Insurance: Mathematics and Economics 58 (2014) 159–167

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Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Explicit solutions of optimal consumption, investment and insuranceproblems with regime switchingBin Zou, Abel Cadenillas ∗

Department of Mathematical and Statistical Sciences, University of Alberta, Canada

h i g h l i g h t s

• 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.

a r t i c l e i n f o

Article history:Received April 2014Received in revised formJune 2014Accepted 19 July 2014

JEL classification:C61E32E44G11G22

Keywords:Economic analysisHamilton–Jacobi–Bellman equationInsuranceRegime switchingUtility maximization

a b s t r a c t

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.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

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-ton’s work can be found in Karatzas (1996), Karatzas and Shreve(1998), Sethi (1997), et cétera. In the traditional models for con-sumption and investment problems, there is only one source of

∗ 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.

E-mail addresses: bzou@ualberta.ca (B. Zou), abel@ualberta.ca (A. Cadenillas).

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 property–liability 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 investor’s ability to consume and therefore reducesthe investor’s expected utility of consumption.

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

160 B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167

optimal 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, Wang’s principle, et cétera.

Moore andYoung (2006) combinedMerton’s optimal consump-tion and investment problem and Arrow’s 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 Young’s work by considering theirproblem in a more general Levy market, and applied the martin-gale approach to obtain explicit optimal strategies for exponentialutility functions.

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.

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).

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 property–liability 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.

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.

2. The model

Consider a complete probability space (Ω, F , P) in which astandard Brownian motionW and an observable continuous-time,stationary, finite-state Markov chain ϵ 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)S×S denotes the strongly irreducible generator of ϵ, where∀ i ∈

S,

j∈S qij = 0, qij > 0 when j = i and qii = −

j=i qij.We consider a financial market consisting of two assets, a bond

with 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) = rϵ(t)P0(t)dt,dP1(t) = P1(t)(µϵ(t)dt + σϵ(t)dW (t)),with initial conditions P0(0) = 1 and P1(0) > 0. The coefficientsri, µi and σi, i ∈ S, are all positive constants.

An investor chooses π = π(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, ϵ(t), X(t)), where X(t) denotes the investor’s wealth attime t . We shall use the short notation Lt to replace L(t, ϵ(t), X(t))if there is no confusion. We use a Poisson process N with intensityλϵ(t), where λi > 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, ∞)×Ω → [0, ∞) and I(t, ω) := It(L(t, ϵ(t, ω), X(t, ω))), 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 investor’s 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) = λϵ(t)(1 + θϵ(t))E[It(Lt)],where the positive constant θi, 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 cétera.

Following Sotomayor and Cadenillas (2009), we assume thatthe Brownian motion W , the Poisson process N and the Markovchain ϵ are mutually independent. We also assume that the lossprocess L is independent of N . We take the P-augmented filtra-tion Ftt≥0 generated by W ,N, L and ϵ as our filtration and de-fine F := σ(∪t≥0 Ft). An investor with triplet strategies u(t) :=

(π(t), c(t), I(t)) has a wealth process X given bydX(t) =

rϵ(t)X(t) + (µϵ(t) − rϵ(t))π(t)X(t) − c(t)

− λϵ(t)(1 + θϵ(t)) · E[It(Lt)]dt

+ σϵ(t)π(t)X(t)dW (t) − (Lt − It(Lt)) dN(t), (1)with initial conditions X(0) = x > 0 and ϵ(0) = i ∈ S.

We define the criterion function J as

J(x, i; u) := Ex,i

+∞

0e−δtU(c(t), ϵ(t))dt

, (2)

where δ > 0 is the discount rate and Ex,i means conditionalexpectation givenX(0) = x and ϵ(0) = i.We assume that for everyi ∈ S, the utility function U(·, i) is C2(0, +∞), strictly increasingand concave, and satisfies the linear growth condition∃K > 0 such that U(y, i) ≤ K(1 + y), ∀ y > 0, i ∈ S.

Besides, we use the notation U(0, i) := limy→0+ U(y, i), ∀ i ∈ S.We define the bankruptcy time as

Θ := inft ≥ 0 : X(t) ≤ 0.Since an investor consumes only when his wealth is strictlypositive, we define

R(Θ) :=

∞

Θ

e−δtU(c(t), ϵ(t))dt =

∞

Θ

e−δtU(0, ϵ(t))dt.

A control u := (π, c, I) is called admissible if utt≥0 is pre-dictable with respect to the filtration Ftt≥0 and satisfies ∀ t ≥ 0

Ex,i

t

0c(s)ds

< +∞, Ex,i

t

0σ 2

ϵ(s)π2(s)ds

< +∞,

Ex,i

Θ

0e−δsU+(c(s), ϵ(s))ds

< +∞,

B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167 161

and 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 ϵ(0) = i is denoted by Ax,i. We study the following problem.

