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Optimal Consumption, Investment andInsurance Problem in Infinite Time Horizon
Bin Zou∗and Abel Cadenillas†
Department of Mathematical and Statistical SciencesUniversity of Alberta
August 2013
Abstract
We incorporate an insurable risk into the classical consumptionand investment problem and intend to find optimal investment, con-sumption and insurance strategies for a risk-averse investor who wantsto maximize her expected utility of consumption in infinite time hori-zon. We obtain explicit optimal strategies for HARA utility functions.Through an economic analysis, we examine the impact of market pa-rameters and the investor’s risk aversion on optimal strategies andhow insurance affects the value function.
Key Words: utility of consumption; deductible insurance; stochasticcontrol; Hamilton-Jabobi-Bellman equation; economic analysis.
1 Introduction
The study of consumption and investment problems in continuous time be-gan with Merton’s seminal paper [5], in which Merton found explicit solutionsto this problem for the first time using classical stochastic control method.
∗Correspondence author: Bin Zou, 632 Central Academic Building, University of Al-berta, Edmonton T6G 2G1 Canada Email: [email protected] Phone: (+1)780-716-7987†Email: [email protected] Phone: (+1)780-492-0572
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Karatzas et al. [3] revisited the same problem in a more rigorous mathe-matical analysis, which overcame several errors in Merton [5], for instance,bankruptcy and infinite value function. Karatzas et al. [4] applied martingalerepresentation method to solve this problem in an incomplete market. Manyinitial contributions to consumption and investment problems can be foundin Sethi’s monograph [10]. Notice that there is only source of uncertainty(risk) in the classical consumption/investment model, namely, the stochasticmovements of stock prices. However, in real life, most investors also faceother random risks apart from the one arisen from investment. For instance,a working employee faces death risk and she can insure against such riskby purchasing life insurance. Therefore, a natural extension to the classicinvestment and consumption framework is to incorporate insurable risks in-to the model and then seek optimal investment, consumption and insurancestrategies.
Early research on optimal insurance problem did not take investmentand/or consumption into consideration and most of them worked in staticmodels. The pioneering work of optimal insurance problem originated fromArrow [1] in medical care. Under the assumptions that premium only de-pends on the expected value of insurance payout and a risk-averse insuredwants to maximize her expected utility, optimal insurance is a deductibleinsurance in a two-period discrete time model. Arrow [2] extended the re-sults to both state-dependent and state-independent utility functions andanalyzed the effect of various factors on optimal insurance, such as benefit-premium ratio and actual probability measure. Mossin [7] obtained similarresults as Arrow [1] and showed that full coverage insurance is never optimalif there is a positive loading in premium. Promislow and Young [9] reviewedoptimal insurance problem (without investment/consumption) and present-ed a general framework to obtain optimal insurance under various premiumprinciples, such as Wang’s premium principle, Swiss premium principle andso on.
For dynamic optimal insurance problem along with investment and con-sumption, please refer to Moore and Young [6] and the references there-in. They successfully solved optimal investment, consumption and insuranceproblem in finite time horizon and random time horizon and found closedform solutions for certain utility functions. Perera [8] studied this problemin a more general Levy processes market and obtained explicit solutions infinite time horizon for exponential utility function.
In this paper, we consider optimal investment, consumption and insurance
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problem for a risk-averse investor who wants to maximize her expected utilityof consumption in infinite time horizon. The main contributions of thispaper are to provide rigorous verification theorems associated with Hamilton-Jacobi-Bellman equations, find sufficient and necessary conditions for optimalinsurance to be no insurance or deductible insurance in dynamic case, andverify the conjectured functions and strategies are indeed the value functionsand optimal strategies.
The structure of this paper is organized as follows. In Section 2, we in-troduce the market model and formulate optimal investment, consumptionand insurance problem. In Section 3, we derive the Hamilton-Jacobi-Bellmanequation to our optimization problem and provide the corresponding verifi-cation theorems. In Section 4, we obtain explicit value functions and optimalstrategies when utility function is of HARA class. In Section 5, we conductan economic analysis to investigate how various factors affect optimal strate-gies and how insurance influences optimal investment and consumption. Ourconclusions are listed in Section 6.
2 The Model
We assume there are two assets in the market for investment. The priceprocess of riskless asset (Bond) S0 is driven by
dS0,t = rS0,tdt, with S0,0 = 1.
The price dynamics of risky asset (stock) S1 is given by
dS1,t = S1,t(µdt+ σdBt), with S1,0 > 0,
where Btt≥0 is a one-dimensional Brownian Motion defined in a standardstochastic basis (Ω,F ,P). We assume µ > r > 0 and σ > 0.
At time t, an investor chooses the proportion of wealth invested in therisky asset, denoted by π(t), and consumption rate c(t). Apart from therisk in trading assets, the investor also faces another insurable risk, whichis assumed to be proportional to the investor’s wealth, denoted by W . Wedenote the loss proportion at time t by L(t), then the insurable risk will beL(t)W (t). We assume L is left continuous in time, ∀ t ≥ 0, L(t) ∈ (0, 1)and the distribution of L(t) does not depend on t, or equivalently, ∀ t 6= s,
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L(t) and L(s) have the same distribution.1 We assume the occurrence of theinsurable risk is governed by a Poisson process N with constant intensityλ. In the market, there are insurance policies available to insure against therisk LW . The investor controls the payout of insurance policy at time tby choosing I(t). For example, if ∆Nt = 1, then the investor suffers a lossof amount LtWt but gets compensation It(LtWt) meanwhile, so the net losswill be LtWt − It(LtWt). Furthermore, we assume that to obtain insuranceprotection, the investor needs to pay premium continuously at the rate
P (t) = λ(1 + θ)E[It(LtWt)],
where θ is a positive constant, known as premium loading in the insuranceindustry. Such extra loading comes from insurance companies’ administrativecost, tax, and profit e.t.c.
