Recursive Calculation of Finite Time Ruin Probabilities Under Interest Force

Recursive Calculation of Finite Time Ruin Probabilities Under Interest Force Rui M R Cardoso and Howard R Waters Abstract In this paper we consider a classical insurance surplus process affected by a constant interest force. We present numerical algorithms for the calculation of finite time ruin probabilities using a discrete time Markov chain to approximate the risk process. Based on this method, upper and lower bounds are also obtained. 1 Introduction We are interested in the following insurance surplus process: U(t) =ue δt + c s t t 0 e δ(t v) ds(v), t 0, where time is measured in some basic unit, which we shall refer to as a year, and: U(t) is the insurer s surplus at time t, u is the insurer s initial surplus, s t δ is the constant force of interest p.a. earned on the insurer s reserves, c is the rate of premium income p.a. for the insurer, = (exp(δt) 1)/δ, so that c s t is the accumulated premium income in the time interval [0,t], and, Department of Actuarial Mathematics & Statistics, Heriot-Watt University, Edinburgh EH14 4AS, Great Britain Support from CMA and Departamento de Matemática, FCT, Universidade Nova de Lisboa and Fundação para a Ciência e a Tecnologia - FCT/POCTI is gratefully acknowledged 1

S(v) is the aggregate claims in [0,v]. We assume that {S(t)} t=0 is a compound Poisson process. The Poisson parameter, and hence the expected number of claims each year, is λ, the (i.i.d.) individual claim amounts have cdf G(x) and pdf g(x) with G(0) = 0, so that all claim amounts are positive. The insurer s rate of premium income can be written c =(1+η)λm 1, where m k is the k th moment about 0 of the individual claim amount distribution and η is the insurer s premium loading factor. We assume, without loss of generality, that: λ =1=m 1 The time to ruin, T, for this process is ined as follows: { inf{t : U(t) < 0} T = if U(t) 0 for all t>0 and the finite time ruin probability, ψ(u, t), is ined as ψ(u, t) =Pr[T t]. (1) Our objective in this paper is to derive recursive numerical algorithms for the (approximate) calculation of, and bounds for, ψ(u, t). In the special case δ =0,{U(t)} t=0 is the classical insurance surplus process and recursive algorithms for the calculation of ψ(u, t) have been developed by De Vylder & Goovaerts (1988) and Dickson & Waters (1991). The key to these algorithms is that the time/amount plane is replaced by a rectangular grid and the continuous (time and amount) surplus process is approximated by a discrete (time and amount) process whose state space is a set of equally spaced points on the amounts axis. The probabilistic properties of this discrete approximating process are easier to derive than the properties of the original process. One of the key simplifying features of this approximating process is that it can be scaled so that in a single unit of time the premium income, and hence the maximum increase in the surplus, is 1. Dickson & Waters (1999) derived recursive algorithms for the calculation of ψ(u, t) for the general model (δ >0) using the discretisation ideas developed by De Vylder & Goovaerts (1988) and Dickson & Waters (1991) for the special case (δ = 0). A complication with this approach was that in any fixed interval of time the maximum increase in the surplus depends not only on the rate of premium income but also on the level of the surplus at the start of the interval. A consequence of this complication was that the resulting algorithms were very calculation-intensive and hence heavy on computer run time. 2

In this paper we use a different approach to derive algorithms to calculate ψ(u, t). Our approach approximates or bounds the continuous surplus process by discrete time Markov chains with countable state spaces, an idea which has been used by Dickson & Gray (1984) and Cardoso & Egídio dos Reis (2002) to calculate ruin probabilities for the classical surplus process, i.e. the special case δ =0. Numerical values for the probability of ultimate ruin, i.e. ψ(u, ), for the general model can be calculated using, for example, methods derived by Sundt & Teugels (1995) or by De Vylder (1996). In the special case of exponentially distributed claim amounts, Segerdahl derived an analytic formula for this probability. See Dickson & Waters (1999) for more details of these results. Numerical algorithms for the calculation of values of ψ(u, t) have been presented by Dickson & Waters (1999) and Brekelmans & De Waegenaere (2001). Exact solutions for special cases have been given by Albrecher et al (2001). In Section 2 we derive algorithms for the calculation of upper and lower bounds for ψ(u, t). We also present an algorithm for the (approximate) calculation of this probability. In Section 3 we describe a truncation procedure which can be used to reduce the number of calculations required to produce the approximation or the bounds, while keeping the error introduced by this procedure within specified limits. In Section 4 we give some numerical examples. 2 The Markov Chain Algorithm 2.1 The accumulated aggregate claim distribution Let h be some small unit of time and let S(h) denote the accumulated aggregate claims in the time interval [0,h], with cdf F (x), so that: [ Nh ] F (x) =P[ S(h) x]=p e δ(h Ti) X i x where: i=1 N h X i T i is the number of claims in [0,h], which has a Poisson distribution with mean λh, is the amount of the i th claim, and, is the time of the i th claim. 3