Problem 2.1. Select an admissible control u∗= (π∗, c∗, I∗) ∈ Ax,i

that maximizes the criterion function J . In addition, find the valuefunction

V (x, i) := supu∈Ax,i

J(x, i; u).

The control u∗ is called an optimal control or an optimal policy.

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 entity’s 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 property–liability in-surance, there is empirical evidence (see for instance Grace andHotchkiss, 1995) that the loading factor θ depends on the regime ofthe economy. Indeed, in those traditional insurance products, λϵ(t)and L(t, ϵ(t, ω), X(t, ω))might be independent of ϵ(t) but θϵ(t) de-pends on ϵ(t).

3. Verification theorems

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.

4. Explicit solutions of value function and optimal strategies

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 investor’swealth, L(t, ϵ(t), X(t)) = ηϵ(t) lt Xt . Here for every i ∈ S, ηi > 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) = −yα, y > 0, α < 0,3. U(y, i) = yα, y > 0, 0 < α < 1.

These utility functions do not depend on the market regime. Theyare C2(0, ∞), strictly increasing and concave, and satisfy the lineargrowth condition with K = 1. In Section 4 of Zou and Cadenillas(2014), we consider in addition a utility function that depends onthe regime of the economy.

4.1. U(y, i) = ln(y), y > 0, ∀ i ∈ S

We denote γi :=12

(µi−ri)2

σ 2i

, r = (r1, r2, . . . , rS)′, γ = (γ1,

γ2, . . . , γS)′, Λi := E

ln1 − ηil +

ηil −

θi1+θi

+

− (1 + θi)

E

ηil −θi

1+θi

+

,

λΛ = (λ1Λ1, λ2Λ2, . . . , λSΛS)′, 1 = (1, 1,

. . . , 1)′S×1, and I the S × S identity matrix. The constant vectorA = (A1, A2, . . . , AS)

′ is defined as the solution of the linear system

(δI − Q )A =

1δ

r + γ +

λΛ − δ1

. (3)

Proposition 4.1. The function v = v(·, ·), given by

v(x, i) =

1δln(δx) + Ai, x > 0

−∞, x = 0

where A = (A1, A2, . . . , AS)′ solves the linear system (3), is the value

function of Problem 2.1. Furthermore, the policy given by

u∗(t) = (π∗(t), c∗(t), I∗(t))

=

µϵ(t) − rϵ(t)

σ 2ϵ(t)

, δX∗

t ,

ηϵ(t) lt −

θϵ(t)

1 + θϵ(t)

+

X∗

t

is an optimal policy of Problem 2.1.

Proof. See the proof of Proposition 4.1 of Zou and Cadenillas(2014).

Example 4.1. In this example, we assume that there are tworegimes in the economy, where regime 1 represents a bull marketand regime 2 represents a bear market. According to French et al.(1987), the stock returns are higher in a bull market, so µ1 > µ2.Hamilton and Lin (1996) found that stock volatility is higher in abearmarket; thusσ1 < σ2. The data of overnight financing rate andtreasury bill rate suggests that the risk-free interest rate is higherin good economy; hence r1 > r2. Haley (1993) found that the un-derwriting margin is negatively correlated with the interest rate,which implies that the loading factor is smaller in a bull market,θ1 < θ2. Norden andWeber (2007) observed that CDS spreads (de-fault risk) are negatively correlated with the stock prices. Equiva-lently, the default risk is higher in a bear market, that is, η1 < η2.

The generator matrix entries become q11 = −Π1, q12 = Π1,q21 = Π2, q22 = −Π2, with Π1, Π2 > 0, so the linear system (3)becomes

(δ + Π1)A1 − Π1A2 =1δ(r1 + γ1 − δ + λ1Λ1)

−Π2A1 + (δ + Π2)A2 =1δ(r2 + γ2 − δ + λ2Λ2)

which gives a unique solution

Ai =Πi(rj + γj − δ + λjΛj) + (δ + Πj)(ri + γi − δ + λiΛi)

δ2(δ + Π1 + Π2),

162 B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167

where i, j = 1, 2 and i = j. From the above expression of Ai, wenotice that only Λi is not directly given by the market. To calculateΛi, we assume that the loss proportion lt does not depend on time tand we discuss the cases that l is constant or uniformly distributedon (0, 1). We further assume θ1

η1(1+θ1)≤

θ2η2(1+θ2)

. If the opposite is

true, then we switch the expressions when calculating Λ1 and Λ2.

