Stochastic control for optimal new business
Christian Hipp
a,∗, Michael Taksar
baUniversity of Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany bSUNY at Stony Brook, Stony Brook, NY, USA
Received 1 June 1998; received in revised form 1 September 1999; accepted 24 November 1999
Dedicated to Etienne de Vylder
Abstract
Given an insurance portfolio, investment in new business is used to minimize the probability of technical ruin for the total position. This is a simple stochastic control problem for which solutions can be characterized and computed when the risk processes for old and new business are modelled by compound Poisson processes. © 2000 Elsevier Science B.V. All rights reserved.
JEL classification: C610, C690
Keywords: Stochastic control; New business; Ruin probability
1. Introduction and summary
Until recently, stochastic control theory and tools have been used in insurance for peculiar problems only (see Martin-Löf (1994) or Brockett and Xia (1995)). Apparently, in insurance there are many control variables which are adjusted dynamically, such as reinsurance, new business, or investment, but the use of stochastic control in this context seems to be rather new. It is the purpose of this paper to show that insurance offers a variety of stochastic control problems which are easily solved with standard control tools such as the Hamilton–Jacobi–Bellman equation. Easily solved means that optimal solutions can be characterized and computed, and smoothness of the value function can be shown. This paper does not provide new results in stochastic control theory, and its results are derived with elementary methods, without referring to the highly developed tool box of this field.
We shall consider the continuous time problem of optimal choice of new business to minimize infinite time ruin probability. The objective function is chosen for simplicity and for the purpose of illustration, other objective functions, such as expected discounted dividends, are possible and can be treated with essentially the same methods. For the expected discounted dividends case in which insurance business is modelled by a diffusion see, e.g. Asmussen and Taksar (1997) or Hoejgaard and Taksar (1998, 1999). In our problem, at each point of timet a proportionb(t )
between 0 and 1 of a certain insurance portfolio can be written, or the intensityb′(t )of acquisition or renewal can be
∗Corresponding author.
E-mail address: [email protected] (C. Hipp)
chosen, and this changes the dynamics of the risk business. The strategyb(t )is chosen predictable, i.e. it depends on all information available before timet. This means that if a claim occurs at timet thenb(t )may not depend on the size of the claim nor on the fact that a claim occurred at timet. Managing some fixed insurance portfolio, the insurer receives additional premia which are proportional tob(t ), and he pays additional claims which occur at a rate proportional tob(t ).
We consider the classical Lundberg process for insurance business (old business)
dR1(t )=c1dt− dS(t ), R1(0)=s,
whereS(t )is a compound Poisson process. So,
S(t )=X1+ · · · +XN (t ),
withN (t )a homogeneous Poisson process with constant intensityλ1. The processN (t )is independent of the claim
sizesX1, X2, . . . which are independent and identically distributed. The numberc1is a fixed premium intensity.
Then the classical infinite time ruin probability
ψ0(s)=P{R1(t ) <0 for somet≥0}
is based on the assumption that the insurer uses a static risk management strategy, i.e. he or she does not adjust risk management decisions in time such as the choice of a reinsurance program, of an investment strategy, or decisions on the volume of new business. For possible new business we consider a second insurance portfolio modelled again by a Lundberg processR2(t )with claims intensityλ2and premium intensityc2which is independent ofR1(t ). The claims inR2(t )areY1, Y2, . . . and have a common distribution for theY’s which will differ from the distribution of theX’s. For a given strategyb(t )for new business, the risk process of the insurer has the following dynamics:
In a short time interval fromttot+h:
• anX-claim occurs with probabilityλ1h+o(h);
• aY-claim occurs with probabilityλ2R0hb(t +u)du+o(h);
• no claims occurs with probability 1−λ1h−λ2R0hb(t+u)du+o(h);
• the amountc1h+c2R0hb(t+u)du+o(h)is received as premium income in the time interval.
Ifbis right continuous attthen
Z h
0
b(t+u)du=b(t )h+o(h).
For increasingb(t )one could also include provisions:
• the amountα(b(t+h)−b(t ))+o(h)is paid for provisions in the time interval.
Furthermore, the speedb′(t )for acquiring new business should be controlled. We shall not do this and try to solve the simpler problem without provision and without speed limit.
