Ruin probability and optimal dividend policy for models with investment

Transcription

1 Ruin probability optimal dividend policy for models with investment PhD Thesis Martin Hunting Department of Mathematics University of Bergen i

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3 Abstract In most countries the authorities impose capital requirements on insurance companies in order to avoid the adverse consequences to society when insurance companies default on claims. Since holding capital is costly, this naturally leads to the problem of deciding how large the risk reserve needs to be, or what is a safe level of liquidity. A common answer is that the probability that the insurance company will default on policyholder claims should not be higher than a certain small level ɛ. An implementation of this policy requires reasonably accurate methods for determining this probability, known as the ruin probability. Rigorous mathematical treatments of the ruin probability problem can be traced at least as far back as the acclaimed doctoral thesis of Filip Lundberg from 193 with the title Approximerad framställning af sannolikhetsfunktionen. Traditionally the focus has been on ruin probability on an infinite time horizon. In these models an insurance company can avoid ruin by allowing its risk reserve to grow toward infinity. At the 15th International Congress of Actuaries in 1957 Bruno de Finetti criticized this approach. In particular he couldn t see why an older company should hold more capital than a younger one bearing similar risks, only because it is older. As an alternative de Finetti formulated what is known as the de Finetti s dividend problem : Maximizing the expected sum of the discounted paid out dividends from time zero until ruin. Since then several papers have presented solutions to this problem for various risk processes. Two of the papers in this thesis, which we denote Paper A Paper B, focus on de Finetti s dividend problem, with the risk process following a general diffusion a jump-diffusion process, respectively. These models are particularly relevant for insurance companies where the premium income is invested in assets with stochastic returns. In keeping with de Finetti s original paper, where ruin probability played a central role, Paper A also discusses solutions of de Finetti s dividend problem under solvency constraints. In the last few decades a growing number of papers have focused on ruin probability on a finite time horizon. For short time spans the assumption that the risk reserve is allowed to grow freely is less spurious. An important tool for calculating the ruin probability on a finite horizon is solving certain partial integro-differential equations PIDEs. The third paper, denoted Paper C, discusses how these PIDEs can be solved numerically. The last paper, denoted Paper D, discusses regularity properties for some of these PIDEs. iii

4 Papers Paper A: L. Bai, M. Hunting J. Paulsen 212. Optimal dividend policies for a class of of growth-restricted diffusion processes under transaction costs solvency constraints. To appear in Finance Stochastics. Paper B: M. Hunting J. Paulsen 212. Optimal dividend policies with transaction costs for a class of jump-diffusion processes. To appear in accepted by, Finance Stochastics. Paper C: M. Hunting 212. A numerical approach to ruin probability in finite time for fitted models with investment. Not submitted, Paper D: M. Hunting 212. Existence of a classical solution of a parabolic PIDE associated with ruin probability. Not submitted. iv

5 Acknowledgements While writing this thesis, I have been employed as a PhD cidate universitetsstipendiat at the Department of Mathematics, University of Bergen, Norway. First of all, I would like to thank my supervisor, Professor Jostein Paulsen, for coping with my foolhardiness, more than once, forcing me back to the right track when I got lost. The other staff students at both the Department of Mathematics in Bergen the Department of Mathematical Sciences at the University of Copenhagen deserve many thanks for being part of an inspiring stimulating environment. Not least would I like to thank my father for invaluable proofreading assistance. Martin Hunting Bergen, June 212 v

6 Contents 1 Ruin probability Cramér-Lundberg model General theory Diffusion approximations Ruin probability in an economic environment Ruin probability in finite time Numerical calculation of ruin probability with investment Optimal dividend policy De Finetti s dividend problem, dividend policy the value of an insurance company Optimal dividend strategies Paper A 4 4 Paper B 76 5 Paper C Paper D 141 1

7 1 Ruin probability 1.1 Cramér-Lundberg model General theory As explained in Chapter 2 in Mikosch 24, the foundations of modern risk theory were laid in 193 by the Swedish actuary Filip Lundberg in his acclaimed thesis, Lundberg 193. Lundberg s major contribution was to introduce a simple model that is capable of describing the basic dynamics of a homogeneous insurance portfolio. There are three assumptions in Lundberg s model: i Claims occur at the Poisson-distributed times τ i, satisfying τ 1 τ 2. In this thesis we will refer to these times as claim times. let λ be the parameter of the Poisson process. ii The i-th claim, arriving at time τ i, results in a claim of size S i. The sequence {S i } constitutes an i.i.d. sequence of non-negative rom variables. In this thesis we will denote the common distribution function of the claim sizes by F x. iii The claim size process {S i } the claim arrival process {τ i } are mutually independent. Based on the above we define the claim number process N t = min {i, 1, : τ i+1 > t}. From the point of view of insurance companies it is common to assume a continuous premium income at a constant rate p. The risk process is then Y t = y + pt S t, t >, where y is the initial capital S t is the total claim amount process N t S t = S i. i=1 Here we follow the convention that i=1 =. If we assume that the waiting times between claims are i.i.d. then S t is referred to as a renewal process. Generalizations of the Cramér-Lundberg model to general i.i.d. waiting times between claims are in the literature referred to as renewal models, or the 2