First,we consider the case inwhich l is constant. Ifηil −

θi1+θi

+

≡ 0, then Λi = ln(1 − ηil). Otherwise, we obtain Λi = − ln(1 +

θi) − ηil(1 + θi) + θi.Second, we consider the case in which l is uniformly distributed

on (0, 1). Ifηil −

θi1+θi

+

≡ 0, then Λi = E [ln(1 − ηil)] =1 −

1ηi

ln(1 − ηi) − 1. Otherwise, through straightforward cal-

culus, we obtain Eln1 − ηil +

ηil −

θi1+θi

+

=

1ηi

− 1

ln(1+θi)−θi

ηi(1+θi)and E

ηil −

θi1+θi

+

=ηi2 +

θ2i2ηi(1+θi)2

−θi

1+θi.

Hence, Λi =

1ηi

− 1ln(1 + θi) −

(ηi(1+θi)−θi)2+2θi

2ηi(1+θi).

4.2. U(y, i) = −yα, y > 0, α < 0, ∀i ∈ S

Let the constants Ai, i ∈ S, satisfy the following non-linearsystem

δ − αri −α

1 − αγi + λi(1 − Λi)

A1−αi − (1 − α)A−α

i

=

j∈S

qijA1−αj , (4)

where Λi := E(1− ηil+ (ηil− νi)

+)α− α(1+ θi)E

(ηil− νi)

+.

We need

δ > maxi∈S

αri +

α

1 − αγi − λi(1 − Λi)

. (5)

Lemma4.1 in Sotomayor and Cadenillas (2009) guarantees that thenon-linear system (4) has a unique positive solution Ai, i ∈ S, if (5)holds.

Proposition 4.2. The function v = v(·, ·), given by

v(x, i) =

−A1−α

i xα, x > 0−∞, x = 0,

where Ai is the unique solution to the non-linear system (4), is thevalue function of Problem 2.1. Furthermore, the policy given by

u∗(t) =

µϵ(t) − rϵ(t)(1 − α)σ 2

ϵ(t),

X∗t

Aϵ(t),ηϵ(t) lt − νϵ(t)

+ X∗

t

is an optimal policy of Problem 2.1.

Proof. See the proof of Proposition 4.2 of Zou and Cadenillas(2014).

Example 4.2. We assume S = 2. To solve the non-linear system(4), we need to find Λi first. In this example, we show how to findΛi when l is constant or is uniformly distributed on (0, 1). Withoutloss of generality, we assume ν1

η1≤

ν2η2. If the opposite holds, we

switch the formulas for Λ1 and Λ2. The results will be used foreconomic analysis in the next section.

First, we consider the case in which l is constant. If (ηil−νi)+

≡

0, then Λi = (1 − ηil)α . Otherwise, we obtain Λi = (1 − νi)α

−

α(1 + θi)(ηil − νi).

Second, we consider the case in which l is uniformly distributedon (0, 1). If (ηil − νi)

+≡ 0, then

Λi = E [(1 − ηil)α] =

−

1ηi

ln(1 − ηi), α = −1

1ηi(1 + α)

(1 − (1 − ηi)1+α), α = −1.

Otherwise, we obtain E[(ηil−νi)+] =

1νiηi

(ηil−νi)dl =(ηi−νi)

2

2ηi, and

when α = −1, E[(1−ηil+ (ηil−νi)+)α] = (1−νi)

−11 −

νiηi

−

1ηiln(1 − νi), and when α = −1, E[(1 − ηil + (ηil − νi)

+)α] =

(1−νi)α1 −

νiηi

−1−νi

ηi(1+α)

+

1ηi(1+α)

. Therefore, if (ηil−νi)+

≡ 0,and α = −1, then

Λi = (1 − νi)−11 −

νi

ηi

−

1ηi

ln(1 − νi) + (1 + θi)(ηi − νi)

2

2ηi;

and if (ηil − νi)+

≡ 0, and α = −1, then

Λi = (1 − νi)α

1 −

νi

ηi−

1 − νi

ηi(1 + α)

+

1ηi(1 + α)

− α(1 + θi)(ηi − νi)

2

2ηi.

4.3. U(y, i) = yα, y > 0, 0 < α < 1, ∀ i ∈ S

Let the constants Ai > 0, i ∈ S, satisfy the non-linear systemδ − αri −

α

1 − αγi + λi(1 − Λi)

A1−αi − (1 − α)A−α

i

=

j∈S

qijA1−αj , (6)

where Λi := E(1− ηil+ (ηil− νi)

+)α− α(1+ θi)E

(ηil− νi)

+.

We need

δ > maxi∈S

αri +

α

1 − αγi

. (7)

Lemma4.2 in Sotomayor and Cadenillas (2009) guarantees that thenon-linear system (6) has a unique positive solution Ai, i ∈ S, if (7)is satisfied.