We assume that Brownian Motion B, Poisson process N and loss propor-tion process L are mutually independent. We take the augmented filtrationgenerated by stochastic processes L, B and N as our filtration Ftt≥0.
Remark 2.1 The above continuous premium setting is called expected valueprinciple and has been commonly used in many research papers, for instance,[6] and [8]. Since the payout will only be triggered when ∆N = 1, so inan ideal world, premium P should satisfy P (t)∆t = E[It(LtWt)1∆Nt=1] =λ∆tE[It(LtWt)], where the second equality comes from the independence as-sumption of B, N and L.
For an investor with triplet strategies ut := (πt, ct, It), the correspondingwealth process W is given by
dWt =(rWt + (µ− r)πtWt − ct − λ(1 + θ)E[It(LtWt)]
)dt
+ σπtWtdBt − (LtWt − It(LtWt))dNt,(1)
with the initial condition W0 = x > 0.2
We define performance function by
J(x;u) := Ex
[∫ ∞0
e−δtU(ct)dt
],
1Due to such assumption, we may write L(t) as L when there is no confusion on timet.
2Although we do not use the notationWu, W is a controlled process and the dependenceof W on u is evident.
4
where Ex means taking conditional expectation given W0 = x, δ > 0 is thediscount factor and utility function U is C2(0,∞), strictly increasing andconcave, and satisfies the linear growth condition: ∃K > 0 such that
U(y) ≤ K(1 + y), ∀ y > 0.
Besides, we denote U(0) = limy→0+
U(y).
Define the bankruptcy time as
Θ := inft ≥ 0 : Wt ≤ 0
and impose Wt = 0 for all t ≥ Θ. Then ∀ t ≥ Θ, we can set ut = 0.With the help of Θ, we can rewrite the performance function as
J(w;u) = Ex
[∫ Θ
0
e−δtU(ct)dt
]+U(0)
δEx[e
−δΘ].
We denote A as the set of all admissible controls and define A as fol-lows: ∀ u = (π, c, I) ∈ A, u is progressively measurable with respect to thefiltration (Ft)t≥0 and satisfies the following conditions, ∀ t ≥ 0
Ex
[∫ t
0
π2sds
]< +∞,
Ex
[∫ t
0
csds
]< +∞ and ct ≥ 0,
Ex
[∫ Θ
0
e−δtU+(ct)dt
]< +∞,
It≥0 is left continuous on time t,
and It ∈ I := I : 0 ≤ I(y) ≤ y,∀y ≥ 0.
Problem 2.1 Find an admissible control u∗ ∈ A such that
V (x) := supu∈A
J(x;u) = J(x;u∗).
V is called the value function and u∗ optimal control or optimal policy.
5
3 Verification Theorems
∀u = (π, c, I) ∈ A0 := (−∞,∞) × [0,∞) × I and ψ : (0,+∞) → R inC2(0,+∞), define the operator Lu by
Lu(ψ(x)) :=(rx+ (µ− r)πx− c− λ(1 + θ)E[I(Lx)])ψ′(x)
+1
2σ2π2x2ψ′′(x)− δψ(x).
Theorem 3.1 Suppose U(0) is finite. Let v ∈ C2(0,∞) be an increasing and
concave function such that v(0) = U(0)δ
. If v satisfies the following Hamilton-Jacobi-Bellman equation
supu∈A0
Luv(x) + U(c) + λE[v(x− Lx+ I(Lx))− v(x)] = 0, (2)
for ∀w > 0 and u∗ = (π∗, c∗, I∗) defined by
u∗t := arg supu∈A0
(Lutv(W ∗
t ) +U(ct) +λE[v(W ∗t −LtW ∗
t + It(LtW∗t ))− v(W ∗
t )])
is admissible, then u∗ is optimal control to Problem 2.1 and v is the corre-sponding value function.
Proof. Define f(t, x) := e−δt(v(x)− U(0)δ
). Applying generalized Ito’s formulato f(t,Wt), we obtain
f(t,Wt) = f(0,W0) +
∫ t
0
e−δs(Lusv(Ws) + U(0))ds
+
∫ t
0
e−δsσπsWsv′(Ws) dBs +
∑0<s≤t
e−δs (v(Ws)− v(Ws−)) ,
where the last summation term can be rewritten as∑0<s≤t
e−δs (v(Ws)− v(Ws−)) =
∫ t
0
e−δsλ (v(Ws − LsWs + Is)− v(Ws)) ds
+
∫ t
0
e−δs (v(Ws− − LsWs− + Is)− v(Ws−)) dMs,
where Mt := Nt−λt is a martingale. Since on the given interval [0, t], Poissonprocess N has up to countable jump points, we can replace Ws− by Ws inthe Riemann integral.
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Let 0 < a < W0 = x < b < ∞ and define two stopping times τa :=inft ≥ 0 : Wt ≤ a and τb := inft ≥ 0 : Wt ≥ b. Take τ = τa ∧ τb, then τis also a stopping time.
