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

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Insurance: Mathematics and Econom

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SAbstract

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 focus

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 companys immediaterisk at any time during the policy term, but the motivationOptimal investment and life insuramaximum

Peter Holm Nielsena,

a PFA Pension, Sundkrogsgade 4bDepartment of Applied Mathematics and Statistics, University of

Received April 2007; received in revised formis on the issue of optimal life insurance rather than optimal

Corresponding author. Tel.: +45 39175876; fax: +45 39175952.E-mail addresses: pei@pfa.dk (P.H. Nielsen), mogens@math.ku.dk

(M. Steffensen).

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

ce strategies under minimum andonstraints

Mogens Steffensenb

K-2100 Copenhagen, Denmarkpenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

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.

Ma16 P.H. Nielsen, M. Steffensen / Insurance:

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.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).

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

combined problem of optimal life insurance and investment(and consumption as well), Campbell (1980), and others.thematics and Economics 43 (2008) 1528

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).

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 1s is denoted by 1d .

2. Setup and basic problem

To formalize the setup we take as given some underlyingprobability space ( ,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 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

t0 (s)ds, t [0, T ].

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

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

dS0(t)/S0(t) = rdt,where r 0 is a fixed constant, and d risky assets with pricedynamics given by

d

dSi (t)/Si (t) = idt +

j=1i jdW j (t), i = 1, . . . , d,

thP.H. Nielsen, M. Steffensen / Insurance: Ma

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