SUSAN M. PITTS ABSTRACT

Transcription

1 A FUNCTIONAL APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL BY SUSAN M. PITTS ABSTRACT A functional approach is taken for the total clai aount distribution for the individual risk odel. Various coonly used approxiations for this distribution are considered, including the copound Poisson approxiation, the copound binoial approxiation, the copound negative binoial approxiation and the noral approxiation. These are shown to arise as zeroth order approxiations in the functional set-up. By taking the derivative of the functional that aps the individual clai distributions onto the total clai aount distribution, new first order approxiation forulae are obtained as refineents to the existing approxiations. For particular choices of input, these new approxiations are siple to calculate. Nuerical exaples, including the well-known Gerber portfolio, are considered. Corresponding approxiations for stop-loss preius are given. KEYWORDS Total clai aount distribution, copound distributions, stop-loss preius, Fréchet derivatives. 1. INTRODUCTION A key task in risk theory is the evaluation of the distribution of the aggregate clai S arising fro a collection of risks over a fixed tie period. Historically, two ain approaches to this are via the individual risk odel and via the collective risk odel. These are considered in Craér (1930) who refers back to earlier work by Lundberg and others. More recent overviews appear in Klugan, Panjer and Willot (1998) and Rolski, Schidli, Schidt and Teugels (1999) The individual risk odel In the individual risk odel, the total aount claied is odelled as the su of the aounts claied on each of the separate risks. Thus if X i is the clai aount for the ith risk, i =1,,, then the aggregate clai is ASTIN BULLETIN, Vol. 34, No. 2, 2004, pp

2 380 SUSAN M. PITTS S = X 1 + +X, where it is usually assued that the X i s are independent but not necessarily identically distributed. If G i is the distribution function of X i, so that G i (x) =P(X i x), we often have G i =(1 q i )1 [0, ) + q i F i, where q i (0,1) is the probability that there is a clai on the ith risk, 1 A is the indicator function of the set A, and F i is the distribution function of a positive rando variable Y i representing the size of clai on the ith risk, given that a clai occurs. Then the distribution function F S of the total clai aount S is F S = G 1 G, where denotes Lebesgue-Stieltjes convolution. Exaple 1. A typical siple exaple has the distribution of Y i concentrated on a single point b i, so that X i is the discrete rando variable 0 with probability 1 q i X i = { bi with probability q i. A concrete exaple of this odel is a group life insurance plan covering lives, where b i and q i are the aount payable on death and the ortality respectively for individual i (see Klugan et al. (1998), ). One of the ain quantities of interest in the individual odel is the total clai aount distribution F S, and we need to handle this both in ters of nuerical evaluation of F S and also in ters of deriving theoretical properties of F S. Although the convolution product F S = G 1 G is a conceptually siple quantity, historically it turned out to be uch easier to deal with the collective risk odel, as descibed below. In addition, the use of the collective risk odel in practice avoids the necessity of obtaining precise statistical estiates of each of G 1,,G fro clais data Approxiations for the individual risk odel In the collective risk odel, we do not keep track of which risk gives rise to which clai. Instead, we odel the total clai aount as the su of a rando nuber N of clais, where the clais Z 1, Z 2, are assued to be independent, identically distributed rando variables with distribution function F Z,say,and with {Z i } assued to be independent of N. Writing T for the total clai aount in the collective risk odel, we have T = Z Z N,

