Volume 13, No. 1 June 1982 A S T I N B U L L E T I N A Journal of the International Actuarial Association :ROTIDE :STNETNOC Peter ter Berg Editorial i Netherlands H. H. REJNAP and G. E TTOMLLIW Rccursions for Compound Distributions 1 LAIROTIDE :DRAOB C S. TAPIERO The Optimal Control of a Jump Mutual Insurance Hans Buhlmann Process 31 Switzerland F. DE VYLDER, M. STREAVOOG and N. ED LIRP Marc Goovaerts Bounds on Modified Stop-Loss Premiums in Case of Belgium Known Mean and Variance of the Risk Variable 32 H. NESNAH-UALMAR ,Iacques Janssen Belgium An Application of Crcdlbdlty Theory to Solvency Margins 73 William S. Jewell ._F KREMER USA Rating of Largest Claims and ECOMOR-Relnsurance TreaDes for Large Portfolios 74 Jan Jung Sweden E. REMERK A Characterization of the Esscher-Transformatlon 75 Ragnar Norberg Norway Book Reviews 16 LeRoy J. Simon USA I IETO LTD EDITORIAL POLICY ASTIN BULLETIN started in 1958 as a journal providing an outlet for actuarml studies in non- hfe insurance Snnce then a well-established non-hfe methodology did result, which is also applicable to other fields of insurance For that reason AS'FIN BULLETIN will pubhsh papers written from any quantitative point of view -- whether actuarial, econometric, engineering, mathematical, statzstlcal, etc -- attacking theoretic and applied problems m any field, faced with elements of insurance and risk. AS'FIN BULLETIN appears twice a year. Each issue consnstmg of about 80 pages MEMBERSHIP ASTIN is a section of the lnternauonal Actuarial Assocmtton (IAA). Membership is open automatically to all IAA members and under certain conditions to non-members also Applications for membership can be made through the National Correspondent or, in the case of countries not represented by a nauonal correspondent, through the Secretary: F E GUASCHI Moorfields House, Moorfields, London EC2Y 9AL England. Members receive ASTIN BULLETIN free of charge SUBSCRIPTION AND BACK ISSUES ASTIN BULLETIN SI pubhshed for ASTIN by Tneto Ltd, Bank House, 8a Hdl Road, Clevedon, Avon BS21 7HH, England All queries and communications concermng subscriptions, including claims and address changes, and concermng back issues should be sent to T~eto Ltd The current subscnptuon or back issue price per volume of 2 Issues including postage is £15.00 Copyrught )c( 1982 T~eto Ltd PRINTEDIN(;RI-ATBRIIAINbyJ W ARROWSMITHLTD BRISTOL EDITORIAL As we all know, it is hard for people outside insurance circles to have an easy opportunity to read our Astin Bulletin. One of the major reasons for this unhappy state of affairs was the lack of promotion outside traditional insurance circles and an appropriate handling of subscriptions and orders for back-issues. As one of the largest journals on quantitative modelling in insurance, we thought it our responsibility to make our journal more accessible for non-actuarial people. In order to achieve all this, it appeared to be necessary to change the printers. As a result of this, we have chosen Tieto Ltd, as the firm which will be responsible for the future publishing and printing of the Astin Bulletin, starting with this ~ssue. This also applies to back-issues. From now on, all quantitative people outside insurance, such as econometrictans, operations researchers, time series and sampling theorists, as well as other statisticians, will have the opportunity to read the Astin Bulletin and to learn about an inspiring field of application where people like Crarn~r and de Finetti have been pioneers: insurance. In the past, most papers published in Astln Bulletin had their emphasis on probability theory, neglecting issues of actual measurement and taking