Optimal Dividends under a Ruin Probability Constraint - University of ...

This equation has a unique positive solution for t (see Gerber and Shiu(1998)), and for the claim amount distributions considered in the next sectionthere is at least one negative solution. In this equation we can think of δ asthe Laplace transform parameter in φ, but we can also think of δ as a forceof interest, and we will use this interpretation in the next section.We now introduce the dividends modification. We assume that when thesurplus process is above level b, the insurer pays dividends to shareholders atrate ĉ. Specifically, we assume in Section 2 that c − ĉ > λµ, as this conditionguarantees that ruin for our modified surplus process does not occur withprobability one. Gerber and Shiu (2005) have considered this modified surplusprocess and, by solving differential equations, have derived expressionsfor the expected present value of dividend income to shareholders when claimshave an exponential or a mixed exponential distribution. Lin and Pavlova(2005) also consider this model and define and study the Gerber-Shiu function.In the next section we will provide an alternative method, based onprobabilistic arguments, of finding explicit solutions for the expected presentvalue of dividend payments to shareholders. We illustrate ideas using exponentialand mixed exponential claim amount distributions, but the techniquecan be applied to other distributions as well.In much of the literature on dividend problems, the focus is on maximisingthe expected present value of dividend payments to shareholders. See, forexample, Gerber and Shiu (2004), Dickson and Waters (2004) and referencestherein. In Section 3, we retain this focus, but impose a constraint. Inparticular, we consider maximising the expected present value of dividendpayments to shareholders subject to a prescribed level of ruin probability. Wepresent some numerical illustrations and make comparisons with situationsin which there is no such constraint. The idea of a constraint is not new.Paulsen (2003) considered a finite time ruin probability constraint while usinga diffusion process to model the insurer’s surplus. His constraint applied fromthe time at which dividends first became payable, whereas we shall apply ourconstraint at time 0.2 DividendsIn what follows we assume that dividends are payable at rate ĉ when thesurplus level is above b, with c − ĉ > λµ, and that if ruin occurs, no furtherdividends are payable. We will adapt the notation of the previous section byintroducing a hat to indicate that the insurer’s net rate of premium incomeis c − ĉ, rather than c. Thus, for example, ˆT u denotes the time of ruin for aclassical surplus process with initial surplus u and premium rate c − ĉ.3

Let V (u, b) denote the expected present value of dividend payments atforce of interest δ. For 0 ≤ u ≤ b, dividends will be payable only if the surplusprocess reaches b without ruin first occurring, so thatV (u, b) = E[exp{−δT u,b }]V (b, b)where T u,b is the time of the first upcrossing of the surplus process throughb from u without ruin occurring. See Gerber (1979, p.147).For u ≥ b, dividends are payable immediately at rate ĉ until the first timethe surplus falls below b (an event which may not occur). As this time isidentical in distribution to ˆT u−b , we can write]V (u, b) = ĉE[ā ˆTu−b+In particular,which gives∫ ∞0∫ be −δt ŵ(u − b, y, t)V (b − y, b)dydt0∫ ∞= ĉ (]) ∫ b1 − E[e −δ ˆT u−b+ e −δt ŵ(u − b, y, t)V (b − y, b)dydtδ00= ĉ (])1 − E[e −δ ˆT u−bδ∫ ∞ ∫ b+V (b, b) e −δt ŵ(u − b, y, t)E[exp{−δT b−y,b }]dydt. (2)V (b, b) = ĉ (1 − EδV (b, b) =0+V (b, b)0[e −δ ˆT 0])∫ ∞0∫ be −δt ŵ(0, y, t)E[exp{−δT b−y,b }]dydt0(])(ĉ/δ) 1 − E[e −δ ˆT 01 − ∫ ∞e ∫ 0 −δt bŵ(0, y, t)E[exp{−δT 0 b−y,b}]dydt .Hence, for 0 ≤ u ≤ b,(])(ĉ/δ) 1 − E[e −δ ˆT 0E[exp{−δT u,b }]V (u, b) =1 − ∫ ∞e ∫ 0 −δt bŵ(0, y, t)E[exp{−δT 0 b−y,b}]dydt . (3)At first sight, it appears that in order to apply formulae (2) and (3) we needto know the functional form of ŵ(u, y, t). However, as we shall see in thesubsequent analysis, this need not be the case.4

