GERBER, GOOVAERTS AND KAAS - PDF document

GERBER, GOOVAERTS AND KAAS 1 The interpretanon of g(u,y) dy serious situation is when ruin occurs. To obtain a quantitative answer we introduce the function G(u, = Pr(T~, -y 0), which is a function of the variables &#x 000;u/ 0 and &#x 000;y/ 0 and ~s the probability that ruin occurs and that the deficit at the time of ruin is less than y. We shall also consider the corresponding density d y) = ~y O(u, y) existence will be shown in Secnon 6. Thus y) dy the probability that ruin occurs and that be between -y and Figure 1). Theorem 12.2 of BOWERS al. tells us that (4) g(0, y) ~' I- P(y) main goal is to explore y) the more interesting case when u is posmve. 2. A FUNCTIONAL EQUATION According to theorem 12.2 of BOWERS al. the probability that the surplus will ever fall below the initial level u and will be between u- x and - + dx it happens for the first time is _X 1- P(x) We use this and the law of total probability to see that i- X .... P(x) Co c ,,, that ~,(u) = oa). the equation for 4'(u) of exercise 11 of BOWERS al. chapter 12) is a special case of (5). THE PROBABILITY AND SEVERITY OF RUIN 153 Differentiating (5) with respect to y, we obtain a functional equauon for g: P(x) dx+ x 1 g(u,y) c o c In the terminology of FELLER (1966), equat)ons (5) and (6) are renewal equations of the defectwe type. Instead of determining g directly, we shall first find its transform y(r, y), which is defined as (7) y(r, y) = r'' g(u, y) du. o, multiply (6) by integrate over u from 0 to oo. On the left-hand side we get we replace the variable the new integration varmble z = u - x we can simplify the resulting double integral on the right-hand side as follows: ~ e "( .... ) g(u x,y) e r' P(x) du Co o X ~o e ~ g(z, y) e rx P(x) dx c o o X ,y(r, y) e" P(x) 0 The second term on the right-hand side can be written as f v e'" 1 - P(u + X e- ~ e" 1 - P(x) Co c y way we obtain from (6) a hnear equation for y(r,y): f °° y) k ~,(r, y) e" P(x) + X ~v e,~ e- - 1 - P(x) 0, c y Its solution ~s e-°' P(x) e" I- P(x) . 0 remaining task is to revert this transform to obtain y). the following we shall look at a famdy of claim amount dlstribunons in which this can be clone m a transparent way. 3. COMBINATIONS OF EXPONENTIAL DISTRIBUTIONS Let us assume that the claim amount distribution ~s a combination of exponential GERBER, GOOVAERTS AND KAAS at has a probability density function of the form (9) = ~ AA3~e -~,', x � O, ~.j positive and (10) + A2 + ... + A,, = 1. the special case when all posltwe, we speak of a mixture of exponential distributions. From (9) it follows that 1- ~ Aje -~,~, �xO. substitute this into (8) and obtain (12) y) = (k/c) j=,~ e-~JY Aj( ~j - r)/ l - (X/c) j=l - r) . the method of partial fractions, we can write thas expression an the following form: 7(r,y)= ~ ~ k=l ..... r. the zeros of the denominator, i.e. the solutions of the equatmn (14) (X/c) '~ 1. is assumed that the n roots are dastmct; of course some may be complex Conditaon (14) is the same as the condition that defines the adjustment coefficient, see exercise 8 of BOWERS al. chapter 12). Thus the adjustment coeffi- cient is one of the roots; without loss of generality we may set r~ = R. The coefficaents be calculated as follows. We multiply ~,(r, y) by (r,. - r) and let --* r,.. we do this m (13), the coefficient of exp(-13jy) is .... (12) we divide the denominator by let .... the denominator vanishes for r = operation gives minus the derivative of the denominator at r .... Thus the coefficmnt of exp(-t3jy) is (15) Cj,,, = - r,.) Ad(~l- r,.) 