We are going to approximate and bound the continuous time surplus process by discrete time processes ined at the time points 0,h,2h,...,sothe smaller the value of h, the better the approximation and the bounds are likely to be. In what follows, it will be convenient for t to be an integer multiple of h, so we will assume that t/h is some positive integer, say, K. For n =0,1,2,..., let p n denote the probability of the event (N h = n). The following result follows from formulae in Karlin & Taylor (1975, pp128-131) and Ross (1996, pp393-396): F (x) = p G n n (x) (2) n=0 where G n is the n th convolution of G, and: h G(x) = 1 G ( xe δ(h u)) du h 0 Note that G is the cdf of X i e δ(h τ i), where τ i is uniformly distributed on (0,h). Now let β be some large positive number. We can approximate G by a discrete distribution function, Gd, and bound it by discrete distribution functions G d and G d, so that: G d (x) G(x) G d (x) with all three of these discrete distributions having masses only at the points 0, 1/β, 2/β,... For example, Gd could be constructed using the method of De Vylder & Goovaerts (1988), and the( two bounding distributions could be constructed by concentrating the mass G( n+1 ) G( ) n) at either (n +1)/β β β or n/β, as appropriate. The larger the value of β, the closer the bounding distributions will be, and the closer the approximating distribution is likely to be, to G. Formula (2) shows that F has a compound Poisson distribution. We can now use the recursion formula due to Panjer (1981), in conjunction with formula (2), replacing G by G d, Gd and G d in turn, to calculate cdfs F d, Fd and F d such that: F d (x) F (x) F d (x) and F d (x) F (x) 4

This generalised version of Panjer s recursion goes back to Boogaert & De Waegenaere (1990) and was used by Brekelmans & De Waegenaere (2001). From the way in which F d ( F d ) has been constructed, it can be seen that a discrete time surplus process ined at the time points 0,h,2h,..., starting at u, having the same rate of premium income and interest on reserves as the continuous time surplus process and for which the accumulated aggregate claims in a time interval of length h has cdf F d ( F d ) will, with probability 1, always be below (above) the surplus process at these time points. 2.2 Construction of the Markov Chains We are going to construct three discrete time Markov chains with countable state spaces which approximate and bound below and above the continuous time surplus process until ruin occurs. These discrete time processes, denoted {Z n } n=0, {Z n } n=0 and {Z n } n=0, will be ined at times 0,h,2h,..., where h is as in Section 2.1. All three chains start from u, the initial value of the surplus process, so that: P[Z 0 = u] =P[Z 0 =u]=p[z 0 =u] = 1 (3) and for all chains zero is an absorbing state (although they are allowed to start from u = 0), so that for n 1: P[Z n+1 =0 Z n =0]=P[Z n+1 =0 Z n =0]=P[Z n+1 =0 Z n =0]=1 From step 1 onwards, the state space for the two bounding chains is a set of non-negative and increasing numbers {x j },j =0,1,... ined as follows: x 0 = 0 x j+1 = x j e δh + c s h for j =0,1,... Note that if the surplus process were to start at x j, then in a time interval of length h it would reach x j+1 if there were no claims in this interval. For i, n 1 and 1 j i + 1, the transition probabilities for {Z n } n=0 are specified as follows: P[Z n+1 =0 Z n =x i ] = 1 F d (x i+1 ) P[Z n+1 = x j Z n = x i ] = F d ((x i+1 x j 1 ) ) F d ((x i+1 x j ) ) To complete the inition of {Z n } n=0, we need to specify the transition probabilities at the first step. If u = x i for some non-negative integer i, these 5

