Optimal investment for insurers
Christian Hipp
∗, Michael Plum
Economics Department and Department of Mathematics, University of Karlsruhe (TH), Postfach 6980, 7500 Karlsruhe 1, Germany Received June 1999; received in revised form February 2000; accepted April 2000
Abstract
We consider a risk process modelled as a compound Poisson process. The ruin probability of this risk process is minimized by the choice of a suitable investment strategy for a capital market index. The optimal strategy is computed using the Bellman equation. We prove the existence of a smooth solution and a verification theorem, and give explicit solutions in some cases with exponential claim size distribution, as well as numerical results in a case with Pareto claim size. For this last case, the optimal amount invested will not be bounded. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Stochastic control; Investment; Ruin probability
1. Introduction and summary
In this paper, stochastic control theory is applied to answer the following question: if an insurer has the possibility to invest part of his surplus into a risky asset, what is the optimal investment strategy to minimize ruin probability? This question has been considered by Browne (1995) for the case that the insurance business is modelled by a Brownian motion with drift, and the risky asset is modelled as a geometric Brownian motion. Without a budget constraint, he arrives at the following surprising result: the optimal strategy is the investment of a constant amount of money in the risky asset, irrespectively of the size of the surplus. We shall show that the answer is different (and more intuitive) for the case when insurance business is modelled by a compound Poisson process. Our optimal invested amountAt, t ≥0, at timethas the following properties: ifT (t ), t≥0, is the surplus process, then
• the amount of moneyAtis a functionA(T (t ))of the current surplus;
• A(0)=0, andA′(s)has a pole at 0;
• the functionA(s)remains bounded for exponential claim sizes, and it is unbounded for heavy-tailed claim size distributions.
The Bellman equation characterizing the value function and the optimal strategy is a second order nonlinear integro-differential equation. We show that this equation has a positive convex solutionV (s)withV (s)→1 when stends to infinity. Then the classical verification argument yields thatV (s)is the value function, and the minimizer A(s)defines the optimal investment strategy via the feedback equation
At =A(T (t−)).
∗Corresponding author.
E-mail address: [email protected] (C. Hipp).
We consider a compound Poisson processS(t ), t ≥0, with Poisson intensityλand claim size distributionQ, and model the risk process of an insurance company (or an insurance portfolio) by
dR(t )=cdt−dS(t ), R(0)=s.
Herecis the (positive) premium intensity for the portfolio. For this type of risk processes, it is usually assumed that there is a positive loading:
c > λµ(Q). (1.1)
Here, we shall not assume this. A possible risk measure for this business is the infinite time ruin probability
ψ0(s)=P{R(t ) <0 for somet ≥0}.
Furthermore, we consider a market index (e.g. a stock index)X(t )which is used for investment. This index is modelled by a geometric Brownian motion, as in the Black and Scholes world:
dX(t )=X(t )(adt+bdW (t )), X(0)=x,
with fixed known parametersa, b >0 and a standard Wiener processW (t ). At timet, the insurer will holdθ (t ) shares of the index, resulting in a technical result
dT (t )=dR(t )+θ (t )dX(t ), T (0)=s. Our aim is to minimize the ruin probability
ψ(s)=P{T (t ) <0 for somet ≥0},
over all possible admissible strategiesθ (t ). Only predictable strategies are admissible; this means in particular that the value of an admissible strategy at timetmay depend on the history of the processesX(u)andR(u)up to time t, but it may not depend on the size of a claim occurring at timet. We shall not consider a budget constraint here.
The result of Browne (1995) is the following. He considers a Brownian motion with drift as risk process:
dR(t )=µdt+σdV (t ),
whereV is the standard Wiener, independent ofW, with µ, σ > 0, and the above Black–Scholes type market index. The (unconstrained) optimal investment strategy is: invest a constant amountAof money into the index, i.e. θ (t ) = A/X(t ), without regarding the current reserveT (t ). With this investment strategy, the technical reserve T (t )satisfies
dT (t )=(µ+aA)dt+σdV (t )+bA dW (t ), T (0)=s, which, withψ (0)=1, yields
ψ(s)=exp
−2 µ+aA σ2+b2A2s
. (1.2)
Minimizing (1.2) givesAas the positive solution of ab2A2+2b2Aµ−aσ2=0.
Notice that in Browne’s model, fors=0, we haveψ(s)=1 for all possible strategies.