2 . the roots have been determined, the coefficients can be calculated easxly from this formula. The inversion of (13) is simple: one verafies that (16) ~ "~ Cjke-~'Ye ..... k=l (7); therefore expression (16) as the desired solution THE PROBABILITY AND SEVERITY OF RUIN These results can also be used to determine the probability of ruin. Since (17) if follows from (16) that (19) = g(u, y) dy, = ~ Ck e ..... , = ~,, (18) can be found in CRAMt~R (1955) and more recently in BOWERS al. in both cases the discussion is limited to mixtures of exponential distributions A related discussion can be found m DICKSON and GRAY (1984). The class of mixtures of exponential distributions is somewhat hmited; for example, of such a distribution is necessarily at 0. On the contrary, the family of combinations of exponential dxstnbutnons is rather rich, though not every choice of A j,/3j gives a probabnhty density function. A subset of this family consnsts of the sums of n independent exponential (/3j) distributed random variables wlth unequal parameters (see FELLER (1966, problem 12 of chapter 1.13)). An elegant proof can be obtained by looking at functions and applying the method of partial fractions. Taking 13j -+/3, one obtains the One may also show that the Gamma distribution with arbitrary values of the non-scale parameter, say n - 6 with 0 6 1, is in the closure of this class. It is sufficient to show that such a Gamma distribution is a mtxture of Gamma distributions wtth non-scale parameter n. The definmon of the Gamma funcnon implies e_t~ls- i -6= -- dt r(6) Using this we may write tlae Gamma (n- 6, 1) density as: -~_ °°(t+ l)"x"-le -(¢+°x t~-~(t+l)-"F(n) dt. 5) 0 J P(n) F(n- 6) F(5) We shall show in the following section that a mixture or combination of Gamma distributions can be handled in quite the same way as a combination of exponential distributions. 4. OF GAMMA DISTRIBUTIONS advantage of considering combinations of exponentials rather than just mix- tures lies in the fact that this class also contains distributions with mode not equal to zero. Another way to include such distributions Is to consider mixtures, or combinations, of Gamma distributions with integer-valued non-scale parameter. GOOVAERTS AND KAAS avoid unnecessarily complicated formulae, we shall limit our discussion to the case where this parameter equals two, but generahzation to other integer values is straightforward. We shall not g~ve all details of the proofs in this section. Note that as the Gamma distribution can be written as a limit of comblnatmns of exponential dlsmbutions, no new situations are added when an exphclt express,on for g(u,y) can be found. Consider the following density function p(x): (20) = k As~jr xe-a'x, x � O, with 13, posinve and (21) + Az + ... + An = I. (8) one obtains after some calculatmn: (22) (k/c) ~ Aj(13, r)-------~(/3fy-f3,yr+2/3,-r) ~ A, 213,-r a=, ,=, (~a-r)2J" applying the method of partial fractions, we can rewrite expression (22) in the following folm: 2n (23) Y(r,Y) = k ~ e-&Y/(rk-r). k=l that in this case we obtain coefficients Cjk depending on y. Here r~, r2, ..., r2,, are the zeros of the denominator of (22), i.e. the solunons of the equation: - r (24) i-(k/c) 2..a Aj- 0. ,=1 (/jS - r) 2- Once more we assume all these roots to be distinct, although the more general case presents no insuperable dlfficulues. One of the roots equals the adjustment coeffiment. In the same way (15) was derived, one has: (/jfy-/jjyr,,,+ 2/3,-r,,,)/ "-'=)_a t A, 3/3,-r,, - (/jj _ r,,,)2 t (13,- r,,,) ~" of (23) leads to 2 n ~ Cjk(y) e-~'Ye ..... k= Again the probability of ruin can be obtained as: 2 ii (27) ~(u)= ~ C~ e ..... , k=l THE PROBABILITY AND SEVERITY OF RUIN 157 ° C~= Cj,(y) e -~jy dy= Aj 3 - 2rk Aj 3/3j- rk j=l 0 - rk) /j=l (/3~ rk) 3" above expositmn can be generalized to other integer values of the non-scale parameter, and also to arbitrary positive real values. ILLUSTRATIONS 1. Suppose that n = 2, A~ = A2 = ~, /3~ = 3, /3, = 7, X = 1 and c= I/3; these are the specifications of example 12.10 of BOWERS et al (1987). The roots of equation (14) are r~ = R = 1 and rz = 6. Then we obtain from (15) the following coefficients: Cii = 9/5 Ct2 = -3/10. Cz, = 3/5 Czz = 9/10. Thus g(u, y) =-9 e- 3,.-,_ + _3 e- 7. .... 3 5 5 10 lntegratton over y gwes 24 l _ 6 u = 35 e- + e which is the result found by BOWERS et al. (1987). 3y_6u + __9 EXAMPLE 2. Let ),. = I, c = 1 and p(x)=12e- 3"-12e -4', x �~ O. This distribution has non-zero mode ln(4/3)= 0.288, and mean 7/12 = 0.583. In this example fit =3,fl2=4,At=4,Az= -3 The roots of (14) arer~=R= land r2 = 5. This leads to the following coefficients: Thus = 3 Ctz = 1 = -3/2 C2z = -3/2. g(u,y)=3e_3Y_U .... 3e_4 ...... +e_3V_su 3 e-4"-5"- 2 2 ' which we can use to obtain e_t, ___ = ~ 3. Let X = I, c = I, and p(x) = 1(5e- A 5u e - 12e-4' + 15e-6'), �x~0. GERBER, GOOVAERTS AND KAAS n = 3, .st = 2,.8z 4,/33 = 6, At = A3 = 5/4, Az = From (14) we find r~ = R = 1 and a pair of conjugate complex roots: = 5 + l, r3 = 5 - i. (15) we get 75 5 20 5 20 = = ~.~-q.- ~1, C13= 68 30 36 42 36 42 C2r- 68' 681; 35 30 35 30 = ; C32- 68 6i'; C33= ~-~ + ~-~ t Substituting all these parameters into (16), we get an answer that is quite accept- able, if the calculations are done in complex mode. Alternatively, we remember that "~ =cos u+l sin //; e-re=cos u-I sin u observe that the coefficients Cj3 are conjugate complex. This way we see that e ..... + Cj3 e- r'" --2e- S"Re(Cs2) + lm(Cj2) u and the answer can be written m the following form: g (u, y) = (75 68) e- - ,, _ e- 4y u ( 15/68) e- 6y u e-ZY-5"l(5/34) cos u + (20/34) sin u e-4Y-4U cos u + (4234) sm ul +e-6 .... 5,(35/34) cos u-(30/34) sin u. From this and (17) we get 65 I~ 1 1 ~(u)=~ e-"- cosu+~sinu . EXAMPLE 4. Suppose we have good estimates of the first three moments of a claim distribution. We want to estimate the d~stribuuon of the severity of ruin using a combination of two Gamma (2,/3) densities, i.e. a distribution with density: = At321e-:S'~x + Az.sz2e-t32~x. determine the unknown parameters of the method of moments, we have to get Al, A2,3t and 32 from the following set of equations: -~- + A2 2 Al+ Az= 1, At .st ~=EX, ~12 ~ 24 24 + A2 ~=EX2, AI "E-Y+ ~ THE PROBABILITY AND SEVERITY OF RUIN Writing A = A i, bj = l/3j and qj = EXS/(j + 1)!, the equatmns can be rewritten in a simpler form as: Ab,+(l-A)b2=q,, Ab2+(l-A)b2=q2, Ab~+(l-A)b3=q3. The first two equations yield A = q~ - b2 b~ - q2- qlb2 bj - b2' ql - b2 Assuming without loss of generality that 3~ 13z, we must have A &#x~ 00;/ 0, otherwise p(x) is negative for large values of x. By the above equanons, this imphes q~ 1132. Note that a value A E 0, 1 is obtained if and only if the ratio Var X/(El X} )2 exceeds the value t corresponding to a Gamma (2,/3) density. Substituting the above expressions in the third equation, we obtain a quadranc function of b2, with the following roots: = (q~qz - ,j 2 - - - 2(q~ 2 - q2) similar system of equations must be solved if one wishes to fit a combination