transition probilities are as above with n = 0. Suppose u (x i 1,x i ). Then for 1 j i +1: P[Z 1 =0] = 1 F d ((ue δh + c s h ) ) P[Z 1 = x j ] = F d ((ue δh + c s h x j 1 ) ) F d ((ue δh + c s h x j ) ) Intuitively, this Markov chain behaves like a discrete time surplus process ined at the time points 0,h,2h,..., starting at u, having the same rate of premium income and interest on reserves as the continuous time surplus process and for which the accumulated aggregate claims in a time interval of length h has cdf F d with the extra features that when this process takes a value in an interval (x j 1,x j ], its value is adjusted upwards to x j and that the process stops, taking the value 0, following the first negative value of the surplus. The Markov chain {Z n } n=0 is ined similarly to {Z n } n=0, the differences being that the cdf F d is used in its inition, rather than F d, and at each step values in the interval [x j 1,x j ) are adjusted downwards to x j 1. Formally, this chain is specified as follows. For i, n 1 and 1 j i +1: P[Z n+1 =0 Z n =x i ] = 1 F d (x i+1 x 1 ) P[Z n+1 = x j Z n = x i ] = F d (x i+1 x j ) F d (x i+1 x j+1 ) If u = x i for some non-negative integer i, the first step transition probabilities are as above with n =0. Ifu (x i,x i+1 ), then for 1 j i +1: P[Z 1 = 0] = 1 F d (ue δh + c s h x 1 ) P[Z 1 = x j ] = F d (ue δh + c s h x j ) F d (ue δh + c s h x j+1 ) The Markov chain {Z n } n=0 is ined as follows. For j =1,2,..., let M j =(x j 1 +x j )/2, so that M j is the mid-point of the interval x j 1 x j, and ine M 0 =0. Forn 1the state space for Z n is the set of real numbers 0,M 1,M 2,... Intuitively, this chain will move from M i to M j if, starting from M i, the surplus process time h later has a value in the interval (x j 1,x j ]. More precisely, for i, n 1 and 1 j i: P[Z n+1 = M i+1 Z n = M i ] = F d ((M i e δh + c s h x i ) ) P[Z n+1 = M j Z n = M i ] = F d ((M i e δh + c s h x j 1 ) ) F d ((M i e δh + c s h x j ) ) P[Z n+1 =0 Z n =M i ] = 1 F d ((M i e δh + c s h ) ) Define the non-negative integer γ(u)tobe0ifu= 0 and to be i if u (x i 1,x i ], for some positive integer i. Then the first step transition probabilities are 6

ined as follows for 1 j i: P[Z 1 =0] = 1 F d ((ue δh + c s h ) ) P[Z 1 = M γ(u)+1 ] = F d ((ue δh + c s h x γ(u) ) ) P[Z 1 = M j ] = F d ((ue δh + c s h x j 1 ) ) F d ((ue δh + c s h x j ) ) For the Markov chain {Z n } n=0 (respectively, {Z n } n=0 and {Z n } n=0), let P and P (u) (respectively P and P (u) and P and P (u) ) denote the matrix of transition probabilities after the first step and the vector of transition probabilities at the first step. For integers n 1 and j 0, let p (n) (u) j (respectively p(n) (u) j and p (n) (u) j ) denote the probability of the event (Z n = M j ) (respectively (Z n = x j ) and (Z n = x j )). For positive integers m and n, let p (n) ij (respectively p (n) and ij p (n) ij ) denote the probability of the event (Z m+n = M j Z m = M i ) (respectively (Z m+n = x j Z m = x i ) and (Z m+n = x j Z m = x i )). It follows from the inition of x 1 that if at any time τ, U(τ) < 0, then U(τ + h) cannot be greater than or equal to x 1. It then follows from our constructions that: (Z K =0) (U(τ)<0 for some τ,0 <τ t) (Z K =0) (Recall that K = t/h.) Hence: p (K) ψ(u, t) p(k) and: p (K) (u )0 ψ(u, t) The probability p (K) is the first entry in the vector P (u) P K 1. A recursive formula for this probability, in the spirit of a formula due to De Vylder & Goovaerts (1988, formula (9)) can be written as follows: γ(u)+1 p (K) =p + j=1 γ(u)+1 p (u) j p (K 1) j 0 = j=0 p (u) j p (K 1) j 0 (4) Similar recursive formulae can be written down for p (K) and p(k). 2.3 Comments on the method In principle, the vectors P (u), P (u), P (u) and matrices P, P, P have infinite dimension. However, since the Markov chains can move up at most one state 7

