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Insurance: Mathematics and Economics 43 (2008) 15–28www.elsevier.com/locate/ime

Optimal investment and life insurance strategies under minimum andmaximum constraints

Peter Holm Nielsena,∗, Mogens Steffensenb

a PFA Pension, Sundkrogsgade 4, DK-2100 Copenhagen, Denmarkb Department of Applied Mathematics and Statistics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Received April 2007; received in revised form September 2007; accepted 18 September 2007

Abstract

We derive optimal strategies for an individual life insurance policyholder who can control the asset allocation as well as the sum insured(the amount to be paid out upon death) throughout the policy term. We first consider the problem in a pure form without constraints (exceptnonnegativity on the sum insured) and then in a more general form with minimum and/or maximum constraints on the sum insured. In both caseswe also provide the optimal life insurance strategies in the case where risky-asset investments are not allowed (or not taken into consideration),as in basic life insurance mathematics. The optimal constrained strategies are somewhat more complex than the unconstrained ones, but the lattercan serve to ease the understanding and implementation of the former.c© 2007 Elsevier B.V. All rights reserved.

IME Subject and Insurance Branch Category: IE30; IE53; IB13

JEL classification: C61; D91; G11

MSC: 49L20; 91B28; 93E20

Keywords: Life insurance; Asset allocation; Optimization

1. Introduction

This paper is concerned with optimal strategies regardinglife insurance (i.e., coverage against death) and investment foran individual policyholder in a life insurance company or apension fund (referred to as the company henceforth). Morespecifically, we consider a life insurance policy comprising lifeinsurance as well as retirement saving during [0, T ], whereT > 0 is a fixed finite time horizon. The policyholder is allowedto choose, in a continuous manner, the sum insured, which isthe sum to be paid out from the company upon death of thepolicyholder, as well as the investment strategy. Our main focusis on the issue of optimal life insurance rather than optimal

∗ Corresponding author. Tel.: +45 39175876; fax: +45 39175952.E-mail addresses: [email protected] (P.H. Nielsen), [email protected]

(M. Steffensen).

0167-6687/$ - see front matter c© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2007.09.007

investment (although, as we shall see, these issues should notbe viewed separately), and we therefore work throughout with asimple and well-known model for the financial market offeringthe assets available for investment.

We impose in general the very realistic constraint that thesum insured must be nonnegative, but as the title of the papersuggests we shall also consider the optimization problem undermore restrictive minimum and maximum constraints in theform of lower and upper boundaries for the sum insured. Themotivation for the company to impose an upper boundary israther obvious, since it puts a limit on the company’s immediaterisk at any time during the policy term, but the motivationfor a lower boundary may not be obvious. However, a lowerboundary is sometimes imposed e.g. in pension schemes thatare mandatory for employees within a certain line of businessin order to ensure that a minimum coverage against death isprovided automatically, i.e., without a specific request fromeach employee.

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16 P.H. Nielsen, M. Steffensen / Insurance: Mathematics and Economics 43 (2008) 15–28

The unconstrained case, i.e., without minimum or maximumconstraints (except nonnegativity), is a special case of theconstrained case since the lower and upper boundaries canbe set to 0 and ∞, respectively. However, for educationalpurposes, and to ease the overall presentation of the results,we have chosen to include the solution of the unconstrainedcase separately (in Section 2), also because the results fromthe unconstrained case play an important role as convenientreferences in the constrained case. The solution of theunconstrained case in itself is not a main contribution of thispaper to the literature, though, since it, from a mathematicalperspective, actually is equivalent to a certain purely financialconsumption/investment problem with a well-known solution(as will be noted).

From a mathematical point of view it is interesting tonote that the unconstrained case can be solved by dynamicprogramming (as is done in this paper), whereas the moregeneral constrained case is quite troublesome (at least) to solveby this approach. The latter case is thus substantially facilitatedby the martingale methodology, emphasizing one of the majorstrengths of this technique, namely that it can lead to (moreor less explicit) solutions in problems with binding constraints,where the dynamic programming approach typically is not easyto apply.

An interesting aspect of the optimal insurance strategy inthe unconstrained case is that although the optimal sum insureddepends heavily on the development of the financial market, itsrange is the entire interval [0,∞) at any time during the policyterm. Thus, the boundaries imposed in the constrained caseare strictly binding whenever they are non-trivial (i.e., strictlypositive and finite, respectively).

In general the investment strategy is taken to beunconstrained (except for technical conditions), i.e., we allowall positions in the risky assets. However, we also provide theoptimal life insurance strategy in the special case of a marketwithout risky assets, or equivalently, under the constraint thatno risky-asset investments (long or short) are allowed. There aretwo main motivations for this “sub-problem”: Firstly, it is morein line with basic life insurance mathematics, where the interestrate is typically assumed constant (or deterministic), see e.g.Møller and Steffensen (2007), so it constitutes an interestingproblem in its own right, at least from an actuarial perspective.Various alternative interpretations of the interest rate (which inthe general setup below is the risk-free money market rate) arethen possible; in particular it may play the role of the so-calledsecond order rate or bonus rate, see Møller and Steffensen(2007). Secondly, the optimal life insurance strategy as suchstands out more clearly and is thus perhaps easier to interpretand analyze. However, we do not provide detailed proofs ofour results pertaining to this case (the proofs are similar to theproofs provided in the general setup; the details are left to theinterested reader).

Studying the optimal demand for life insurance for aneconomic agent dates back to Yaari (1965) and has beenfollowed up by Richard (1975), who was the first to study thecombined problem of optimal life insurance and investment(and consumption as well), Campbell (1980), and others.

The problem variations studied in the literature concernwhether some or all of the processes regarding investment,life insurance, and consumption are considered as decisionprocesses, whether the agent has non-capital (wage) income,and whether the problems are solved in discrete or continuoustime. More recent contributions to this body of literature areprovided by Chen et al. (2006), who allow for stochasticincome, Hong and Rıos-Rull (2007), who take a family pointof view and also take social security into account, and Yeand Pliska (2007), who study a problem close to the onestudied by Richard (1975) and also provide a nice survey ofthe literature. The main contribution of the present paper is thesolution of the continuous-time problem where all processes aredecision processes (as in Richard (1975)), and with constraintson the life insurance decision. To the knowledge of the authors,the problem with such constraints has not been consideredpreviously in the literature. Dynamic utility optimization isstudied recently in the context of non-life insurance by Mooreand Young (2006). Another related body of research concernsoptimal investment with the objective of minimizing thelifetime ruin probability (and generalizations); this constitutesa relevant personal finance problem in the absence of lifeinsurance, see e.g. Bayraktar and Young (2007).

The remainder of the paper is organized as follows: Section 2introduces the general setting and the basic optimizationproblem, and the unconstrained case is treated. In Section 3 wesolve the problem in the general constrained case; this sectionthus contains the main results of the paper. Section 4 concludes.

Some basic notations: All vectors are column vectors. Thetransposed of a vector or matrix a is denoted by a′. Thed-dimensional vector of 1’s is denoted by 1d .

2. Setup and basic problem

To formalize the setup we take as given some underlyingprobability space (Ω ,F,P), on which all random variablesintroduced in the following are defined.

We consider a policyholder with a life insurance policyissued at time 0 and terminated at a fixed finite time horizonT > 0. Let τ be a nonnegative random variable representingthe (random) time of death of the policyholder. For t ∈ [0, T ],the mortality intensity of τ is given by a continuous functionµ : [0, T ] → [0,∞), which means that

P (τ ≤ t) = 1 − e−∫ t

0 µ(s)ds, t ∈ [0, T ].

The (conditional) distribution of τ on (τ > T ) is irrelevant inthis paper.