In the interval [0, t ∧ τ ], both v′ and W are bounded, so ∀ s ∈ [0, t ∧ τ ],|e−δsv′(Ws)| ≤ M for some positive constant M and Ws ≤ b. Hence, by thedefinition of admissible control, we have
Ex
[∫ t∧τ
0
(e−δsσπsWsv′(Ws))
2ds
]≤M2b2σ2Ex
[∫ t∧τ
0
(πs)2ds
]<∞,
which implies
Ex
[∫ t∧τ
0
e−δsσπsWsv′(Ws)dBs
]= 0.
Furthermore, integrand e−δs (v(Ws− − LsWs− + Is)− v(Ws−)) is left con-tinuous and bounded in [0, t ∧ τ ], so by Theorem 11.4.5 in Shreve [11],
Ex
[∫ t∧τ
0
e−δs (v(Ws− − LsWs− + Is)− v(Ws−)) dMs
]= 0.
Therefore, by taking conditional expectation and applying the HJB equa-tion (2), we obtain
Ex[f(t ∧ τ,Wt∧τ )] ≤ v(x)− U(0)
δ− Ex
[∫ t∧τ
0
e−δs(U(cs)− U(0))ds
].
Let a ↓ 0 and b ↑ ∞, then τa → Θ and τb → ∞, so τ → Θ, andt ∧ τ → Θ when t → ∞. Along with the continuity of f , we get, whena ↓ 0, b ↑ ∞, t→∞
f(t ∧ τ,Wt∧τ )→ f(Θ, 0) = e−δΘ(v(0)− U(0)
δ
)= 0.
Taking limit for the above inequality yields
0 ≤ v(x)− U(0)
δ− Ex
[∫ Θ
0
e−δs(U(cs)− U(0))ds
].
Since utility function U(x) satisfies the linear growth condition, it follows
Ex
[∫ Θ
0
e−δs(U(cs)− U(0))ds
]≤ K1Ex
[∫ Θ
0
csds
]<∞, K1 > 0.
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Therefore, due to the finiteness of U(0), we have
v(x) ≥ Ex
[∫ Θ
0
e−δs(U(cs)− U(0))ds
]+U(0)
δ
= Ex
[∫ Θ
0
e−δsU(cs)ds
]+U(0)
δ(E[e−δΘ]− 1) +
U(0)
δ
= Ex
[∫ Θ
0
e−δsU(cs)ds
]+U(0)
δE[e−δΘ] = J(x;u), ∀x > 0, u ∈ A
and the equality will be achieved when u = u∗.
The previous verification theorem depends on the assumption that U(0) isfinite, while U(x) is a strictly increasing function and so U(0) 6= +∞. Thenext verification theorem deals with the case when U(0) = −∞.
Theorem 3.2 Suppose v(0) = U(0)δ
= −∞. Theorem 3.1 still holds.
Proof. Define g(t, x) := e−δtv(x). By following a similar argument, we obtain
Ex [g(t ∧ τ,Wt∧τ )] = v(w) + Ex
[∫ t∧τ
0
e−δsLusv(Ws)ds
]+ Ex
[∫ t∧τ
0
λe−δsE (v(Ws − LsWs − Is)− v(Ws)) ds
]≤ v(w)− Ex
[∫ t∧τ
0
e−δsU(cs)ds
].
Since we assume Ex
[∫ Θ
0e−δtU+(ct)dt
]< +∞ and U(x) satisfies the lin-
ear growth condition, we have Ex
[∫ t∧τ0
e−δsU(cs)ds]< +∞. So the above
inequality can be rearranged as
v(w) ≥ Ex [g(t ∧ τ,Wt∧τ )] + Ex
[∫ t∧τ
0
e−δsU(cs)ds
].
From the assumption that v is increasing on (0,∞) and v(0) = U(0)δ
, weget
Ex [g(t ∧ τ,Wt∧τ )] ≥ Ex
[e−δ(t∧τ)U(0)
δ
]= Ex
[∫ ∞t∧τ
e−δsU(0)ds
].
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By letting a ↓ 0, b ↑ ∞ and t → ∞ and applying the dominated conver-gence theorem and monotone convergence theorem, we obtain
v(w) ≥ Ex
[∫ ∞Θ
e−δsU(0)ds
]+ Ex
[∫ Θ
0
e−δsU(cs)ds
]= J(w;u),
and the equality holds when u = u∗.
4 Explicit Solutions for HARA Utility Func-
tion
In this section, we obtain explicit solutions to the HJB equation (2) for threespecific HARA utility functions. To such purpose, we first conjecture candi-dates for the value function and optimal control, then verify our conjecture.
We rewrite the HJB equation (2) as follows
rxv′(x) + maxπ
[(µ− r)πxv′(x) +1
2σ2π2x2v′′(x)] + max
c[U(c)− cv′(x)]
+ λmaxI
[E(v(x− Lx+ I(Lx)))− (1 + θ)E(I(Lx))v′(x)] = (δ + λ)v(x).
By the assumption that v′ > 0, v′′ < 0, we obtain candidates for optimalinvestment and consumption as follows
π∗ := arg sup[(µ− r)πxv′(x) +1
2σ2π2x2v′′(x)] = −(µ− r)v′(x)
σ2xv′′(x),
c∗ := arg sup[U(c)− cv′(x)] = (U ′)−1(v′(x)).
The following lemma and theorem provide the criteria for optimal insur-ance I∗.