in a single time interval, only a finite number of entries in each vector can have non-zero values. For example, P (u) can have non-zero values in only its first (γ(u) + 2) entries and P (u) P n can have non-zero entries only in its first (γ(u) + 2 + n) entries. In essence, our method replaces the continuous time surplus process by discrete time processes constrained to take values within a countable set. Expressed in this way, this is similar to the method of Dickson & Waters (1999). An important difference between the two methods is that the countable set of values used by Dickson & Waters (1999) was equally spaced whereas in our method the intervals (M j M j 1 ) and (x j x j 1 ) are increasing with j, so that the discretisation by amount becomes coarser as the surplus increases. A coarser discretisation results in fewer states for our processes and hence faster computations, whereas a finer discretisation results in greater accuracy. Our method is intuitively appealing in that it uses a finer discretisation for values of the surplus more likely to lead to ruin. The larger the value of β and the smaller the value of h, the more accurate our approximations are likely to be. These two parameters can be set independently, but in our numerical examples we have always set: h = 1 (1 + η)β so that the premium income in a time interval of length h is 1, before accumulating with interest. This relationship means that an increase in β automatically results in a decrease in h. Once the parameter h has been set, the value of x j, j =1,2,..., cannot be greater than: x j 1 e δh + c s h If it were, Z n, starting from x j 1, could never reach x j and so this process could never exceed its starting value. 3 Truncation of the numerical algorithms In their paper, De Vylder & Goovaerts (1988) show how the number of calculations in their recursive formula corresponding to our formula (4) can be reduced by a truncation procedure in such a way that the error introduced can be controlled. We can use these ideas in connection with formula (4). We will do this in two stages. 8

3.1 Truncation - Stage 1 Let ɛ (0 <ɛ<1) be fixed. For each positive integer i, ine the integer valued function α(i) as follows: { } j α(i) = min j : p ik ɛ Similarly, for the initial surplus u, we ine α((u)) as follows: { } j α((u)) = min j : p (u) k ɛ We ine ɛ p (u)j and ɛ p ij for i =1,2,... and j =0,1,... as follows: k=0 k=0 ɛp (u) j ɛp (u) j ɛp ij ɛp ij = 0 if j<α((u)) = p (u) j if j α((u)) = 0 if j<α(i) = p ij if j α(i) Intuitively, for each state i, and the initial surplus u, we are setting to zero the probability of reaching a state if the sum of the probabilities of reaching that, or a lower, state is less than ɛ. We now ine ɛ p (n) ij and ɛ p (n) recursively for n =2,3,... as follows: ɛp (n) i 0 ɛp (n) = = i+1 j=α(i) γ(u)+1 j=α((u)) ɛp ij ɛ p (n 1) j 0 ɛp (u) j ɛ p (n 1) j 0 Calculating ɛ p (n) (u) j should be quicker than calculating p(n) (u) j since the summation for the former is, possibly, over a shorter range of values. Our first result tells us that using ɛ p (n) (u) j in place of p(n) (u) j introduces an error which can be quantified and which grows only linearly with n. RESULT 1: For n, i =1,2,... we have: 0 p (n) i 0 ɛp (n) i 0 nɛ (5) 0 p (n) ɛp (n) nɛ (6) 9

Proof: Consider first the case n =1. Ifα((u)) = 0, then ɛ p = p. On the other hand, if α((u)) > 0, then p <ɛand ɛ p = 0. In either case, formula (6) holds. The proof of formula (5) for n = 1 is similar. Now suppose the result holds for some positive integer n. We have: ɛp (n+1) = p (n+1) = γ(u)+1 j=α((u)) γ(u)+1 j=0 ɛp (u) j ɛ p (n) j 0 p (u) j p (n) j 0 We have: p ɛ p by construction p (u) j ɛ p (u) j by construction p (n) j 0 ɛ p (n) j 0 by assumption Hence: Also: p (n+1) ɛ p (n+1) = p (n+1) γ(u)+1 j=α((u)) ɛ p (n+1) p (u) j [p (n) j 0 ɛp (n) nɛ + ɛ =(n+1)ɛ j 0 ]+ α((u)) 1 j=0 p (u) j p (n) j 0 Hence, formula (6) holds by induction. The proof of formula (5) is similar. 3.2 Truncation - Stage 2 We can extend the ideas of the previous section by, additionally, setting to zero the probability of going from any state i to 0 if this is less than ɛ. We proceed recursively as follows. For n, i =1,2,...: ɛ p i0 ɛ p i 0 ɛ p (n+1) i 0 ɛ p (n+1) i 0 = ɛ p i 0 if this is ɛ = 0 otherwise i+1 = ɛp ij ɛ p (n) j0 j=α(i) = 0 otherwise if this is ɛ 10