Let W = (W1, . . . ,Wd)′ be a d-dimensional standard

Brownian Motion (d ∈ N) stochastically independent of τ . Thefinancial market is assumed to be frictionless and to consist of arisk-free money market account with price dynamics given by

dS0(t)/S0(t) = rdt,

where r ≥ 0 is a fixed constant, and d risky assets with pricedynamics given by

dSi (t)/Si (t) = αi dt +

d∑j=1

σi j dW j (t), i = 1, . . . , d,

P.H. Nielsen, M. Steffensen / Insurance: Mathematics and Economics 43 (2008) 15–28 17

where α = (α1, . . . , αd)′

∈ Rd , and σ = (σi j )i, j=1,...,d ∈

Rd×d is non-singular. The filtration generated by W andaugmented by the null-sets of F is denoted by FW

=

(FWt )t∈[0,T ]. Let

Λ(t) = e−λ′W (t)− 12 ‖λ‖2t , t ∈ [0, T ],

where λ = σ−1(α − r1d) is the market price of riskvector. It is well-known that the equivalent martingale measureof the market, denoted by Q, is unique and given by theRadon–Nikodym derivative

dQdP

= Λ(T ),

and that the process W Q defined by W Q(t) = W (t) + λt, t ∈

[0, T ], is a standard Brownian motion under Q.The policyholder pays an initial lump sum premium π0 ≥ 0

(at time 0) as well as continuous premiums at a fixed rate π ≥ 0during [0, T ] as long as he is alive (of course, π0 and π cannotboth be 0 at the same time). We disregard policy expenses,and the premiums could thus be interpreted as net premiums.The overall problem consists in choosing optimal life insuranceand investment strategies. Formally, a life insurance strategy,resp. an investment strategy, is an FW -adapted measurablestochastic process S = (S(t))t∈[0,T ], resp. w = w((t))t∈[0,T ],taking values in [0,∞), resp. Rd . For a given pair (S, w) ofstrategies, S(t) denotes the sum insured to be paid upon deathat time t , and w(t) is the vector of amounts invested in therisky assets at time t . Disregarding technicalities at this point,the corresponding reserve V S,w

= (V S,w(t))t∈[0,T ], which isset aside as a liability by the insurance company as long asthe policyholder is alive, is assumed to be governed by thedynamics

V S,w(0) = π0, (2.1)

dV S,w(t) = (r V S,w(t)+ w′(t)(α − r1d)+ π

−µ(t)[S(t)− V S,w(t)])dt + w′(t)σdW (t) (2.2)

=

(r V S,w(t)+ π − µ(t)

[S(t)− V S,w(t)

])dt

+w′(t)σdW Q(t), t ∈ (0, T ). (2.3)

It is straightforward to verify that if the policyholder is aliveat time t , t ∈ [0, T ), the reserve is given by the retrospectiveformula

V S,w(t) = π0G(t)+

∫ t

0G(t)G(s)−1

[π − µ(s)S(s)

+w′(s)(α − r1d)]ds

+

∫ t

0G(t)G(s)−1w′(s)σdW (s) (2.4)

= π0G(t)+

∫ t

0G(t)G(s)−1

[π − µ(s)S(s)]ds

+

∫ t

0G(t)G(s)−1w′(s)σdW Q(s), (2.5)

where

G(t) = e∫ t

0 (r+µ(s))ds, t ∈ [0, T ].

If risky-asset investments are not allowed, i.e., w(t) ≡ 0, ∀t ∈

[0, T ], then the third term in (2.5) drops out and renders (2.5)as a retrospective formula for the reserve, which is well-knownfrom basic life insurance mathematics.

We stress that only the sum insured, S(t), is paid outin the event of death at time t ∈ [0, T ]. The policy issubsequently taken out of force, and the liability is set to0. The reserve V S,w(t) goes to the insurance company andshould therefore not be interpreted as the policyholder’s wealth.Correspondingly, the terminal reserve, V S,w(T −), is paid out attime T only upon survival of the policyholder.

In the following, when we refer to the reserve, we alwaysimplicitly mean the reserve given that the policyholder is alive.In actuarial terminology our reserve is thus the state-wisereserve corresponding to (the policyholder being in) the statealive. Our reserve process is thus “purely financial” in the sensethat it is FW -adapted, i.e., independent of τ .

Remark 2.1. It should be noted that the reserve processis not financed by the premium payments and the invest-ment strategy; as (2.2) shows, the actuarial risk premium,µ(t)

[S(t)− V S,w(t)

]dt (which is negative if S(t) < V S,w(t)),

is (continuously) deducted from the reserve and thus “paid” tothe company.

Remark 2.2. To avoid confusion we briefly elaborate on howthe setup can be interpreted: The reserve is maintained andinvested by the company, but the policyholder is allowed tochoose (in a continuous way) the risky-asset allocations as wellas the size of the sum insured. An alternative interpretation isthat the company chooses the life insurance and investmentstrategies to be carried out but does so on behalf of thepolicyholder. The mathematical contents of the paper do notdepend on the exact interpretation, though, and the readeris free to choose whichever one he or she prefers. Note,though, that our setup differs from the one in Richard (1975):It is assumed there that the consumer (policyholder) investshis wealth himself and in particular leaves his wealth as alegacy upon death (in addition to the insurance sum paid outby the insurance company), whereas we assume (as stressedabove) that the reserve goes to the company upon death. (Thepolicyholder is compensated for this by the term µ(t)V S,w(t)dtin the reserve dynamics (2.2).)

Remark 2.3. We have defined the reserve through thedynamics (2.1) and (2.2), with the solution given by (2.4), ratherthan as a conditional expected present value of future payments(benefits less premiums), as is common actuarial practice. Sinceit is an actuarial convention to define the reserve with respectto strictly future payments only, the reserve at time T is 0,which is why we consider the left-hand limit V S,w(T −) ratherthan V S,w(T ) as the lump sum benefit at time T . See alsoRemark 2.5, where we demonstrate that the reserve actuallyfulfills a prospective formula.

We allow the reserve to become negative, as long as it doesnot exceed (in absolute value) the actuarial present value of the


future premium payments, i.e., as long as

V S,w(t)+ g(t) ≥ 0, ∀t ∈ [0, T ], (2.6)

where

g(t) = π

∫ T

tG(t)G(s)−1ds, t ∈ [0, T ]. (2.7)

Allowing for negativity of the reserve is admittedly at odds withthe usual constraint in life insurance that the reserve should benonnegative, which is imposed mainly because the policyholderusually has the right to cancel his future premium paymentsat any time during the policy term. However, requiringnonnegativity in this paper would substantially increase themathematical difficulty and complexity of the consideredoptimization problems, and we therefore choose to be contentwith the milder constraint (2.6). In other words, we assume thatthe policyholder sticks to the premium plan stipulated in thepolicy. Note, though, that (2.6) in particular implies

V S,w(T −) ≥ 0.

We define the set E ⊆ [0, T )× R as

E = (t, v) ∈ [0, T )× R : v + g(t) ≥ 0.

We may thus interpret E as the state space of the process(t, V S,w(t))t∈[0,T ). The closure of E is given by

E = (t, v) ∈ [0, T ] × R : v + g(t) ≥ 0.

Definition 2.4. A pair (S, w) of life insurance and investmentstrategies is said to be admissible if the corresponding reserveprocess V S,w as given by (2.4) (or (2.5)) is well-defined on[0, T ), has a well-defined left-hand limit V S,w(T −) at T , andobeys (2.6), and if

EQ(∫ T

0G(t)−1w′(t)σdW Q(t)

)= 0. (2.8)

The set of such pairs is denoted by A′.

The technical condition (2.8) is equivalent to the mildcondition that under Q the process Zw(·) given by

Zw(t) =

∫ t

0G(s)−1w′(s)σdW Q(s), t ∈ [0, T ],

is a true martingale (it is a local martingale for any investmentstrategy w with

∫ T0 ‖w′(u)σ‖

2 < ∞, a.s., and also asupermartingale if (2.6) is fulfilled, see e.g. Karatzas and Shreve(1998)). The condition ensures that for any (S, w) ∈ A′ theactuarial equivalence principle is satisfied under Q, i.e.,

EQ(∫ T

0G(t)−1(µ(t)S(t)− π)dt + G(T )−1V S,w(T −)

)= π0. (2.9)

To see this, insert V S,w(T −) from (2.5) in the expression on theleft-hand side of (2.9) and perform a few calculations to obtain

EQ(∫ T


)+ π0,

then use (2.8) to obtain (2.9).