8 Sparre-Andersen model. The time τ when the process falls below zero for the first time is called ruin time, The probability of eventual ruin is then τ = inf {t > : Y t < } ψy = P τ < Y = y, y >. In Section 1.3 we consider the probability that τ T. An important result concerning renewal processes of the above type is given in, for example, Proposition in Mikosch 24. This result says that if we assume that then Eτ 1 < ES 1 <, ES 1 peτ 1 implies that τ < with probability 1 for every initial capital y. Any sensible premium policy would therefore satisfy the condition ES 1 < peτ 1, known as the net profit condition. In the following we will assume that this condition holds let ρ = p Eτ 1 ES The quantity ρ is often referred to as the safety loading. In both Mikosch 24 Asmussen 2 there are extensive discussions of ruin probability results in the Cramér-Lundberg model. To better underst these results we first review some of the definitions used in these two books. Definition The survival probability sometimes referred to as the nonruin probability is defined as Definition Let φy = 1 ψy. Z 1 = S 1 pτ 1, assume that the moment-generating function of Z 1 exists in some neighborhood around. If a unique positive solution h of the equation Ee hs 1 pτ 1 = exists it is called the adjustment coefficient or Lundberg coefficient. 3

9 In the literature equation is known as the Lundberg equation, a distribution whose moment-generating function exists around the origin is generally referred to as being light-tailed. In the important special case of the exponential distribution with parameter β it is shown in Example in Mikosch 24 that the adjustment coefficient γ is given as γ = β λ p Definition A function Lx is said to be slowly varying if Lcx lim x Lx = 1, for all c >. Definition A positive rom variable S its distribution are said to be regularly varying with tail index α if for some α the right tail of the distribution has the representation where L is a slowly varying function. P S > x = Lxx α, Definition A positive rom variable S its distribution are said to be subexponential if, for a sequence S i of i.i.d. rom variables with the same distribution as S, the following relation holds: For all n 2 : n P S j > x = P max S i > x 1 + o 1 as x. i=1,...,n j=1 Definition Define F x = 1 F x, y F s y = ES 1 1 F xdx, F s y = 1 F s y It is well known that all subexponential distributions are heavy-tailed. It is shown in Section in Mikosch 24 that every regulary varying distribution is a subexponential distribution. Furthermore, it is shown there that if a distribution has a density f, then a sufficient criterion for the distribution to be regulary varying is that, for some tail index δ >, fcx lim x fx = cδ, for all c >. 4

10 For i.i.d. rom variables X 1,..., X n with common distribution function F x we will denote the cumulative distribution of the sum n j= X j by F n x. For general claim distributions no closed form formula is known for the ruin probability in the Cramér-Lundberg model. However, under some not very restrictive conditions, the ruin probability can be expressed as a solution of an integral equation. This is indicated in the result below, which is the same as Lemma in Mikosch 24. Theorem Consider the Cramér-Lundberg model with safety loading ρ > expected claim size ES 1 <. In addition assume that the claim size distribution F has a density. Then the survival probability satisfies the integral equation φy = ρ 1 + ρ + 1 y F xφy xdx ρ ES 1 In the above F x = 1 F x is the common tail distribution of the claims. While for general claim distributions does not give very much qualitative information, it can be used as a basis for numerical computation. Moreover, for the case of exponential distributions with parameter β, it can be shown see e.g. Example in Mikosch 24 that the exact ruin probability is given by ψy = 1 ρ 1 + ρ e β 1+ρ y In chapter VIII in Asmussen 2 there is a discussion of ruin probability for a wider class of claims distributions, known as phase-type distributions. A distribution F is said to be of phase-type if F is the distribution of the lifetime of a terminating Markov process {J t } with finitely many states time homogeneous transition rates. This class includes, for example, the exponential distribution, the hyper-exponential distribution a mixture of a finite number of exponential distributions the Erlang distribution Gamma distribution with an integer shape parameter as special cases. The tail distribution F x of a phase-type distribution can be shown to be of 5

11 the form F x Cx k e ηx, where C η are positive constants k is a non-negative integer. For claim distributions of this type an exact formula for the ruin probability is given in Theorem VIII.2.1 in Asmussen 2. In the same chapter of that book an example is given on how that formula can be applied to a mixture of two exponential distributions. For general light-tailed claim distributions the following result is well known see e.g. Proposition II.1.1 in Asmussen 2. Lemma Let let N t S t = S i pt k=1 ξy = S τ y be the overshoot at the time of ruin. Make the following assumptions: } a For some c >, {e c S t is a martingale. b S t a.s. on {τ = }. Then t ψy = e cy E [e cξy τ < ] It can be shown see e.g. Example II.1.2 in Asmussen 2 that if the adjustment } coefficient γ exists, then under the Cramér-Lundberg model {e γ S t is a martingale. Furthermore, since ξy it then immediately t follows that ψy e γy, whenever the conditions of the Lemma hold. The formula is known as the Cramér-Lundberg inequality. Moreover, this formula the memoryless property of the exponential distribution provide an alternative method for deriving the identity This is done in Example 1.3 in Asmussen 2. As mentioned above the exponential distribution is a light-tailed distribution. For many situations it is more appropriate to assume that the claim sizes follow a heavy-tail distribution. The most important heavy-tail distributions in insurance belong to the class of subexponential distributions, defined in Definition For this class of distributions there is no known exact formula, but the asymptotic result below is given, for example, as Theorem IX.3.1 in Asmussen 2. 6

12 Theorem Let υ = ES 1 Eτ 1. Assume the Cramér-Lundberg model stardized such that p = 1. In addition assume that υ < 1, that ES 1 < that the integrated tail distribution F s defined in of the claim size distribution is a subexponential distribution. Then, asymptotically, ψy Diffusion approximations υ 1 υ F s y A rather different way of obtaining approximations of the ruin probability is by fitting diffusion processes to approximate the compound Poisson process in the Cramér-Lundberg model. In Section XI.1 in Asmussen 2 there is a short discussion on ruin probability under a diffusion model with drift µy variance σ 2 x >. This kind of model can be expressed as letting Y t = y + P t, where P t is the continuous solution of the stochastic differential equation dy t = µ Y t + σ 2 Y t dy t. The appeal of this approach is that under mild conditions the exact ruin probability has the closed-form solution stated in the result below, which is the same as Theorem XI.1.1 in Asmussen 2. Theorem Let x µz sx = exp 2 σ 2 z dz. Assume that µx σ 2 x are continuous functions, that σ 2 x > for x > that Then < ψy < 1 for all y > ψy = 1 szdz < Conversely, if fails then ψy = 1 for all y >. y szdz szdz As described in Section IV.5 in Asmussen 2, the simplest diffusion approximation is to let µy σ 2 y be two positive constants fitted to the first two moments of the compound Poisson process the desired premium rate p. In this model the ruin probability is given as ψy = exp 2 µ σ y. 2 7

