter-minal wealth coincides with the benchmark max{S(T ), K}, whenp = . (In this example we set K = 0.85). We adopted thebenchmarking function (52) fromSection 4.2with = 0.4 and dif-ferent values of q. Fig. 3 shows plots of the terminal payoff for q =2,6. Observe that the payoff for the benchmark investor alwaysequals Y (0) = 0.34 when the terminal stock price S(T ) = 0.

6. Conclusion

In this paper we considered the optimal portfolio allocationproblem for insurers and pension funds by using a benchmark-

ing approach. Our approach tolerates but progressively penalizesunderperformance, and progressively rewards outperformance. Aatics and Economics 46 (2010) 317327

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

terminal stock price

term

inal

pay

off v

alue

Optimal terminal payoffs and benchmark values as a function of stock price

p = 0.3p = 2p = 4p = infinity/benchmarkstandard power utility (p = 4)density of S(T)

Fig. 1. Plots of the optimal terminal wealth for the benchmark investor as afunction of closing stock price S(T ) for p = 0.3,2,4,. Also shown is theterminal wealth profile associated with a standard power utility maximizer whenthe risk-aversion parameter p = 4.

Fig. 2. Performance of the benchmark investor (p = 0.3,2,4,) relative tothe usual power utility maximizer (p = 4).

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

terminal stock price

term

inal

pay

off v

alue

Optimal payoffs for benchmark investor (alpha=0.4)

q = 2q = 6benchmarkusual utility maximizerlognormal payoff density

Fig. 3. Plot of terminal payoff for benchmark investors with q = 2,6 and = 0.4 as a function of the terminal stock price. Superimposed are plots of thebenchmark as well as the density function for the terminal value of the stock.

general solution under general market models, benchmarks, andconcave benchmarking functions is presented, and insights to theimpact of benchmarking to the optimal portfolio are obtained. Adetailed example particularly applicable to pension funds and in-surers is analyzed, and closed form solutions to the optimal port-folio is derived. Future work include the relaxation of some of theassumptions made in this paper. In particular, it would be of in-

terest to consider models with jumps and other sources of marketincompleteness, including portfolio constraints.

A.E.B. Lim, B. Wong / Insurance: Mathem

Acknowledgements

This work is supported in part by the Berkeley-NUS RiskManagement Institute, the NSF CAREER Award DMI-0348746, andthe NSF Award DMI-0500503. The opinions, findings, conclusionsand recommendations expressed in this material are those of theauthors and do not necessarily reflect the views of the NationalScience Foundation. The support of Andrew Lim from the ColemanFung Chair in Financial Modelling is also acknowledged. Carefuland detailed reading and suggestions by an anonymous referee arealso appreciated.

Appendix. Technical proofs

A.1. Technical results accompanying Theorem 7

Lemma 10. Assume that the condition (32) is satisfied. ThenX() isnon-increasing and continuous on (0,), and strictly decreasing on(0, ), where

X(0+) := lim0X() = , (72)

X() := limX() = E[H0(T )Y (T )], (73)

= sup { > 0 : X() > X()} . (74)Proof. Recall that H0(T )Y (T ) is strictly positive. As () is non-increasing it follows that X() is also non-increasing. Continuityand (72)(74) follow by the applications of the monotone conver-gence and dominated convergence theorems.

Corollary 11. For (0, ),X() has a strictly decreasing inverseG : (X(),) onto(0, ) (75)such that

X(G()) = ; (X(),). (76)

A.2. Derivation of the optimal portfolio

We now present the derivations for the optimal portfolio con-sidered in Section 5.The following identities will be of help in the computation

via change of measure techniques. Firstly note that we have thefollowing decomposition (cf. Karatzas and Shreve (1998, Section3.6.6))

(H0(t))pp1 = mp(t)p(t), (77)

wheremp(t) is as defined in (65), and

p(t) = exp{p1 pW (t)

12

(p1 p

)22t

}, (78)

with p() a strictly positive martingale. In particular, we candefine a measure P1 equivalent to P by

dP1dP

FT

= p(T ). (79)

Under P1 we have, by Girsanovs theorem, a Brownian motionW1() defined by

W1(t) = W (t)(p1 p

)t. (80)Subsequently S(T ) is log-normal with parametersatics and Economics 46 (2010) 317327 325(ln(S(0))+

(+ p

1 p ( r)12 2)T , 2T

)(81)

under P1.Similarly, we can decompose

(H0(t))pp1 (S(t))

pp1 = S pp1 (0)p(t)p(t), (82)

where p(t) is defined in (66), and

p(t) = exp{p1 p ( )W (t)

12

(p1 p ( )

)2t

},

(83)

with p() a strictly positive martingale. Define a measure P2equivalent to P by

dP2dP

FT

= p(T ). (84)

Under P2wehave byGirsanovs theorem, a BrownianmotionW2()defined by

W2(t) = W (t)(p1 p

)( )t. (85)

Subsequently S(T ) is log-normal with parameters(ln(S(0))+

(+ p

1 p ( r 2) 1

2 2)T , 2T

)(86)

under P2.Finally, recall that the equivalent local martingale measure Q is

defined by

dQdP

FT

= Z0(T ) (87)

with Brownian motionWQ () defined byWQ (t) = W (t)+ t. (88)As P , P1, P2, and Q are all equivalent measures, we can usemea-

sure transformation techniques to evaluate expectations under themost convenient measure(s) in the following.