Since the equivalence principle is satisfied under Q (ratherthan some technical actuarial measure), the policy may beviewed as a unit-linked policy. In particular, the policyholderhas no rights to any bonus benefits.

Remark 2.5. In general, as a similar calculation shows, (2.8)implies that for any (S, w) ∈ A′ the reserve fulfills theprospective formula

V S,w(t) = EQ(∫ T

tG(t)G(s)−1(µ(s)S(s)− π)ds

+ G(t)G(T )−1V S,w(T −)

∣∣∣∣FWt

), (2.10)

for every t ∈ [0, T ). As noted in Remark 2.3, it is actuarialpractice to define the reserve as a conditional expected presentvalue of (strictly) future payments (benefits less premiums),and (2.10) shows that our definition is in accordance with thispractice.

Since the (conditional) expected value is taken under themarket-induced measure Q, we refer to this value as theactuarial market value. It could be argued that the companyshould be compensated somehow for the pure policy risk, i.e.,the uncertainty regarding the time of death of the policyholder,even though this compensation should be small due to thediversification effect, which is at the heart of life insurance.One could work with a “market mortality rate” to be used in thereserve dynamics, but the optimization objective should in anycase involve the true mortality rate. This approach was taken byRichard (1975). However, since this issue is not in focus in thispaper we disregard it for simplicity.

The following lemma will be useful in the following, but itis also interesting in its own right. It corresponds to the well-known result from mathematical finance stating (roughly) thatany contingent claim can be perfectly hedged in a completemarket, see e.g. Karatzas and Shreve (1998), Theorem 3.3.5.However, for completeness, and to facilitate the proof ofProposition 3.1, we provide a full proof (similar to the proofin Karatzas and Shreve (1998)).

Lemma 2.6. Let S be a life insurance strategy, and let X be anonnegative FW

T -measurable random variable such that

EQ(∫ T

0G(t)−1(µ(t)S(t)− π)dt + G(T )−1 X

)= π0.

Then there exists an investment strategy w such that (S, w) isadmissible and V S,w(T −) = X.

Proof. Consider the nonnegative Q-martingale M defined by

M(t) = EQ(∫ T

0G(t)−1(µ(t)S(t)− π)dt + G(T )−1 X

∣∣∣∣FWt

),

0 ≤ t ≤ T .

According to the martingale representation theorem (e.g.Karatzas and Shreve (1991), Theorem 3.4.15 and Problem3.4.16), there exists a progressively measurable, Rd -valued


process ψ = (ψ(t))t∈[0,T ] satisfying∫ T

0 ‖ψ(t)‖2dt < ∞, a.s.,and

M(t) = π0 +

∫ t

0ψ ′(s)dW Q(s), 0 ≤ t ≤ T .

Now, define a nonnegative process Y = (Y (t))t∈[0,T ] by

Y (t) = EQ(∫ T

tG(t)G(s)−1(µ(s)S(s)− π)ds

+ G(t)G(T )−1 X

∣∣∣∣FWt

)= G(t)

(M(t)−

∫ t

0G(s)−1(µ(s)S(s)− π)ds

),

0 ≤ t ≤ T,

so that in particular Y (0) = M(0) = π0. Ito’s lemma implies

dY (t) = (r + µ(t))Y (t)dt

+ G(t)[ψ ′(t)dW Q(t)− G(t)−1(µ(t)S(t)− π)dt

]= (rY (t)+ π − µ(t) [S(t)− Y (t)]) dt

+w′(t)σdW Q(t),

where

w(t) = G(t)(σ ′)−1ψ(t), t ∈ [0, T ]. (2.11)

By comparing with (2.1)–(2.3) we see that Y (t) =

V S,w(t), ∀t ∈ [0, T ), and we have Y (T ) = Y (T −) =

V S,w(T −) = X, a.s. To complete the proof we need to checkthat (S, w) is admissible. From the definition of Y it easilyfollows that (2.6) is satisfied. Finally, since M is a martingalewe have

EQ(∫ T


)= EQ

(∫ T

0ψ ′(t)dW Q(t)

)= 0,

so (2.8) is fulfilled.

We assume that the policyholder’s preferences can be statedin terms of a CRRA utility function u : [0,∞) → [−∞,∞)

with relative risk-aversion coefficient δ > 0, that is,

u(x) =

x1−δ/(1 − δ), if δ ∈ (0,∞) \ 1,

log(x), if δ = 1,x > 0,

and u(0) = limx0 u(x). We consider the optimizationobjective

max(S,w)∈A

E(

e−∫ τ

0 β(s)dsu(S(τ ))1(τ<T )

+ K e−∫ T

0 β(s)dsu(V S,w(T −))1(τ≥T )

), (2.12)

where β : [0, T ] → R is a continuous time preference function,K > 0 is a weighting parameter, and A ⊆ A′ is the set ofadmissible strategies (S, w) such that

E(

min(0, u(S(τ )))1(τ<T ) + K min(0, u(V S,w(T −)))1(τ≥T )

)> −∞. (2.13)

From the objective it is seen that the policyholder has utilityfrom both the life insurance coverage (in the event of death)and the terminal payoff (in the event of survival).

The subjective time preference function β determines thepolicyholder’s preferences concerning life insurance coverageat different points in time in relation to each other (and inrelation to the terminal payoff). We allow β to be negative,which would correspond to attaching more weight to lifeinsurance late in the policy term than early in the policyterm. We also allow β to vary over time, so that virtually anyweighting preferences can be modelled.

The parameter K > 0 can be viewed as a measure of thepolicyholder’s preferences concerning terminal payoff versuslife insurance: If K > 1, then the (utility of the) terminal payoffis given more weight than the (utility of the) life insurance, andvice versa if K < 1. They are given equal weight if K = 1. Onecould allow K to be 0, meaning that the policyholder wouldonly want life insurance. It is fairly straightforward to analyzethe case K = 0 along the same lines as the analysis carriedout in the following, but since it requires special treatmentwe choose to omit it to avoid blurring the picture. However,although we do not prove it, the optimal strategies in the caseK = 0 are in fact the natural generalizations of the optimalstrategies that we find below. The reverse situation, i.e., with nodesire for life insurance, is not in focus here (cf. the openingremarks of this section), and, moreover, has a well-knownsolution.

Of course, the factor e−∫ T

0 β(s)ds in the terminal payoff termcould be absorbed in K , but we have chosen the objective in theform above in order to separate the time preferences from thegeneral life insurance versus terminal payoff preferences in amore clean fashion.

Now, in terms of the mortality intensity the objective can berewritten as

max(S,w)∈A

E(∫ T

0D(t)µ(t)u(S(t))dt + K D(T )u(V S,w(T −))

),

(2.14)

where

D(t) = e−∫ t

0 (β(s)+µ(s))ds, t ∈ [0, T ].

The following condition (which must be checked) will ensurethat the problem is well-posed:

sup(S,w)∈A

E(∫ T

0D(t)µ(t)u(S(t))dt + K D(T )u(V S,w(T −))

)< ∞. (2.15)

Proposition 2.7. Let S : E → [0,∞) and w : E → Rd begiven by

S(t, v) =v + g(t)

h(t), (t, v) ∈ E, (2.16)

w(t, v) =v + g(t)

δ(σ ′)−1λ, (t, v) ∈ E, (2.17)


where, for t ∈ [0, T ],

h(t) =

∫ T

te−

∫ st (κ(y)+µ(y))dyµ(s)ds

+ K 1/δe−∫ T

t (κ(y)+µ(y))dy, (2.18)

with

κ(t) = (β(t)− (1 − δ)r)/δ − (1 − δ)‖λ‖2/(2δ2). (2.19)

Then the pair (S, w) given by(S(t), w(t)

)=

(S(t, V S,w(t)), w(t, V S,w(t))

), t ∈ [0, T ],

belongs to A and is optimal. In other words, S and w are theoptimal strategies in feedback form.