13 Let γ be the adjustment coefficient as before. As explained in Section IV.6 the simplest diffusion approximation can be refined via so-called correction terms. This leads to the following approximation: ψy exp γ [ y + L F γ 3L F γ ] Here L F γ L F γ are, respectively, the second the third derivative of the Laplace transform of the claim size distribution evaluated at the point γ. We will return to the corresponding finite time approximation in Section Ruin probability in an economic environment In the classical Cramér-Lundberg model premium income is modeled as a constant rate that does not earn any interest. Neither the claims nor the premium income is subject to inflation. One of the earlier papers to feature an interest rate is Harrison In this model it is assumed that the risk reserve is invested in a risk free bank savings account, continuously earning interest at a constant rate r. Further, let the sum N t n=1 S n be the compound Poisson process from the Cramér- Lundberg model let P t = pt N t n=1 S n. In what follows we will refer to P y = y + P t as the surplus generating process. With this notation the content of the account at time t is written or, equivalently where Y t = e rt y + t e rt s dp s, Y t = e rt [y + Z t ], t, Z t = t e rs dp s, t. It is shown in Harrison 1977 that Z = lim t Z t exists is finite almost surely. A formula for the characteristic function of Z i.e. Ee iuz is given. Furthermore it is shown in Theorem 2.3 in Harrison 1977 that ψy = H y [ ], E H Y τ τ < 8

14 where H is the distribution function of Z. For general distributions may look more like a reformulation of the problem than a solution. However, in Harrison is used to derive more explicit ruin formulas for a few specific claim size distributions, including the exponential distribution. The ruin probability is then given as ψy = λp e 1 βx + 1 rx r dx y p p + e 1 λ βx + rx p λp 1. r dx This classical result is also found in Segerdahl Another kind of model that is also considered in Harrison 1977 assumes that the surplus generating process is a diffusion process, P y t = y + µt + σw t, where µ σ are positive constants, W t is a stard Brownian motion y is the initial value. For this it is shown that ψy = 1 Φ ay + b, 1 Φb where a = 2ī σ, b = a µ 2 ī function. Φ is the stardized normal distribution Taylor 1979 is one of the earliest papers to consider the effect inflation may have on premium income claims size distribution. This paper is notable for its conclusion that probability of ruin is nondecreasing with increasing inflation. Some bounds for ruin in finite horizon are also given in that paper. We will return to those bounds in Section 1.3. In Delbaen Haezendonck 1987 the authors incorporate both interest inflation in their models. Both the interest force the inflation force which we denote by r ī, respectively are assumed to be constant. In this model the n th claim size is of size eīτn S n the premium density at time t is peīt. More formally this model can be written as the stochastic equation dy t = pe rt dt + īy t dt e rt S Nt dn t. 9

15 The present value Ỹt of Y t can be written as Ỹ t = y + p t N t e r īu du e r īτn S n As before, y in the above is the initial reserve. The most relevant result in this paper for this thesis is that the probability for eventual ruin can be written as the solution of the integro-differential equation ψ y = n=1 λ p + r ī y ψy λ p + r ī y E [ψ y S 1], where ψx = 1 for x <. As before, λ is the intensity of the Poisson process. Two other papers which take inflation into account are Waters 1983 Paulsen In Waters 1983 the risk process is considered in discrete time. At a time t n, n 1, 2,..., the reserve is given as Y tn = y + n c n X k. Here c is a constant greater than 1 the X k s are i.i.d. variables with finite expectation a continuous common distribution function such that k=1 P X 1 < >. In this model it is implied that premiums claims are affected by the same inflation factor c. With no more than the conditions given above it is shown that in this model ultimate ruin is certain, i.e. ψ y = 1, for every y >. This might seem to imply that avoiding ruin requires increasing premiums by an amount greater than the the increase in claim size. However the paper also shows that eventual ruin is not certain if the risk reserve Y earns interest. The model in Paulsen 1993 is preferably explained in 5 steps. The first step is that the surplus generating process P y t = y + P t is assumed to be a semimartingale with initial value y. The second step is that the inflation generating process I is assumed to be a semimartingale with I =, while the level of inflation Ī is given as the solution of dīt = Īt di t. 1

16 Here Ī = 1. As explained in Paulsen 1993 it then follows that at time t Ī t = exp {I t 12 } Ic, I c t Π s t 1 + I s e Is. Here I c, I c is the predictable quadratic variation of the continuous martingale part I c of the semimartingale I. If I is discontinuous at time t then I t is the jump I t + It. Otherwise I t =. The third step is that the inflated surplus process P y at time t is given as P y t = y + t Ī s dp y s. The fourth step is that the surplus is assumed to be continuously invested in stochastic assets. The return on investment process R is assumed to be a semimartingale with R =. In terms of nominal units the total risk process Ȳ t is given as the solution of dȳt = d P y t + Ȳt dr t, where Ȳ = y. The last step is that the risk process in terms of real units at time t is given as Y t = Ī 1 t Ȳ t, where Y = y. At time t let R t = exp { } R t 1 2 Rc, R c t, where R c, R c is the predictable quadratic variation of the continuous martingale part R c of the semimartingale R. It is shown in Paulsen 1993 that Y can also be written as where U = Ī R 1. Y t = U 1 t y + t U s dp s, Inspection of above shows that the important quantity is not so much the interest rate or the inflation rate, as the difference between the two. This is often called the real interest rate. This is a consequence of the rate of inflation the rate of interest being constant in those equations, which are taken from Delbaen Haezendonck In Paulsen 1993 it is shown that the so-called real interest rate retains its importance as long as either R I or I is a continuous deterministic process. Thus, in these cases inflation can be accounted for by focusing on the inflation-adjusted rate of return, rather than the nominal rate of return. 11

