# Explicit solutions of optimal consumption, investment and insurance problems with regime switching

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<ul><li><p>Insurance: Mathematics and Economics 58 (2014) 159167Contents lists available at ScienceDirect</p><p>Insurance: Mathematics and Economics</p><p>journal homepage: www.elsevier.com/locate/ime</p><p>Explicit solutions of optimal consumption, investment and insuranceproblems with regime switchingBin Zou, Abel Cadenillas Department of Mathematical and Statistical Sciences, University of Alberta, Canada</p><p>h i g h l i g h t s</p><p> We obtain explicit solutions for simultaneous optimal consumption, investment and insurance problems. The solution depends strongly on the regime of the economy. In our model, optimal insurance is either no insurance or deductible insurance. We determine the conditions under which it is optimal to buy insurance in our model. We calculate the advantage of buying insurance in our model.</p><p>a r t i c l e i n f o</p><p>Article history:Received April 2014Received in revised formJune 2014Accepted 19 July 2014</p><p>JEL classification:C61E32E44G11G22</p><p>Keywords:Economic analysisHamiltonJacobiBellman equationInsuranceRegime switchingUtility maximization</p><p>a b s t r a c t</p><p>Weconsider an investorwhowants to select his optimal consumption, investment and insurance policies.Motivated by new insurance products, we allow not only the financial market but also the insurableloss to depend on the regime of the economy. The objective of the investor is to maximize his expectedtotal discounted utility of consumption over an infinite time horizon. For the case of hyperbolic absoluterisk aversion (HARA) utility functions, we obtain the first explicit solutions for simultaneous optimalconsumption, investment, and insurance problems when there is regime switching. We determine thatthe optimal insurance contract is either no-insurance or deductible insurance, and calculate when it isoptimal to buy insurance. The optimal policy depends strongly on the regime of the economy. Throughan economic analysis, we calculate the advantage of buying insurance.</p><p> 2014 Elsevier B.V. All rights reserved.1. Introduction</p><p>In the classical consumption and investment problem, a risk-averse investor wants to maximize his expected discounted utilityof consumption by selecting optimal consumption and investmentstrategies. Merton (1969) was the first to obtain explicit solutionsto this problem in continuous time. Many generalizations to Mer-tons work can be found in Karatzas (1996), Karatzas and Shreve(1998), Sethi (1997), et ctera. In the traditional models for con-sumption and investment problems, there is only one source of</p><p> Correspondence to: Central Academic Building 639, Department of Mathemat-ical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G2G1. Tel.: +1 780 492 0572; fax: +1 780 492 6826.</p><p>E-mail addresses: bzou@ualberta.ca (B. Zou), abel@ualberta.ca (A. Cadenillas).</p><p>http://dx.doi.org/10.1016/j.insmatheco.2014.07.0060167-6687/ 2014 Elsevier B.V. All rights reserved.risk that comes from the uncertainty of the stock prices. But in reallife, apart from the risk exposure in the financial market, investorsoften face other random risks, such as propertyliability risk andcredit default risk. Thus, it is more realistic and practical to extendthe traditional models by incorporating an insurable risk. When aninvestor is subject to an additional insurable risk, buying insuranceis a trade-off decision. On one hand, insurance can provide the in-vestorwith compensation and then offset capital losses if the spec-ified risk events occur. On the other hand, the cost of insurancediminishes the investors ability to consume and therefore reducesthe investors expected utility of consumption.</p><p>The initial optimal insurance problemstudies an individualwhois subject to an insurable risk and seeks the optimal amount ofinsurance under the utility maximization criterion. Using the ex-pected value principle for premium, Arrow (1963) found that the</p><p>http://dx.doi.org/10.1016/j.insmatheco.2014.07.006http://www.elsevier.com/locate/imehttp://www.elsevier.com/locate/imehttp://crossmark.crossref.org/dialog/?doi=10.1016/j.insmatheco.2014.07.006&domain=pdfmailto:bzou@ualberta.camailto:abel@ualberta.cahttp://dx.doi.org/10.1016/j.insmatheco.2014.07.006</p></li><li><p>160 B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159167optimal insurance is deductible insurance in discrete time. Promis-low and Young (2005) reviewed optimal insurance problems(without investment and consumption). They proposed a generalmarketmodel and obtained explicit solutions to optimal insuranceproblems under different premium principles, such as varianceprinciple, equivalent utility principle, Wangs principle, et ctera.</p><p>Moore andYoung (2006) combinedMertons optimal consump-tion and investment problem and Arrows optimal insurance prob-lem in continuous time. They found explicit or numerical solutionsfor different utility functions (although they did not verify rig-orously that the obtained strategies were indeed optimal). Per-era (2010) revisited Moore and Youngs work by considering theirproblem in a more general Levy market, and applied the martin-gale approach to obtain explicit optimal strategies for exponentialutility functions.