Finally, ine: and for n =1,2,...: ɛ p = ɛ p as before ɛ p (n+1) = γ(u)+1 j=α((u)) ɛp (u) j ɛ p (n) j 0 RESULT 2: For n, i =1,2,... we have: 0 p (n) i 0 ɛ p (n) i 0 2nɛ (7) 0 p (n) ɛ p (n) 2nɛ (8) Proof: Consider first the case n = 1. Formula (8) follows from formula (6) since: ɛ p = ɛ p Formula (7) is proved using Result 1 as follows: p i 0 ɛ p i 0 ɛ p i 0 p i 0 ɛ p i 0 = p i 0 ɛ p i 0 + ɛ p i 0 ɛ p i 0 ɛ + ɛ =2ɛ Now suppose the result holds for some positive integer n. We will show that formula (8) holds for n + 1, and hence, by induction, for all n. The proof of formula (7) is similar. p (n+1) = ɛ p (n+1) = γ(u)+1 j=0 γ(u)+1 j=α((u)) p (u) j p (n) j 0 ɛp (u) j ɛ p (n) j 0 We have: p ɛ p by construction p (u) j ɛ p (u) j by construction p (n) j 0 ɛ p (n) j 0 by assumption Hence: p (n+1) ɛ p (n+1) 11

Also: p (n+1) ɛ p (n+1) = γ(u)+1 j=α((u)) p (u) j [p (n) j 0 ɛ p (n) 2nɛ + ɛ 2(n +1)ɛ j 0 ]+ α((u)) 1 j=0 p (u) j p (n) j 0 3.3 Comments on the truncation In this section we showed how to reduce the number of calculations in the computation of approximations to the ruin probability and, consequently, the computational effort by using a truncation procedure. This procedure can also be applied in the case of the lower and upper bounds obtaining similar results to the ones presented. From the results derived we can also control the error due to this truncation. Furthermore, we conclude that the Markov chain methods presented are stable. In Figures 1 and 2 we compare the approximations of ψ(u, t) without the truncation procedure (solid line) and with the truncation procedure (dashed line). The dotted line gives the values of the latter approximations plus Kɛ. We considered β = 100 and η =0.1. In Figure 1 the individual claim amounts are exponentially distributed with mean 1 and in Figure 2 they follow a Pareto(3,2) distribution. In the first case 30 t 40, u = 5 and δ =0.1. For the second one 35 t 40, u = 10 and δ =0.075. For both cases we set ɛ =0.1 10 7 (so, at most, the error is 0.44 10 4 ). We observe that the two approximations are close. Of course, we can control the closeness of the values, but the point is that the difference between the two approximations is much smaller than the error bound. We found this pattern in other cases studied. 4 Examples We will now present numerical results in cases where the individual claim amount have an exponential distribution with mean 1 or a Pareto(3,2) distribution. In all examples we considered η =0.1 and β = 100. For the case δ = 0 we do not present any values although the approximations computed by the Markov chain method have the same accuracy as the approximations given by the algorithm of Dickson & Waters (1991, section 8). The values obtained for different combinations of u, δ and t are summarized in the following tables. The lower and upper bounds are denoted 12

0.09370 0.09375 0.09380 0.09385 without truncation with truncation truncation plus error 30 32 34 36 38 40 Figure 1: Approximations to ψ(5,t) with and without truncation for exponential(1) claim size distribution, with δ = 0.1 13

0.04700 0.04705 0.04710 0.04715 without truncation with truncation truncation plus error 35 36 37 38 39 40 Figure 2: Approximations to ψ(10,t) with and without truncation for Pareto(3,2) claim size distribution, with δ = 0.075 14

by Lu and Ul, respectively, and the approximations denoted by A. Where it is possible we compare these numbers with the values found in Dickson & Waters (1999), denoted by DW, and in Brekelmans & De Waegenaere (2001). These latter authors also split the time horizon into smaller intervals of equal length and they derived an algorithm to determine a lower and a upper bound, denoted by LB and UB, respectively, for ψ(u, t) by assuming that premium income is received, respectively, at the beginnning and at the end of each interval. Then by averaging these bounds they get an approximation (AVG) to the ruin probability. They also get simulated values (SIM) of the ruin probabilities (see Brekelmans & De Waegenaere (2001, subsection 4.1)) In tables 1 and 2, the values from Brekelmans & De Waegenaere (2001) were calculated with a step size h =0.01 while in tables 4 and 5, h =0.009. Note that the values produced by Dickson & Waters (1999) and the ones obtained by our methods were calculated considering β = 100, i.e., a step size h = 0.00909091. Table 1: Approximations to, and bounds for, ψ(0,t) - exponential claims δ t =1 t=5 t=10 t=20 t=40 Lu Ul 0.4582 0.4652 0.7108 0.7200 0.7736 0.7837 0.8154 0.8265 0.8406 0.8529 0.01 A 0.4633 0.7166 0.7797 0.8219 0.8476 DW 0.4615 0.7151 0.7785 0.8209 0.8467 LB UB 0.4596 0.4629 0.7014 0.7040 0.7538 0.7560 0.7806 0.7825 AVG 0.4612 0.7027 0.7549 0.7816 0.05 SIM 0.4613 0.7033 0.7556 0.7821 Lu Ul 0.4573 0.4642 0.6980 0.7074 0.7498 0.7603 0.7760 0.7877 0.7836 0.7962 A 0.4615 0.7033 0.7556 0.7823 0.7904 DW 0.4598 0.7019 0.7544 0.7812 0.7893 Lu Ul 0.4554 0.4623 0.6814 0.6910 0.7204 0.7312 0.7329 0.7447 0.7341 0.7461 0.1 A 0.4594 0.6867 0.7264 0.7393 0.7407 DW 0.4577 0.6852 0.7250 0.7380 0.7393 15