17 In most of Paulsen 1993 it is also assumed that the vector process X = P, I, R is a stochastic process with stationary independent increments, with a finite number of jumps on each finite interval. Furthermore, it is assumed that the first component the surplus generating process P is independent of the two other components. Under these assumptions it follows see Krylov 22 that X has the representation X t = āt + C W t + V t. Here ā is a constant vector, W is a three-dimensional Brownian motion process, V is a three-dimensional compound Poisson process, independent of W C is a 3 3 matrix with the property C C σ tr P 2 = σi 2 cσ I σ R. cσ I σ R σr 2 Here c 1 is the correlation between the continuous part of the inflation process the return on investment process. In addition σ P, σ I, σ R are non-negative constants, tr means transposed the Brownian motion vector W. Furthermore, it is assumed that the first component P of V is independent of the other two components, I R. In terms of the components of X this gives N P,t P t = pt + W P,t S P,n, I t = īt + W I,t + R t = rt + W R,t + n=1 N I,t n=1 N R,t n=1 S I,n S R,n Here W P, W I, W R tr = C W, N P, N I N R are three Poisson processes with intensities λ, λ I λ R respectively, N P is independent of N I, N R. Also the summs in each sum are i.i.d., {S P,n } N P,t n=1 the jumps { } NI,t { } NR,t SI,n, SR,n are independent. Moreover it is assumed that n=1 n=1 P SI,1 1 = P SR,1 1 =. As explained in Paulsen 1993, this leads to that at time t { Ī t = exp ī 1 } 2 σ2 I t + W I,t Π N I,t n=1 1 + S I,i 12

18 { R t = exp r 1 } 2 σ2 R t + W R,t Π N R,t n=1 1 + S R,i. Also given is the unified process for inflation return on investment, U t = exp { α U t + σ U W U,t } Π N I,t n=1 1 + S I,i Π N R,t j= S, R,j where α U = r ī+ 1 2 σ2 I σ2 R, σ2 U = σ2 I 2cσ Iσ R +σ 2 R, W U is a Brownian motion. Here we follow the convention that Π i=1 = 1. Most of the implications for ruin probability discussed in Paulsen 1993 are easier to grasp if Π N I,t n=1 1 + S I,n Π N R,t n=1 1 + S R,n are assumed to be independent, which we do for the rest of our discussion of that paper. With this assumption the only dependence between P, I R is by means of the correlation c between the Brownian motion processes W I W R. It also follows from Lemma 2.1 in Paulsen 1993 that Π N I,t n=1 1 + S I,i Π N R,t n= S R,i can be written as V = Π N U,t n=1 S U,n, where N U is a Poisson process with intensity λ U = λ I + λ R. Furthermore, the S U,n s are i.i.d. independent of N U, the S U,n s have the common distribution F U s = λ I F I s + λ R 1 1 F R λ U λ U s A key result in Paulsen 1993 is Theorem 3.1, which gives that, with the assumptions made above, the process Z t = t U s dp s is a semimartingale. Continuing our assumption that Π N I,t n=1 1 + S I,i Π N R,t n=1 1 + S R,i are independent, also assuming that r ī + cσ I σ R σ 2 R + λ U 1 ES U,1 >, then lim t Z t = Z exists converges almost surely in L 1. In Theorem 3.2 in Paulsen 1993 it is shown that the probability of eventual ruin is given by H y ψy = E [ H Y τ ], τ < where H is the distribution function of Z τ is the ruin time. formula is similar to the formula discussed earlier. This 13

19 As pointed out in Bankovsky et al. 211, some additional conditions are needed for the results in Paulsen 1993 to hold. These conditions are given in Remark 23 in Bankovsky Sly 29. However, it is clear from these conditions that these problems can be avoided by assuming that p. It follows from Theorem 3.4 in Paulsen 1993 that the distribution H can be derived from a certain integro-differential equation. We state this result below. Theorem Let Ψu = E [ e ius P,1] be the characteristic function of S P,1. Assume that that Ψ u du <, E [ln S U,1 ] <. In addition assume that either σ P = σ U = or that 1 σ 2 2 P + σuz 2 2 H z + + λ U H Ψu du <, uψu du <. Then the distribution function H of Z is twice continuously differentiable is the solution of p + α U σ2 U z H z λ U + λ Hz z s df U s + λ H z + s df s = Here α U is still r ī+ 1 2 σ2 I σ2 R, λ is still the intensity of the claims process, F is still the claim size distribution. Asymptotic boundary conditions are lim z Hz = lim z Hz = 1. If σu 2 = σ2 P =, then H is the continuously differentiable solution of From the identity it is obvious that ψy H y H

