# Optimal investment and life insurance strategies under minimum and maximum constraints

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<ul><li><p>Insurance: Mathematics and Econom</p><p>nc</p><p>,</p><p>, DCo</p><p>SAbstract</p><p>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</p><p>JEL classification: C61; D91; G11</p><p>MSC: 49L20; 91B28; 93E20</p><p>Keywords: Life insurance; Asset allocation; Optimization</p><p>1. Introduction</p><p>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 focus</p><p>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.</p><p>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 companys immediaterisk at any time during the policy term, but the motivationOptimal investment and life insuramaximum</p><p>Peter Holm Nielsena,</p><p>a PFA Pension, Sundkrogsgade 4bDepartment of Applied Mathematics and Statistics, University of</p><p>Received April 2007; received in revised formis on the issue of optimal life insurance rather than optimal</p><p> Corresponding author. Tel.: +45 39175876; fax: +45 39175952.E-mail addresses: pei@pfa.dk (P.H. Nielsen), mogens@math.ku.dk</p><p>(M. Steffensen).</p><p>0167-6687/$ - see front matter c 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2007.09.007ics 43 (2008) 1528www.elsevier.com/locate/ime</p><p>ce strategies under minimum andonstraints</p><p>Mogens Steffensenb</p><p>K-2100 Copenhagen, Denmarkpenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark</p><p>eptember 2007; accepted 18 September 2007for 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.</p></li><li><p>Ma16 P.H. Nielsen, M. Steffensen / Insurance:</p><p>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).</p><p>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.</p><p>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).</p><p>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.Mller 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 Mller 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).</p><p>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 the</p><p>combined problem of optimal life insurance and investment(and consumption as well), Campbell (1980), and others.thematics and Economics 43 (2008) 1528</p><p>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 Ros-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).</p><p>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.</p><p>Some basic notations: All vectors are column vectors. Thetransposed of a vector or matrix a is denoted by a. Thed-dimensional vector of 1s is denoted by 1d .</p><p>2. Setup and basic problem</p><p>To formalize the setup we take as given some underlyingprobability space ( ,F,P), on which all random variablesintroduced in the following are defined.</p><p>We consider a policyholder with a life insurance policyissued at time 0 and terminated at a fixed finite time horizonT > 0. Let be a nonnegative random variable representingthe (random) time of death of the policyholder. For t [0, T ],the mortality intensity of is given by a continuous function : [0, T ] [0,), which means thatP ( t) = 1 e</p><p> t0 (s)ds, t [0, T ].</p><p>The (conditional) distribution of on ( > T ) is irrelevant inthis paper.</p><p>Let W = (W1, . . . ,Wd) be a d-dimensional standardBrownian Motion (d N) stochastically independent of . Thefinancial market is assumed to be frictionless and to consist of arisk-free money market account with price dynamics given by</p><p>dS0(t)/S0(t) = rdt,where r 0 is a fixed constant, and d risky assets with pricedynamics given by</p><p>d</p><p>dSi (t)/Si (t) = idt +</p><p>j=1i jdW j (t), i = 1, . . . , d,</p></li><li><p>thP.H. Nielsen, M. Steffensen / Insurance: Ma</p><p>where = (1, . . . , d) Rd , and = (i j )i, j=1,...,d Rdd is non-singular. The filtration generated by W andaugmented by the null-sets of F is denoted by FW =(FWt )t[0,T ]. Let(t) = eW (t) 12 2t , t [0, T ],where = 1( 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 theRadonNikodym derivative</p><p>dQdP</p><p>= (T ),</p><p>and that the process WQ defined by WQ(t) = W (t) + t, t [0, T ], is a standard Brownian motion under Q.</p><p>The policyholder pays an initial lump sum premium pi0 0(at time 0) as well as continuous premiums at a fixed rate pi 0during [0, T ] as long as he is alive (of course, pi0 and pi 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</p><p>V S,w(0) = pi0, (2.1)dV S,w(t) = (rV S,w(t)+ w(t)( r1d)+ pi</p><p>(t)[S(t) V S,w(t)])dt + w(t)dW (t) (2.2)=(rV S,w(t)+ pi (t)</p><p>[S(t) V S,w(t)</p><p>])dt</p><p>+w(t)dWQ(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</p><p>V S,w(t) = pi0G(t)+ t0G(t)G(s)1[pi (s)S(s)</p><p>+w(s)( r1d)]ds+ t0G(t)G(s)1w(s)dW (s) (2.4)</p><p>= pi0G(t)+ t0G(t)G(s)1[pi (s)S(s)]ds</p><p>+ t0G(t)G(s)1w(s)dWQ(s), (2.5)</p><p>whereG(t) = e t0 (r+(s))ds, t [0, T ].ematics and Economics 43 (2008) 1528 17</p><p>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.</p><p>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 policyholders wealth.Correspondingly, the terminal reserve, V S,w(T), is paid out attime T only upon survival of the policyholder.</p><p>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 .</p><p>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,(t)</p><p>[S(t) V S,w(t)] dt (which is negative if S(t) < V S,w(t)),</p><p>is (continuously) deducted from the reserve and thus paid tothe company. </p><p>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 (t)V S,w(t)dtin the reserve dynamics (2.2).) </p><p>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</p></li><li><p>a18 P.H. Nielsen, M. Steffensen / Insurance: M</p><p>future premium payments, i.e., as long as</p><p>V S,w(t)+ g(t) 0, t [0, T ], (2.6)where</p><p>g(t) = pi Tt</p><p>G(t)G(s)1ds, t [0, T ]. (2.7)Allowing for negativity of the rese...</p></li></ul>

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