Proof. We show the proposition by use of the dynamicprogramming approach (see e.g. Fleming and Rishel (1975)).We define the value function J by

J (t, v) = sup(S,w)∈A

E(∫ T

tD(s)µ(s)u(S(s))ds

+ K D(T )u(V S,w,v(T −))

),

for (t, v) ∈ E, where V S,w,v(·) is the reserve processcorresponding to the initial value V S,w,v(t) = v and withdynamics as in (2.2), and where A is as above with obviousmodifications (strictly speaking, A now depends on (t, v), butwe shall not go into details). Moreover, we set J (T, v) =

K D(T )u(v), v > 0.The HJB equation is given by

0 = sup(S,w)∈[0,∞)×Rd

(D(t)µ(t)u(S)+

∂ J

∂t

+∂ J

∂v

[(r + µ(t))v + w′(α − r1d)+ π − µ(t)S

]+

12∂2 J

∂v2 ‖w′σ‖2),

where the argument (t, v) on J has been suppressed to easenotation, and with the boundary condition

J (T, v) = K D(T )u(v).

Now, by making the (qualified) guess that J has the form

J (t, v) = f (t)u(v + g(t)), (t, v) ∈ E, (2.20)

where g and f are unknown functions, we obtain (througha series of tedious calculations, which are carried out inAppendix A) a solution of this form with g given by (2.7) andf given by

f (t) = D(t)h(t)δ, t ∈ [0, T ]. (2.21)

From this solution and the HJB equation it easily follows thatthe candidate optimal strategies are given in feedback formby (2.16) and (2.17). In order to check the candidates foradmissibility we consider the process A S,w given by

A S,w(t) = V S,w(t)+ g(t), t ∈ [0, T ]. (2.22)

Using (2.2) with S and w inserted, and the fact that

g′(t) = (r + µ(t))g(t)− π, t ∈ (0, T ),

leads to

dA S,w(t) = A S,w(t)(

r + µ(t)[1 − h(t)−1

]+ ‖λ‖2δ−1

)dt

+ A S,w(t)δ−1λ′dW (t). (2.23)

Thus, for t ∈ [0, T ) we have

A S,w(t) = (π0 + g(0))

× e∫ t

0

(r+µ(s)

[1−h(s)−1]

+‖λ‖2(2δ−1)/(2δ2))ds+δ−1λ′W (t),

which is well-defined, since h is bounded away from zero on[0, t] for every t ∈ [0, T ).

This means in particular that V S,w(t) is well-defined on[0, T ). The left-hand limit V S,w(T −) is well-defined, since h isbounded away from zero on [0, T ]. Moreover, (2.6) is satisfiedsince

A S,w(t) ≥ 0, ∀t ∈ [0, T ].

This, in turn, allows us to conclude that S(·) is nonnegativethroughout [0, T ]. Now, using that w(t) = A S,w(t)δ−1(σ ′)−1λ

and the fact that A S,w(t) is lognormally distributed for everyt ∈ [0, T ) it is straightforward to check that

EQ(∫ T

0‖w(t)σ‖

2dt

)< ∞,

which ensures that (2.8) is fulfilled. Thus, the candidates areadmissible.

Finally, since π0 and π cannot both be 0 we have π0+g(0) >0, and since A S,w(t) is lognormally distributed for every t ∈

[0, T ) we can conclude that (2.13) is satisfied, so (S, w) ∈ A.By similar arguments, the condition (2.15) is met.

We conclude that (S, w) is the optimal pair of strategies.

Remark 2.8. From a purely mathematical point of viewthe problem is equivalent to a certain optimal invest-ment/consumption problem, which has a well-known solution.To see this, rewrite (2.2) as

dV S,w(t) = ([r + µ(t)] V S,w(t)+ w′(t)(α − r1d)+ π

−µ(t)S(t))dt + w′(t)σdW (t), t ∈ (0, T ).

Now, put r(t) = r + µ(t) and c(t) = µ(t)S(t), t ∈ [0, T ], andinterpret V S,w(t) as the wealth of an investor who can invest ina risk-free money market account with (deterministic) interestrate r(·) and risky assets as above, and who receives incomecontinuously at the fixed rate π and consumes at the rate c(·).Assuming that the investor’s optimization objective is given by

max(c,w)∈A

E(∫ T

0D(t)u(c(t)/µ(t))dt + K D(T )u(V S,w(T −))

),

(where A of course corresponds to A), implying in particularthat the mortality risk is disregarded, leads to the exact samevalue function and thus analogous solutions for the optimalstrategies as obtained above.


The optimal investment strategy w is similar to the oneobtained by Merton (1971) for an investor who has fixed non-capital income but ignores his mortality risk; in this case thetotal amount invested in the risky assets is proportional to thesum of the current wealth and the present value of the futurenon-capital income. In the present problem the total amountinvested in the risky assets is proportional to the sum of thereserve and the actuarial present value of the future premiums.In both cases the risky-asset amounts are distributed among therisky assets in proportions given by the vector (σ ′)−1λ, andthe risky-asset amounts are proportional to the reciprocal of therelative risk-aversion coefficient δ.

The optimal life insurance strategy S is such that at anytime t the sum insured equals A S,w(t), the sum of the currentreserve and the actuarial present value of future premiums,divided by h(t). The latter can be interpreted as a subjectiveactuarial present value of an endowment insurance that pays out1 upon death before time T or K 1/δ at time T upon survival; thesubjectivity having to do with the fact that κ(·), which dependson δ and β(·), is used for discounting. In the special case withequally weighted life insurance and terminal payoff (K = 1),no time preferences (β = 0), and logarithmic utility (δ = 1),we get h(t) = 1, ∀t ∈ [0, T ], such that S(t) = A S,w(t),corresponding to exact insurance protection of the total assets.Since h is decreasing in β and increasing in K , subjectivediscounting (β > 0) and/or reduced weight on terminal payoffs(K < 1) will lead to a lower protection and vice versa. Thedependence of h on δ is not as transparent. However, if β(·) ≥ 0and K = 1, we have h(t) < 1 unless δ is close to 0, i.e.,the optimal sum insured is greater than A S,w(t) unless thepolicyholder has very low risk aversion in this case. This mayseem somewhat counterintuitive and may indicate that a largerweight K on the terminal payoff or a negative time preferencefunction β(·) may be more in line with “normal” preferences.

To see how S(·) develops over time when the optimalstrategies are used we derive the dynamics of S using Ito’sformula and the relation

S(t) =A S,w(t)

h(t), t ∈ [0, T ),

where A S,w is given by (2.22). From (2.23) and the fact that

h′(t) = (κ(t)+ µ(t))h(t)− µ(t), t ∈ (0, T ),

we get

dS(t) =dA S,w(t)

h(t)−

A S,w(t)

h(t)2h′(t)dt

= S(t)(

r + µ(t)[1 − h(t)−1

]+ ‖λ‖2δ−1

)dt

+ S(t)δ−1λ′dW (t)

− S(t)(κ(t)+ µ(t)− h(t)−1µ(t))dt

= S(t)(

r − κ(t)+ ‖λ‖2δ−1)

dt + S(t)δ−1λ′dW (t)

= S(t)((r − β(t)+ ‖λ‖2/2)δ−1

+ ‖λ‖2/(2δ2))

dt

+ S(t)δ−1λ′dW (t).

Thus, we have

S(t) = S(0)e∫ t

0 (r−β(s)+‖λ‖2/2)δ−1ds+δ−1λ′W (t),

t ∈ [0, T ). (2.24)

In particular, S is a geometric Brownian motion if β is constant.Note that S depends on K only through the initial point S(0),i.e., both the growth rate and the volatility of S are independentof K . Moreover, it is interesting to note that S does not dependon the shape of the mortality intensity µ(·), but only on itsoverall level (through S(0)), that is, for any given t ∈ [0, T ],S(t) does not depend on µ(t). Note also that the range of Sis the entire interval (0,∞). In the next section we study theproblem under minimum and maximum constraints on S.