20 In the most basic situation, λ = no jumps in the claims process, we get equality in Here, if S P,1 has an increasing failure rate, i.e. P S P,1 > u + v S P,1 > u P S P,1 > u, for u, v >, then ψy H y E [HS P,1 ]. On the other h if σ 2 P = S P,1 has a decreasing failure rate then ψy H y E [HS P,1 ] The most basic situation with jumps is when the jumps are exponentially distributed σ P =, in which case we get equality in An asymptotic result for the finite horizon ruin probability is given in Proposition Perhaps it is because deterministic inflation can be accounted for by considering the real interest rate that very few papers after Paulsen 1993 have included a separate inflation process. In the rest of the thesis we will tacitly make the assumption that inflation is indeed a continuous deterministic function that the interest rate is the real interest rate. An alternative approach could be to consider the process U given in as the real stochastic return on investment process. This might be a topic for further research. In this thesis our only result regarding stochastic inflation is the asymptotic formula in Proposition in the next section. Other than not including explicit inflation the assumptions in the later paper, Paulsen Gjessing 1997, are similar to the assumptions in Paulsen Since in this model there is no I process to worry about it is more convenient to write the surplus generating process P at time t as N P,t P t = pt + σ P W P,t S P,i, t Similarly, the investment return process R is written as i=1 N R,t R t = rt + σ R W R + S R,i i=1 15

21 Here W P W R are independent Brownian motion processes that are also independent of the compound Poisson processes. As before all the jumps are i.i.d. independent of the Poisson processes N P,t N R,t. Lastly, N P,t N R,t are independent. The risk process is then given as the solution of the stochastic differential equation Y t = y + P t + which for time t has the solution Y t = R t y + t t Y s dr s, R 1 s dp s Here R t is given as { R t = exp r 1 } 2 σ2 R t + σ R W R,t Π N R,t n=1 1 + S R,n, t. As well as the assumption that F R = P 1 + S R,1 =, it is also assumed that both S P,1 S R,1 have finite expectation. Under these assumptions it is shown that the risk process Y has the same distribution as Ỹ, where Ỹ t = y + + t t p + rỹs ds + N R,t Ỹ s d i=1 S R,i. t N σp 2 + P,t σ2 RỸ s 2 dw s It is also shown that the infinitesimal generator for Ỹ is given by i=1 S P,i Agy = 1 2 σ 2 P + σ 2 Ry 2 g y + p + ry g y + λ g y x gy df x + λ R g y 1 + x gy df R x The result in Paulsen Gjessing 1997 that is most relevant for this thesis is Theorem 2.1 part i, which we state below. The proof of this result is based on the generator A given above. 16

22 Theorem Assume that gy is bounded twice continuously differentiable on y, with a bounded first derivative there, where we at y = mean the right-h derivative. If gy solves subject to the asymptotic boundary condition Agy = λ F y on y >, lim gy =, y, if σ P >, the boundary condition g = 1 holds, then for every y. ψy = gy In the paper Yuen et al. 24 it is shown that a smooth solution of exists provided the following conditions are satisfied: i σ P =. ii S P,1 S R,1 have finite first two moments, the distribution functions F F R are three times continuously differentiable, the limits F +, F +, F +, F R 1 +, F R 1 + F R 1 + all exist. iii 2r VarS P,1 >, λ + λ R 2r + VarS P,1 > the net profit condition p λes P,1 > is satisfied. Some alternative sufficient conditions for the existence of a smooth solution of are given in Paulsen et al. 25. Here it is shown that if λ R = i.e. no jumps in the return on investment process R, then a smooth solution exists, provided the distribution function F is four times continuously differentiable on [,, for some c >, F xx c is bounded for every x >. A third set of sufficient conditions is found in Yuen Wang 25. A few examples are given in Paulsen Gjessing 1997 where the equation can be explicitly solved. One of the examples is the case when σ P = σ R = λ R = S P,1 is exponentially distributed with parameter β. Another example is when σ P = σ R = λ R = F is a mixture of two exponential distributions. For this case the solution of is more complex takes the form of an integral representation. 17

23 A generalization of the first example in Paulsen Gjessing 1997 is to let the risk process Y t take the form Y t = y + t N P,t q Y s ds S P,i Here q is a continuous function the claim sizes are still exponentially distributed with parameter β. It is shown in Dassios Embrechts 1989 that in this case where Qx = x qs 1 ds. ψy = y i=1 e βx+λqs dx qx 1 + e βx+λqs dx, λ qx In Bankovsky et al. 211 there is a discussion of more general risk processes of the form t Y t = e y ξt + e ξs dη s, t where ξ t, η t t is a bivariate Lévy process. The models defined in are special cases of , with η = P ξ t = r 1 N R,t 2 σ2 R t + σ R W R,t + ln 1 + S R,n, t. For models of type Bankovsky et al. 211 derive the theorem below. n=1 Theorem Suppose that the following conditions hold: a ψy > for every y. b There exists w > such that Ee wξ 1 = 1. c There exists ɛ > c, d > 1 with = 1 such that c d E [ e ] [ max1,w+ɛcξ 1 < E η 1 max1,w+ɛd] <. d The distribution of ξ 1 is spread out, i.e. has a convolution power with an absolutely continuous component. Then there exists a constant C such that asymptotically ψy Cy w. 18

24 The result above tells us that under mild conditions, the eventual ruin probability decays like a power law even if the claim distribution is lighttailed. As an example consider the models defined in with λ R =. Assume that the claim size distribution has moments of all orders that r > 1 2 σ2 R. A calculation then shows that the theorem holds with w = 2 r 1. Similar conclusions can be drawn from Klüppelberg Kostadinova σr 2 28, Kalashnikov Norberg 22 Frolova et al. 22. As we shall see in the next section, the ruin probability in finite time is not quite as gravely affected by moderately risky investments as is the case for the eventual ruin probability. In most of the papers that include a return on investment process the return is assumed to be a constant real interest force. With this assumption the risk process is of the type , where qx is a linear function. In Klüppelberg Stadtmüller 1998 it is shown that if the claim size distribution is regularly varying with index α > 1, then asymptotically ψy λ αr F y. The most noteworthy with this estimate is that it implies that the ruin probability decays as the tail distribution F y, rather than as the integrated tail distribution F xdx. Lastly, we mention that the paper Paulsen y 1998 offers a fairly extensive survey of other results for eventual ruin. Newer results for eventual ruin are discussed in Paulsen 28. That paper also discusses ruin in finite time, which is the topic of the next section. 1.3 Ruin probability in finite time In this section we discuss the probability that ruin as defined in Section 1.1 occurs within a finite time T. We will denote this probability by ψ y, T. Unfortunately the known results for ruin in a finite time horizon are generally even less explicit than the results for eventual ruin. The focus here is either on approximations or on results that can be seen as a basis for numerical computation. Consider again the classical Cramér-Lundberg compound Poisson model, defined in Section 1.1. For a stardized model with premium rate p = 1 stard exponentially distributed claims, assume that the net profit condition is satisfied, which in this simple model just means that λ < 1. 19