Table 2: Approximations to, and bounds for, ψ(5,t) - exponential claims δ t =1 t=5 t=10 t=20 t=40 Lu Ul 0.0133 0.0140 0.0946 0.1002 0.1707 0.1816 0.2548 0.2728 0.3231 0.3487 0.01 A 0.0136 0.0973 0.1760 0.2637 0.3359 DW 0.0135 0.0970 0.1757 0.2633 0.3354 LB UB 0.0126 0.0127 0.0778 0.0781 0.1259 0.1264 0.1627 0.1632 AVG 0.0127 0.0780 0.1262 0.1630 0.05 SIM 0.0127 0.0780 0.1263 0.1631 Lu Ul 0.0123 0.0130 0.0755 0.0805 0.1218 0.1308 0.1565 0.1698 0.1686 0.1844 A 0.0126 0.0778 0.1261 0.1629 0.1762 DW 0.0126 0.0778 0.1260 0.1628 0.1761 Lu Ul 0.0112 0.0119 0.0566 0.0608 0.0794 0.0862 0.0887 0.0971 0.0897 0.0983 0.1 A 0.0115 0.0586 0.0826 0.0927 0.0938 DW 0.0115 0.0587 0.0827 0.0928 0.0939 Table 3: Approximations to, and bounds for, ψ(10,t) - exponential claims δ t =1 t=5 t=10 t=20 t=40 Lu Ul 0.0002 0.0004 0.0077 0.0086 0.0250 0.0278 0.0589 0.0662 0.1001 0.1145 0.01 A 0.0003 0.0081 0.0263 0.0625 0.1072 DW 0.0003 0.0081 0.0264 0.0625 0.1072 Lu Ul 0.0002 0.0003 0.0046 0.0052 0.0114 0.0130 0.0188 0.0218 0.0219 0.0258 0.05 A 0.0003 0.0049 0.0122 0.0202 0.0237 DW 0.0003 0.0049 0.0122 0.0202 0.0238 Lu Ul 0.0002 0.0003 0.0025 0.0029 0.0044 0.0052 0.0054 0.0064 0.0055 0.0066 0.1 A 0.0002 0.0027 0.0048 0.0059 0.0060 DW 0.0002 0.0027 0.0048 0.0059 0.0060 16

Table 4: Approximations to, and bounds for, ψ(0,t) - Pareto claims δ t =1 t=5 t=10 t=20 t=40 LB UB 0.4182 0.4227 AVG 0.4204 0.025 SIM 0.4209 Lu Ul 0.4155 0.4239 0.6542 0.6645 0.7202 0.7312 0.7642 0.7760 0.7878 0.8005 A 0.4211 0.6604 0.7267 0.7709 0.7950 DW 0.4191 0.6587 0.7253 LB UB 0.4172 0.4217 AVG 0.4195 0.05 SIM 0.4199 Lu Ul 0.4148 0.4231 0.6468 0.6572 0.7061 0.7173 0.7398 0.7519 0.7516 0.7645 A 0.4201 0.6528 0.7124 0.7466 0.7587 DW 0.4181 0.6512 0.7110 LB UB 0.4162 0.4207 AVG 0.4185 0.075 SIM 0.4188 Lu Ul 0.4139 0.4222 0.6393 0.6497 0.6920 0.7034 0.7168 0.7291 0.7221 0.7349 A 0.4190 0.6453 0.6984 0.7236 0.7292 DW 0.4170 0.6436 0.6969 LB UB 0.4152 0.4196 0.6339 0.6389 0.6806 0.6856 AVG 0.4174 0.6364 0.6831 0.1 SIM 0.4179 0.6375 0.6844 Lu Ul 0.4130 0.4213 0.6318 0.6423 0.6782 0.6896 0.6959 0.7082 0.6981 0.7107 A 0.4180 0.6377 0.6846 0.7027 0.7051 DW 0.4160 0.6360 0.6830 17