into account real market conditions which belonged to economics. From now on, however, we may expect papers to be more balanced with regard to theory and measurement as well as the activities formed by specification, estimation, testing and decision making. Last but not least, we are glad to inform you that the Boleslaw Monic Fund Foundation will award prizes of Dfl. 2500, Dfl. 1500 and Dfi. 1000 for the three best articles published in the (re)insurance press, in the period 1982-1983, on the subject of the current problems facing non-life reinsurers and their solutions. So, by having your papers on that subject published in journals like Astin Bulletin, you will automatically become part of the competition for these three prizes, the decision of which will be announced in the early part of 1984. P. TER BERG pv JOURNALOF TIM SERIES ANALYSIS A JOURNAL SPONSORED BY EHT BERNOULLI SOCIETY ROF MATHEMATICAL STATISTICS AND PROBABILITY This is rapidly becoming the foremost journal in the field of Time Series Analysis and related studies. Volume 3 1982 subscriptions (4 issues) £25.00 (US$55.00) Individuals subscribing for their own use may do so by sending a personal cheque for £8.50 (US$19.00). Recent papers include: K. S. Lii, K. N. Helland and M. Rosenblatt: Estimating Three-dimensional Energy Transfer in Isotropic Turbulence T. Ozaki: The Statistical Analysis of Perturbed Limit Cycle Processes Using Nonlinear Time Series Models Jack H. W. Penm and R. D. Terrell. On the Recursive Fitting of Subset Autoregressions Peter Praetz: The Market Model, CAPM and Efficiency in the Frequency Domain M. N. Bhattacharyya: Lydia Pinkham Data Remodelled C. W. J. Granger: Acronyms in Time Series Analysis (ATSA) T. Hasan: Nonlinear Time Series Regression for a Class of Amplitude Modulated Cosinusoids B. G. Quinn and D. .F Nicholls: Testing for the Randomness of Autoregressive Coefficients H. Tong: A Note on Using Threshold Autoregressive Models for Multi-Step-Ahead Prediction of Cyclical Data If you feel the journal might be of interest, please send for a sample copy to: I IETO LTD Bank House, a8 Hill Road, Cievedon, Avon 12SB 7HH, dnalgnE RECURSIONS FOR COMPOUND DISTRIBUTIONS* BY H. H. PANJER AND G. E. WILLMOT University of Waterloo, Ontario, Canada I. INTRODUCTION Various methods for developing recursive formulae for compound distributions have been reported recently by PANJER (1980, including discussion), PANJER (1981), SUNDT and JEWELL (1981) and GERBER (1982) for a class of claim frequency distributions and arbitrary claim amount distributions. The recurslons are particularly useful for computational purposes since the number of calcula- tions required to obtain the distribution function of total claims and related values such as net stop-loss premiums may be greatly reduced when compared with the usual method based on convolutions. In this paper a broader class of claims frequency distributions Is considered and methods for developing recursions for the corresponding compound distribu- tions are examined. The methods make use of the Laplace transform of the density of the compound distribution. 