Thenso that∫ b0e αy m(y)dy =∫ b0e αy ( (α + ρ)e ρy − (α − R)e −Ry) dy= e (α+ρ)b − e (α−R)b ,L(b) = (α + ρ)e ρb − (α − R)e −Rb −( ) ( )= ρ + ˆR e ρb + R − ˆRe −Rb( ) (eα − ˆRρb − e −Rb)and henceV (u, b) = ĉ ˆRδα(α + ρ)e ρu − (α − R)e( ) ( ) −Ru.ρ + ˆR e ρb + R − ˆR e −RbConsider now the case when u ≥ b. Writing ϕ(u−b) = Eterm in formula (2) involvesϕ(u − b) =∫ ∞0e −δt ŵ(u − b, t)dt = ϕ(0)e − ˆR(u−b) .[e −δ ˆT u−b], the firstSee, for example, Gerber and Shiu (1998, equation (3.16)). Then for u ≥ b,V (u, b) = ĉ (])1 − E[e −δ ˆT u−bδ∫ ∞∫ b+ e −δt m(b − y)ŵ(u − b, t)dt f(y) V (b, b)dy00 m(b)= ĉ(b, b)(1 − ϕ(u − b)) + ϕ(u − b)Vδ m(b) α ( e ρb − e −Rb) .In particular,givesso thatV (b, b) = ĉ(b, b)(1 − ϕ(0)) + ϕ(0)Vδ m(b) α ( e ρb − e −Rb)V (b, b)m(b) α ( e ρb − e −Rb) = 1 (V (b, b) − ĉ )ϕ(0)δ (1 − ϕ(0)) ,V (u, b) = ĉϕ(u − b)(1 − ϕ(u − b)) +δ ϕ(0)= ĉδ()1 − e − ˆR(u−b)+ e − ˆR(u−b) V (b, b).6(V (b, b) − ĉδ (1 − ϕ(0)) )

For 0 ≤ u ≤ b, the value of b that maximises V (u, b) is the value of b thatminimises L(b). Thus∂(∂b L(b) = ρ + ˆR) (ρe ρb − R − ˆR)Re −Rband setting this equal to 0 gives the optimal level, b ∗ , as(b ∗ = 1 R − ˆR)Rρ + R log (ρ + ˆR) . (5)ρWe remark that the formulae for V (u, b) and b ∗ can be found in Gerber andShiu (2005).Although the exponential distribution is the most straightforward claimamount distribution to consider, one advantage it offers is that it is the basisof De Vylder’s (1978) approximation, and Højgaard (2002) has shownthat this approximation can successfully be applied to an optimal dividendsproblem. As the solution to the problem considered in Section 3 is considerablyeasier to deal with for exponential claims than for other claim amountdistributions, De Vylder’s approximation has a certain appeal.2.2 Mixed exponential claimsLet us now consider the case when the claim amount distribution is a mixtureof two exponentials withf(x) = pαe −αx + qβe −βxwhere p > 0 and p + q = 1.Following the arguments in Gerber (1979, pp.147-148) we find that, as inthe previous case,E[exp{−δT u,b }] = m(u)m(b)where we now havem(u)m(b) = ̟0(δ, b)e ρu + ̟1(δ, b)e −R 1u + ̟2(δ, b)e −R 2uwhere ρ > 0, −R 1 < 0 and −R 2 < 0 (which all depend on δ) are the solutionsof Lundberg’s fundamental equation (1), i.e.λ + δ − ct = λpαα + t + λqββ + t .7