Proposition 4.3. The function v(x, i) = A1−αi xα, x ≥ 0, where Ai is

the unique solution to the non-linear system (6), is the value functionof Problem 2.1. Furthermore, the policy given by

u∗(t) :=

µϵ(t) − rϵ(t)(1 − α)σ 2

ϵ(t),

X∗t

Aϵ(t), (ηϵ(t) lt − νϵ(t))

+X∗

t

is an optimal policy of Problem 2.1.

Proof. See the proof of Proposition 4.3 of Zou and Cadenillas(2014).

Example 4.3. We assume S = 2. We notice that the non-linearsystems (4) and (6) are identical except that α is negative in (4)while α ∈ (0, 1) in (6). Hence, we obtain Λi in the same way as Λiin Example 4.2.

5. Economic analysis

In this section, we analyze the impact ofmarket parameters andthe investor’s risk aversion on the optimal policy, and how insur-ance affects the expected total discounted utility of consumption

B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167 163

Fig. 1. Optimal consumption to wealth ratio when α < 0.

(the value function). To conduct the economic analysis, we assumethat there are two regimes in the economy, like in Examples 4.1–4.3: regime 1 represents a bull market while regime 2 represents abear market.

5.1. Impact of market parameters and risk aversion on the optimalpolicy

According to the results obtained in Section 4, we write the op-timal proportion invested in the stock in a uniform expression

π∗

t =1

1 − α

µϵ(t) − rϵ(t)σ 2

ϵ(t), (8)

where α = 0 when U(y, i) = ln(y). During any given regime,the optimal investment proportion in the stock π∗ is constant, andonly depends on market parameters (expected excess return overvariance) and the investor’s risk aversion parameter α. The depen-dency of π∗ on market parameters is evident. Through empiricalresearch, French et al. (1987) find that the expected excess returnover variance is higher in good economy. Therefore, in a bull mar-ket, investors should invest a greater proportion of their wealth onthe stock.

Expression (8) shows that π∗ is inversely proportional to therelative risk aversion 1 − α, so low risk-averse investors (withgreater α) will invest a higher proportion of their wealth on thestock.

For all three cases, the optimal consumption rate process is pro-portional to thewealth process and such ratio κ(t) :=

c∗(t)X∗(t) is given

by

κ(t) =

δ, if U(y, i) = ln(y), α = 0;1

Aϵ(t), if U(y, i) = −yα, α < 0;

1Aϵ(t)

, if U(y, i) = yα, 0 < α < 1.

Since κ(t) is positive in all three cases, investors will consume pro-portionallymorewhen they becomewealthier. To examine the de-pendency of the optimal consumption towealth ratio κ(t) onα, weseparate our discussion into the following three cases: α = 0, α <0, and 0 < α < 1.

Formoderate risk-averse investors (α = 0), κ(t) is constant re-gardless of the market regimes, so moderate risk-averse investorsconsume the same proportion of their wealth in both bull and bearmarkets.

For high risk-averse investors (α < 0), their optimal consump-tion to wealth ratio is given by 1/Ai, i = 1, 2, where A can be ob-tained from the system (4). To find a numerical solution to the sys-tem (4), we set market parameters as µ1 = 0.2, µ2 = 0.15, r1 =

0.08, r2 = 0.03, σ1 = 0.25, σ2 = 0.6, θ1 = 0.15, θ2 = 0.25, η1 =

0.8, η2 = 1, λ1 = 0.1, λ2 = 0.2, Π1 = 6.04, Π2 = 6.4, andδ = 0.15 (for the convenience of citation thereafter, we denotethe choice of market parameters here as Parameter Set I). Noticethat these parameters satisfy the technical condition (5). We drawgraphs in Fig. 1 for the optimal consumption to wealth ratio when−1 < α < 0 and l = 0.3, l = 0.5, and l = 0.7. We see thatthe optimal consumption to wealth ratio is an increasing functionof α. Thus, the higher the risk tolerance, the higher the proportionof consumption over wealth. For the above parameter values, wefind 1/A1 > 1/A2, which can be seen in Fig. 1. Hence investorsshould allocate a higher proportion of their wealth to consump-tion in a bull market. For any chosen investor (fixed α), she/he willbehave more conservatively by reducing the proportion spent inconsumption when facing larger losses (greater l). This behaviorwas not noticed in Sotomayor and Cadenillas (2009), because theydid not incorporate an insurable loss in their model. Besides, froma mathematical point of view, the ratios all converge to 0.15 whenα approaches 0, which is exactly the same optimal consumption towealth ratio when α = 0 (δ = 0.15).

