Insurance: Mathematics and Economics 27 2000 215–228
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 amount A
t
, t ≥ 0, at time t has the following properties: if T t, t ≥ 0, is the surplus process, then
• the amount of money A
t
is a function AT t of the current surplus; • A0 = 0, and A
′
s has a pole at 0; • the function As 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 solution V s with V s → 1 when
s tends to infinity. Then the classical verification argument yields that V s is the value function, and the minimizer As defines the optimal investment strategy via the feedback equation
A
t
= AT t−.
∗
Corresponding author. E-mail address: christian.hippwiwi.uni-karlsruhe.de C. Hipp.
0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 4 9 - 4
216 C. Hipp, M. Plum Insurance: Mathematics and Economics 27 2000 215–228
We consider a compound Poisson process St , t ≥ 0, with Poisson intensity λ and claim size distribution Q,
and model the risk process of an insurance company or an insurance portfolio by dRt
= c dt − dSt, R0
= s. Here c is the positive premium intensity for the portfolio. For this type of risk processes, it is usually assumed that
there is a positive loading: c λµQ.
1.1 Here, we shall not assume this. A possible risk measure for this business is the infinite time ruin probability
ψ s
= P {Rt 0 for some t ≥ 0}. Furthermore, we consider a market index e.g. a stock index Xt which is used for investment. This index is
modelled by a geometric Brownian motion, as in the Black and Scholes world: dXt
= Xta dt + b dW t, X0
= x, with fixed known parameters a, b 0 and a standard Wiener process W t . At time t , the insurer will hold θ t
shares of the index, resulting in a technical result dT t
= dRt + θt dXt, T 0
= s. Our aim is to minimize the ruin probability
ψs = P {T t 0 for some t ≥ 0},
over all possible admissible strategies θ t . Only predictable strategies are admissible; this means in particular that the value of an admissible strategy at time t may depend on the history of the processes Xu and Ru up to time
t , but it may not depend on the size of a claim occurring at time t . 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:
dRt = µ dt + σ dV t,
where V is the standard Wiener, independent of W , with µ, σ 0, and the above Black–Scholes type market index. The unconstrained optimal investment strategy is: invest a constant amount A of money into the index, i.e.
θ t = AXt, without regarding the current reserve T t. With this investment strategy, the technical reserve
T t satisfies dT t
= µ + aA dt + σ dV t + bA dW t, T 0
= s, which, with ψ0
= 1, yields ψs
= exp −2
µ + aA
σ
2
+ b
2
A
2
s .
1.2 Minimizing 1.2 gives A as the positive solution of
ab
2
A
2
+ 2b
2
Aµ − aσ
2
= 0. Notice that in Browne’s model, for s
= 0, we have ψs = 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, then ψs = ψ
s is the ruin probability in the classical risk process for which in case 1.1
ψ =
λµQ c
1.
C. Hipp, M. Plum Insurance: Mathematics and Economics 27 2000 215–228 217
If for surplus s = 0 there is a positive investment in the asset, then the risk in the asset will produce an immediate
ruin and so 1
= ψ0 ψ 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 for δs = 1 − ψs is derived as follows: consider a short
interval [0, dt ] of length dt in which θ shares are held. Then there will be a claim of size Y ∼ Q with probability
λ dt , and if Y s, then the risk process continues not leading to ruin with probability δs − Y . If no claim occurs
in the time interval, then there will be no ruin in the future with probability δs
+ c dt + θxa dt + θxb dW t, and this will happen with probability 1
− λ dt. Taking expectations, we obtain with Itô’s lemma δs
= δs + {λE[δs − Y − δs] + c + aθxδ
′
s +
1 2
b
2
θ
2
x
2
δ
′′
s } dt,
s 0. Introducing A
= θx and maximizing, we obtain the Bellman equation for our problem: sup
A
{λE[δs − Y − δs] + c + aAδ
′
s +
1 2
b
2
A
2
δ
′′
s } = 0,
s 0. 1.3
The supremum exists whenever δ
′′
s 0, and in this case A
= As = − a
b
2
δ
′
s δ
′′
s .
1.4 The function δs will satisfy 1.3 only if δs is strictly concave, strictly increasing, twice continuously differen-
tiable, and satisfies δs → 1 for s → ∞. In this case, the optimal investment strategy will be
θ t =
AT t −
Xt −
, where As is the optimizer 1.4. In this argument, we assume that δs 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