Lemma 4.1
(a) I∗(y0) = 0, where y0 = l0x, l0 ∈ (0, 1) if and only if
(1 + θ)v′(x) ≥ v′(x− y0) = v′((1− l0)x). (3)
(b) 0 < I∗(y0) < y0, where y0 = l0x, l0 ∈ (0, 1) if and only if
(1 + θ)v′(x) = v′(x− y0 + I∗(y0)). (4)
9
Proof. We use the notation y0 := l0x for l0 ∈ (0, 1) in what follows. First,we show that I∗(y0) 6= y0, ∀ y0. Assume to the contrary that ∃ y0 such thatI∗(y0) = y0. We consider I(y) := I∗(y) − ε i(y), y = lx, l ∈ (0, 1), wherei(y) = 1 when l0 − η < l ≤ l0 + η and 0 otherwise. Here we take ε and ηsmall enough such that I ∈ I. Since I∗ is the maximizer, we have
E[v(x−Lx+I(Lx))−v(x−Lx+I∗(Lx))]−(1+θ)v′(x)E[I(Lx)−I∗(Lx)] ≤ 0.
By applying Taylor’s expansion and dividing by ε on both hand sides ofthe inequality, we shall get
(1 + θ)v′(x)E[i(y)] ≤ E[v′(x− Lx+ I∗(Lx))i(y)], when ε→ 0+.
Taking η → 0+ and using the mean value theorem for integration, weobtain
(1 + θ)v′(x) ≤ v′(x− y0 + I∗(y0)) = v′(x),
which does not hold since v′(x) > 0 and θ > 0.To prove the sufficient condition in case (a), we consider I ′(y) := I∗(y) +
ε i(y) where i(y) has the same definition as above. Then a similar analysisgives
(1 + θ)v′(x) ≥ v′(x− y0).
To prove the sufficient condition in case (b), notice that for small enoughε and η, both I(y) and I ′(y) are in the set of I, so we have (1 + θ)v′(x) ≤v′(x − y0 + I∗(y0)) and (1 + θ)v′(x) ≥ v′(x − y0 + I∗(y0)) at the same timeand then obtain equality.
To prove the necessary condition in case (a), we assume to the contrarythat I∗(y0) > 0. Then by the sufficient condition in case (b), we have
(1 + θ)v′(x) = v′(x− y0 + I∗(y0)) < v′(x− y0),
which is a contradiction to the given condition.The above argument also applies to proof of the necessary condition in
case (b).
Theorem 4.1
(a) Optimal insurance is no insurance I∗(Lx) = 0 when
(1 + θ)v′(x) ≥ v′((1− ess sup(L))x
). (5)
10
(b) Optimal insurance is deductible insurance I∗(Lx) = (Lx − d)+ whenthere exists d ∈ (0, x) such that
(1 + θ)v′(x) = v′(x− d). (6)
Proof. Case(a): Assume there exists y0 = l0x, l0 ∈ (0, 1) such that
0 < I∗(y0) < y0.
Then by Lemma 4.1, we have
(1 + θ)v′(x) = v′(x− y0 + I∗(y0)).
Define N := ω ∈ Ω : L(ω) > ess sup(L). It is evident that P (N) = 0.If l0 ≤ ess sup(L), we obtain (1− ess supL)x < x− y0 + I∗(y0) and then
v′((1− ess supL)x) > v′(x− y0 + I∗(y0)) = (1 + θ)v′(x),
which is a contradiction to the condition (5). Hence I∗(Lx) = 0 on set N c.Next we need to show that if two insurance policies I1 and I2 only differ
on a negligible set, then h(I1) = h(I2), where h(I) := E[v(x−Lx+I(Lx))]−(1+θ)v′(x)E[I(Lx)]. This result is evident since the integration of a boundedfunction over a negligible set is zero.
Therefore, I∗(Lx) = 0 almost surely.Case(b): First, by the strict concavity of v, if such d exists, it is also
unique.∀ 0 < l0 ≤ d
x< 1, we have
v′((1− l0)x) ≤ v′(x− d) = (1 + θ)v′(x).
Then by Lemma 4.1, I∗(l0x) = 0 = (l0x− d)+.∀ dx< l0 < 1, we have
v′((1− l0)x) > v′(x− d) = (1 + θ)v′(x),
which implies 0 < I∗(l0x) < l0.By Lemma 4.1, we get
(1 + θ)v′(x) = v′(x− l0x+ I∗(l0x)) = v′(x− d).
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Then recall the strict monotonicity of v, we obtain
I∗(l0x) = l0x− d = (l0x− d)+, and I∗(Lx) = (Lx− d)+.
To complete the proof, we need to show that either (5) or (6) will hold.Assume the condition (5) does not hold, then
v′(x) < (1 + θ)v′(x) < v′((1− ess supL)x) ≤ v′(0),
where v′(0) := limx→0+ v′(x). Since v′ is continuous and strictly decreasing
on [0, x], there exists a unique d ∈ (0, x) such that (1 + θ)v′(x) = v′(x− d).
4.1 U(y) = ln(y), y > 0
When U(y) = ln(y), ln(y) ≤ 1 + y, ∀ y > 0, so we can take K = 1 toguarantee the linear growth condition will be hold. In this case, a solutionto the HJB equation (2) is given by
v(x) =1
δln(δx) + A,
where A is constant.Differentiating v(x) yields v′(x) = 1
δxand v′′(x) = − 1
δx2. Then candidates
for optimal portfolio and consumption are given by
π∗ =µ− rσ2
and c∗ = δx.