For low risk-averse investors (0 < α < 1), the optimal con-sumption to wealth ratio is given by 1/Ai, i = 1, 2, where 1/Ai canbe calculated from the system (6). We set market parameters tobe µ1 = 0.2, µ2 = 0.15, r1 = 0.15, r2 = 0.1, σ1 = 0.4, σ2 =

0.6, θ1 = 0.15, θ2 = 0.25, η1 = 0.8, η2 = 1, λ1 = 0.1, λ2 =

0.2, Π1 = 6.04, Π2 = 6.4, and δ = 0.2 (denoted as ParameterSet II). For these parameters’ values, the corresponding technicalcondition (7) is satisfied. Fig. 2 shows the optimal consumption towealth ratio when l = 0.3, 0.5, and 0.7. Similar to the previouscase, we also observe that the optimal consumption to wealth ra-tio is an increasing function ofα. However, contrary to the previouscase, we have 1/A1 < 1/A2 when 0 < α < 1. This means low risk-averse investors (0 < α < 1) spend a smaller proportion of theirwealth on consumption in a bull market. We notice that for verylow risk-averse investors (α close to 1), the optimal consumptionto wealth ratio is even greater than 1, meaning they finance con-sumption by borrowing.

By comparing all three cases, we conclude that investors withhigh risk tolerance (α is large) consume a large proportion of theirwealth in every market regime. However, investors’ consumptiondecision depends on the market regimes, and investors withdifferent risk aversion attitudes behave differently in bull and bearmarkets.

The optimal insurance for all three utility functions is de-ductible insurance and is given by

I∗t =

ηϵ(t) l − 1 + (1 + θϵ(t))

−1

1−α

+

X∗

t .

164 B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167

Fig. 2. Consumption to wealth ratio when 0 < α < 1.

We observe that, for each fixed regime, the optimal insurance isproportional to the investor’s wealth X∗. We note that it is optimalto buy insurance if and only if ηϵ(t) l − 1 + (1 + θϵ(t))

−1

1−α > 0, orequivalently if and only if

ηϵ(t)l > 1 − (1 + θϵ(t))−

11−α . (9)

Thus, it is optimal to buy insurance if and only if, relative to theother variables, ηϵ(t) is large, l is large, θϵ(t) is small, and α is small(we recall thatα ∈ (−∞, 1)). That is, it is optimal to buy insuranceif the insurable loss is large, the cost of insurance is low, and theinvestor is very risk averse. It is surprising that the variable λϵ(t)does not appear explicitly in this expression. Our explanation isthat λϵ(t) is implicitly incorporated in X∗

t , so λϵ(t) is important aswell to determine the optimal insurance.

If it is optimal to buy insurance, or equivalently the condition(9) is satisfied, then I∗t =

ηϵ(t) l − 1 + (1 + θϵ(t))

−1

1−α

X∗t . Thus,

as expected, the optimal insurance is proportional to ηϵ(t) and l.Furthermore,

∂ I∗t∂θϵ(t)

= −

1

1 − α(1 + θϵ(t))

−2−α1−α

X∗

t < 0,

∂2I∗t∂θ2

ϵ(t)=

2 − α

(1 − α)2(1 + θϵ(t))

2α−31−α

X∗

t > 0.

Hence, the optimal insurance is a decreasing and convex functionof θ . The decreasing property means that, as the premium loadingθ increases, it is optimal to reduce the purchase of insurance.The convexity indicates the amount of reduction in insurancedecreases as the premium loading rises.

In addition, if it is optimal to buy insurance (when (9) holds),then

∂ I∗t∂α

= −

1

(1 − α)2ln(1 + θϵ(t))(1 + θϵ(t))

−1

1−α

X∗

t < 0,

∂2I∗t∂α2

=

ln(1 + θϵ(t))

(1 − α)3

ln(1 + θϵ(t))

1 − α− 2

(1 + θϵ(t))

−1

1−α

X∗

t .

Hence, the optimal insurance is a decreasing function of α, whichimplies the higher the risk tolerance, the smaller the amount spenton insurance. We observe ∂2I∗t

∂α2 and ln(1+θϵ(t))

1−α− 2 have the same

sign. Recall that θ is the premium loading, which usually does notexceed 100%. So when α ≤ 0, we have ∂2I∗t

∂α2 < 0. This indicates thatfor high andmoderate risk-averse investors (α ≤ 0), the reductionin insurance is more significant when α is greater. If 0 < α < 1,we find that ∂2I∗t

∂α2 < 0 when α < α and ∂2I∗t∂α2 > 0 when α > α,

where α := 1 −12 ln(1 + θϵ(t)). So for low risk-averse investors

(0 < α < 1), the magnitude of reduction in insurance depends onthe risk aversion attitude.

5.2. Impact of insurance on the value function

In this subsection, we want to calculate the advantage ofbuying insurance for investors who face a random insurable risk.To achieve this objective, we first assume that some investorscannot access the insurance market. We then calculate the valuefunction with the constraint of no insurance, denoted by V1(x, i),and compare V1(x, i) with V (x, i) (the value function of theunconstrained Problem 2.1).