of two exponential distributions to three given moments, or to a given mean, mode and variance. Another necessary condition for p(x) to be non-negative is that either p(0) &#x~ 00; 0, or p(0) = 0 and p'(0) &#x~ 00;1 0 must hold. By fitting moments, this condi- tion ~s sometimes violated, as can be seen by taking a distribution with mean and variance 6, and third central moment 36. To make comparison possible with results previously obtained, assume that the moments of the distribution to be estimated are those of an exp(l) distribution, so mean and variance are 1, and the third central moment equals 2. We obtain the following values for the parameters of p(x): Ai=A2=½, 31= 3- ,13 =1.268, B2=3+,13=4.732. The mode of this dlstributmn equals 0.235. Taking X = 1, and c= 2, we find the following roots for (24): r~ = 0.506 (This is the adjustment coefficient. For this value of the premium rate c, the adjustment coefficient of an exp(l) distribution equals 0.5.) r2 = 1.765, r3 = 3.544, r4 = 5.685. The coefficients Cjk(y) for use in (26) can be obtained as: C{t (y) = 0. 147y + 0.066, C~z(y) = - 0.099y - 0.054, C{3(y) = 0.629y + 0.663, C4 (y) = 0.506y - 0.424, C~l(y) = 0.218y + 0.458, C~z(y) = 0.158y - 0. 193, C:~3 (y) = - 0.088y - 0.031, C~4(y) = 0.029y + 0.016. GOOVAERTS AND KAAS probability of rum, obtained from (27) and (28) is: ~,(u) = 0.517e- 0 5o6, _ 0.070e- t 365,+ 0.089e-3 5441, _ 0.036e-5 685, The probability of ruin corresponding to the exp(I) distribution and this value of c equals -° 51'. maximum deviation of the ruin probability obtained with the approximating combination of Gammas and the exponential(l) ruin probability is 0.004. 6. DIRECT METHOD (5) and (6) are defective renewal equations and can be solved (at least in principle) without the use of transforms. With the notation (29) 1 - P(x) can write equation (5) as S y) = G(u- x, y) h(x) dx + h(x) dx. I1 successive substitution we obtain first the following formal solution: (30) y)= h*"(x) h(z) dz dx. 11=0 II-- I" rigorous proof follows from the following interpretation (combined with the law of total probability): -- I -.-y h(z) dz dx the probability of the event that the nth record low of the surplus process is between u - x and - x + dx that rum occurs wIth the following record low, such that the deficit is less than y; see theorem 12.2 of BOWERS al Expression (30) shows that indeed a density derivatives we obtain (31) h*~(x)h(u- y) dx. n~O the following section we shall illustrate the application of (31) in a particular If we set y = oo in (30), we obtain a well-known representation for the probablhty of ruin (the so-called "convolution formula"). THE PROBABILITY AND SEVERITY OF RUIN UNIT CLAIM AMOUNTS that all claims are of size one. Thus, by (29), h(x) = XJc ff 0 x l; 0 otherwise. We can write thts defective probability density as h (x) = af(x), where oe= X/c and f(x) is the umform (0, 1) density There is an exphclt expressmn for the n-fold convolunon of f: (32) f'*"(x) (n 1)~ ,=o This formula can be found m FELLER (1966, theorem 1, I 9), and a very elegant derivation is gaven by SH1U (1985). We prefer to write it as 7- (- I)J(x-j)'{. (n-j)!jt ax d=o Then = Z n=O ,,=0 a=o (n l)ty! the order of summation, we obtain more simply: d (-c~)a (x- J)+, (rTZT) ! (x-J)~. -a (33) h*"(x) -dx j! n=O = ~, (- oe) j ( x _ j )g+e ~( ' -J)- =-- ~ dX j=o Note that this is m fact a fimte sum, as terms with j � x vanish. If we substitute this expressmn m (31), the integrauon can be limited from x= (u + y- 1)+ to x= u, where h(u- x + y)= u. The resulting integral is trivial; for u �t 0 and 0 y 1, we obtain (34) g(u, y) Od ~ (--o~)J (U- g j)+e~("- j) + J! (.