Solving (1 + θ)v′(x) = v′(x − d) gives d = θ1+θ
x ∈ (0, x). Therefore, byTheorem 4.1, candidate for optimal insurance is
I∗(Lx) = (Lx− d)+ = (L− θ
1 + θ)+x.
By plugging candidate strategies into the HJB equation (2), we obtain
r
δ+
(µ− r)2
2δσ2− 1 +
λ
δΛ = δA,
where Λ := E[ln(1− L+ (L− θ1+θ
)+)]− (1 + θ)E(L− θ1+θ
)+.So constant A is determined by
A =1
δ2
[r − δ +
(µ− r)2
2σ2+ λΛ
]. (7)
12
Proposition 4.1 The function defined by
v(x) :=
1
δln(δx) + A, x > 0
−∞, x = 0
where A is given by (7), is the value function of Problem 2.1. Furthermore,the control given by
u∗t := (µ− rσ2
, δW ∗t , (Lt −
θ
1 + θ)+W ∗
t )
is optimal control to Problem 2.1.
Proof. By the definition of v, we have v(0) = U(0)δ
= −∞ and v(x) is strictlyincreasing and concave on (0,∞). By the construction of A, v satisfies theHJB equation (2). Therefore, by Theorem 3.2, v(x) is the value function ofProblem 2.1.
Under the control u∗ given above, the SDE (1) becomes
dW ∗t = (r − δ +
(µ− r)2
σ2)W ∗
t dt− λ(1 + θ)E[I∗t ]dt+µ− rσ
W ∗t dBt
− (Lt − (Lt −θ
1 + θ)+)W ∗
t dNt.
Since we cannot solve the above SDE explicitly, we then drop the premiumpart and consider an upper bound process Y of W ∗ with dynamics
dYtYt
= (r − δ +(µ− r)2
σ2)dt+
µ− rσ
dBt − d
N(t)∑i=1
(Lτi − (Lτi −θ
1 + θ)+)
,
where τi is the hitting time of the i-th jump of Poisson process N .Given Y0 = W ∗
0 = x, we solve to obtain Y
Yt = xe(r−δ+ (µ−r)2
2σ2)t+µ−r
σBt
N(t)∏i=1
(1− Lτi + (Lτi −
θ
1 + θ)+
).
13
Apparently, we have W ∗t ≤ Yt, ∀ t ≥ 0. So we obtain
Ex
[∫ t
0
c∗sds
]= δEx
[∫ t
0
W ∗s ds
]≤ δEx
[∫ t
0
Ysds
]= xδ
∫ t
0
e(r−δ+ (µ−r)2
σ2)sλsE(1− L+ (L− θ
1 + θ)+)ds
≤ xδλ
∫ t
0
s e(r−δ+ (µ−r)2
σ2)sds < +∞,
and
Ex
[∫ ∞0
e−δt ln+(c∗t )dt
]≤ Ex
[∫ ∞0
e−δt ln+(δYt)dt
]≤ Ex
[ ∫ ∞0
e−δt(| ln(δx)|+ |r − δ +
(µ− r)2
σ2|t+
µ− rδ|Bt|+
N(t)∑i=1
| ln(1− Lτi + (Lτi −θ
1 + θ)+)|
)dt].
Since B(t) is normally distributed with mean 0 and variance t, we get
Ex[|Bt|] =√
2tπ
. Due to the independence assumption of N and L, we have
Ex
N(t)∑i=1
| ln(1− Lτi + (Lτi −θ
1 + θ)+)|
= λtEx| ln(1− L+ (L− θ
1 + θ)+)|
≤ λt ln(1 + θ).
Therefore, we can claim
Ex
[∫ ∞0
e−δt ln+(c∗t )dt
]≤ 1
δ| ln(δw)|+ 1
δ2|r − δ +
(µ− r)2
σ2|
+
∫ ∞0
e−δt
(√2
π
µ− rσ
t12 + λ ln(1 + θ)t
)dt
= constant +µ− r√
2δ3σ+λ ln(1 + θ)
δ2< +∞.
Besides, it’s obvious that
Ex
[∫ t
0
σ2(π∗s)2ds
]=
(µ− r)2t
σ2< +∞,
14
and I∗ is left continuous in time satisfying
0 ≤ I∗t (Ltx) = (Lt −θ
1 + θ)+x ≤ Ltx.
Therefore, we conclude u∗t := (µ−rσ2 , δW
∗t , (Lt − θ
1+θ)+W ∗
t ) is indeed anadmissible control, and thus optimal control to Problem 2.1.
In Proposition 4.1, we obtain the value function v in explicit expressionwith A determined by (7), which depends on the distribution of L. In thenext two example, we calculate A in closed form.
4.1.1 Loss proportion L is a constant
If L = l ∈ (0, θ1+θ
], then
A =1
δ2
[r − δ +
(µ− r)2
2σ2+ λ ln(1− l)
].
If L = l ∈ ( θ1+θ
, 1), then
A =1
δ2
[r − δ +
(µ− r)2
2σ2− λ ln(1 + θ)− λl(1 + θ) + λθ
].
4.1.2 Loss proportion L is uniformly distributed on (0, 1)
In this example, a straightforward calculus gives
E[ln(1− L+ (L− θ
1 + θ)+)] = − θ
1 + θ,
E(L− θ
1 + θ)+ =
∫ 1
θ1+θ
(l − θ
1 + θ)dl =
1
2(1 + θ)2.