25 From Proposition IV.1.3 in Asmussen 2, we then have that φ y, t = λe 1 λy 1 π π f 1 θ f 2 θ dθ, f 3 θ where { f 1 θ = λ exp 2 } λt cos θ 1 + λ T + y λ cos θ 1, f 2 θ = cos y λ sin θ cos y λ sin θ + 2θ f 3 θ = 1 + λ 2 λ cos θ. The result can be easily extended to more general Cramér-Lundberg models with exponentially distributed claim sizes. As an intermediate step we first show how the ruin probability for a model with a general premium rate p can be expressed in terms of the ruin probability for a model with a stardized premium rate p = 1. The last step is to obtain ruin probabilities for a general claim counting process intensity λ a general exponential parameter β, as well as a general premium rate p. Now, let ψ y, T, p, λ, β be the probability of ruin as a function of the parameters p, λ β as well as y T. Since the ruin time, inf t> we see that { } N t y + pt S n < n=1 = inf t> { Nt y p + t n=1 S n p < y ψ y, T, p, λ, β = ψ, T, 1, λ, pβ p This stardizes the premium rate. To stardize the parameter of the exponential distribution we can use the transformation suggested in Proposition IV.1.3 in Asmussen 2. This yields ψ y, T, p, λ, β = ψ βy, pβt, 1, λ pβ, 1. Any general Cramér-Lundberg model with exponentially distributed claim sizes can thus be reduced to the stardized model treated in Proposition IV.1.3 in Asmussen 2. }, 2

26 For general distributions it is shown in Pervozvansky Jr that if the premium income is invested with constant real interest force r the claim size distribution has a continuously differentiable density f, then φy, t = 1 ψy, t is the solution of the following partial integro-differential equation PIDE: φ y, t t = p + ry φ y, t y Here the solution is subject to the initial condition y λφ y, t + λ φ y z, t fzdz φ y, = 1 on y > the asymptotic condition lim y,t φ y, t = 1. Let u x IG x; ζ; u = 1 Φ ζ + e 2ζu Φ u ζ x, x where Φ is the distribution function of the normal distribution. Let L F γ L F γ be as in Section 1.1. As discussed in Section 1.1 above, more thoroughly in Section IV.5 in Asmussen 2, a diffusion model with correction terms can be used to approximate the Cramér-Lundberg model. In finite time this approach leads to the approximation T δ1 φ y, T IG y + δ 2 2 y ; 1 2 γy; 1 + δ 2, y where T is the time horizon, γ is the adjustment coefficient, δ 1 = λl F γ δ 2 = L F γ. 3L F In Asmussen Højgaard 1999 it is discussed how γ the ruin probability for general renewal models can be approximated by the formula For regularly varying claim distributions Theorem 4.1 in Hult Lindskog 211, in combination with Example 3.5 in Hult Lindskog 211, can be used to obtain asymptotic formulas for the ruin probability for a fixed finite time horizon for models with investment. Below we have formulated a simplified form of their Theorem 4.1 to fit with models with a surplus generating process P of the form given in , a continuous investment process R of the form given in , a risk process Y of the form given in

27 Theorem Assume that the claim size distribution is regularly varying with index α denote the fixed finite time horizon by T. a In the case of the Cramér-Lundberg model the probability of ruin before time T is asymptotically given by ψ y, T λt F y. b Consider a risk process of the form given in Make the following extra assumptions: i Either λ R = or for some δ > E 1 + S R α+δ <. ii θ = 1 2 σ2 R α 2 + α αr + λ R E 1 + SR α 1. Then the probability of ruin before time T is asymptotically given by ψ y, T 1 e θt 1 λ θ F y. Proof. As explained in Theorem 3 in Paper C, this follows from Theorem 4.1 Example 3.5 in Hult Lindskog 211. Remark: Assume that λ R > let τ R,1, τ R,2,..., denote the jump times of the R process. Let X i = ln R τr,i + ln RτR,i for i 1, 2,...,. The condition E 1 + S R α < corresponds to Ee αx 1 <. In other words the results in the theorem above only hold if the jumps of the logreturns of the investment process are light-tailed. Below we give a few wellknown examples of such models. Example Assume that λ R > that each X i is normal distributed with parameters µ σ 2 as in the Merton model, see Merton Then for every α > θ = 1 2 σ2 R α 2 + α αr + λ R exp αµ + 12 α2 σ 2 1. Example Assume that λ R > that the jumps of the log-returns are as in the Kou model see Kou 22, i.e. obey an asymmetric exponential probability distribution with density f X x = qβ 1 1 x> e β 1x + 1 q β 2 1 x< e β 2 x for some q, 1. Assume that β 1, β 2 > that β 2 > α. Then θ = 1 2 σ2 R α 2 + α q αr + λ R 1 + α + 1 q β 1 1 α 1. β 2 22