Table 5: Approximations to, and bounds for, ψ(5,t) - Pareto claims δ t =1 t=5 t=10 t=20 t=40 LB UB 0.1912 0.1920 AVG 0.1916 0.025 SIM 0.1917 Lu Ul 0.0255 0.0261 0.1171 0.1212 0.1879 0.1956 0.2581 0.2703 0.3052 0.3216 A 0.0258 0.1190 0.1916 0.2640 0.3132 DW 0.0258 0.1190 0.1915 LB UB 0.1649 0.1656 AVG 0.1653 0.05 SIM 0.1654 Lu Ul 0.0249 0.0255 0.1068 0.1107 0.1619 0.1689 0.2058 0.2162 0.2235 0.2361 A 0.0251 0.1085 0.1653 0.2108 0.2296 DW 0.0251 0.1085 0.1652 LB UB 0.1421 0.1427 AVG 0.1424 0.075 SIM 0.1426 Lu Ul 0.0242 0.0249 0.0971 0.1008 0.1393 0.1456 0.1651 0.1738 0.1710 0.1806 A 0.0245 0.0988 0.1422 0.1692 0.1756 DW 0.0245 0.0989 0.1423 LB UB 0.0239 0.0240 0.0900 0.0904 0.1227 0.1232 AVG 0.0240 0.0902 0.1229 0.1 SIM 0.0239 0.0903 0.1230 Lu Ul 0.0236 0.0243 0.0884 0.0919 0.1201 0.1257 0.1346 0.1418 0.1365 0.1440 A 0.0239 0.0900 0.1227 0.1380 0.1401 DW 0.0239 0.0901 0.1228 Table 6: Approximations to, and bounds for, ψ(10,t) - Pareto claims δ t =1 t=5 t=10 t=20 t=40 Lu Ul 0.0051 0.0053 0.0297 0.0307 0.0573 0.0600 0.0935 0.0991 0.1232 0.1322 0.025 A 0.0052 0.0302 0.0586 0.0962 0.1276 DW 0.0052 0.0302 0.0586 Lu Ul 0.0050 0.0051 0.0257 0.0267 0.0446 0.0468 0.0627 0.0666 0.0707 0.0758 0.05 A 0.0051 0.0262 0.0456 0.0646 0.0731 DW 0.0051 0.0262 0.0457 Lu Ul 0.0048 0.0050 0.0224 0.0232 0.0351 0.0368 0.0437 0.0464 0.0456 0.0487 0.075 A 0.0049 0.0228 0.0359 0.0450 0.0471 DW 0.0049 0.0227 0.0359 Lu Ul 0.0047 0.0048 0.0195 0.0203 0.0279 0.0294 0.0319 0.0338 0.0324 0.0344 0.1 A 0.0047 0.0198 0.0286 0.0328 0.0333 DW 0.0047 0.0198 0.0286 18

Table 7 shows approximations to ψ(u, t) for large values of t considering the individual claim amounts exponentially distributed with mean 1. The values in column (t = ) are exact and provided by Segerdahl s formula. Table 7: Approximations to ψ(u, t) for large values of t - exponential claims u δ t =50 t= 100 t = 200 t = 300 t = 400 t = 500 t = 0.01 0.8529 0.8620 0.8640 0.8641 0.8641 0.8641 0.8639 0 0.05 0.7909 0.7912 0.7912 0.7910 0.1 0.7407 0.7407 0.7404 0.01 0.3528 0.3837 0.3907 0.3911 0.3911 0.3911 0.3909 5 0.05 0.1771 0.1775 0.1775 0.1776 0.1 0.0938 0.0938 0.0941 0.01 0.1199 0.1453 0.1516 0.1519 0.1519 0.1519 0.1521 10 0.05 0.0240 0.0241 0.0241 0.0241 0.1 0.0060 0.0060 0.0060 Albrecher et al (2001) investigated when it is suitable to represent the probability of survival, 1 ψ δ (u, t), as gamma series. In particular, they derived exact analytical solutions for exponentially distributed claim sizes with mean 1/θ and for integer values of λ/δ. From their paper, second example in Section 3, we find that, for λ =2δ, with ψ(u, t) =1 b 0 (t) (1 e θu )b 1 (t) (1 e θu θue θu )b 2 (t) (9) b 0 (t) = θ2 c 2 D+e R1t (δ 2 θc + D(δθc + δ 2 ) δ 3 )+e R2t ( δ 2 θc + δ 3 + D(δθc + δ 2 )) D(θ 2 c 2 +2δθc +2δ 2 ) b 1 (t)= δ(2dθc e R1t ( 4δ 2 δθc + Dθc) e R2t (4δ 2 + δθc + Dθc)) D(θ 2 c 2 +2δθc +2δ 2 ) b 2 (t)= δ2 (e R1t (2θc +3δ+D) 2D+e R2t ( 2θc 3δ + D)) D(θ 2 c 2 +2δθc +2δ 2 ) where R 1 = 1 2 (2θc +3δ D), R 1 = 1 2 (2θc +3δ+D) and D = δ(4θc + δ). In Table 8 and Figures 3 to 5 we compare the values from our algorithms with the exact ones provided by (9). In this case we considered δ =0.5 and θ = 1. We can observe that the approximations are close to the exact values of the ruin probability. 19