Under the constraint of no insurance, the dynamics of thewealth process X1 is given bydX1(t) =

rϵ(t)X1(t) + (µϵ(t) − rϵ(t))π(t)X1(t) − c(t)

dt

+ σϵ(t)π(t)X1(t)dW (t) − LtdN(t).Here, X1(0) = x and the insurable loss L(t) = ηϵ(t) l(t)X1(t).

We then formulate the constrained problem as follows.

Problem 5.1. Select an admissible policy u∗

1 := (π∗

1 , c∗

1 ) thatmaximizes the criterion function J , defined by (2). In addition, findthe value function

V1(x, i) := supu1∈A1

J(x, i; u1).

For every u1 = (π1, c1) ∈ A1, π1 and c1 need to satisfy all theconditions that π and c satisfy, where (π, c, I) ∈ Ax,i. For anyu1 = (π1, c1) ∈ A1, we have (π1, c1, I ≡ 0) ∈ Ax,i. Therefore,V (x, i) ≥ V1(x, i) for all x > 0 and i ∈ S. Theorem 5.1 of Zou andCadenillas (2014) is a verification theorem to solve Problem 5.1.

5.2.1. U(y) = ln(y), y > 0Under logarithmic utility, the value function to Problem 5.1 is

given by

v1(x, i) =1δln(δx) + ai,

where the constants ai satisfy the following linear system

riδ

+γi

δ+

λi

δΥi − 1 = δai −

j∈S

qijaj, (10)

with Υi defined by Υi := E[ln(1 − ηil)]. To compare the valuefunctions v and v1, we assume that there are two regimes (S = 2)in the economy. Thus,

ai =Πi(rj + γj − δ + λjΥj) + (δ + Πj)(ri + γi − δ + λiΥi)

δ2(δ + Π1 + Π2),

B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167 165

Fig. 3. Increase ratio of the value function when loss proportion is constant.

where i, j = 1, 2 and i = j. We then calculate

v(x, i) − v1(x, i) =Πiλj(Λj − Υj) + λi(δ + Πj)(Λi − Υi)

δ2(δ + Π1 + Π2), (11)

where i, j = 1, 2 and i = j.To facilitate our scenario analysis, we assume θ1

η1(1+θ1)≤

θ2η2(1+θ2)

and l is either constant or uniformly distributed on (0, 1).Case 1. First, we consider the case in which l is constant. In this

case, Υi = ln(1 − ηil), i = 1, 2.

(i) Let us consider first the situation in which optimal insuranceis no insurance for both regimes. From Example 4.1, we noticewhen the optimal insurance I∗ is no insurance, we have Λi =

ln(1 − ηil), i = 1, 2. Then, we obtain Λi − Υi = 0 for bothregimes. Hence v(x, i) = v1(x, i) for all x > 0 and i = 1, 2.

(ii) Now, let us consider the situation in which optimal insuranceis strictly positive in at least one regime. In this situation, wemust have at least one Λi in the form of Λi = − ln(1 + θi) −

ηil(1 + θi) + θi. Without loss of generality, we assume I∗ > 0in regime 1, or equivalently, η1l −

θ11+θ1

> 0. Then,

Λ1 − Υ1 = − ln(1 + θ1) − η1l(1 + θ1) + θ1 − ln(1 − η1l)> −η1l(1 + θ1) − ln(1 − η1l),

where the second inequality comes from− ln(1+θ1)+θ1 > 0.Considerw(l) := −η1l(1+θ1)−ln(1−η1l).We havew(0) = 0and

w′(l) =η1

(1 − η1l)(1 + θ1)

η1l −

θ1

1 + θ1

> 0.

This implies w(l) > 0 for all l ∈ (0, 1), and then Λ1 − Υ1 > 0.Togetherwith the result above, we can claim that Λ2−Υ2 ≥ 0.Hence, regardless of the optimal insurance I∗ in regime 2, wehave v(x, i) > v1(x, i) for both regimes according to (11).Even when I∗(x, i = 2, l) = 0, buying insurance in regime 1increases the value function in regime 2, which is a surprisingresult.

To further study the advantage of buying insurance, we definethe increase ratio of the value function by

m(x, i) :=

V (x, i) − V1(x, i)V1(x, i)

, i = 1, 2,

where V (x, i) and V1(x, i) are the value functions to Problems 2.1and 5.1, respectively. Without loss of generality, we assume x =

1δ

(such an assumption makes the constant 1δln(δx) be 0). Hence, we

have V (x, i) = v(x, i) = Ai and V1(x, i) = v1(x, i) = ai, i = 1, 2.Then we obtain for i = 1, 2m(x, i)

=Πiλj(Λj − Υj) + λi(δ + Πj)(Λi − Υi)

|Πi(rj + γj − δ + λjΥj) + (δ + Πj)(ri + γi − δ + λiΥi)|.