28 A similar result is also valid for the models in Paulsen 1993 with stochastic inflation as indicated below: Proposition Assume that the claim size distribution is regularly varying with index α. Let U t, α U, σ U, S U,n, λ I, λ R, λ U, S I,n, S R,n, N I,t, N R,t, N U,t, F I, F R F U be as in Assume that Π N I,t n=1 1 + S I,n Π N R,t n=1 1 + S R,i are independent. Furthermore, assume that U t is a strictly positive process that, for some δ >, Let ES α+δ U,1 <. θ = 1 2 α2 σ 2 U αα U + λ U ES α U,1 1. Assume that θ. Then asymptotically ψ y, T 1 e θt 1 λ θ F y, where the the distribution function F U x of the common distribution of the S U,1 s is given by F U s = λ I F I s + λ R 1 1 F R λ U λ U s Proof. This follows from Theorem 4.1 Example 3.5 in Hult Lindskog 211, by considering making the appropriate time changes as in Theorem 3 in Paper C. It follows from Lemma 2.1 in Paulsen 1993 that the distribution of S U,1 is given by In Wang et al. 212 the result in Theorem for the classical case σ P = r = σ R = λ R = is generalized to a general renewal process defined in Section 1.1 above. Assume that the waiting times between claims are i.i.d. with finite expectation λ 1, ES 1 < the claims satisfy Then asymptotically ψy, T lim lim z 1 x λ p λes 1 F xz F x = 1. y+t p λes1 y F udu. In Wang et al. 212 it is shown that this asymptotic formula holds even for certain kinds of dependence between claim sizes. Yet another generalization of this result is found in Chen et al. 211, this time to a model where the risk process is a bivariate renewal process. 23

29 In the case when σ P = σ R = λ R = r > constant force of interest model, it is shown in Tang 25 that the asymptotic formula above holds even if the surplus generating process P is generalized to the form N t P t = y + Ct S n, where Ct can be any nondecreasing right-continuous stochastic process. Theorem the generalization in Wang et al. 212 only provide approximate ruin probabilities for large values of the initial capital for a special class of distributions. As a basis for numerical calculations it would be useful to also have a formula for the exact ruin probability in finite time for investment models having more general distributions. For models with a constant force of interest the ruin probability can be calculated by solving the PIDE We want to obtain a PIDE for the ruin probability that is valid for stochastic investments. Let the integrodifferential operator A be as in Theorem In Paulsen 28 it is stated that the ruin probability should be the solution of the following partial integro-differential equation PIDE: n=1 ψ y, t t subject to the initial condition = Aψ y, t + λ F y, ψ y, =, y >, the asymptotic boundary condition lim ψ y, t =. y If the diffusion parameter σ P is positive, we have the extra boundary condition ψ, t = 1. A sufficient condition for the ruin probability to satisfy the above PIDE is that there exists a classical solution of 1.3.6, i.e. a solution that is bounded smooth on the interior of the domain. This statement can be proved using arguments similar to those in the proof of Theorem 2.1 in Paulsen Gjessing

30 Now let us look at the PIDE for the case that λ R =. Let L be the differential operator defined by Lhy, t = 1 2 σ2 P + σry hy, t hy, t + p + ry, y 2 y let A be the integro-differential operator A hy, t = Lhy, t + λ y hy z, tdf z λhy, t. Here A is the same as the operator A defined in , but with λ R =. In Paper D we consider the equation 1.3.6, under the assumptions that λ R = σ P >, that either σ R > or r = σ R =, that the tail of the claim size distribution satisfies the bound F x C 1 + x c, x, for some positive constants C c. Under these assumptions it is shown in Paper D that a solution of exists that is smooth for y, t away from the origin. In particular, a smooth solution exists even if the claim size distribution is discrete. Since the coefficients are all smooth the integrodifferential operator A is linear, this might seem like a trivial result. There are, however, a number of reasons, listed below, why this is not the case. The domain is unbounded. Some literature, in particular on PDEs, discuss problems with unbounded domain. In general, however, these treatises require at least that the coefficients of the space derivatives of second order be bounded. In our case the only coefficient of a second order derivative is 1 2 σ2 P + σ 2 Ry 2, which is obviously not bounded for y,, when σ R >. Violation of compatibility condition. The initial condition dictates that lim y ψy, =, whereas the boundary condition dictates that lim s ψ, s = 1. The initial condition the boundary condition are thus incompatible, any solution of must hence be discontinuous at the origin. This violates the requirement that a classical solution must be continuous at the boundary. 25

31 Asymptotic boundary condition. In addition to the difficulties mentioned above we need to verify that, for any s, t], lim y ψy, s =. The upshot of this is that stard theory does not immediately ensure existence uniqueness of a solution of equation It turns out that by far the biggest problem is that the domain is unbounded that in the general case, when σ R >, the coefficients grow to infinity. To get around this problem we first consider solutions ψ κ y, t of on a truncated domain with the more stard boundary condition ψκ, t =, for some κ >. To get around the problem with the singularity at the origin we consider the truncated solution ψ κ y, t as a sum of the three functions ψ 1,κ y, t,..., ψ 3,κ y, t. Here ψ 1,κ y, t is a solution of ψ 1,κ y, =, y, κ, ψ 1,κ, t = 1, t [, 1], ψ 1,κ κ, t =, t [, 1], ψ 1,κ y,t t = 1 2 σ2 P 2 ψ 1,κ y,t y 2 ψ 2,κ y, t is a solution of ψ 2,κ y, =, y, κ, ψ 2,κ, t =, t [, 1], Here + p ψ 1,κy,t y, y, t, κ, 1]. ψ 2,κ κ, t =, t [, 1], ψ 2,κ y,t Lψ t 2,κ = H 1,κ y, t, y, t, κ, 1]. H 1,κ y, t = 1 2 σ2 Ry 2 2 ψ 1,κ y, t + ry ψ 1,κ y, t λψ 2 y 2 1,κ y, t y + λ y Finally, ψ 3,κ y, t is a solution of ψ 3,κ y, =, y, κ, ψ 3,κ, t =, t [, 1], ψ 1,κ y z, t df z + λ F y. ψ 3,κ κ, t =, t [, 1], ψ 3,κ y,t A t ψ 3,κ y, t = H 2,κ y, t, y, t, κ, 1]