3 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 381 with T =0 ifn = 0. Thus T is a rando su and has a copound distribution, with distribution function F T = k = 0 3 P(N = k)fz k, where F 0 =1 [0, ). One popular collective odel approxiation to the individual odel is given by a copound Poisson approxiation where N has a Poisson distribution with ean l and F Z is F, where l = qi and F = 1 qifi. (1.1) l We use the distribution function F T of the resulting collective total clai aount T as an approxiation to the distribution function F S of the individual total clai aount S. Another approxiation uses a copound binoial approxiation where N has a binoial distribution with index, so that N + Bin(,p) with p = 1 qi, and F Z = F as above. (1.2) A third collective risk odel is a copound negative binoial approxiation, where N has a negative binoial distribution, N + NB(,1/(1+p)), with k 1 p P N k, k,,,..., k + - ] = g = d n c = + p d n (1.3) 1 + p (so that N has probability generating function G N (z) =(1+p pz) ), where p = qi / as in (1.2) and F Z = F again. All these three approxiating collective risk odels have the ean of the approxiating total clai aount exactly equal to the ean of the total clai aount S in the individual risk odel, but they all have variances that are larger than var(s). See Rolski et al. (1999) 4.6 and the references therein for further discussion of these approxiations, and see Kuon, Radtke and Reich (1993) for different choices of paraeter values in the collective risk odels that ensure that both the eans and the variances atch those of the individual odel. An alternative copound Poisson approxiation has N Poisson distributed with ean l = qi / ^1 - q i h and F Z = ^qi / ^1 - q i hh Fi / l (see Kornya (1983), and also Hipp (1986)). A third possible copound Poisson approxiation has Poisson ean l = log 1 q - e^ - ihand F Z = - loge^1 -qih Fi / l (see Klugan et al (1998), ). All the above coonly found collective risk odels have N distributed as Poisson, binoial or negative binoial, and hence are such that Panjer recursion (see Panjer (1981)) can be used to calculate the copound distribution. An alternative nuerical ethod of calculation of copound distributions is the fast Fourier transfor algorith (see Ebrechts, Grübel and Pitts (1993)), whose use is not restricted to those N-distributions suitable for Panjer recursion.

4 382 SUSAN M. PITTS For oderate portfolio sizes, direct calculation ethods ay be useful for F S in the individual risk odel. These direct ethods include the use of generating functions, the De Pril algorith (see De Pril (1986), (1989)), and the fast Fourier transfor algorith (see Charles (2002)). Coparing the use of the fast Fourier transfor algorith for the individual odel and for the collective odel, we see that in general we need transfor steps for direct nuerical evaluation of F S for the individual odel (if the F i s are all different), then ultiplication of the transfors, and finally one inverse transfor step. In contrast, for the copound Poisson approxiation we need first to find F in (1.1), then one transfor step, then an exponentiation step and one inverse, ie fewer transfors. A uch sipler approxiation is obtained by approxiating F S by a noral distribution function (see Klugan et al. (1998) ). If i = (Y i ) and s 2 i = var(y i ) (assued finite), then the approxiating noral distribution has ean and variance given by = ] Sg = q and s = var] Sg = _ q s + q ^1 - qh i. (1.4) S i i S i i i i i This approxiation is easy to calculate, although it ay fail to capture iportant aspects, such as skewness, of F S. Many papers have considered approxiations and calculation techniques for the individual risk odel, soe of which have already been entioned. A coon thee in the literature is to derive error bounds for these approxiation in ters of, for exaple sup F T (x) F S (x) (De Pril and Dhaene (1992), Dhaene and Sundt (1997)). An iportant paper for our work is Hipp (1986), who gives higher order copound Poisson approxiations. These are discussed below in A functional approach In this paper, we adopt a siple functional fraework that is appropriate for both the individual odel and for the approxiating odels. In this set-up, the existing approxiations described in 1.2 arise as zeroth order approxiations, and we then refine these to obtain new naturally arising first order approxiations which require very little extra atheatical effort beyond finding an appropriate functional fraework. The approxiations obtained are in principle approxiation forulae, rather then nuerical approxiations, although they can be used to obtain nuerical values. To adopt a functional fraework eans that we regard the risk odel as a ap F that takes the input, for exaple, (G 1,,G ), onto an output quantity of interest, such as F S. We show below that each of the collective risk and noral approxiations can be written as the result of applying F to various input (a 1,,a ). Then, by showing that F is differentiable (in an appropriate sense) at (a 1,,a ), we obtain first order correction ters for the various