</p><p>In traditional financial modeling, themarket parameters are as-sumed to be independent of general macroeconomic conditions.However, historical data and empirical research show that themarket behavior is affected by long-term economic factors, whichmay change dramatically as time evolves. Regime switching mod-els use a continuous-time Markov chain with a finite-state spaceto represent the uncertainty of those long-term economic factors.</p><p>Hamilton (1989) introduced a regime switching model to cap-ture themovements of the stock prices and showed that the regimeswitching model represents the stock returns better than themodel with deterministic coefficients. Thereafter, regime switch-ing has been applied to model many financial and economic prob-lems (see for instance Sotomayor and Cadenillas, 2009, for somereferences).</p><p>In the insurance market, insurance policies can depend on theregime of the economy. In the case of traditional insurance, theunderwriting cycle has been well documented in the literature.Indeed, empirical research provides evidence for the dependenceof insurance policies underwriting performance on external eco-nomic conditions (see for instance Grace and Hotchkiss, 1995; Ha-ley, 1993 on propertyliability insurance; and Chung and Weiss,2004 on reinsurance). In the case of non-traditional insurance, byinvestigating the comovements of credit default swap (CDS) andthe bond/stockmarkets, Norden andWeber (2007) found that CDSspreads are negatively correlated with the price movements of theunderlying stocks and such cointegration is affected by the corpo-rate bond volume.</p><p>In this paper, we use an observable continuous-time finite-state Markov chain to model the regime of the economy and al-low both the financial market and the insurance market to dependon the regime. Our objective is to obtain simultaneously optimalconsumption, investment and insurance policies for a risk-averseinvestor who wants to maximize his expected total discountedutility of consumption over an infinite time horizon. We extendSotomayor and Cadenillas (2009) by including a random loss in themodel and an insurance policy in the control. The most importantdifference between the model of Moore and Young (2006) and ourpaper is that they do not allow regime switching, while we allowregime switching in both the financial market and the insurancemarket.</p><p>2. The model</p><p>Consider a complete probability space (, F , P) in which astandard Brownian motionW and an observable continuous-time,stationary, finite-state Markov chain are defined. Denote byS = {1, 2, . . . , S} the state space of this Markov chain, whereS is the number of regimes in the economy. The matrix Q =(qij)SS denotes the strongly irreducible generator of , where i S,</p><p>jS qij = 0, qij > 0 when j = i and qii = </p><p>j=i qij.</p><p>We consider a financial market consisting of two assets, a bondwith price P0 (riskless asset) and a stock with price P1(risky asset),respectively. Their price processes are driven by the following dy-namics:dP0(t) = r(t)P0(t)dt,dP1(t) = P1(t)((t)dt + (t)dW (t)),with initial conditions P0(0) = 1 and P1(0) > 0. The coefficientsri, i and i, i S, are all positive constants.</p><p>An investor chooses = {(t), t 0}, the proportion ofwealth invested in the stock, and a consumption rate process c ={c(t), t 0}.We assume that the investor is subject to an insurableloss L(t, (t), X(t)), where X(t) denotes the investors wealth attime t . We shall use the short notation Lt to replace L(t, (t), X(t))if there is no confusion. We use a Poisson process N with intensity(t), where i > 0 for every i S, to model the occurrence of thisinsurable loss. In the insurancemarket, there are insurance policiesavailable to insure against the loss Lt . We further assume that theinvestor can control the payout amount I(t), where I(t) : [0, ) [0, ) and I(t, ) := It(L(t, (t, ), X(t, ))), or in short,I(t) = It(Lt). For example, if N(t0) = 1, then at time t0 theinvestor suffers a loss of amount Lt0 but receives a compensation ofamount It0(Lt0) from the insurance policy, so the investors net lossis Lt0 It0(Lt0). Following the premium setting used in Moore andYoung (2006) (the famous expected value principle), we assumeinvestors pay premium continuously at the rate P given byP(t) = (t)(1 + (t))E[It(Lt)],where the positive constant i, i S, is known as the load-ing factor in the insurance industry. Such extra positive loadingcomes from insurance companies administrative cost, tax, profit,et ctera.</p><p>Following Sotomayor and Cadenillas (2009), we assume thatthe Brownian motion W , the Poisson process N and the Markovchain are mutually independent. We also assume that the lossprocess L is independent of N . We take the P-augmented filtra-tion {Ft}t0 generated by W ,N, L and as our filtration and de-fine F := (t0 Ft). An investor with triplet strategies u(t) :=((t), c(t), I(t)) has a wealth process X given bydX(t) =</p><p>r(t)X(t) + ((t) r(t))(t)X(t) c(t)</p><p> (t)(1 + (t)) E[It(Lt)]dt</p><p>+ (t)(t)X(t)dW (t) (Lt It(Lt)) dN(t), (1)with initial conditions X(0) = x > 0 and (0) = i S.