Therefore, constant A will be
A =1
δ2
[r − δ +
(µ− r)2
2σ2− λ(1 + 2θ)
2(1 + θ)
].
15
4.2 U(y) = −yα, α < 0
Similarly, we can take K = 1 regarding the linear growth condition whenU(y) = −yα. In this scenario, a solution to the HJB equation (2) has theform
v(x) = −A1−αxα, A > 0,
from which we obtain
v′(x) = −αA1−αx−(1−α) and v′′(x) = α(1− α)A1−αx−(2−α).
We then find candidates for optimal portfolio and consumption as follows
π∗ = −µ− rσ2x
v′(x)
v′′(x)=
µ− r(1− α)σ2
,
c∗ = (U ′)−1(v′) =x
A> 0.
Solving (1 + θ)v′(x) = v′(x−d) gives unique d = (1−γ)x ∈ (0, x), where
γ := (1+θ)−1
1−α ∈ (0, 1). Therefore, by Theorem 4.1, the uniform expressionof optimal insurance is
I∗(Lx) = (Lx− d)+ = (L+ γ − 1)+x.
From the HJB equation (2), we can solve to get A as
A =1− α
δ − αr − α(µ−r)22(1−α)σ2 + λ(1− Λ′)
, (8)
where Λ′ := E(1− L+ (L+ γ − 1)+)α − α(1 + θ)E(L+ γ − 1)+.Assume the cumulative distribution function of L is FL, then
1− Λ′ = 1−∫ 1−γ
0
(1− l)αdFL −∫ 1
1−γγαdFL + α(1 + θ)
∫ 1
1−γ(l + γ − 1)dFL
≥ 1− γαFL(1− γ)− γα(1− FL(1− γ)) + αγ(1 + θ)(1− FL(1− γ))
≥ 1− γα + αγ(1 + θ).
So in order to guarantee the existence of a strictly increasing and concavevalue function, we impose the following condition for the discount rate δ
δ > αr +α(µ− r)2
2(1− α)σ2− λ(1− γα + αγ(1 + θ)). (9)
Due to the technical condition (9) above, we have A > 0.
16
Proposition 4.2 The function defined by
v(x) :=
−A1−αxα, x > 0
−∞, x = 0
where A is given by (8), is the value function of Problem 2.1. Furthermore,the control u∗ given by
u∗t := (µ− r
(1− α)σ2,W ∗t
A, (Lt + γ − 1)+W ∗
t )
is optimal control to Problem 2.1.
Proof. By definition, v(0) = U(0)δ
= −∞ and v(x) is a strictly increasing and
concave function. By the construction of A, v satisfies the HJB equation (2).Therefore, by Theorem 3.2, v(x) is the value function of Problem 2.1.
Under the optimal control u∗, the corresponding wealth process is
dW ∗t = (r +
(µ− r)2
(1− α)σ2− 1
A)W ∗
t dt− λ(1 + θ)E[I∗t ]dt+µ− r
(1− α)σW ∗t dBt
− (Lt − (Lt + γ − 1)+)W ∗t dNt.
Similarly, we consider an upper bound process Z of W ∗ defined by
dZtZt
=
(r +
(µ− r)2
(1− α)σ2
)dt+
µ− r(1− α)σ
dBt.
Given Z0 = W ∗0 = x, the above SDE has a unique solution
Zt = x exp
((r +
(1− 2α)(µ− r)2
2(1− α)2σ2
)t+
µ− r(1− α)σ
Bt
).
In the next step, we verify candidate control is admissible.
Ex
[∫ t
0
c∗sds
]≤ 1
AEx
[∫ t
0
Zsds
]=x
A
∫ t
0
e
(r+
(µ−r)2
(1−α)σ2
)sds < +∞.
Ex
[∫ ∞0
e−δtU+(c∗t )dt
]= Ex
[∫ ∞0
e−δt(−(c∗tA
)α)+dt
]= 0.
17
Furthermore,
Ex
[∫ t
0
σ2(π∗s)2ds
]=
(µ− r)2t
(1− α)2σ2< +∞.
I∗ is left continuous in time and satisfies
0 ≤ I∗t (Ltx) = (Lt + γ − 1)+x ≤ Ltx.
So u∗t := ( µ−r(1−α)σ2 ,W
∗t /A, (Lt + γ − 1)+W ∗
t ) is an admissible control andthen optimal control to Problem 2.1.
4.2.1 Loss proportion L is a constant
If L = l ∈ [0, 1− γ], then
A =1− α
δ − αr − α(µ−r)22(1−α)σ2 + λ(1− (1− l)α)
.
If L = l ∈ (1− γ, 1], then
A =1− α
δ − αr − α(µ−r)22(1−α)σ2 + λ(1− γα + α(1 + θ)(l + γ − 1))
.
4.2.2 Loss proportion L is uniformly distributed on (0, 1)
In this example, we calculate
E(L+ γ − 1)+ =
∫ 1
1−γ(l + γ − 1)dl =
1
2γ2,
and
E(1− L+ (L+ γ − 1)+)α =
1 + αγ1+α
1 + αif α 6= −1,
1− ln(γ) if α = −1.
Therefore, we obtain when α 6= −1,
A =1− α
δ − αr − α(µ− r)2
2(1− α)σ2+ λ
[α
1 + α(1− γ1+α) +
1
2α(1 + θ)γ2
] ,and when α = −1,
A =1− α
δ − αr − α(µ− r)2
2(1− α)σ2+ λ
[ln(γ) +
1
2α(1 + θ)γ2
] .