We also point out that the optimal investment and lifeinsurance strategies can be considered independent in the sensethat the optimal risky-asset allocation and optimal sum insuredat any given time are independent.

We round off this section by briefly considering thecase where no risky-asset investments are allowed (cf. theintroduction). In this case the optimal life insurance strategyis formally exactly as in Proposition 2.7, the only adjustmentbeing that the second term of κ(·) (see (2.19)) drops out, so thatκ(t) = (β(t)− (1 − δ)r)/δ. Thus, (2.24) becomes

S(t) = S(0)e∫ t

0 (r−β(s))/δds, t ∈ [0, T ).

In particular we can make the fairly reasonable observation thatS(·), which is deterministic in this case, is increasing (resp. de-creasing) whenever r > β(t) (resp. r < β(t)). Moreover, theslope of S(·), be it negative or positive, is less severe for a highrisk averter (with large δ) than for a low risk averter. A highrisk averter thus seeks to smooth out the sum insured over time,whereas a low risk averter takes a more aggressive view andtries to shape the sum insured according to his time preferences.

3. Minimum and maximum constraints

In this section we consider the problem of Section 2 under(possibly time-dependent) minimum and maximum constraintson the sum insured S(·). Thus, we assume that we are given two(measurable) functions s : [0, T ] → [0,∞) and s : [0, T ] →

[0,∞] such that s(t) ≤ s(t), ∀t ∈ [0, T ], and such that the lifeinsurance strategy is subject to the constraint

s(t) ≤ S(t) ≤ s(t), ∀t ∈ [0, T ]. (3.1)

We denote by A(s, s) the set of admissible strategies (S, w)such that S satisfies (3.1), and such that (2.13) is fulfilled.Thus, we still consider the optimization objective (2.12) (withA replaced by A(s, s), of course). Note that condition (2.15)automatically is met in this section, since A(s, s) ⊆ A.

To ensure that A(s, s) is non-empty we impose here thenatural assumption that

π0 + g(0) ≥

∫ T

0G(t)−1µ(t)s(t)dt,

i.e., the actuarial present value at time 0 of the total premiumpayments must exceed the actuarial present value of theminimum life insurance coverage.


The following proposition, which is based on the martingalemethodology developed by Karatzas et al. (1987) and Coxand Huang (1989, 1991), provides the optimal strategies ina somewhat abstract form. Subsequently we characterize theoptimal strategies, derive formulas for them in explicit form(Proposition 3.2), and analyze them.

First, a few preliminaries. Let I : (0,∞) → (0,∞) denotethe inverse of the derivative of u, i.e.,

I (ξ) = ξ−1/δ, ξ > 0, (3.2)

and define for each t ∈ [0, T ],

It (ξ) = max[s(t),min (I (ξ), s(t))

], ξ > 0.

Let

H(t) = Λ(t)G(t)−1 D(t)−1, t ∈ [0, T ],

and let ξ > 0 be the constant satisfying

EQ(∫ T

0G(t)−1 [µ(t)It

(ξH(t)

)− π

]dt

+ G(T )−1 I(ξK −1 H(T )

))= π0. (3.3)

Such a ξ clearly exists, since I is continuous and maps (0,∞)

onto (0,∞).

Proposition 3.1. The optimal life insurance strategy is given by

S(t) = It(ξH(t)

), t ∈ [0, T ], (3.4)

and the optimal investment strategy is given by

w(t) = G(t)(σ ′)−1ψ(t), t ∈ [0, T ],

where ψ is the integrand in the stochastic integral represen-tation M(t) = π0 +

∫ t0 ψ

′(s)dW Q(s) of the Q-martingale Mdefined for t ∈ [0, T ] by

M(t) = EQ(∫ T

0G(t)−1(µ(t)S(t)− π)dt

+ G(T )−1 I(ξK −1 H(T )

) ∣∣∣∣FWt

).

Proof. By maximizing the function x 7→ u(x) − ξ x, x ≥ 0,where ξ > 0 is a fixed constant, it is easily verified that

u(x) ≤ u(I (ξ))− ξ(I (ξ)− x), ∀x ≥ 0, ξ > 0. (3.5)

Similarly, it can be verified that

u(x) ≤ u(It (ξ))− ξ(It (ξ)− x), ∀t ∈ [0, T ],

x ∈ [s(t), s(t)], ξ > 0. (3.6)

Now, any (S, w) ∈ A(s, s) fulfills (2.9), which can be writtenas

E(∫ T

0Λ(t)G(t)−1(µ(t)S(t)− π)dt

+ Λ(T )G(T )−1V S,w(T −)

)= π0,

or, equivalently,

E(∫ T

0Λ(t)G(t)−1µ(t)S(t)dt + Λ(T )G(T )−1V S,w(T −)

)= π0 + g(0). (3.7)

From (3.5)–(3.7) it follows that for any (S, w) ∈ A(s, s) andany ξ > 0 we have

E(∫ T

0D(t)µ(t)u(S(t))dt + K D(T )u

(V S,w(T −)

))≤ E

(∫ T

0D(t)µ(t)u

(It (ξΛ(t)G(t)−1 D(t)−1)

)dt

)− E

(∫ T

0µ(t)ξΛ(t)G(t)−1

×

[It (ξΛ(t)G(t)−1 D(t)−1)− S(t)

]dt

)+ E

(K D(T )u

(I (ξK −1Λ(T )G(T )−1 D(T )−1)

))− E

(ξΛ(T )G(T )−1

[I (ξK −1Λ(T )G(T )−1 D(T )−1)

− V S,w(T −)])

= E(∫ T

0D(t)µ(t)u

(It (ξΛ(t)G(t)−1 D(t)−1)

)dt

)− E

(∫ T

0µ(t)ξΛ(t)G(t)−1 It (ξΛ(t)G(t)−1 D(t)−1)dt

)+ E

(K D(T )u

(I (ξK −1Λ(T )G(T )−1 D(T )−1)

))− E

(ξΛ(T )G(T )−1 I (ξK −1Λ(T )G(T )−1 D(T )−1)

)+ ξ(π0 + g(0)).

In particular we have

E(∫ T

0D(t)µ(t)u(S(t))dt + K D(T )u

(V S,w(T −)

))≤ E

(∫ T

0D(t)µ(t)u

(It (ξH(t))

)dt

+ K D(T )u(

I[ξK −1 H(T )

]))= E

(∫ T

0D(t)µ(t)u

(S(t)

)dt

+ K D(T )u(

I[ξK −1 H(T )

])),

since

π0 + g(0) = EQ(∫ T

0G(t)−1µ(t)It

(ξH(t)

)dt

+ G(T )−1 I[ξK −1 H(T )

])= E

(∫ T

0µ(t)Λ(t)G(t)−1 It

(ξH(t)

)dt

+ Λ(T )G(T )−1 I[ξK −1 H(T )

]).


Since (S, w) was arbitrary, our candidate optimal life insurancestrategy is thus given by S, and our candidate optimalinvestment strategy is the one for which the terminal reserveis given by

X := I(ξK −1 H(T )

). (3.8)

Now, it follows from Lemma 2.6 that there exists an investmentstrategy w such that (S, w) is admissible, and V S,w(T −) = X .Moreover, from the proof of Lemma 2.6, in particular (2.11), itfollows that w = w. To complete the proof we therefore onlyneed to show that (S, w) ∈ A(s, s), which amounts to checking(2.13) since admissibility of (S, w) follows from Lemma 2.6 (asalready noted), and (3.1) clearly holds. However, since H(t) islognormally distributed for each t ∈ [0, T ], (2.13) is clearlysatisfied.