5 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 383 approxiations, giving rise to the new refined first order approxiations. Higher derivatives potentially lead to higher order approxiations. However, the first order ones here are particularly siple. This functional approach has previously been used for stochastic odels in renewal theory (Grübel (1989), Politis and Pitts (1998)), queueing theory (Grübel and Pitts (1992)), and ion channel odels (Pitts (1998), Pitts (1999)). The structure of the paper is as follows. In 2 we give the functional fraework for the ap F, and the relevant differentiability result. In 3, we apply the general first order correction ters to the copound Poisson, binoial and negative binoial collective risk odel approxiations, and also to the noral approxiation. 4 contains nuerical exaples, and stop-loss preius are considered in 5. 6 contains soe iscellaneous coents. 2. THE CONVOLUTION PRODUCT MAP In this section we specify the doain and codoain for the convolution product ap F, and we find its derivative. We work in the space A of finite signed easures on (,B), where B is the Borel s-field of. We note that A is contained in the space of finite coplex easures on (,B), and inherits any of the properties of this space (see Rudin (1986), chapter 6). For a in A, we write 3 a for the non-negative easure given by a (E) = sup ae ( i ), the supreu being taken over all partitions {E i } i =1 of E in B, and we write a for the total variation nor of a, given by a = a ( ). Then (A, ) is a Banach space (see Dunford and Schwartz (1958), III 7.4). The convolution of a and b in A is a * b, where for E in B, a * b(e )= # a (E x)b(dx), and it is easy to check that the nor inequality is satisfied, i.e. a * b a b. Let d b be the eleent of A that assigns a unit ass to b. Then d 0 is the identity eleent of A. Now let A = A A, and for (x 1,,x ) in A, let (x 1,,x ) =. Let the convolution product ap F: A A be defined by F(x 1,, x i x )=x 1 * * x. We soeties write % = 1a i i for the convolution product a 1 * * a. It should be clear fro the context whether refers to the nor in A or in A, and whether the sybol Π refers to convolution of eleents of A or to ultiplication of nubers. We show below that F is Fréchet differentiable and we give its Fréchet derivative. In general, a ap C : U B 2, where U is an nonepty open subset of B 1, and where (B i, i ),,2 are Banach spaces, is Fréchet differentiable at a U if there exists a linear bounded ap C a : B 1 B 2 such that for all e >0, there exists d > 0 such that h 1 < d iplies C(a + h) C(a) C a (h) 2 e h 1. (2.1) Then C a is the Fréchet derivative of C at a ( 2.1 in Chapter 1 of Cartan (1971)). Proposition 2. The convolution product ap F is Fréchet differentiable at (a 1,, a ) for all (a 1,,a ) A, with Fréchet derivative

6 384 SUSAN M. PITTS (,..., ) 1 = 1 % F a a ^h,..., h h hi * a, (2.2) j j i for (h 1,,h ) in A. Proof. For a =(a 1,,a ) and h =(h 1,,h ) in A, we have F] a + hg - F] ag - hi * aj = ^ai + hih - ai - hi * a % % % % j j i j i = hi * hi * aj hi. i1< i2 1 2 % j i1, i2 % This is at ost % % h h h a h i1 i2 i1< i2 j i1, i2 # j < c 2 a + c 3 h a h F i (2.3) For h 1, this is bounded above by h 2 (1 + a ). Given e > 0, for h in{e /((1 + a ) ),1}, we have that (2.3) is bounded above by e h, so that (2.1) holds, and the derivative of F is as in (2.2). The ap F a is linear. If h 1, then F a (h) % h a # a i j i j - 1, so that F a is bounded. Hence the ap F is Fréchet differentiable at a with Fréchet derivative F a there. 3. APPLICATIONS TO THE INDIVIDUAL RISK MODEL For the individual risk odel, suppose the distribution G i of the ith individual clai X i corresponds to eleent x i of A, i =1,,. Then the total clai aount distribution F S corresponds to s in A given by s = F(x 1,,x ). Using notation as in 1.1, if y i in A corresponds to the distribution of Y i, then we set x i =(1 q i )d 0 + q i y i A, i =1,,. For Exaple 1 we have x i =(1 q i )d 0 + q i d bi A, i =1,,. We approxiate F(x 1,,x ) by F(a 1,,a ) where a i A, i =1,,, and the a i s are chosen so that F(a 1,,a ) is easily found. We call this the zeroth order approxiation. We refine this by adding a first order correction ter