18
4.3 U(y) = yα, 0 < α < 1
When U(y) = yα, the linear growth condition is satisfied for K = 1. We finda solution to the HJB equation (2) given by
v(x) = A1−αxα, A > 0.
Then we obtain candidates for optimal investment and consumption asfollows
π∗ =µ− r
(1− α)σ2> 0, and c∗ =
x
A> 0.
From the equation (1 + θ)v′(x) = v′(x − d), we can solve to obtain d =
(1− γ)x, where γ = (1 + θ)−1
1−α , so candidate for optimal insurance is
I∗(Lx) = (Lx− d)+ = (L+ γ − 1)+x.
From the HJB equation (2), we shall get A = A, where A is given by(8). Since α > 0 in this case, we have 1 − Λ′ > 0. Therefore, the technicalcondition we need here is
δ > αr +α(µ− r)2
2(1− α)σ2, (10)
which guarantees A > 0.
Proposition 4.3 The function defined by v(x) = A1−αxα, x ≥ 0, whereA = A, is the value function of Problem 2.1. Furthermore, the control u∗
given by
u∗t := (µ− r
(1− α)σ2,W ∗t
A, (Lt + γ − 1)+W ∗
t )
is the corresponding optimal control to Problem 2.1.
Proof. U(0) = 0 is finite and v(0) = U(0)δ
= 0. By definition, v is a strictlyincreasing and concave function. Besides, the construction of A ensures thatv satisfies the HJB equation (2). Therefore, according to Theorem 3.1, v(x)is the value function of Problem 2.1.
We use the same upper bound process Zt defined in the proof Proposition4.2 to verify candidate control is admissible. By following the same argument,we get
Ex
[∫ t
0
c∗sds
]<∞.
19
Besides,
Ex
[∫ ∞0
e−δtU+(c∗t )dt
]≤ Ex
[∫ ∞0
e−δt(ZtA
)αdt
]=xα
Aα
∫ ∞0
e−(δ−αr− α(µ−r)2
2(1−α)σ2
)tdt
=xα
Aα1
δ − αr − α(µ−r)22(1−α)σ2
<∞.
Furthermore,
Ex
[∫ t
0
σ2(π∗s)2ds
]=
(µ− r)2t
(1− α)2σ2< +∞
and left continuous I∗ also satisfies
0 ≤ I∗t (Ltx) = (Lt + γ − 1)+x ≤ Ltx.
Therefore, u∗t := ( µ−r(1−α)σ2 ,
W ∗t
A, (Lt + γ − 1)+W ∗
t ) is an optimal policy toProblem 2.1.
4.3.1 Loss proportion L is a constant
In this case, A has the exactly same formula as A in the previous section.
4.3.2 Loss proportion L is uniformly distributed on (0, 1)
In this example, we find A given by
A =1− α
δ − αr − α(µ−r)22(1−α)σ2 + λ[ α
1+α(1− γ1+α) + 1
2α(1 + θ)γ2]
.
5 Economical Analysis
In Section 5.1, we analyze how market parameters and investors’ risk aver-sion affect optimal investment, consumption and insurance strategies. InSection 5.2, we discuss the impact of insurance on the value function andconsumption.
20
5.1 Impact of market and risk aversion on optimal con-trol
According to the results obtained in Section 4, we can rewrite optimal in-vestment proportion π∗ in the following uniform expression
π∗t =µ− r
(1− α)σ2,
where α = 0 when U(y) = ln(y).From this expression, we can conclude that all investors will allocate more
proportion of their capital to the risky asset when economy is booming, inwhich the expected excess return over variance is higher than that wheneconomy is in recession. Since π∗ is positively related to the risk aversiontolerance α, investors with higher risk tolerance (less risk-averse investors)will spend proportionally more on the risky asset comparing with those withlower risk tolerance.
Optimal consumption rate c∗ is proportional to the wealth process W ∗ forall three utility functions and the proportion (consumption to wealth ratio)is given by
c∗(t)
W ∗t
= δ, when U(y) = ln(y);
c∗(t)
W ∗t
=1
A, when U(y) = −yα, α < 0;
c∗(t)
W ∗t
=1
A, when U(y) = yα, 0 < α < 1.
In order to analyze how risk aversion affects optimal consumption towealth ratio, we conduct a numerical analysis. We take µ = 0.07, σ = 0.3,r = 0.05, θ = 0.25, λ = 0.1, δ = 0.15. Since the expected value of a randomvariable uniformly distributed on (0, 1) is 0.5, so we consider two cases: lossproportion L is constant L = 0.5 and L is uniformly distributed on (0, 1).We find optimal ratio in those two cases almost coincides, so in both Figure 1and Figure 2, we only draw for the case of L = 0.5, but depict the differenceof optimal ratio between two cases.
From these two graphs, we conclude that optimal consumption to wealthratio has a positive correlation with α, so a less risk-averse investor will con-sume proportionally more compared with a more risk-averse investor. Math-ematically, when α approaches 0, we observe that the optimal consumption
21
Figure 1: Optimal consumption to wealth ratio when α < 0
Figure 2: Optimal consumption to wealth ratio when 0 < α < 1
to wealth ratio will converge to 0.15, which is δ (optimal consumption towealth ratio when α = 0).
For all three HARA utility functions considered in Section 4, optimalinsurance can be written as I∗t = (LtW
∗t − dt)
+ with deductible dt beingproportional to wealth W ∗
t . We have such proportion given as
κ :=dtW ∗t
= 1− (1 + θ)−1
1−α ,
where α = 0 for logarithm utility function U(y) = ln(y).