As already mentioned, the optimal strategies as given inProposition 3.1 are somewhat abstract and thus not easy toimplement directly. We shall therefore now derive a fairlyexplicit characterization of the strategies.

From the definition of M (see Proposition 3.1) and (2.7),(3.2), (3.4) and (3.8), we get

M(t) = M1(t)− M2(t)+ M3(t)

+

∫ t

0G(s)−1µ(s)S(s)ds − g(0), t ∈ [0, T ],

where M1, M2, and M3 (which are not martingales) are definedby

M1(t) = EQ(∫ T

tG(s)−1µ(s)

(ξH(s)

)−1/δds

∣∣∣∣FWt

)+ EQ

(G(T )−1 K 1/δ ( ξH(T )

)−1/δ∣∣∣FW

t

),

M2(t) = EQ(∫ T

tG(s)−1µ(s)

[(ξH(s)

)−1/δ

− s(s)]+

ds

∣∣∣∣FWt

),

M3(t) = EQ(∫ T

tG(s)−1µ(s)

[s(s)

−(ξH(s)

)−1/δ]+

ds

∣∣∣∣FWt

).

Now, from the proof of Lemma 2.6, or by direct inspection of(2.10) and the definition of M , it follows that

V S,w(t) = G(t)

(M(t)−

∫ t

0G(s)−1 (µ(s)S(s)− π

)ds

)= G(t) [M1(t)− M2(t)+ M3(t)] − g(t),

t ∈ [0, T ),

that is,

V S,w(t)+ g(t) = G(t)[M1(t)− M2(t)+ M3(t)],

t ∈ [0, T ).

This formula shows that at any time t ∈ [0, T ) the totalliabilities can be decomposed into three parts as

L1(t)− L2(t)+ L3(t),

where L i (t) = G(t)Mi (t), i = 1, 2, 3. From the definitionof Mi , i = 1, 2, 3, it is seen that L1(t) is the actuarialmarket value of the liability to pay (a) the sum (ξH(s))−1/δ

in the event of death at time s ∈ (t, T ) and (b) the terminalpayoff K 1/δ (ξH(T ))−1/δ at time T upon survival; L2(t) isthe actuarial market value of an asset returning the surplus[(ξH(s))−1/δ

− s(s)]+ in the event of death at time s ∈ (t, T );and L3(t) is the actuarial market value of the liability to pay thedeficit [s(s)− (ξH(s))−1/δ

]+ in the event of death at time s ∈

(t, T ). In loose terms, L1 accounts for what could be termedthe “free” liabilities, and L2 and L3 account for two hedgingderivatives making up for the surplus or deficit in relation tothe constraint that the sum insured at any time t ∈ [0, T ) mustbelong to [s(s), s(s)].

From an overall perspective the optimal investment strategycan thus be characterized as follows: At any time t ∈ [0, T ) thetotal assets, A S,w(t) = V S,w(t) + g(t) (the sum of the reserveand the actuarial present value of the future premiums), can bedecomposed into three parts as

A S,w(t) = A S,w1 (t)− A S,w

2 (t)+ A S,w3 (t),

corresponding to the decomposition of the total liabilities,

so that A S,wi (t) = L i (t), i = 1, 2, 3. Each of these parts

should then be invested so as to hedge the “purely financial”part of the corresponding liability. Just as the reserve inSection 2 was not financed by the premium payments (and theinvestment strategy), as was noted in Remark 2.1, the liabilities

L i , i = 1, 2, 3 cannot be financed by A S,wi , i = 1, 2, 3 (and

the corresponding investment strategies) — the actuarial riskpremiums need to be deducted. This is why only the “purelyfinancial” part should be hedged.

The optimal life insurance strategy is to set the sum insuredat time t equal to (ξH(t))−1/δ if this value exceeds s(t) anddoes not exceed s(t). Otherwise it should be set equal to therelevant boundary.

The following proposition provides the optimal investmentand life insurance strategies in the explicit form. First, however,we introduce some notation.

For 0 ≤ t < s ≤ T and y ∈ (0,∞), let F(t, s, y) denotethe price at time t of a European call option with maturity sand strike price s(s) on an underlier worth y at time t withprice dynamics corresponding to a mutual fund with fixedproportions given by δ−1λ in the risky assets,

dY (s) = Y (s)(

r + δ−1‖λ‖2

)dt + Y (s)δ−1λ′dW (s),

s ∈ (t, T ], (3.9)

i.e.,

F(t, s, y) = yΦ(

d(t, s, y))− e−r(s−t)s(s)

×Φ(

d(t, s, y)− ‖λ‖/δ√

s − t),


with Φ denoting the standard normal cumulative distributionfunction, and

d(t, s, y) =δ

‖λ‖√

s − t

(log

(y

s(s)

)+

(r +

‖λ‖2

2δ2

)(s − t)

)(with the interpretation d(t, s, y) = ∞ and Φ(∞) = 1 ifs(s) = 0 and ‖λ‖ > 0 or if ‖λ‖ = 0 and y ≥ s(s), and similarlyd(t, s, y) = −∞ and Φ(−∞) = 0 if s(s) = ∞ and ‖λ‖ > 0or if ‖λ‖ = 0 and y < s(s)).

We define F(t, s, y) and d(t, s, y) correspondingly (withs(s) replaced by s(s)).

Proposition 3.2. We have

L1(0) = ξ−1/δh(0), (3.10)

where h is given by (2.18). The constant ξ can be calculated(numerically) from the identity

π0 + g(0) = ξ−1/δK −1/δe−∫ T

0 (κ(y)+µ(y))dy

+

∫ T

0G(t)−1µ(t)s(t)dt

+

∫ T

0e−

∫ t0 µ(s)dsµ(t)F

(0, t, ξ−1/δe−

∫ t0 κ(y)dy

)dt

−

∫ T

0e−

∫ t0 µ(s)dsµ(t)F

(0, t, ξ−1/δe−

∫ t0 κ(y)dy

)dt.

(3.11)

The dynamics of L1 are given by

dL1(t) = L1(t)

(r +

1δ‖λ‖2

+ µ(t)[1 − h(t)−1

])dt

+ L1(t)1δλ′dW (t), (3.12)

so that

L1(t) = L1(0)e∫ t

0 (r+µ(s)[1−h(s)−1]+‖λ‖2(2δ−1)/(2δ2))ds+δ−1λ′W (t),

t ∈ [0, T ).

The optimal investment strategy is given by

w(t) = L1(t)

(1 +

∫ Tt e−

∫ st (κ(y)+µ(y))dyµ(s)η(t, s)ds

h(t)

)

×1δ(σ ′)−1λ, t ∈ [0, T ), (3.13)

where, for 0 ≤ t < s < T ,

η(t, s) = Φ(

d(t, s, L1(t)h(t)−1e−

∫ st κ(y)dy)

)−Φ

(d(t, s, L1(t)h(t)

−1e−∫ s

t κ(y)dy))

− 1.

The optimal life insurance strategy is given by

S(t) = max[

s(t),min(

L1(t)

h(t), s(t)

)]. (3.14)

Remark 3.3. By comparing with (2.23) we see that thedynamics of L1 are exactly as the dynamics of the total asset

process A S,w in the unconstrained case treated in Section 2(the initial values are different, though). Thus, the first asset

component A S,w1 should always be distributed among the risky

assets in proportions given by the vector (σ ′)−1λ with risky-asset amounts proportional to the reciprocal of the relativerisk-aversion coefficient δ. The actuarial risk premium, which

is deducted continuously from L1 (or A S,w1 ), is given by

µ(t)L1(t)[1 − h(t)−1] at time t, t ∈ (0, T ]. From (3.15) below

it follows that this risk premium satisfies

µ(t)L1(t)[1 − h(t)−1

]= µ(t)

[L1(t)−

(ξH(t)

)−1/δ],

as it should (cf. the interpretation of L1(t) in the discussionabove).

The total allocation in the risky assets is formed by

multiplying the risky-asset allocation of A S,w1 by the factor(

1 +

∫ Tt e−


h(t)

).