The coefficients {̟i(δ, b)} 2 i=0 satisfy the conditions1 = ̟0(δ, b)e ρb + ̟1(δ, b)e −R1b + ̟2(δ, b)e −R2b ,0 = ̟0(δ, b)α + ρ + ̟1(δ, b)+ ̟2(δ, b),α − R 1 α − R 20 = ̟0(δ, b)β + ρ + ̟1(δ, b)β − R 1+ ̟2(δ, b)β − R 2,where the final two conditions are just a special case of equation (A9) ofGerber and Shiu (2005).Considering first the case when 0 ≤ u ≤ b, in the numerator of formula(3) for V (u, b) we have]E[e −δ ˆT 0= λ ( pc − ĉ α + ˆρ +q )β + ˆρ(see Dickson (2005, p.188)) where ˆρ > 0 is the positive solution of Lundberg’sfundamental equation with premium rate c − ĉ. To find the denominator offormula (3) for V (u, b) we shall find the double integral by treating it as aspecial case of the double integral in formula (2). The starting point is thefunction φ, defined in Section 1. From Gerber and Shiu (1998), this functionsatisfies the equationwherecφ ′ (u) = (δ + λ)φ(u) − λω(u) =∫ ∞0∫ u0(6)φ(x)f(u − x)dx − λω(u) (7)e −sy f(u + y)dy= pαα + s e−αu +qββ + s e−βu .By applying standard arguments (see Gerber (1979, p.117)) we find thatφ(u) = k 0 (δ, s)e ρu + k 1 (δ, s)e −R 1u + k 2 (δ, s)e −R 2u . (8)As ρ > 0 and lim u→∞ φ(u) = 0, we must have that k 0 (δ, s) = 0. Next, wecan obtain k 1 (δ, s) and k 2 (δ, s) by inserting expression (8) for φ into equation(7). We obtain the equalities1α + s1β + s= k 1(δ, s)α − R 1+ k 2(δ, s)α − R 2,= k 1(δ, s)β − R 1+ k 2(δ, s)β − R 2,8

which givek 1 (δ, s) =αγ 1 (δ)α + s + γ β2(δ)β + s ,k 2 (δ, s) =ασ 1 (δ)α + s + σ β2(δ)β + s ,whereγ 1 (δ) = (α − R 1)(α − R 2 )(β − R 1 ),α(R 2 − R 1 )(α − β)γ 2 (δ) = − (α − R 1)(β − R 1 )(β − R 2 ),β(R 2 − R 1 )(α − β)σ 1 (δ) = − (α − R 1)(α − R 2 )(β − R 2 ),α(R 2 − R 1 )(α − β)σ 2 (δ) = (α − R 2)(β − R 1 )(β − R 2 ).β(R 2 − R 1 )(α − β)Thusφ(u) = ( γ 1 (δ)e −R 1u + σ 1 (δ)e −R 2u ) αα + s+ ( γ 2 (δ)e −R 1u + σ 2 (δ)e −R 2u ) ββ + s , (9)so that for u ≥ 0,where for i = 1, 2,w(u, y, t) = η 1 (u, t)αe −αy + η 2 (u, t)βe −βy∫ ∞0e −δt η i (u, t)dt = γ i (δ)e −R 1u + σ i (δ)e −R 2u .In the context of finding the expected present value of dividend payments,we need to evaluate∫ ∞0∫ be −δt ŵ(u − b, y, t)m(b − y)dydt09