A.2.1. Proof of (64)Wewill need to evaluate the expected value in the denominator

of the optimal wealth (62). Specifically, we have

E[(H0(T ))

pp1 (max(S(T ), K))

pp1]

= E[(H0(T ))

pp1

pp1 S(T )

pp1 1S(T )>K

]+ E

[(H0(T ))

pp1

pp1 K

pp1 1S(T )K

]= pp1

(Spp1 (0)p(T )EP2 [1S(T )>K ] + K

pp1mp(T )EP1 [1S(T )K ]

)= pp1

(Spp1 (0)p(T )(N(c2,p(0, T )))

+ K pp1mp(T )N(c1,p(0, T )))

(89)

where c1,p(0, T ) and c2,p(0, T ) are as defined in (67) and (68)respectively, and N() is the distribution function of a standardnormal random variable.

A.2.2. Proof of (69)Consider now the determination of the portfolio allocationpi().There are a number of methods available. In the following we willutilize a convenient martingale technique, and in particular the

326 A.E.B. Lim, B. Wong / Insurance: Mathem

BismutElworthy formula (Bismut, 1984, Elworthy and Li, 1994,see also Qin (2008))which in this particular instance even providesus with a closed form solution. In more complicated settings theBismutElworthy formula can also be used to derive numericalmethods that can be applied to determine pi() (See also relatedresults in Fourni et al. (1999), Broadie and Glasserman (1996) andBoyle et al. (2008)).For 0 t T , the value process X() in (63) has an associated

optimal allocation in the stock, pi() ofx

E[(H0(T ))

pp1 (Y (T ))

pp1]p(t)S(t), (90)

where p() represents the delta (cf. Bermin (2002)) of acontingent claim with terminal payoff

(H0(T ))1p1 (Y (T ))

pp1 . (91)

Observe that (H0(T ))1p1 (Y (T ))

pp1 is a function ofW (T ), and hence

of S(T ). It follows that the value function for this claim can berepresented as V (S(), ), withV (S(0), 0) = E

[(H0(T ))

pp1 (Y (T ))

pp1], (92)

and

V (S(t), t) = 1H0(t)

E[(H0(T ))

pp1 (Y (T ))

pp1Ft]

= (H0(t))1p1 E

[(H0(T )H0(t)

) pp1(Y (T ))

pp1

Ft]. (93)

It is helpful to also recall the equivalent representation

V (S(t), t) = EQ[S0(t)S0(T )

(H0(T ))1p1 (Y (T ))

pp1Ft] . (94)

Consider firstly the determination ofp(0). Direct calculationsusing techniques analogous to (77)(86) shows that

EQ

[1

(S0(T ))2(H0(T ))

2p1 (Y (T ))

2pp1]K

(W (T )+ T )sT

]s=S(0)

+ E[(H0(T ))

pp1

pp1 K

pp1 1S(T )K

(W (T )+ T )sT

]s=S(0)

. (97)

Hence we are interested in, firstly

E[(H0(T ))

pp1

pp1 S(T )

pp1 1S(T )>K

(W (T )+ T )S(0)T

]p p= p1 S p1 (0)p(T )S(0)T

(EP2 [1S(T )>K (W2(T ))]atics and Economics 46 (2010) 317327

+((

1 p)(p1 p

))TEP2 [1S(T )>K ]

)=

pp1 S

pp1 (0)p(T )S(0)T

(n(c2,p(0, T ))

T

+((

1 p)(p1 p

))TN(c2,p(0, T ))

), (98)

where n() is the density function of a standard normal. Similarly,the second term in (97), can be calculated by measure transforma-tion techniques as

E[(H0(T ))

pp1

pp1 K

pp1 1S(T )K

(W (T )+ T )S(0)T

]=

pp1 K

pp1mp(T )

S(0)T

(EP1 [1S(T )KW1(t)]

+(

1 pT)EP1 [1S(T )K ]

)=

pp1 K

pp1mp(T )

S(0)T

( n(c1,p(0, T ))

T

+(

1 pT)N(c1,p(0, T ))

). (99)

Notice that by the properties of the normal density function it canbe shown algebraically that

Kpp1mp(T )n(c1,p(0, T )) = S

pp1 (0)p(T )n(c2,p(0, T )), (100)

and hence we have, on substitution,

p(0)S(0) = pp1 e

p2T2(1p)2

S pp1 (0)e p(r+ 12 2

)T

(1p)2

(

(1 p) p1 p

)N(c2,p(0, T ))

+ K pp1 eprT

(1p)2(

(1 p))N(c1,p(0, T ))

. (101)The allocation for time t can be derived by applying calcula-tions analogous to (96)(101) to the representation of V (S(t), t)in (93)(94). Specifically, by considering time horizon T t , andconditioning on Ft , we have

p(t)S(t) = pp1 e

p2(Tt)2(1p)2 (H0(t))

1p1

S pp1 (t)e p(r+ 12 2

)(Tt)

(1p)2

(

(1 p) p1 p

)N(c2,p(t, T ))

+ K pp1 epr(Tt)(1p)2

(

(1 p))N(c1,p(t, T ))

. (102)The optimal portfolio (69) follows by combining (64), (90) and(102).

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A benchmarking approach to optimal asset allocation for insurers and pension fundsIntroductionFinancial marketPrimary securitiesPortfolios, budget constraint, and hedging

Benchmark and benchmarking functionBenchmarkBenchmarking function

Optimization problem and solution: General casePower benchmarking functionPower benchmarking function with a minimum threshold ratioExponential benchmarking function

A portfolio insurance benchmarkOptimal portfolio in a market with constant coefficientsNumerical illustrations

ConclusionAcknowledgementsTechnical proofsTechnical results accompanying Theorem 7Derivation of the optimal portfolioProof of (64)Proof of (69)

References