We emphasize that this is the optimal total risky-assetallocation. Alternatively, as the above characterization suggests,one can consider the three asset components separately. Therisky-asset allocations of each component can then be derivedfrom the proof below.

As in the unconstrained case we see that the optimal risky-asset allocation and the optimal sum insured at any given timet are independent of one another as both only depend on L1(t)and t . However, since the optimal investment strategy dependson the upper and lower boundaries and is designed specificallyso as to hedge against the “otherwise optimal” sum insuredL1(·)/h(·) being outside the boundaries, we stress that the twostrategies should not (from an overall perspective at least) beviewed completely separately.

Proof. We begin by finding explicit formulas for L i (t), i =

1, 2, 3. We have, for s ∈ [t, T ],(ξH(s)

)−1/δ=

(ξΛ(t)G(s)−1 D(s)−1

)−1/δ

× e1δ

∫ st λ

′dW (y)+ 12δ ‖λ‖

2(s−t)

=

(ξΛ(t)G(s)−1 D(s)−1

)−1/δ

× Y (s)e−

(r+

δ−12δ2

‖λ‖2)(s−t)

,

where

Y (s) = exp((

r +2δ − 1

2δ2 ‖λ‖2)(s − t)+

1δ

∫ s

tλ′dW (y)

),

s ∈ [t, T ].

Now, Y = (Y (s))s∈[t,T ] is the value process of a constant-proportions portfolio with Y (t) = 1 and wealth dynamics givenby (3.9). Therefore,

EQ(

e−r(s−t)Y (s)∣∣∣FW

t

)= Y (t) = 1, ∀s ∈ [t, T ],

so


M1(t) =

∫ T

tG(s)−1µ(s)

(ξΛ(t)G(s)−1 D(s)−1

)−1/δ

× e1−δ

2δ2‖λ‖2(s−t)ds

+ G(T )−1 K 1/δ(ξΛ(t)G(T )−1 D(T )−1

)−1/δ

× e1−δ

2δ2‖λ‖2(T −t)

.

A few more calculations (left to the reader) show that

M1(t) =(ξH(t)

)−1/δG(t)−1h(t),

so that

L1(t) =(ξH(t)

)−1/δh(t), t ∈ [0, T ), (3.15)

from which (3.10) follows in particular.To derive an explicit formula for L2(t) we first note that M2

can be recast as

M2(t) =

∫ T

tG(s)−1µ(s)

× EQ([(

ξH(s))−1/δ

− s(s)]+ ∣∣∣∣FW

t

)ds.

Now, it follows from the calculations above and theBlack–Scholes formula (for the price of a European call option)that

EQ([(

ξH(s))−1/δ

− s(s)]+ ∣∣∣∣FW

t

)= er(s−t)F (t, s, ρ(s,Λ(t))) , s ∈ [t, T ],

where, for (s, x) ∈ [t, T ] × (0,∞),

ρ(s, x) =

(ξ xG(s)−1 D(s)−1

)−1/δe−

(r+

δ−12δ2

‖λ‖2)(s−t)

.

Thus, we have

M2(t) =

∫ T

tG(s)−1µ(s)er(s−t)F (s, ρ(t, s,Λ(t))) ds.

Now, a few easy calculations show that

ρ(s,Λ(t)) =(ξH(t)

)−1/δe−

∫ st κ(y)dy, s ∈ [t, T ],

where κ(·) is given by (2.19), so L2(t) can be written (using(3.15)) as

L2(t) =

∫ T

te−

∫ st µ(y)dyµ(s)

× F(

t, s, L1(t)h(t)−1e−

∫ st κ(y)dy

)ds. (3.16)

Finally, M3(t) can be written as

M3(t) =

∫ T

tG(s)−1µ(s)

×

[s(s)− EQ

( (ξH(s)

)−1/δ∣∣∣FW

t

)]ds

+

∫ T

tG(s)−1µ(s)

× EQ([(

ξH(s))−1/δ

− s(s)]+ ∣∣∣∣FW

t

)ds,

so from the above calculations we obtain (with an obviousnotation)

L3(t) =

∫ T

tG(t)G(s)−1µ(s)s(s)ds

− L1(t)h(t)−1∫ T

te−


+

∫ T

te−

∫ st µ(y)dyµ(s)

× F(

t, s, L1(t)h(t)−1e−

∫ st κ(y)dy

)ds. (3.17)

From (3.15)–(3.17), and the identity

L1(0)− L2(0)+ L3(0) = π0 + g(0),

we easily obtain (3.11).Now, it is easily verified from (3.15) and the well-known

formula

dΛ(t) = −Λ(t)λ′dW (t),

that L1 has the dynamics given by (3.12).As for the (diffusion part of the) dynamics of L2 (see (3.16)),

it is well-known (or fairly easy to show) that ∂F/∂y(t, s, y) =

Φ(d(t, s, y)), so for fixed t ∈ [0, T ] and fixed s ∈ [t, T ], thederivative of the function

x 7→ F(t, s, xh(t)−1e−∫ s

t κ(y)dy), x ∈ (0,∞),

is given by

Φ(

d(t, s, xh(t)−1e−∫ s

t κ(y)dy))

h(t)−1e−∫ s

t κ(y)dy,

x ∈ (0,∞).

As a function of s ∈ [t, T ] this derivative is bounded (for eachfixed x ∈ (0,∞)). We may therefore differentiate under theintegral sign in the expression for L2(t), which, combined withthe dynamics for L1, leads us to conclude that

dL2(t) = L2(t)rdt + σL2(t)δ−1

‖λ‖2dt

−µ(t)

[(L1(t)h(t)

−1− s(t)

)+

− L2(t)

]dt

+ σL2(t)δ−1λ′dW (t),

where the diffusion coefficient process σL2 is given by

σL2(t) =L1(t)

h(t)

∫ T

te−

∫ st (κ(y)+µ(y))dyµ(s)

×Φ(

d(t, s, L1(t)h(t)−1e−

∫ st κ(y)dy)

)ds.

Similarly, we get (see (3.17))

dL3(t) = L3(t)rdt + σL3(t)δ−1

‖λ‖2dt

−µ(t)

[(s(t)− L1(t)h(t)

−1)+

− L3(t)

]dt

+ σL3(t)δ−1λ′dW (t),

where the diffusion coefficient process σL3 is given by


σL3(t) = −L1(t)

h(t)

(∫ T

te−


)+

L1(t)

h(t)

∫ T

te−

∫ st (κ(y)+µ(y))dyµ(s)

×Φ(

d(t, s, L1(t)h(t)−1e−

∫ st κ(y)dy)

)ds.

From the diffusion coefficient processes σL2 and σL3 one candeduce how the risky-asset allocations pertaining to the secondand third parts of the total assets should be. Summing up, wesee that the integrand ψ in the stochastic integral representationM(t) = π0 +

∫ t0 ψ

′(s)dW Q(s) of the Q-martingale M is givenby

ψ(t) = L1(t)

(1 +

∫ Tt e−


h(t)

)× G(t)−1δ−1λ, t ∈ [0, T ],

from which it easily follows that the optimal investmentstrategy is given by (3.13).

Finally, from (3.15) we have

L1(t) =(ξH(t)

)−1/δh(t) = I

(ξH(t)

)h(t),

and thereby

S(t) = It(ξH(t)

)= max

[s(t),min

(L1(t)

h(t), s(t)

)],

as asserted.

Remark 3.4. Since

L1(t) = V S,w(t)+ g(t)+ L2(t)− L3(t),

the optimal life insurance strategy can also be expressed as

S(t) = max

[s(t),min

(V S,w(t)+ g(t)+ L2(t)− L3(t)

h(t), s(t)

)].

(3.18)

If s(·) ≡ 0 and s(·) ≡ ∞, then L2(·) ≡ L3(·) ≡ 0, and S is asin Proposition 2.7.