To analyze the impact of the insurable loss on the ratiom, we keepl as a variable and choose Parameter Set I but δ = 0.2. Notice thatfor the chosen parameters, our assumption θ1

η1(1+θ1)= 0.16 <

θ2η2(1+θ2)

= 0.2 is satisfied. Since we assume I∗ > 0 in regime 1,l ∈ (0.16, 1). We draw the graph of the increase ratio of the valuefunction in Fig. 3. As expected, the advantage of buying insuranceincreases when the insurable loss becomes larger in both regimes.But surprisingly, we find that buying insurance benefits investorsmore in a bull market, especially when the insurable loss is large.

Case 2. Second, we consider the case in which l is uniformlydistributed on (0, 1). In this case, Υi =

10 ln(1 − ηil)dl = (1 −

1ηi

)

ln(1 − ηi) − 1, i = 1, 2.(i) First, we consider the situation in which optimal insurance is

no insurance for both regimes. In this scenario, it is obvious thatΛi = Υi and then v(x, i) = v1(x, i), for all x > 0 and i = 1, 2.

(ii) Second, we consider the situation in which optimal insuranceis strictly positive in at least one regime. Again we assumeI∗ > 0 in regime 1. Then,

Λ1 − Υ1 =

1η1

− 1ln((1 + θ1)(1 − η1)) + 1

−(η1(1 + θ1) − θ1)

2+ 2θ1

2η1(1 + θ1).

Here Λ1 − Υ1 depends on the premium loading θ and lossintensity η in regime 1. To investigate such dependency, weconduct a numerical calculation. Notice that η1 must satisfythe condition η1 ≥

θ11+θ1

. We draw the difference Λ1 − Υ1

in Fig. 4 when θ1 = 0.01, 0.1, 0.2, 0.5, 0.8, 0.99. We observethat Λ1 − Υ1 is strictly positive and therefore v(x, i) > v1(x, i)for both regimes, which is consistent with our findings in theprevious case. Furthermore, as θ increases (which means thecost of insurance policy increases), the difference of Λi − Υibecomes smaller, so the benefit of purchasing insurance policydecreases accordingly. Investors gain more advantage frominsurance when the insurable loss becomes larger (that is, theloss intensity η increases).

166 B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167

Fig. 4. Λi − Υi .

Table 1Ai − ai when loss proportion l ∈ (

ν2η2

, 1).

α l v(x, 1) − v1(x, 1) v(x, 2) − v1(x, 2)

l = 0.30 7.7176 × 10−5 7.4304 × 10−5

−0.01 l = 0.60 9.8981 × 10−4 9.5315 × 10−4

l = 0.90 0.0041 0.0039

l = 0.15 0.0015 0.0015−0.5 l = 0.35 0.0797 0.0781

l = 0.50 0.3010 0.2961

l = 0.20 0.1675 0.1653−1 l = 0.30 0.8116 0.8036

l = 0.40 2.6969 2.6841

l = 0.08 0.2454 0.2441−2 l = 0.10 0.9421 0.9381

l = 0.12 2.2418 2.2344

5.2.2. U(y) = −yα, α < 0The value function to Problem 5.1 is given by

v1(x, i) = −a1−αi xα,

where the positive constants ai satisfyδ − αri −

α

1 − αγi + λi(1 − Υi)

a1−αi − (1 − α)a−α

i

=

j∈S

qija1−αj , (12)

with Υi := E(1 − ηil)α

. Comparing with the value function we

found in Section 4.2, we have v(x, i)− v1(x, i) = −(A1−αi − a1−α

i )x.We assume that there are two regimes in the economy and

the loss proportion l is constant. We skip the trivial case of I∗ ≡

0, in which v(x, i) = v1(x, i) in both regimes. We then carryout a numerical calculation to study the non-trivial case, that isI∗(x, i, l) > 0 in at least one regime.

To solve the systems (4) and (12) numerically, we chooseParameter Set I but δ = 0.25. For the chosen parameters, it ismore reasonable to consider the case when l ∈ (

ν2η2

, 1) (since bothν1η1

and ν2η2

are small). In Table 1 we calculate v(x, i) − v1(x, i) forvarious values of α (when calculating v(x, i) − v1(x, i), we takex = 1). The result clearly confirms that v(x, i) > v1(x, i) in bothregimes. We also observe that the advantage of buying insuranceis greater for investors with higher risk aversion. The size of theinsurable loss l affects the advantage of buying insurance as well.When the insurable loss increases (loss proportion l increases),buying insurance will give investors more advantage. We obtainv(1, 1)− v1(1, 1) > v(1, 2)− v1(1, 2), meaning buying insuranceis more advantageous in a bull market.