1.1. HJB equation for optimal new business
In the following we deal with functionsδ0(s),δ(s)andδn(s)which are zero fors <0 and which satisfy equations
holding fors≥0. Forδ0(s)=1−ψ0(s)the survival probability without new business we have
0=λ1E[δ0(s−X)−δ0(s)]+c1δ0′(s). (1.1) This follows by considering the two distinct cases:
• there is no claim in the interval [0,dt] which happens with probability 1−λ1dt, and we are left with a surplus of sizes+c1dt.
Averaging over all possible claim sizes we arrive at the equation
δ0(s)=λ1dt E[δ0(s−X)]+(1−λ1dt )δ0(s+c1dt )+o(dt ).
We see thatδ0(s)has a right derivativeδ0′(s), and we obtain Eq. (1.1). If the distribution ofXis continuous, then δ0(s)has a continuous derivative, and the above equation holds in the usual sense.
For the survival probability with new business we obtain in exactly the same way — withb =b(0)given and assuming thatb(t )is right continuous att =0 — the equation
0=λ1E[δ(s−X)−δ(s)]+λ2bE[δ(s−Y )−δ(s)]+(c1+c2b)δ′0(s),
or, more explicitly
δ(s)=δ(s)+dt{λ1E[δ(s−X)−δ(s)]+λ2bE[δ(s−Y )−δ(s)]+(c1+bc2)δ′(s)} +o(dt ).
An optimal choice for b (which might be seen as the proportion written in the interval [0,dt]) is obtained by
maximizing the bracket, where the set of possibleb’s must be restricted to 0≤b≤1:
0= sup
0≤b≤1
{λ1E[δ(s−X)−δ(s)]+λ2bE[δ(s−Y )−δ(s)]+(c1+bc2)δ′(s)}. (1.2)
This is the Hamilton–Jacobi–Bellman equation for our problem. An optimal strategy is derived from a solution
(δ(s), B(s))of this Hamilton–Jacobi–Bellman equation for all state variabless, whereB(s)is a measurable selection of a point at which the maximum is attained. Each such solution if any has the following properties:
λ1E[δ(s−X)−δ(s)]+λ2B(s)E[δ(s−Y )−δ(s)]+(c1+B(s)c2)δ′(s)=0, (1.3)
and for arbitrary values 0≤b≤1 we have
λ1E[δ(s−X)−δ(s)]+λ2bE[δ(s−Y )−δ(s)]+(c1+c2b)δ′(s)≤0. (1.4)
2. Computation of the optimal strategy
Since the right-hand side of (1.2) is a linear function ofb, the supremum is always attained at one of the extreme points of [0,1], namely at the point
B(s)=
1 ifλ2E[δ(s−Y )−δ(s)]+c2δ′(s)
0 ifλ2E[δ(s−Y )−δ(s)]+c2δ′(s) >0,
<0 (2.1)
(ifλ2E[δ(s−Y )−δ(s)]+c2δ′(s)=0, thenB(s)can be chosen in an arbitrary way). Using this we can eliminate the sup from the HJB equation: either
0=λ1E[δ(s−X)−δ(s)]+λ2E[δ(s−Y )−δ(s)]+(c1+c2)δ′(s) (2.2)
or
0=λ1E[δ(s−X)−δ(s)]+c1δ′(s). (2.3)
So (1.2) is equivalent to the equation
δ′(s)=min
g1(s, δ)+g2(s, δ)
c1+c2 ,
g1(s, δ) c1
g1(s, δ)=λ1E[δ(s)−δ(s−X)], g2(s, δ)=λ2E[δ(s)−δ(s−Y )].
This equation has a smooth solution:
Proposition 2.1. If the distributions of X and Y are continuous, then Eq. (2.4) has a smooth solutionδ(s)with δ(s)→1 fors→ ∞. The solutionδ(s)is continuously differentiable and nondecreasing.
Proof. Define a sequenceδn(s)viaδ0(s)the ruin probability without new business forn=0, and by the recursion
δn′+1(s)=min
Furthermore, ifδn(s)is continuous in all pointss6=0 then — by continuity of the distributions ofXandY — the
functionδ′n+1(s)is continuous. Therefore the sequenceδn′(s)converges to a nonnegative continuous functiong(s), the sequenceδn(s)converges to the nondecreasing functionu(s)=δ0(0)+R0sg(t )dt, souis differentiable with
derivativeu′(s)=g(s). This implies thatuis a solution of (2.4), and the norming
δ(s)= u(s)
u(∞)
yields the solution in the proposition.