For a compound Poisson model for the claims process, the optimal investment strategy is completely different. If in this model we do not invest money into the index, i.e.θ (t )=0, thenψ(s)=ψ0(s)is the ruin probability in
the classical risk process for which in case (1.1)
ψ0(0)=
If for surpluss=0 there is a positive investment in the asset, then the risk in the asset will produce an immediate ruin and so
1=ψ (0) > ψ0(0),
which cannot be optimal. So in the compound Poisson risk process case, the form of the optimal investment strategy cannot be constant.
Consumers or supervisory authorities will use ruin probability as objective function. Shareholders would prefer other objective functions such as expected discounted dividends. Other optimization problems would be: optimiza-tion of reinsurance programs (see Hoejgaard and Taksar, 1998; Schmidli, 1999) or of new business (see Hipp and Taksar, 2000).
1.1. Bellman equation
The computation of the optimal investment strategy is based on the Bellman equation (see Fleming and Soner, 1993, p. 12, Eq. (5.3′)). This integro-differential equation forδ(s)=1−ψ(s)is derived as follows: consider a short interval [0,dt] of length dt in whichθshares are held. Then there will be a claim of sizeY ∼Qwith probability λdt, and ifY < s, then the risk process continues not leading to ruin with probabilityδ(s−Y ). If no claim occurs in the time interval, then there will be no ruin in the future with probability
δ(s+cdt+θxa dt+θxb dW (t )),
and this will happen with probability 1−λdt. Taking expectations, we obtain with Itô’s lemma δ(s)=δ(s)+ {λE[δ(s−Y )−δ(s)]+(c+aθ x)δ′(s)+12b2θ2x2δ′′(s)}dt, s >0. IntroducingA=θ xand maximizing, we obtain the Bellman equation for our problem:
sup
A {
λE[δ(s−Y )−δ(s)]+(c+aA)δ′(s)+12b2A2δ′′(s)} =0, s >0. (1.3)
The supremum exists wheneverδ′′(s) <0, and in this case
A=A(s)= −a b2
δ′(s)
δ′′(s). (1.4)
The functionδ(s)will satisfy (1.3) only ifδ(s)is strictly concave, strictly increasing, twice continuously differen-tiable, and satisfiesδ(s)→1 fors→ ∞. In this case, the optimal investment strategy will be
θ (t )=A(T (t−)) X(t−) ,
whereA(s)is the optimizer (1.4). In this argument, we assume thatδ(s)is a smooth function. As this is not an essential assumption, we shall start from a smooth solution of (1.3) and use the following verification theorem.
2. Verification theorem
Here, we shall show that from Bellman’s equation, we obtain a strategy which dominates all admissible strategies.
investment strategy for which the reserve processT (s, θ, t )is defined on 0 ≤ t < ∞, then the corresponding survival probabilityδ(s)satisfies
δ(s)≤δ∗(s), s≥0,
with equality forθ∗(t )=A∗(T (t−))/X(t ).
Proof. Assume thatδ′∗(s0)=0, ands0is the smallest positive value with this property. Then
0=sup
A {
λE[δ∗(s0−Y )−δ∗(s0)]+12A2δ′′∗(s0)},
which impliesδ∗′′(s0)≤0, and thus
E[δ∗(s0−Y )−δ∗(s0)]≥0;
but this contradictsδ∗(y) < δ∗(s0)for 0 < y < s0. Let nowθ (t )be an arbitrary admissible strategy. We write
T (s, θ, t )for the process with this investment strategyθ (t )and initial surplus s. Fix a positiveε to be chosen arbitrarily small later. Letτ∗andτ be the times of first ruin forT (s+ε, θ∗, t )andT (s, θ, t ), respectively, where θ∗(t )=A∗(T (s, θ∗, t−)). Letθ1(t )be the strategy
θ1(t )=θ (t )+
ε2 X(t ).
Writeτ1andδ1(s)for the corresponding ruin time and the survival probability using strategyθ1(t )when starting at
s+ε, respectively. We have
T (s+ε, θ1, t )=T (s, θ, t )+ε+aε2t+bε2W (t ),
P{τ1< τ} ≤P{ε+aε2t+bε2W (t ) <0 for somet} =exp
−2a
2
b2ε
.