for u ≥ b. This can be written as==∫ ∞0∫ ∞+0∫ ∞0∫ be −δt (ˆη1 (u − b, t)αe −αy + ˆη 2 (u − b, t)βe −βy) m(b − y)dydt0e −δtˆη 1 (u − b, t)dt∫ be −δtˆη 2 (u − b, t)dt0∫ bαe −αy m(b − y)dy0βe −βy m(b − y)dy() ∫ bˆγ 1 (δ)e − ˆR 1 (u−b) + ˆσ 1 (δ)e − ˆR 2 (u−b)() ∫ b+ ˆγ 2 (δ)e − ˆR 1 (u−b) + ˆσ 2 (δ)e − ˆR 2 (u−b)0αe −αy m(b − y)dy0βe −βy m(b − y)dy.The integrals in the above expression are straightforward to evaluate as m isa linear combination of three exponential functions.Then for 0 ≤ u ≤ b, we have all the components required to find]V (u, b).For u ≥ b, we require an additional component, namely E[e −δ ˆT u−b. This iseasily obtained from the above workings by noting thatφ(u)| s=0= E [exp{−δT u }I(T u < ∞)] ,so that by formulae (4) and (9), we haveE[exp{−δT u }] = (γ 1 (δ) + γ 2 (δ)) e −R 1u + (σ 1 (δ) + σ 2 (δ)) e −R 2u .We remark that, unlike in the case of exponential claims, it does not seempossible to obtain a closed form solution for the level b ∗ that maximises theexpected present value of dividend payments for given values of u and ĉ.However, it is possible to find this level numerically.2.3 An alternative approachIn the previous subsection, we derived expressions for V (u, b) both for0 ≤ u ≤ b and u ≥ b. If we are interested only in the former case, there isan alternative, more straightforward approach to finding the double integralin formula (3).From Gerber and Shiu (1998, equation (2.6)) we know thatφ(0) = λ c∫ ∞∫ ∞e −ρt e −s(y−t) f(y)dydt0t10

where ρ > 0 is the unique positive solution of Lundberg’s fundamental equation(1), and hence depends on δ. Consider a more general mixed exponentialdistribution than in the previous subsection, sayf(x) =n∑w i α i exp{−α i x}i=1where {w i } n i=1are positive weights which sum to 1. Thenφ(0) = λ c= λ c= λ cn∑∫ ∞ ∫ ∞w i α i e −ρt e −s(y−t) e −αiy dydti=10n∑∫ ∞w i α ii=1n∑α i 1w is + αi=1i0te −(ρ−s)t e−(s+α i)tdts + α iρ + α i.Thus, we havewhere for i = 1, 2, ..., n,w(0, y, t) =n∑η i (t)α i exp{−α i y}i=1∫ ∞0e −δt η i (t)dt = λ cw iρ + α i.In the case when n = 2, we thus obtain slightly simpler coefficients thanin the previous subsection, as we haveγ 1 (δ) + σ 1 (δ) = (α − R 1)(α − R 2 )α(α − β)= λ cpα + ρandγ 2 (δ) + σ 2 (δ) = − (β − R 1)(β − R 2 )β(α − β)= λ cqβ + ρ .We remark that these last two equalities also lead to formula (6) (allowingfor the different premium rate).3 Dividends under a constraintIn this section we turn our attention to the expected present value of dividendpayments subject to a ruin probability constraint. We motivate the problem11