The strategy maximizing survival probability will beb∗(t )=B(R∗(t−)), whereR∗(t )is the risk process resulting from this strategyb∗(t ). We shall now show that this strategy is indeed optimal.
Theorem 2.2. If the distributions of X and Y are continuous, then there exists a strategy, namelyb∗(t ), for new
business which maximizes survival probability.
Proof. Above we have seen that the HJB equation admits a smooth solution (δ(s), B(s)), i.e.δ(s)is continuously differentiable on [0,∞),B(s)given by (2.1) is a measurable selection of a value at which the supremum is attained, and the functionδ(s)satisfies
0≤δ(s)≤1, and lim
s→∞δ(s)=1.
We shall use the two relations (1.3) forB(s)and (1.4) for an arbitrary actionb. Letδ(s)ˆ be the survival probability using an arbitrary predictable strategy 0 ≤ ˆb(t ) ≤ 1, andδ∗(s)the survival probability with strategy b∗(t ) =
B(R∗(t−)), whereR(t )ˆ andR∗(t )are the surplus processes resulting from strategiesb(t )ˆ andb∗(t ). Letτ∗andτˆ
be the ruin times for the processesR∗(t )andR(t )ˆ , respectively. Consider the stochastic processes
ˆ
V (t )=δ(R(tˆ ∧ ˆτ )), t ≥0.
The dynamics of the two processesR∗(t )andR(t )ˆ are as follows: in the small interval [t, t+h]
• the processR∗(t )has no jump with probability
1−λ1h−λ2
ForR(t )ˆ we just have to replace *-objects by ˆ-objects. This implies that (a) the processV∗(t )is a martingale according to (1.3), while (b)V (t )ˆ is a supermartingale according to (1.4).
(a) Lets≥0 be such that
λ2E[δ(s−Y )−δ(s)]+c2δ′(s)=0.
Then for arbitrary accumulation pointAof
1
(2.5) is an immediate consequence of (1.3). (b) For each accumulation pointAof
which yields the supermartingale property. We have to consider the stopped processes since the two relations (1.3) and (1.4) hold fors≥0 only. Using (a) and (b) we obtain that fort≥0
EV (t )ˆ ≤ ˆV (0)=δ(s)=V∗(0)=EV∗(t ). (2.6)
On{τ∗= ∞}we haveR∗(t )→ ∞, and therefore
V∗(t )→1{τ∗=∞}.
By dominated convergence this implies
lim
t→∞EV ∗(t )=
P{τ∗= ∞} =δ∗(s).
Similarly,
ˆ
δ(s)=P{ ˆτ = ∞} = lim
t→∞EV (t ).ˆ
With (2.6) we obtain
ˆ
δ(s)≤δ∗(s),
and this optimal survival probability is attained using strategyb∗(t )=B(R∗(t−)). This implies that the strategy
b∗(t )is optimal in the class of all predictable strategiesb(t )ˆ which are bounded by 0 and 1.
Notice that for most HJB equations the value function has to be convex and smooth. This is not obvious for value functions which are ruin probabilities: the classical ruin probabilities with discrete claim size distributions are neither convex nor smooth, they are not differentiable. Also our optimal ruin probabilities with new business are not necessarily convex.
The qualitative behaviour of the optimal strategy is best visible at the points = 0, i.e. when the surplus has dropped to zero and the insurer is very close to ruin. The choiceB(0)=0 orB(0)=1 depends on the value of
δ′(0)computed with one of the two equations:
0=λ1δ(0)+λ2δ(0)−(c1+c2)δ′(0),
0=λ1δ(0)−c1δ′(0).
We haveB(0)=1 iff the first equation leads to a smallerδ′(0), i.e. iff
λ1δ(0) c1
> λ1δ(0)+λ2δ(0) c1+c2
,
i.e.
λ1 c1
> λ2 c2 .