</p><p>We define the criterion function J as</p><p>J(x, i; u) := Ex,i</p><p>+</p><p>0etU(c(t), (t))dt</p><p>, (2)</p><p>where > 0 is the discount rate and Ex,i means conditionalexpectation givenX(0) = x and (0) = i.We assume that for everyi S, the utility function U(, i) is C2(0, +), strictly increasingand concave, and satisfies the linear growth conditionK > 0 such that U(y, i) K(1 + y), y > 0, i S.Besides, we use the notation U(0, i) := limy0+ U(y, i), i S.</p><p>We define the bankruptcy time as := inf{t 0 : X(t) 0}.Since an investor consumes only when his wealth is strictlypositive, we define</p><p>R() :=</p><p>etU(c(t), (t))dt =</p><p>etU(0, (t))dt.</p><p>A control u := (, c, I) is called admissible if {ut}t0 is pre-dictable with respect to the filtration {Ft}t0 and satisfies t 0</p><p>Ex,i</p><p> t0</p><p>c(s)ds</p><p>< +, Ex,i</p><p> t0</p><p> 2(s)2(s)ds</p><p>< +,</p><p>Ex,i</p><p> 0</p><p>esU+(c(s), (s))ds</p><p>< +,</p></li><li><p>B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159167 161and I(t) It := {I : 0 I(Y ) Y , where Y is Ft-measurable}.The set of all admissible controls with initial conditions X(0) = xand (0) = i is denoted by Ax,i. We study the following problem.</p><p>Problem 2.1. Select an admissible control u = (, c, I) Ax,ithat maximizes the criterion function J . In addition, find the valuefunction</p><p>V (x, i) := supuAx,i</p><p>J(x, i; u).</p><p>The control u is called an optimal control or an optimal policy.</p><p>Moore and Young (2006) also incorporated an insurable riskinto the consumption and investment framework. However, theydid not consider a regime switching model, or equivalently theyassumed that there is only one regime in the economy. Neverthe-less, the insurable risk and the coefficients of the financial mar-ket most likely depend on the regime of the economy. Hence, inthe above regime switching model, we assume that the insurancemarket (insurable loss and insurance performance) and the finan-cial market are regime dependent. Furthermore, we assume thatthese two markets depend on the same regime. We mention threeexamples below. First, the assumption that the financial marketand the insurancemarket depend on the same regime is supportedby the bailout case of AIG (see Sjostrom, 2009 for details) and thefinancial derivatives in the insurance industry. Before the crashof the U.S. housing market in 2007, many investors, banks andfinancial institutions bought obligations constructed from mort-gage payments or made loans to the housing agencies. To insureagainst the credit risk that the obligations or loans may default,they purchased credit default swap (CDS) contracts from insurancecompanies like AIG. In a CDS contract, the buyer makes periodicpayments to the seller, and in return, receives the par value of theunderlying obligation or loan in the event of a default. Apparently,the credit default risk insured by CDS contracts is negatively corre-lated with the reference entitys stock performance (see Nordenand Weber, 2007 for empirical evidence). Second, generated bythe financial engineering onderivatives, insurance companies havecreated numerous equity-linked products, such as equity-linkedlife insurance (see Hardy, 2003 for more details on such insur-ance policy). If the insured of an equity-linked life insurance policysurvives to the expiration, then the beneficiary receives invest-ment benefit that depends upon the market value of the referenceequity. Hence, equity-linked life insurance and its reference eq-uity are affected by the same long-term economic factors. Third,even in traditional insurance products like propertyliability in-surance, there is empirical evidence (see for instance Grace andHotchkiss, 1995) that the loading factor depends on the regime ofthe economy. Indeed, in those traditional insurance products, (t)and L(t, (t, ), X(t, ))might be independent of (t) but (t) de-pends on (t).</p><p>3. Verification theorems</p><p>In Zou and Cadenillas (2014) we develop two verification theo-rems (Theorems 3.1 and 3.2) to obtain the optimal solution to Prob-lem 2.1. Those verification theorems provide a sufficient conditionof optimality.</p><p>4. Explicit solutions of value function and optimal strategies</p><p>In this section,we obtain explicit solutions to optimal consump-tion, investment and insurance problems when there is regimeswitching in the economy. We assume the utility function is ofHARA type and the insurable loss L is proportional to the investorswealth, L(t, (t), X(t)) = (t) lt Xt . Here for every i S, i > 0measures the intensity of the insurable loss in regime i, and for ev-ery t 0, lt denotes the loss proportion at time t . We assume thatlt is Ft-measurable and lt (0, 1) for all t 0.We consider three utility functions of HARA class:1. U(y, i) = ln(y), y > 0,2. U(y, i) = y, y > 0, < 0,3. U(y, i) = y, y > 0, 0 < < 1.</p><p>These utility functions do not depend on the market regime. Theyare C2(0, ), strictly incre...</p></li></ul>