in the following way. Suppose that the insurer is not paying dividends andhas an initial surplus which gives a ruin probability below a regulated level(e.g. the regulated level is 1%, and the insurer’s initial surplus, u, is suchthat ψ(u) = 0.005). Then the insurer is in a position to pay dividends toshareholders and still satisfy the regulations on ruin probability. For example,for u such that ψ(u) = 0.005, there is a set of pairs of dividend rate andthreshold that result in a ruin probability of 0.01. Our objective is then tofind the optimal pair, where optimal means that the expected present valueof dividend payments is maximised, subject to the constraint on the ruinprobability. We will use the notation ψ(u, b) to denote the ultimate ruinprobability when the dividend threshold is b.Our aim here is not to derive explicit solutions for optimal levels. Indeed,it is not clear that this is possible, even in the simplest situation (exponentialclaims). Rather we wish to investigate the effect of the constraint, and so allresults presented in this section have been obtained numerically, using thesoftware Mathematica. Ruin probabilities when dividends are payable havebeen calculated using results given in Dickson (1991).As a first illustration, let us consider the case when the claim amountdistribution is exponential with mean 1. We set λ = 1 and δ = 0.001, so thatwe can think of this as a force of interest of 10% per annum with 100 claimsexpected per annum. Now suppose that the premium is calculated with a20% loading, and that u = 30.7 so that ψ(u) = 0.005 (using the well-knownformula for ψ in this case). Let us further suppose that the insurer is subjectto a regulated ruin probability of ε = 0.01.In this situation, we find that the optimal combination of b and ĉ suchthat ψ(u, b) = 0.01 is b # = 49.1 and ĉ # = 0.1912, giving V (u, b # ) = 153.76.Let us compare this with two other scenarios, under each of which the insureris not subject to a ruin probability constraint. First, suppose that the insurerselects the optimal level of dividends, ĉ # = 0.1912. Then, by formula (5),the optimal threshold is b ∗ = 27.11, giving V (u, b ∗ ) = 166.46 and ψ(u, b ∗ ) =0.167. Second, suppose that the insurer simply elects to maximise V (u, b)subject to the constraint c−ĉ ≥ λµ. Then the optimal threshold is ¯b = 27.96,the optimal dividend rate is ¯c = 0.2, V (u, ¯b) = 170.5, and ψ(u, ¯b) = 1.Looking at these numbers, we can see that in the first case, for the samerate of dividend payment, the relaxation of the ruin probability constrainthas produced about an 8% increase in the expected present value of dividendpayments, but the ruin probability has increased by a factor of almost 17.The second change results in about an 11% increase in the expected presentvalue of dividend payments, while the ruin probability has increased by afactor of 100.Table 1 shows results for a range of scenarios, labelled (A) to (G). In12

Scenario u ¯b ¯c V (u, ¯b) ψ(u, ¯b)(A) 57.23 26.82 0.1 87.54 1(B) 30.70 27.96 0.2 170.50 1(C) 21.82 25.48 0.3 263.99 1(D) 49.61 26.82 0.1 84.20 1(E) 49.61 26.82 0.1 84.20 1(F) 57.23 15.01 0.1 46.39 1(G) 57.23 9.24 0.1 31.88 1Table 3: Levels under the constraint c − ĉ ≥ λµ.of u, there is no relationship between the threshold and initial surplus asthere is in Table 1. In each scenario the expected present value of dividendpayments is greater than in Table 1, since b ∗ is the optimal threshold.In Table 3 we see that the optimal dividend level is always equal to thepremium loading factor, which results in certain ruin. The expected presentvalue of dividend payments is greater under each scenario than in Table 2,and hence under Table 1.Thus, the changes as we move from Table 1 to Table 3 result in bothincreased ruin probabilities and increased expected present values of dividendpayments. Scenario (C) is perhaps the most interesting. Here we see thatthe expected present values in Tables 2 and 3 are respectively 3% and 4.2%greater than in Table 1, while the ruin probabilities have increased by factorsof 9.9 and 100. It is interesting that relaxing the ruin probability constraintbenefits the shareholders so little in this scenario.As a second illustration, we consider the situation when the claim amountdistribution ( is a mixed exponential distribution with f(x) = 2 3 (2e−2x ) +1 1 e−x/2) so that µ = 1. Table 4 shows values for the same scenarios as in3 2Table 1, and we observe a similar pattern as in Table 1. However, in this casethe calculations are not as straightforward as those for Table 1. In particular,the solutions of Lundberg’s fundamental equation do not exist in a neat closedform as they do when the claim amount distribution is exponential. Table 5shows the values we obtain when we use De Vylder’s approximation to thesurplus process, resulting from the same simple calculations that produceTable 1. These approximations are remarkably good, suggesting that incases where explicit solutions for V (u, b) exist, at the very least the use ofDe Vylder’s approximation provides an excellent indication of the levels ofthe optimal pair (b # , ĉ # ) and V (u, b # ).14