2. THE CLAIM FREQUENCY DISTRIBUTION Consider the class of claim frequency distributions which has the property that successive probabilities may be written as the ratio of two polynomials. For convenience we write the polynomials in terms of descending factorial powers. For obvious reasons, only distributions on the non-negative integers are con- sidered. Hence, successive probabilities of claim frequencies are written as (Oro+Cr tn +ot2n12)+ • • ")pn = (~o+ril(n -- 1) (1) +/32(n - 1)12)+ • • ")P,-1, n = 1, 2, 3 ..... where n~k~=n(n-1)... (n-k +1). Note that the coefficients are only specified up to a multiplication constant. The class of probability distribution satisfying (1) is very broad and includes the following distributions: 1. Binomial: ao=0, al =q, rio=N-p, tir =-P. * This research was supported by the Natural Sciences and Engineering Research Councd of Canada. The authors are indebted to an anonymous referee for a number of comments that greatly xmproved this paper ASTIN BULLETIN loV ,31 oN 1 2 PANJER AND WILLMOT 2. Poisson: e -A/~, n Pn = n! 0~0 0) al= 1, 3o=h, 31 =0. : 3. Negative Binomial: a + n - 1) p, = p"q'~ n 0~O = 0) al = 1, 30=Pa, 1=p. 4. Hypergeometric: N M-N Or0 0) al = M-N-m + l, tO 2 ~- I~ : /3, = -(N + m - 1), 32=1. 5. Hyper-Poisson: nO p,=C (h + n - 1) (") ao=h -1, al = 1, 3o =0, 31 =0. 6. Logarithmic: _0n'4-1 P" (n + l)In(1-O) ao = 1, al = 1, flo = 0, fit = 0. 7. Waring: A( - a)(a + n - 1) (") P"= " (h+n) )1+"~ ao=h, al=l, 3o=a, 1=1. 8. Polya-Eggenberger (Negative Hypergeometric): N()o( -a -b ao=O, al=N+b-1, gO 2 ~" --1 30=aN, ~l=N-a-1, 32 = -1. RECURSIONS 3 9. Generalized Waring Distribution: F(a +n) F(b +a) F(a +b)F(a +n) P" n!F(a) F(b)F(a) F(a+b+a+n) ao=0, al=a+b+a, O~2=1 Bo=a(a+l), fll=a+~--2, fl2=1. The binomial, Poisson and negative binomial distributions have a natural appeal for actuaries in connection with contagion models (cf. BUHLMANN 1970). It is well known that the negative binomial also arises as a mixed Poisson distribution with a gamma mixing function. The Polya-Eggenberger (Negative Hypergeometric) arises as a mixed binomial with the parameter p mixed in accordance with a beta mixing function. This can be seen from the following (cf. SKELLAM (1948)): oI ~ N'x . N-,~ F(a +b) a-1 b-1 "P = ~ n }P q a~F )F('~ p q dp n)1-n-N+b()1-n+a( N n (\n-a)(N-n-b ) The generalized Waring distribution can arise as a mixed Poisson (cf. IRWIN, 1965, 1968, SEAL, 1978 or GOOVAERTS and VAN WOUWE 1981) or more simply as a mixed negative binomial with a beta mixing function. This can be seen from the following: ~p = fo~(a+n-l) p,q~ F(a +b) pa-lqb-1 n r(a)r(b) dp F(a +n) F(b +a) F(a +b)F(a +n) n!F(a) F(b)F(a) F(a+b+ce+n) " It is easy to construct other members of this family by appropriately choosing the coefficients in the polynomials m (1). Such claim frequency distributions may have finite range or infinite range. One obtains a finite range when inter alia the polynomial on the right-hand side of (1) has its smallest positive root at an integer. Many references about the listed distributions can be found in JOHNSON and KOTZ (1969). 3. TRANSFORM RELATIONSHIPS It is assumed that claim sizes are posmve so that the claim size distribution F(x) has support only on the non-negative real axis. It is further assumed that 4 PANJER AND WlLLMOT claim frequencies and amounts are mutually independent. Under these assump- tions the distribution function of aggregate claims s~ given by (2) G(x)= ~ pnF*"(x), x~O. rt=O Let jZ(t) and if(t) denote the Laplace (or Laplace-Stieltjes) transforms (3) (t) = Eve-'X = f e-~X dF(x) d and (4) ~,(t) = Ece-'X = j e-'X dG(x). Further let P(s) denote the probability generating function of claim frequencies (5) P(s)=EpsN = ~ p,s'. rico Then it is easily shown that (6) if(t) = P((t)). Successive differentiation of (5) results in (7) P'(s)=Y. ns "-' p,~ P"(s ) = ~Y n IZ~s" -2p, ptk(s ) = ~ n(t)s"-kp, where the square subscript brackets indicate derivatives. If the polynommls in (1) are of order k, then (8) ~,n ~'C p. = ~,(n - 1) l'C p.