This means that close to ruin new business is written irrespectively of the mean claim size of new business. Even nonprofitable business (i.e.λ2EY> c2)will be written in order to collect premia, and this money will be used to
pay the next claim. If the company survives, then at some large surplussthe (possibly nonprofitable) new business has to be sold (B(s)=0), and this will not be possible in real life.
2.1. Optimal new business without selling
which are bounded by 0 and 1. This problem is harder than the above problem, a characterization of the optimal strategyb(t )via a Hamilton–Jacobi–Bellman equation is not straightforward. For the sake of simplicity we assume that
Xhas a positive densityf (x).
For numerical computation it is sufficient to consider strategiesb(t )which take a finite number of values. To make the presentation simple we assume for the moment thatb(t )may take the two values 0 and 1 only. Letδ(s)be the maximal survival probability based on strategiesb(t )which are nondecreasing and take values 0 and 1. Then
δ(s)≥δ1(s), s≥0,
whereδ1(s)is the classical ruin probability with strategyb(t )≡1. Fixs0 >0 and assume that the optimal value δ(s0)is attained by a strategyb(t )(depending ons0) withb(0)=1. Then this strategy equalsb(t )≡1, and
δ(s0)=δ1(s0).
With stochastic calculus we obtain that
λ1E[δ(s0−X)−δ(s0)]+λ2E[δ(s0−Y )−δ(s0)]+(c1+c2)δ′(s0)=0,
λ1E[δ1(s0−X)−δ1(s0)]+λ2E[δ1(s0−Y )−δ1(s0)]+(c1+c2)δ1′(s)=0,
and hence
0=λ1E[δ1(s0−X)−δ(s0−X)]+λ2E[δ1(s0−Y )−δ(s0−Y )]+(c1+c2)(δ1′(s0)−δ′(s0)).
Sinceδ′(s0)≥δ1′(s0)(for otherwiseδ(s) < δ1(s)for somes > s0close tos0) we have
δ1(s)=δ(s), s≤s0,
which implies that for alls≤s0the optimal survival probabilityδ(s)is produced with the strategyb(t )≡1. Notice
that we may assume right continuity of all strategiesb(t )here by monotonicity, which implies the existence of right derivatives forδ(s).
Using similar arguments, we can compute the optimal strategy for our discretized problem as follows: For given discretization step size1we consider possible valuesw =0, 1,21, . . . ,1 for strategiesb(t )and compute the maximal survival probabilityδ(s, w)over all nondecreasing strategiesb(t )taking values in the set{0, 1,21, . . . ,1}
withb(0) ≥ w. We first compute the survival probabilitiesδ1(s)andδ0(s) for which b(t ) ≡ 1 or b(t ) ≡ 0, respectively, which are classical ruin probabilities. We start withδ(s,1)=δ1(s). We then computeδ(s,1−1)and its optimal strategyb(t,1−1)by complete search over the set of strategies of the following form:
b(t )=
1 as soon asR(t−)≤s0,
1−1 untilR(t−)≤s0,
withs0≥0. The functionsδ(s, k1)and their optimal strategiesb(t, k1)are derived recursively by complete search in the set
b(t )=
b(t, (k+1)1) as soon asR(t−)≤s0, k1 untilR(t−)≤s0, s0≥0.
Fig. 1. New business with and without selling.
3. Numerical example
We consider an exponential claim size distribution for both, new and old business. We takeλ1=λ2=1,c1=2, c2 = 8, and the means of XandY are 1 and 103, respectively. Fig. 1 shows the survival probabilities with new
business(b(t )=1)and without(b(t )=0), the optimal survival probability with selling, and the optimal survival probability without selling. With selling, the optimal strategy hasB(s)=1,s <3.06, andB(s)=0 fors≥3.06. Without selling the optimal strategy is again bang–bang, it isB(s)=1 up tos=0.5, andB(s)=0 for largers.
References
Asmussen, S., Taksar, M., 1997. Controlled diffusion models for optimal dividend payout. Insurance: Mathematics and Economics 20, 1–15. Brockett, P., Xia, X., 1995. Operations research in insurance: a review. Transactions of the Society of Actuaries XLVII, 7–80.
Hoejgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal 81, 166–180. Hoejgaard, B., Taksar, M., 1999. Controlling risk exposure and dividents payout schemes: insurance company example. Mathematical Finance,
9, 153–182.