Scenario c δ u ψ(u) ε V (u, b # ) b # ĉ #(A) 1.1 0.001 87.29 0.005 0.01 51.87 111.77 0.0812(B) 1.2 0.001 47.49 0.005 0.01 143.28 72.31 0.1870(C) 1.3 0.001 34.17 0.005 0.01 241.42 57.04 0.2897(D) 1.1 0.001 75.61 0.01 0.025 55.10 93.07 0.0813(E) 1.1 0.001 75.61 0.01 0.05 63.90 76.34 0.0815(F) 1.1 0.002 87.29 0.005 0.01 21.14 100.01 0.0682(G) 1.1 0.003 87.29 0.005 0.01 12.77 92.22 0.0575Table 4:claims.Optimal levels and expected present values, mixed exponentialScenario c δ u ψ(u) ε V (u, b # ) b # ĉ #(A) 1.1 0.001 87.25 0.005 0.01 51.88 111.72 0.0812(B) 1.2 0.001 47.41 0.005 0.01 143.32 72.20 0.1870(C) 1.3 0.001 34.07 0.005 0.01 241.52 56.85 0.2898(D) 1.1 0.001 75.58 0.01 0.025 55.11 93.04 0.0813(E) 1.1 0.001 75.58 0.01 0.05 63.90 76.32 0.0815(F) 1.1 0.002 87.25 0.005 0.01 21.14 99.98 0.0683(G) 1.1 0.003 87.25 0.005 0.01 12.77 92.20 0.0575Table 5: De Vylder approximations to values in Table 4.4 Concluding remarksAlthough our analysis in Section 2 involved the defective density w(u, y, t)we did not need to find its explicit form, although we did identify its generalform for the examples considered. It is possible to identify the explicit formof w(u, y, t) for certain claim amount distributions, particularly when u = 0,and this will be discussed in a forthcoming manuscript.References[1] De Vylder, F. (1978) A practical solution to the problem of ultimate ruinprobability. Scandinavian Actuarial Journal, 114-119.[2] Dickson, D.C.M. (1991) The probability of ultimate ruin with a variablepremium loading - a special case. Scandinavian Actuarial Journal, 75-86.[3] Dickson, D.C.M. (2005) Insurance Risk and Ruin. Cambridge UniversityPress, Cambridge.15

[4] Dickson, D.C.M. and Waters, H.R. (2004) Some optimal dividends problems.ASTIN Bulletin 34, 49-74.[5] Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S.Huebner Foundation, Philadelphia, PA.[6] Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. NorthAmerican Actuarial Journal 2, 1, 48-72.[7] Gerber, H. U. and Shiu, E. S. W. (2004) Optimal dividends: analysiswith Brownian motion. North American Actuarial Journal 8, 1, 1-20.[8] Gerber, H.U. and Shiu, E.S.W. (2005) On optimal dividend strategies inthe compound Poisson model. Unpublished manuscript.[9] Højgaard, B. (2002) Optimal dynamic premium control in non-life insurance.Maximising dividend pay-outs. Scandinavian Actuarial Journal,225-245.[10] Lin, X.S. and Pavlova, K. (2005) The compound Poisson risk model witha threshold dividend strategy. Unpublished manuscript.[11] Paulsen, J. (2003) Optimal dividend payouts for diffusions with solvencyconstraints. Finance and Stochastics 7, 457-473.David C M DicksonCentre for Actuarial StudiesDepartment of EconomicsUniversity of MelbourneVictoria 3010Australiaemail: dcmd@unimelb.edu.auSteve DrekicDepartment of Statistics andActuarial ScienceUniversity of WaterlooWaterloo, OntarioCanada N2L 3G1email: sdrekic@math.uwaterloo.ca16