-~, n = 1, 2 ..... I 1 Multiplying the n th equation of (8) by s" and summing all equation results in ao(P(s )-po) + ~o lsP'(s ) + a 2s2 p"(s ) +" ' • + C~ksk pk (s ) (9) = s floP(s) + ~ lsP'(s) + B2s 2P"(s) +" " " + ~ks kpk(S). Successive differentiation of (6) results in (10) ~(t) = P(jZ(t)) ~'(t) = Jr'(t)P'(Jr(t)) ~"(t) = J~'(t)e'(p(t)) + jr'(t)Ze"(jr(t)) ~'"(t) = "'(t)e'(f(t))+ 3"(t)lr'(t)e"((t)) + P(t)3P"'((t)) RECURSIONS 5 ~r'3(t ) = ?t'3(t)p'(?(t)) + {4D~"(t)Dr(t) + 3?'(t)2}P"(?(t)) +6?'(t)f'(t)zP'"((t)) + ?(t)4P4l(t). Solving the successive equations for P, P', P", P"' and pI4~ respectively yields (dropping the argument t) (11) 1~ • e'(f) = ~' f,2. p,,(f) = p,3. p,,,(f) = ~,,,_ 3#=,~"- #3, - 3#22,g ' T,4 , p41(f) = ~4 6#2,~'"- 4#3, - 15##,if"- #4, - I0#2,#3, + 15#~,~' where fD3(/) #" (t) - ~~t ) ' TO obtain the desired relationship of the various transforms, the equations (11) are substituted into (9) with s =f(t). This results in (for k = 4): (12) ~o(~ - po) + a,#o,~' + a2#~o,Eg"- #z,ff ' +o~3hol{g "3 -.t -3h21g ° -. -h31-3#~ff'}+aaho,{g " °4 -4 -6h2~g " -m -4#3,- 15#~,#"-#4,- I02,#3, + 15#231g t} + +terms similar to the above in/33,/34}. Equation (12) will be used to develop recursions in the next section. The corresponding results for k = 1, 2 or 3 are obtained by setting the approprmte coefficients a, and/3, to zero 4. DEVELOPING RECURSIONS It is assumed that the claim size distribution F(x), with support on (0, oo), is of either the continuous type or the discrete type with jumps on the positive integers. In order to distinguish the two cases, the measure ~ will refer to Lebesgue or counting measure on (0, oo) as is done by SUNDT and JEWELL (1981). To set up a recurslon, equation (12) (or similar equation for k <4) is used. One of ~, if', if", if'" or ~t43 from the left-hand side of (12) is isolated and the resulting equation is inverted. Note that these exists one higher power of T on the right-hand side. Upon inversion, the result may revolve an auxiliary function h(x) whose transform #(t) is a function of the transforms h(t) is a function of the transforms/~,(t). The success of any of the resultant recursions depends on the ability to identify these auxiliary functions. Different possible values of k are now considered separately. Special attention is paid to special cases when certain of the coefficients a, are zero. 6 PANJER AND WILLMOT Case I: k = 1 Equation (12) reduces to 'g~o~t,cO+)op- .}'~,o~t~lf+ (13) so(~ =/z{flog Special Case I(a): So = 0 Isolating ~' from the left-hand side of (13) results in (14) ¢ = 1~o/'¢ + ~,:~'}/s ~ = {/3opo' + ~oP~ - po + ~, :~'}/s, Upon reversion and division by -x, one obtains the recursion g(x)=lfloPof(x)+flo Io.x)xY-f(y)g(x-y)dcz(y) (15) +fl~I ,~,o x-y f(Y)g(x-y) which reduces to g(x)=fl°P° f(x)+ I fllx +(fl°-fll)Y f(y)g(x-y)dtz(y). (16) S I (0,x) S IX This is the recursion obtained by PAN~ER (1981) and applies to the Poisson, binomial, negative binomial and geometric claim frequency distributions. Special Case I(b): s~ = 0 In this case, the claim frequencies satisfy n(~lf+olf{ - (17) sop,, = 1)}p,-1. flo+fl~N=O, If there exists a positive integer N such that then the claim frequency distribution has fimte range. In this case, (17) leads to p,=(fl°l"(n fl'~'"' (18) +Bo) Po, n = 1,2 ..... N. \S01\ If there exists no such N, the condition that ~ p, = 1 implies that rio +/31nl < leo for sufficiently large n ; and so fl i = 0 and the geometric distribution follows. The geometric distribution was considered in Special Case I(a) above. In order to develop a recursion for the distribution (18), equation (12) reduces to (19) s o(ff - Po) = :~0(ff - p0) + B 0P0 + fl,/~o~ if'}. Upon division by so and inversion one obtains g(x)=fl°P° f(x)+fl° f(y)g(x-y)d~(y) SO 0eO dl0.x) ~os I o x~ (g)y(h)y-x( )y(~d)y-