32 Here y H 2,κ y, t = λψ 2,κ y, t + λ ψ 2,κ y z, t df z. In the general case with σ R > the coefficients grow to infinity. Therefore we choose the following change of variables: x = ln 1 + y. This leads to a formulation where the coefficients are bounded independently of the value of κ. This makes it easier to obtain suitable bounds on the partial derivative of ψ 3,κ y, t. The last step is to obtain the solution of the equation as a limit of a sequence of functions {ψ κn y, t} n=, where κ n. In addition to existence of a smooth solution of 1.3.6, Paper D also establishes the following: If the tail distribution F x satisfies F x C 1 + x c, x, for some positive constants C c, then for some constant C c, depending on c, ψ y, T C c y c In particular we conclude from this that if the moment generating function of the claim size distribution exists in a neighborhood around, then for an arbitrarily large c there exists a constant C c such that holds. This is markedly different from the asymptotic behavior of the corresponding eventual ruin probability discussed in Section Numerical calculation of ruin probability with investment Very few of the results concerning the ruin probability are easily analyzable closed form solutions. Even when relatively simple asymptotic results are known, such as the approximations given in Theorem 1.3.1, it is still desirable to make numerical computations. This is because those approximations are often not very good for moderate, or even quite large, values of the initial reserve. As a consequence much attention has been paid to numerical calculations of ruin probability, especially for the classical Cramér-Lundberg model. Most of these are based on either matrix computation see e.g. Asmussen Rolski 1991 or recursive formulas see e.g Vylder 1999, Dufresne Gerber 1994, Dickson Waters 1991 especially Steenackers Goovaerts 1991 Dickson Waters It is not, however, clear how these methods can be applied to models that include a stochastic return on investment process of the type defined in

33 In Paulsen et al. 25 there is a discussion on using numerical methods to calculate the eventual ruin probability. Instead of using recursive formulas the focus in that paper is solving the integro-differential equation IDE , assuming λ R =. In Paper C there is a discussion of finite-difference methods for solving the equivalent partial integro-differential equation PIDE for ruin in finite time. Numerical examples are given for models fitted to different investment strategies different data. Integrals are evaluated using precalculated Gaussian quadrature rules, while the numerical differentiation is much like that in Halluin et al. 25. The claim process is fitted to a classic dataset of Danish fire insurance claims. The investment strategies considered are investing in U.S. treasury bills of 3 month maturity, U.S. Treasury bonds or in American stocks. For each of these investment strategies a geometric Brownian motion model GBM is fitted to historical data. We calculated ruin probabilities for several GBM investment models for some jump-diffusion investment models. For the jump-diffusion investment models a few GBM investment models we used parameter values from Ramezani Zeng 27. The main finding is that for data covering the period the models fitted to investments in stocks lead to a ruin probability that is about twice as high as the ruin probabilities with the Cramér-Lundberg model or with the models fitted to investments in bonds. In contrast to this example we find that, when the models are fitted to returns of the S&P 5 for , the resulting ruin probabilities are slightly lower than for investments in bonds or the Cramér-Lundberg model. The results reflect quantify the relatively high volatility of stock prices since year 2. 2 Optimal dividend policy 2.1 De Finetti s dividend problem, dividend policy the value of an insurance company As described in Avanzi 29, during the first part of the twentieth century actuarial literature focused on minimizing the probability of ruin on an infinite time horizon in the Cramér-Lundberg model discussed in Section 1.1. On an infinite time horizon this assumes that insurance companies let their liquidity reserve grow without limit. Bruno de Finetti couldn t see why an 28

34 older company should hold more capital than a younger one bearing similar risks, only because it is older. As described in Avanzi 29 the goal of Bruno de Finetti was to propose an alternative formulation that would be sufficiently realistic tractable to study the practical problems regarding risk reinsurance translation in Avanzi 29. This led Bruno de Finetti to formulate in de Finetti 1957 the optimal dividends problem for an insurance company: Maximizing the expected sum of the discounted paid out dividends from time zero until ruin, often referred to as de Finetti s dividend problem. The rationale behind maximizing the expected sum of the discounted paid out dividends is based on the idea that the value of the company is given by this sum. It is remarked in Gordon 1959 that: The hypothesis that the investor buys the dividend when he acquires a stock seems intuitively plausible, because the dividend is literally the payment stream that he expects to receive. An alternative hypothesis discussed in Gordon 1959 is that the investor buys income per share, referred to as The Earnings Hypothesis. Based on some data on price earnings for companies in four different industries it is concluded in Gordon 1959 that the dividend hypothesis is correct regardless of whether the earnings hypothesis is correct. The only point at issue is whether the dividend hypothesis is unnecessary. A seemingly different opinion is expressed in the classical paper Miller Modigliani There it is argued that... there are no financial illusions in a rational perfect economic environment. Values there are determined solely by real considerations, in this case the earning power of the firm s assets its investment policy, not by how the fruits of the earning power are packaged for distribution. More precisely, the analysis in Miller Modigliani 1961 is done in discrete time with the following assumptions: In perfect capital markets, no buyer or seller... is large enough for his transactions to have an appreciable impact on the then ruling price. All traders have equal costless access to information about the ruling price all other relevant characteristics of shares. No... other transaction costs are incurred... there are no tax differentials either between distributed undistributed profits or between dividends capital gains. 29