Furthermore,
T (s+ε, θ1, t )→ ∞ on the set{τ = ∞}. (2.1)
We observe that
δ∗(T (s+ε, θ1, t∧τ1)) (2.2)
is a local supermartingale, while
δ∗(T (s+ε, θ∗, t∧τ∗)) (2.3)
is a local martingale. Sinceδ∗(s)is bounded and convergent fors→ ∞, the process (2.2) is a supermartingale, and (2.3) is a martingale. Therefore, for allt >0
δ∗(s+ε)≥Eδ∗(T (s+ε, θ1, t∧τ1)).
Now lett → ∞. Sinceδ∗(u)→1 foru→ ∞, we obtain with (2.1)
δ∗(s+ε)≥ lim
t→∞Eδ∗(T (s+ε, θ1, t∧τ1))1(τ=∞}=P{τ = ∞andτ1= ∞} ≥δ(s)−P{τ1< τ}
≥δ(s)−exp
−2a
2
b2ε
Nowε→0 yields δ∗(s)≥δ(s). On the other hand,
δ∗(s+ε)=Eδ∗(T (s+ε, θ∗, t∧τ∗)) which, witht tending to infinity, implies
δ∗(s+ε)≤P{τ∗= ∞}.
3. Existence of a solution
Ifδ(s)is a solution of (1.3), then for a constantα, the functionαδ(s)is a solution, too. Hence, we may fixδ(0) and replace the resulting functionδ(s)byαδ(s)if it is necessary to achieve the propertyδ(s) → 1 fors → ∞. FromA(0)=0, we deriveδ′(0)=δ(0)λ/c >0. We shall use the norming
δ′(0)=1.
Substituting the maximizingA(s), we obtain the equation
λ
Substitutingλandcbyλb2/a2and cb2/a2, respectively, and denoting these new constants with the same symbols λandc, we obtain the standard form
λ
we transform this equation into
δ′′(s)
Hence our Bellman equation is equivalent to the following problem foru=δ′:
u′(s)
Using the above transformationδ′=uand the conditionδ′(0)=δ(0)λ/c, we immediately obtain from Theorem 3.1, the following corollary.
Corollary 3.2. If Q has a locally bounded density, there exists a positive, strictly increasing and strictly concave solutionδ∈C2(0,∞)∩C1[0,∞)of the Bellman equation satisfying
Proof of Theorem 3.1. (a) First we prove the existence of a solutionuwith the asserted properties on [0, ε2] for someε >0. Via the transformation,v(s)=u(s2), we obtain the equivalent problem
v′(s)φ[v](s)=v(s)2, v(0)=1, (3.2)
is a closed subset. A lengthy (but elementary) calculation shows that the operatorT defined by
T (v)(s):=1+
is sufficiently small. Therefore, Banach’s Fixed-Point Theorem provides the existence of a fixed pointv ∈ Dε,M
ofT and thus, of a solution of (3.2) on [0, ε]. For reasons of continuity, we havev >0 andv′ <0 on [0, ε] after (possibly) further reduction ofε. Moreover, v(s) = 1−s/√c+o(s)ass → 0. Thus,u(s) = v(√s)has the corresponding properties on(0, ε2].
we have limk→∞ψ[u](sk)=0 and thus, usingu(b)=0, holds for alls∈(0, b+η]. The proof is by Banach’s Fixed-Point Theorem again, this time using the Banach space (C[b, b+η],k · k∞), the closed subset
Elementary calculations show that T mapsDη,ρ into itself and is a contraction (onDη,ρ) if bothη andρ are
sufficiently small. The unique fixed pointv ∈ Dη,ρ ofT obtained from Banach’s Fixed-Point Theorem provides
the desired unique extension ofu. Ifηis chosen sufficiently small, (3.3) holds on(0, b+η].
(d) To prove the existence of a solutionuon [0,∞)with the properties asserted in the theorem, letu(0)denote some solution on [0, ε2] provided by (a). Due to(u(0))′<0 and (3.1),u(0)satisfies (3.3) on(0, ε2]. Now denote the supremum property ofb∗. Thus,b∗= ∞, which proves the desired existence statement.
(e) To show the additional assertion, letHhave a finite integral over [0,∞), and letK:=R∞
0 H (s)ds. SinceH
is moreover a decreasing function, standard results provide
u∞>0, we obtain from (3.1) that limk→∞ψ[u](sk)= −∞, contradicting the fact thatψ[u]≥ −λK−c. Thus,
u∞=0.