Once again we briefly consider the case where no risky-asset investments are allowed (cf. the introduction). In this casethe optimal life insurance strategy is formally exactly as in(3.14), with the only adjustments being that the second termof κ(·) drops out (as in the unconstrained case), i.e., κ(t) =

(β(t)− (1 − δ)r)/δ, and that L1(·) reduces to

L1(t) = L1(0)e∫ t

0

(r+µ(s)

[1−h(s)−1])ds, t ∈ [0, T ).

Moreover, L2(·) and L3(·) become

L2(t) =

∫ T

0e−

∫ t0 µ(s)dsµ(t)

×

(ξ−1/δe−

∫ t0 κ(y)dy

− s(t)e−r t)+

dt,

L3(t) =

∫ T

0e−

∫ t0 µ(s)dsµ(t)

×

(s(t)e−r t

− ξ−1/δe−∫ t

0 κ(y)dy)+

dt.

The constant ξ can be calculated numerically from

π0 + g(0) = ξ−1/δK −1/δe−∫ T

0 (κ(y)+µ(y))dy

+

∫ T

0G(t)−1µ(t)s(t)dt

+

∫ T

0e−

∫ t0 µ(y)dyµ(t)

(ξ−1/δe−

∫ t0 κ(y)dy

− s(t)e−r t)+

dt

−

∫ T

0e−

∫ t0 µ(y)dyµ(t)

(ξ−1/δe−

∫ t0 κ(y)dy

− s(t)e−r t)+

dt.

Furthermore, if r > β(t), ∀t ∈ [0, T ], and s(·) is constant, thenthe expression (3.18) for the optimal life insurance strategy canbe reduced to

S(t) = max

[s,min

(V S,w(t)+ g(t)+ L2(t)

h(t), s(t)

)]since L3(·) decreases to 0 exactly at the (deterministic) pointin time where the (deterministic) process [V S,w(·) + g(·) +

L2(·) − L3(·)]/h(·) hits the lower boundary s. In particular, ifwe also have that the upper boundary is “large” in the sense thatit always exceeds [V S,w(·) + g(·) + L2(·)]/h(·) (for this it issufficient that s(t) ≥ L1(0)h(0)−1e

∫ t0 (r−κ(y))dy, ∀t ∈ [0, T ]),

then L2(·) ≡ 0, and we get the nice expression

S(t) = max

[s,

V S,w(t)+ g(t)

h(t)

],

which is directly comparable with the optimal unconstrainedlife insurance strategy in feedback form, see (2.16).

Similarly, it can be shown that if r < β(t), ∀t ∈ [0, T ], s(·)is constant, and s(·) is sufficiently small, then

S(t) = min

[V S,w(t)+ g(t)

h(t), s

].

4. Concluding remarks

We briefly summarize our results, focusing on the structuralforms of the optimal strategies. In the unconstrained case theoptimal investment strategy corresponds to that obtained byMerton (1971) in the case with non-capital income, with thereserve interpreted as the current financial wealth, and theactuarial present value of the future premiums interpreted as thepresent value of future income (human capital). The optimallife insurance strategy is to let the sum insured at any timet ∈ [0, T ] equal A(t)/h(t), where A(t) is the sum of the currentreserve and the actuarial present value of the future premiumsand thus can be interpreted as the total assets, and where h(·)is a certain function depending on the individual preferencesof the policyholder (see Section 2 for a closer analysis). Inparticular, the range of the sum insured is the entire interval(0,∞). Moreover, the investment and life insurance strategiescan be viewed as being independent in the sense that at anygiven time, the optimal risky-asset allocation and the optimalsum insured are independent.


In the constrained case with lower and upper boundariess(·) and s(·) on the sum insured, the total assets should bedecomposed into three parts as

A(·) = A1(·)− A2(·)+ A3(·).

The first component, A1(·), corresponds to the total assetprocess of the unconstrained case: It should be investedaccording to the same (constant) risky-asset proportions as inthe unconstrained case, and it forms the basis for the optimallife insurance strategy in the sense that the sum insured atany time t ∈ [0, T ] should be set to A1(t)/h(t) unlessA1(t)/h(t) < s(t) or A1(t)/h(t) > s(t), in case of whichit should be set to the relevant boundary. The other twoasset components, A2(·) and A3(·), act as hedging instrumentsensuring that the optimal insurance strategy can be financed (inthe sense of the actuarial equivalence principle; see Remark 2.1and the discussion following Definition 2.4). Thus, A3(·) is(the value of) a continuum of European put options writtenon A1(·)/h(·) with strike s(·) paying off the deficit [s(t) −

A1(t)/h(t)]+ in the event of death at time t , and A2(·), inwhich the policyholder holds a short position, is (the value of)a continuum of European call options written on A1(·)/h(·)with strike s(·) paying off the surplus [A1(t)/h(t) − s(t)]+ inthe event of death at time t . The amount allocated to the firstcomponent is determined by the budget constraint

π0 + g(0) = A1(0)− A2(0)+ A3(0).

In contrast to the unconstrained case, the optimal investmentand life insurance strategies should not be viewed separatelysince the optimal investment strategy depends on the upper andlower boundaries and is specifically designed so as to hedgeagainst the “otherwise optimal” sum insured A1(·)/h(·) beingoutside the boundaries.

Appendix. Derivation of (2.21)

In this appendix we provide the derivation of (2.21) in thecase δ 6= 1. The case δ = 1 is left to the interested reader.

We have u′(x) = x−δ, x > 0, and with J in the form (2.20)we have

∂ J/∂t = f ′(t)u(v + g(t))+ f (t)u′(v + g(t))g′(t),

∂ J/∂v = f (t)u′(v + g(t)),

∂2 J/∂v2= f (t)u′′(v + g(t)).

The maxima S and w of the expression in the parenthesis inthe HJB equation is therefore found as the solutions to theequations

0 = D(t)µ(t)S−δ− f (t)(v + g(t))−δµ(t),

0 = f (t)(v + g(t))−δ(α − r1d)− δ f (t)(v + g(t))−δ−1σσ ′w,

that is,

S = ( f (t)/D(t))−1/δ (v + g(t)),

w =(v + g(t))

δ(σσ ′)−1(α − r1d) =

(v + g(t))

δ(σ ′)−1λ.

Note that S is indeed a maximum, since −δS−δ−1 < 0, S > 0,and that w is a maximum if (and only if) f (t)(v+ g(t))−δ−1 >

0, since σσ ′ is positive definite.The HJB equation thus becomes

0 = D(t)µ(t) ( f (t)/D(t))(δ−1)/δ (v + g(t))1−δ/(1 − δ)

+ f ′(t)(v + g(t))1−δ/(1 − δ)+ f (t)(v + g(t))−δg′(t)

+ f (t)(v + g(t))−δ

×

((r + µ(t))v +

(v + g(t))

δ‖λ‖2

+ π

)− f (t)(v + g(t))−δµ(t) ( f (t)/D(t))−1/δ (v + g(t))

−12δ f (t)(v + g(t))−δ−1 (v + g(t))2

δ2 ‖λ‖2

= (v + g(t))−δA(t)+ (v + g(t))1−δB(t),

where

A(t) = f (t)g′(t)+ f (t) (−(r + µ(t))g(t)+ π) ,

B(t) = D(t)µ(t) ( f (t)/D(t))(δ−1)/δ /(1 − δ)

+ f ′(t)/(1 − δ)+ f (t)(

r + µ(t)+ ‖λ‖2/δ)

− f (t)µ(t) ( f (t)/D(t))−1/δ

− f (t)‖λ‖2/(2δ).

Now, for a solution to the HJB equation of the form (2.20) wemust necessarily have the boundary conditions

g(T ) = 0,

f (T ) = K D(T ).

It is now straightforward to check that with the functions gand f given by (2.7) and (2.21), respectively, the function Jgiven by (2.20) solves the HJB equation. Note that the conditionf (t)(v + g(t))−1−δ > 0 is fulfilled for every (t, v) ∈ E .

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