22
Notice that the deductible/wealth ratio is a positive constant, indepen-dent of time t, which implies the demand for insurance drops when the in-vestor’s wealth is increasing. The deductible/wealth ratio κ only depends onpremium loading θ and risk aversion α. Since
∂κ∂θ
=1
1− α(1 + θ)−
2−α1−α > 0,
the increase of premium loading θ will lead to the decrease of insurancedemand.
As∂κ∂α
= ln(1 + θ)(1− α)−2(1 + θ)−1
1−α > 0,
less risk-averse investors will spend less proportion of their wealth on insur-ance.
5.2 Impact of insurance on value function
Since optimal investment proportion π∗ is a constant, independent of wealth,and then independent of consumption and insurance, which is due to ourchoice of HARA utility functions. The performance function J we consider isthe expected value of consumption, so we can analyze how insurance affectsoptimal consumption by examining the impact of insurance on the valuefunction.
To conduct a comparison analysis, we impose a constraint of no insuranceto Problem 2.1. With this constraint, the wealth process is given by
dWt = (rWt + (µ− r)πtWt − ct)dt+ σπtWtdBt − LtWtdNt,
with W (0) = x.We then obtain the associated HJB equation as
supu∈(−∞,∞)×[0,∞)
Guv(x) + U(c) + λE[v(x− Lx)− v(x)] = 0,
where ∀ v ∈ C2(0,∞), the operator G is defined by
Gu(v(x)) := (rx+ (µ− r)πx− c)v′(x) +1
2σ2π2x2v′′(x)− δv(x).
23
5.2.1 U(y) = ln(y), y > 0
We find the corresponding value function is given by
v(x) =1
δln(δx) + A,
where
A =1
δ2
(r − δ +
(µ− r)2
2σ2+ λE[ln(1− L)]
).
If loss proportion L is constant and L = l ≤ θ1+θ
, then by the analysis inSection 4, we have I∗ = 0. So in this case, v(x) = v(x).
If loss proportion L is constant and L = l > θ1+θ
, define
g(l) := A− A =λ
δ2
(ln(
1
1 + θ) + θ − l(1 + θ)− ln(1− l)
).
Since g′(l) > 0 for l > θ1+θ
and g( θ1+θ
) = 0, we obtain A − A > 0 and thenv(x) > v(x).
If L is uniformly distributed on (0, 1), then A − A = λδ2
(1 − 1+2θ2(1+θ)
) > 0,
and so we have v(x) > v(x).
5.2.2 U(y) = yα, 0 < α < 1, y > 0
In this case, the value function to constrained problem is
v(x) = A1−αxα, A > 0,
where constant A is given by
A =1− α
δ − αr − α(µ−r)22(1−α)σ2 + λ(1− E(1− L)α)
.
Since 0 < α < 1, the condition (10) also guarantees that A > 0.If loss proportion L is constant and L = l ≤ 1 − γ, then I∗ = 0 and so
v(x) = v(x).If loss proportion L is constant and L = l > 1− γ, define
u(l) := α(1 + θ)(l + γ − 1) + (1− l)α − γα.
24
Then u′(l) = α[(1 + θ) − (1 − l)−(1−α)] < 0 for l > 1 − γ, which impliesu(l) < u(1 − γ) = 0, ∀ l > 1 − γ and thus A > A > 0. So we obtainv(x) > v(x).
If L is uniformly distributed on (0, 1), then[α
1 + α(1− γ1+α) +
1
2α(1 + θ)γ2
]−(
1− 1
1 + α
)= −α(1− α)
2(1 + α)(1 + θ)−
1+α1−α < 0,
which implies A > A and then v(x) > v(x).Based on the results obtained above, we can claim that investors with op-
tion to purchase insurance coverage will achieve a greater value function whenno insurance is not optimal. To examine how insurance affects consumption,we can define value function as V (t, w) = sup E[
∫∞te−δsU(cs)ds|Wt = w].
By following a similar argument, we can obtain v(t, x) ≥ v(t, x), where equal-ity only holds when I∗ = 0. Since utility function U is strictly increasing, weconclude c∗(t) > c∗(t) as long as I∗ 6= 0.
6 Conclusion
This paper considers a risk-averse investor who faces a random insurablerisk apart from the risk involved in the investment. The investor wants tomaximize her expected utility of consumption in infinite time horizon bychoosing investment proportion in the risky asset, consumption rate andinsurance policy.
We obtain the value function and optimal strategies in explicit formsfor three HARA utility functions. Through an economic analysis, we ob-serve that optimal investment proportion in the risky asset is constant forHARA utility functions, and will be greater when economy is booming. Lessrisk-averse investors will spend proportionally more on the risky asset com-pared with more risk-averse investors. Optimal consumption is proportionalto wealth, and we find such proportion is positively correlated with the in-vestor’s risk aversion parameter α, which means less risk-averse investorsspend a greater proportion of their wealth on consumption. We claim Ar-row’s finding for optimal insurance in [1] still holds in continuous time, thatis, optimal insurance is a deductible insurance. Furthermore, we conclude
25
that the increase of wealth, premium loading θ or risk aversion parameter αcan contribute to the decrease of insurance demand. According to a compar-ative analysis, we find having the option to purchase insurance will enableinvestors to achieve a greater value function, and then to spend more inconsumption.
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