7 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 385 % F( a,..., a ) x1 - a1,..., x - a = xi - ai a, 1 ^ h ^ h * j j i which leads to the first order approxiation % F(x 1,,x ) F(a 1,,a )+ ^xi - aih * aj. (3.1) j i If all the a i s are the sae, and equal to a say, then the zeroth order approxiation is a *. The first order correction ter is then ( ) F,..., - 1 ( a a) ^x1 - a,..., x - ah = d xin * a * - a *, (3.2) and the first order approxiation is *( - 1) * F^x1,..., xh. d xin * a - ] -1g a. (3.3) We apply these results to the various approxiations introduced in 1.2. For each case, we show how the approxiation can be written in the for of a zeroth order approxiation F(a 1,,a ), for suitably chosen a i s, and then we find the for of the resulting first order approxiation The copound Poisson approxiation We consider the copound Poisson approxiation given in (1.1), with l = q i and F Z = F = ( q i ) 1 q i F i. Given the relative siplicity of (3.3) copared to (3.1), we approxiate the copound Bernoulli distribution x i = (1 q i ) d 0 + q i y i by a copound Poisson distribution a with Poisson paraeter q i / and with clais size distribution y in A given by y = ( q i ) 1 q i y i (so that y corresponds to F in (1.1)). Then a * is an approxiation to x 1 * * x, and oreover, a * is the convolution of copound Poisson distributions and hence is itself a copound Poisson distribution, with Poisson paraeter l = q i and the clai size distribution y, so that a * is the copound Poisson approxiation in (1.1). Thus the approxiation in (1.1) is F(a,, a), and so fits into our functional fraework. We note that the copound Poisson approxiation can be written as exp(l(y d 0 )) in A, where for a A, 3 exp a = k = 0 a *k /k, so that the above relationship says that exp(l(y d 0 )) = (exp (l(y d 0 )/)) *. We apply (3.3) to obtain the first order approxiation

8 386 SUSAN M. PITTS ( - 1) d xin * a - ] -1g a * * *( - 1) * = 7^ ^1- qihh d0 + ^ qiyiha * a - ] -1g a (3.4) ( -1) ( -1) = ] - lga + ly * a - ] -1ga * * *. The convolution product a *( 1) is a copound Poisson distribution with Poisson paraeter (1 1 ) q i and clai size distribution y. If the zeroth order approxiation a * has already been found, then the first order correction ter involves evaluation of the copound Poisson distribution a *( 1) (which is as easy to find as the zeroth order approxiation) and it also involves evaluation of one potentially difficult convolution ly * a *( 1). Roughly, writing cop(poiss(l),f) to denote a copound Poisson distribution with Poisson paraeter l and clai size distribution F, we obtain, F S ( q i )cop(poiss(( 1)l/),F) + ( qf i i ) cop(poiss(( 1)l/),F) ( 1) cop(poiss(l),f). For illustration purposes only, we can apply this expression to Exaple 1 where F i = d bi, although we note that in this case exact calculation of s is avaiable via De Pril recursion (De Pril (1986)). Write s k for P(S = k), and write cop (Poiss (l),f) k for the probability that the given copound Poisson rando variable takes the value k. Then s k ( q i )cop(poiss(( 1)l/),F) k + q i cop(poiss(( 1)l/),F) k bi ( 1) cop(poiss(l),f) k. In ters of nuerical calculation via transfors, we suppose we already have the fast Fourier transfor of y and a * in yfft and zerofft respectively. To calculate the first order approxiation in (3.4), we need basically one exponentiation step and one ultiplication step to find (( l)+lyfft)exp(( 1)l/(yfft 1)) ( 1)zerofft, and then one inverse transfor. An alternative representation of the zeroth order copound Poisson approxiation as a convolution product is to approxiate the copound Bernoulli factor x i =(1 q i )d 0 + q i y i by the copound Poisson distribution a i, with Poisson paraeter q i and with clai size distribution y i. Then a 1 * * a is the sae zeroth order copound Poisson distribution given in (1.1). The first order approxiation is given by (3.1). Although each % j iaj is a copound Poisson