To prove (ii), we estimate, assuming thatuandH are decreasing,
0<−ψ[u](s)≤λ
so that (i) and (3.4) yield lims→∞ψ[u](s)=0. From de l’Hospital’s rule and (3.5), we therefore obtain
lim
To show (iii), we estimate similarly
0<−√sψ[u](s)≤λ√s
4. Unboundedness ofA(s)A(s)A(s)
We now consider the process without claims: letZ(0)=s, and
Recall that by Kolmogorov’s inequality, forε >0,
P (U > ε)≤E
and so it will be violated ifY is heavy-tailed, e.g. ifY has a log-normal or a Pareto distribution.
5. More explicit solutions
5.1. Exponential claim sizes
Here, we shall consider the case of a claim size distribution which is exponential,Q=Exp(θ ). Changing the monetary unit, we can achieveθ =1. With
v(x)=u(x)ex, Eq. (3.1) is rewritten as
Dividing byv(x)gives us
(−λ+c−12)f2(x)−cf3(x)= 12(f′(x)−f (x)). (5.1)
Solving forf′(x)and introducing the notation
α= 1
For later reference, we notice that this yields the following differential equation forw(s)=1/f2(s): w′(s)=4c(1−γ1
with some constantk, or
right-hand side is never zero, so is the left-hand side, and hence
f (s) > γ1,
Fig. 1. Integral ofu(x)forc=0.5, . . . ,9.5.
Notice thatu(0)=1 andλδ(0)=cδ′(0)together imply
δ(s)=δ(0)
1+λ c
Z s
0
u(x)dx
,
which gives
δ(0)=
1+λ c
Z ∞
0
u(x)dx
−1
.
The values for (5.4) are given in Fig. 1 forcfrom 0.5 to 9.5.
5.2. Claims larger thans
Here, we consider the case that the surplussis smaller than any possible claim: P{Y ≤s} =0.
Then the Bellman equation reads
δ′′(s)(−λδ(s)+cδ′(s))= a
2
2b2(δ′(s)) 2.
Dividing both sides byδ(s)2, we obtain withw(s)=(λ−cδ′(s)/δ(s))2 −δ′′(s)
δ(s)
p
w(s)= a
2
2b2c2(λ−
p
Using
w′(s)= −2cpw(s)
δ′′(s)
δ(s) − 1 c2(λ−
p
w(s))2
we arrive at
w′(s)=
a2
b2c+
2 c
p
w(s)
(λ−pw(s))2. (5.5)
With our initial conditionw(0)=0, this differential equation can easily be solved numerically.
6. Numerical examples
Example 6.1. We consider the case of an exponential distribution Exp(1)forX, and the parameters areλ =1, c=2,a =b=1. In Fig. 2, we show the optimal strategyA(s)together with the survival probabilities with and without optimal investment. We first computed the function√w(s)=A(s)with Eq. (5.2), thenδ′(s)andδ(s)from
g(s)−δ(s)+2δ′(s)= −12δ′(s)A(s), g′(s)=g(s)−δ(s), g(0)=0, and δ(∞)=1. (6.1) The difference in survival probabilities seems to be small, but notice that, fors=9.43, we have a ruin probability of 0.001 with optimal investment, while the ruin probability is 0.008 without investment.
Fig. 3. Optimal investment for Pareto and continuous Pareto claims.
Example 6.2. We now consider a claim size distribution which is Pareto with the densities:
(a) h(x)= 3
x4, x >1, (b) h(x)= 6
5(x−1), 1< x <2, h(x)= 6
5(x−1)− 4, x
≥2.
The parameters arec=2,λ=a =b=1. Fig. 3 gives the optimal investmentA(s)for the two densities, where both curves coincide up tos=1, and the smooth curve corresponds to the smooth density. Here, the functionδ(s) was computed from
δ′′(s)(E[δ(s−Y )−δ(s)]+cδ′(s))=12(δ′(s))2
fors >1, which is a quite unstable algorithm since solving forδ′′(s)whensis large gives the ratio of two numbers close to zero. Fors≤1, we first solve the differential equation (5.5) and then use the relation
δ′(s)=1 c(λ−
p
w(s))δ(s).
According to Section 4,A(s)will go to infinity, but apparentlyA(s)may be decreasing for smalls.
References
Fleming, W.H., Soner, M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer, New York.
Hipp, C., Taksar, M., 2000. Stochastic control for optimal new business. Insurance: Mathematics and Economics 26, 185–192.
Hoejgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal, 166–180. Schmidli, H., 1999. Optimal proportional reinsurance policies in a dynamic setting. Research Report 403. Department of Theoretical Statistics,