+ y_ i _j)4e.(,, + &#x 000;.-,_.). j=o J! COMPUTER IMPLEMENTATION of the algorithm suggested m Section 3 on a computer involves mainly elementary operations on polynomials. To solve (14), however, we must GERBER, GOOVAERTS AND KAAS have a routine to compute all roots, real as well as complex, of a real polynomial. Any textbook on numerical mathematics contains material on this; see for instance STOER (1972). Also, any library of numerical routines such as the NAG or the IMSL hbrary provides adequate software. One may also consult the ACM algorithms. Note that only for n �/ 5 may the need to iteratlvely compute complex roots arise. One of the roots is the adjustment coefficxent, so at least one of the roots IS real. In case all coefficients positIve, one may show that all roots of (14) are real and non-negative. In this case, simpler algorithms will suffice, for instance the Newton-Maehly algorithm described in STOER (1972, pp. 220--221). An algorithm to compute complex roots of real polynomials that can be pro- grammed easily, even using an electronic spreadsheet, is the method of Balrstow. For a motivation of the method, see STOER (1972, pp. 226--227). ItS main advan- tage is that no complex arithmetic is revolved. A disadvantage is that convergence cannot be guaranteed PRESS al. recommend a two-step procedure: first find approximations to all roots and then "polish" the roots found using Bair- tow's method. This method works as follows. First write (14) as the following polynomial equation: (AI) + a~r "-L + ... + a.-~r+ Next determine a quadratic divisor 2 + pr + q, p2_ 4q 0, as follows. Choose a starting point (q, p) and calculate the vector (B0, B~ ..... B.) by means of the following recurslve scheme: Bo: ao, = al - pBo, B2 = a2 - pBi - qBo, t = a.- t - pB.-2 - qB,,- 3, B. = a. - pB._ l - qB._ 2 compute the vector C~ .... C.-l) follows: Co = B0, = Bi - pCo, C2 = B2 - pCi - qCo, z = B.- 2 - pC.-3 - qC.-4, I = - pC.- 2 - qC.- 3. the auxiliary quantities C2.-2 - Cn-lCn-3, P = Q = BnC.-2 THE PROBABILITY AND SEVERITY OF RUIN 163 we find the next approximation (q, p) as: (A5) p:= + P/D, + Q/D. restart the algorithm with these values of q and p untd the old and new values of q and p differ by less than the prescribed precisIon. A divisor z + pr + q the left-hand side of (14) gives two complex conjugate roots. Hav- ing dlv~ded out this factor, run the algorithm again to determine the other roots. REFERENCES BOWERS, N L , GERUER, H U , HtCKM,XN, J C , JONES, D A and NESBtTT, C J (1987) Mathemattcs, of Actuaries CraMEr, H (1955) Collective risk theory -- A survey of the theory from the point of view of the theory of stochastic processes Jubdee Volume of Skandta. D C M and GRAY, J R (1984) Exact solutions for rum probabdly m the presence of an absorbing upper bamer Actuartal Journal, FEH.Er, W (1966) lntroductton to Probabdtty Theory and tts Apphcatzons 2 Wdey PRESS, W H , Ft.ANNE~Y, B P TEUKOI_SKY, A and T (1986) Rectpes -- The Art of Sctenttfic Computing Umvers~ty Press, Cambrtdge SHiU, E S W (1985) of Umform Dtstrtbutton Report, UmversJty of Mamtoba SroEr, J (1972) m dte Numertsche Mathemattk I Berhn HANS GERBER des H.E C., Swttzerland. Umversltd de Lausanne, CH-IOI5 Lausanne-Dorzgn y, J. GOOVAERTS Umversttett Leuven, B-3000 Leuven, Belgtum. lnstttuut voor Aktuartele Wetenschappen, KAAS van Amsterdam, Jodenbreestraat 23, NL-IOll NH Amsterdam, Netherlands