9 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 387 distribution, it has a different Poisson paraeter and a different clai size distribution for each i. Thus this first order approxiation is ore coplicated to evaluate than (3.4). For nuerical evaluation via transfors, this would need different exponentiations and different counterparts of yfft. The sae procedure can be applied to the other copound Poisson approxiations in 1.2 to get first order approxiations. Hipp (1986) obtains first and higher order approxiations, together with error bounds, for copound Poisson approxiations. Each of these approxiations is itself a (generalised) copound Poisson easure. For the zeroth order copound Poisson approxiation in (1.1), Hipp s corresponding next higher order approxiation in our notation is the eleent h (1) of A given by h (1) () 1 () 1 =exp_ l _ y - d ii, (3.5) 0 where l (1) = 2 qi fqi + 2 p, y (1) = 1 2 q q y q y l * 2 () 1 b_ i + i i i - i i l. (3.6) This is different fro the first order approxiations we obtained above. While (3.5) turns out to be uch ore accurate than (3.3) in the exaples considered below in 4, we note that it is intrinsically ore coplicated than our approxiations. Calculation of (3.6) involves coputation of y *2 i for i =1,,. For soe distributions this is easy, for exaple if y i = d bi then y i *2 = d 2bi,but for general y i s, calculation via fast Fourier transfor ethods involves finding the transfor of each y i and squaring it. In suary, the first order approxiation in (3.3) is cruder but often sipler than Hipp s corresponding approxiation. Furtherore, our approxiation technique is also applied to copound binoial, copound negative binoial and noral approxiations below, whereas Hipp s ore accurate expansions are only for copound Poisson approxiations. Siilar coents apply to the expansions discussed in Cekanavicius and Wang (2003) The copound binoial approxiation The copound binoial approxiation given in (1.2) approxiates x 1 * * x by ((1 p)d 0 + py) *, where p = qi / and y = qy i i / qi. It is iediate that this approxiation is an -fold convolution product and that it fits into our functional fraework. This copound binoial approxiation uses the sae copound Bernoulli distribution a =(1 p)d 0 + py to approxiate each copound Bernoulli factor x i, so that x 1 * * x is approxiated by a *. We note that

10 388 SUSAN M. PITTS x q d 1 qy i = ^ - ih 0 + i i = ^1 - ph d + py = a. 0 Hence, by (3.2), the first order correction ter for the copound binoial approxiation is d * ( - 1) e xin - ao * a = 0, and so in this case the zeroth and first order approxiations are the sae The copound negative binoial approxiation The copound negative binoial approxiation in (1.3) has oent generating function M T (t) =(1+p pm Y (t)), where M Y (t) is the oent generating function of a rando variable with distribution function F. Hence M T (t) =a(t), where a(t) =(1+p pm Y (t)) 1 is the oent generating function of a copound geoetric distribution. This copound geoetric distribution corresponds to an eleent a, say, of A. Hence the copound negative binoial approxiation can be written as the output a * of the convolution product ap. Using (3.2), the first order correction ter is ( - 1) ^^ - qihd0 + ^ qihyh * a - a * * and a *( 1) here is a copound negative binoial distribution with N NB ( 1,1/(p +1)) and clai size distribution y. Thus this expression is relatively easy to evaluate with only one potentially difficult convolution., 3.4. The noral approxiation We apply the above technique to the Noral approxiation in 1.2. Write f (,s 2 ) for a N(,s 2 ) distribution with ean and variance s 2. Then, taking a i to be f ( (X i ), var(x i )), where (X i )=q i i = Xi say, and var(x i )=q i si 2 + q i (1 q i ) i 2 = sx 2 i say, we have the zeroth order approxiation F(a 1,,a )=f( S, ss 2 ), ie the usual noral approxiation to the distribution of S. We write \i for and j i X s2 j \i for s 2 j i X, and obtain fro (3.1) the first order approxiation j