5.2.3. U(y) = yα, 0 < α < 1We find the corresponding value function to Problem 5.1 given

by

v1(x, i) = a1−αi xα,

where the constants ai satisfy the system (12) with 0 < α < 1.From the discussion in Section 4.3, we obtain v(x, i) − v1(x, i) =

(A1−αi − a1−α

i )xα . We then follow all the assumptions made inSection 5.2.2 including x = 1 and conduct a numerical analysisby choosing Parameter Set II. In this numerical example, we haveν1η1

≤ν2η2

< 1 when α ∈ (0, 0.8672], ν2η2

≤ν1η1

< 1, whenα ∈ (0.8672, 0.9132], and ν2

η2≤ 1 <

ν2η2

whenα ∈ (0.9132, 1).Weconsider the first scenario: ν1

η1≤

ν2η2

< 1 since it includes most lowrisk-averse investors. We are interested in the case of I∗ > 0 in atleast one regime. For the chosen parameters, we find ν1

η1is so small

that the case of l ∈ (0, ν1η1

) is rare. So we further assume constantloss proportion l ∈ (

ν1η1

,ν2η2

]. Notice thatwhen l ∈ (ν1η1

,ν2η2

], we haveI∗(x, 1, l) > 0 but I∗(x, 2, l) = 0.

From solving the non-linear systems (6) and (12), we drawv(x, i) − v1(x, i) for l = lM :=

12 (

ν1η1

+ν2η2

) and l = lm :=ν2η2

− 0.01in Fig. 5. It is obvious that v(1, i)− v1(1, i) > 0 in both regimes. Ashave seen in the previous cases, the benefit of buying insurancein a bull market strictly outperforms that in a bear market. Wealso observe the surprising result that the difference of the valuefunctions (advantage of buying insurance) is not an increasingfunction of α, which is different from the result in Section 5.2.2.But the difference is a concave function of α.

6. Conclusions

We have considered simultaneous optimal consumption, in-vestment and insurance problems in a regime switching modelwhich enables the regime of the economy to affect not only thefinancial but also the insurance market. A risk-averse investor fac-ing an insurable risk wants to obtain the optimal consumption, in-vestment and insurance policy that maximizes his expected totaldiscounted utility of consumption over an infinite time horizon.

We have obtained the first versions of verification theoremsfor simultaneous optimal consumption, investment and insuranceproblems when there is regime switching. We have also obtainedexplicitly the optimal policy and the value function when the util-ity function belongs to the HARA class.

The optimal proportion of wealth invested in the stock is con-stant in every regime and is greater in a bull market regardless

B. Zou, A. Cadenillas / Insurance: Mathematics and Economics 58 (2014) 159–167 167

Fig. 5. v(1, i) − v1(1, i) when l ∈ (ν1η1

,ν2η2

].

of the investor’s risk aversion attitude. We observe that investorswith high risk tolerance invest a large proportion of wealth in thestock.

The optimal consumption to wealth ratio is a strictly increasingfunction of the investor’s risk aversion parameter (α). Moderaterisk-averse investors (α = 0) consume at a constant proportionin both regimes. High risk-averse investors (α < 0) consume ahigher proportion of their wealth in a bull market. In contrast, lowrisk-averse investors (0 < α < 1) consume proportionally morein a bear market.

The optimal insurance is proportional to the investor’s wealthand such proportion depends on the premium loading θ and the in-vestor’s risk aversion parameter α. As the loading θ increases, thedemand for insurance decreases. This decrease of the demand forinsurance is more significant when θ is small. We observe that in-vestors who are very risk tolerant (that is, investors with large α)spend a small amount of wealth in insurance. For high and mod-erate risk-averse investors (α ≤ 0), the amount of reduction ininsurance is greater when α is far away from 0. However, low risk-averse investors (0 < α < 1) reduce the amount of insurance indifferent magnitudes that depend on the value of α.

We have also obtained the conditions under which it is optimalto buy insurance and analyzed their dependence on the differentparameters.

We have calculated the advantage of buying insurance. Basedon a comparative analysis, we find that the value function V (x, i)to Problem 2.1 is strictly greater than the value function V1(x, i)to Problem 5.1 when the optimal insurance is not equal to 0 in allregimes. We also observe that the advantage of buying insuranceis greater in a bull market. Investors who face a large random lossgain more benefits from purchasing insurance.

Acknowledgments

We are grateful to the referees for suggestions to improve thepaper. The work of B. Zou and A. Cadenillas was supported by the

Natural Science and Engineering Research Council of Canada Grant194137-2010. The results of this paper were presented at the 2ndIndustrial-Academic Workshop on Optimization in Finance andRisk Management, Fields Institute for Research in MathematicalSciences.

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