Table 8: Approximations to, and bounds for, ψ(u, t), with δ =0.5 - exponential claims u t =1 t=5 t=10 t=20 Lu Ul 0.4382 0.4450 0.5628 0.5726 0.5643 0.5742 0.5643 0.5742 0 A 0.4421 0.5686 0.5701 0.5701 Exact 0.4417 0.5679 0.5694 0.5694 Lu Ul 0.0056 0.0062 0.0093 0.0105 0.0094 0.0105 0.0094 0.0106 5 A 0.0058 0.0098 0.0099 0.0099 Exact 0.0058 0.0098 0.0098 0.0098 Lu Ul 0.000054 0.000064 0.000098 0.000118 0.000099 0.000119 0.000099 0.000120 10 A 0.000059 0.000107 0.000108 0.000108 Exact 0.000058 0.000106 0.000107 0.000107 0.0 0.2 0.4 0.6 0.8 L-u A Exact U-l 0 5 10 15 20 Figure 3: Approximations to, and bounds for, ψ(0,t) with δ =0.5 - exponential(1) claims 20

0.0 0.005 0.010 0.015 L-u A Exact U-l 0 5 10 15 20 Figure 4: Approximations to, and bounds for, ψ(5,t) with δ =0.5 - exponential(1) claims 21

0.0 0.00005 0.00010 0.00015 L-u A Exact U-l 0 5 10 15 20 Figure 5: Approximations to, and bounds for, ψ(10,t) with δ =0.5 - exponential(1) claims 22

5 References Albrecher, H., Teugels, J. L., and Tichy, R. F. (2001). On a gamma series expansion for the time-dependent probability of collective ruin. Insurance: Mathematics and Economics, 29:345 355. Boogaert, P. and De Waegenaere, A. (1990). Macro-economic version of a classical formula in risk theory. Insurance: Mathematics and Economics, 9:155 162. Brekelmans, R. and De Waegenaere, A. (2001). Approximating the finitetime ruin probability under interest force. Insurance: Mathematics and Economics, 29:217 229. Cardoso, R. M. R. and Egídio dos Reis, A. D. (2002). Recursive calculation of time to ruin distributions. Insurance: Mathematics and Economics, 30:219 230. De Vylder, F. E. (1996). Advanced risk theory. Editions de l Université de Bruxelles. De Vylder, F. E. and Goovaerts, M. J. (1988). Recursive calculation of finite time survival ruin probabilities. Insurance: Mathematics and Economics, 7:1 8. Dickson, D. C. M. and Gray, J. R. (1984). Approximations to ruin probability in the presence of an upper absorbing barrier. Scandinavian Actuarial Journal, 105-115. Dickson, D. C. M. and Waters, H. R. (1991). Recursive calculation of finite time survival probabilities. Astin Bulletin, 21:199 221. Dickson, D. C. M. and Waters, H. R. (1999). Ruin probabilities with compounding assets. Insurance: Mathematics and Economics, 25:49 62. Karlin, S. and Taylor, H. M. (1975). A first course in stochastic processes. Academic Press, San Diego, 2nd edition. Panjer, H. H. (1981). Recursive calculation of a family of compound distributions. Astin Bulletin, 12:22 26. 23

Ross, S. M. (1996). Stochatic processes. Wiley, New York, 2nd edition. Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance: Mathematics and Economics, 16:7 22. 24