11 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 389 s = ^x,..., x h. x * f( \i, s\i) ( 2 1)f( S, ss 2 ). (3.7) F 1 i As we have seen for copound distribution approxiations, siplifications often occur when a 1 = =a = a, say. If we take a = f( S /, s 2 S /), then the zeroth order approxiation F(a,,a) =a * corresponds to the sae Noral approxiation N( S, s 2 S ) in (1.4). The first order approxiation is s x f 1, d in * b S ssl - ] -1g f _ S, ssi. (3.8) The above two approaches to the Noral approxiation give the sae zeroth order approxiations, but different first order approxiations. The expression (3.8) is easier to evaluate, since it has just one potentially awkward convolution ( x i) * f ((1 1 ) S,(1 1 )ss 2 ), whereas (3.7) has such convolutions. If we apply (3.7) and (3.8) to Exaple 1 we obtain respectively the siple approxiation forulae s. ^1 - qihf( \i, s\i) 2 + q i f( \i + b i, s\i) ( 2 1)f( S, ss 2 ), and s. d ^1- qhn f 1- i,_ 1- is i i S q f 1- i + b,_ 1- is i - ] -1g f_, s i. i S S i S S S 4. NUMERICAL EXAMPLES Table 1 gives Gerber s portfolio of 31 policies (page 35 in Gerber (1979); see also Hipp (1986), Kuon et al. (1993)). Here, each x i is of the for x i =(1 q i )d 0 + q i d bi, as in Exaple 1. TABLE 1 GERBER S PORTFOLIO OF 31 POLICIES No. of b policies q

12 390 SUSAN M. PITTS TABLE 2 ZEROTH ORDER S (0) AND FIRST ORDER S (1) APPROXIMATIONS FOR GERBER PORTFOLIO. S: TRUE TOTAL CLAIM AMOUNT DISTRIBUTION; SUFFIX p DENOTES COMPOUND POISSON, b DENOTES COMPOUND BINOMIAL AND nb DENOTES COMPOUND NEGATIVE BINOMIAL. (0) (1) (0) (1) (0) (1) s s p s p s b = s b s nb s nb Our approach allows us to find the corresponding first order correction ters for each of the copound Poisson, copound binoial and copound negative binoial approxiations given in (1.1), (1.2) and (1.3) respectively. Table 2 shows the true s, the copound Poisson zeroth and first order approxiations s (0) p and s (1) p, the copound binoial zeroth order approxiation s (0) b (this is the sae as the first order approxiation, see 3.2), and the zeroth and first order copound negative binoial approxiations s (0) nb and s (1) nb. For each approxiation, Table 2 shows the ass given to k = 0, 1,,19. These are all calculated using the fast Fourier transfor. The errors for these approxiations are given in the first row of Table 3. For an approxiation s (i), write S (i) for the ass given by this approxiation to the interval (,x]. The second row of Table 3 gives sup x S (i) (x) F S (x). Our theory only concerns -errors, but we include the sup-error since other papers (in particular Hipp (1986)) refer to this distance easure. Fro Table 3, we see that for Poisson and negative binoial cases, going fro zeroth to first order approxiations leads to a reduction in both types of error. We see that the copound binoial is the best of the zeroth order approxiations for both types of error. Further, the copound Poisson and copound negative binoial first order approxiations are siilar to each

13 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 391 TABLE 3 GERBER PORTFOLIO: ERRORS FOR THE ZEROTH AND FIRST ORDER COMPOUND POISSON, COMPOUND BINOMIAL AND COMPOUND NEGATIVE BINOMIAL APPROXIMATIONS Poisson Binoial Negative Binoial zeroth first zeroth/first zeroth first -error sup-error other and also siilar to the zeroth order (= first order) copound binoial approxiation, for both errors. For -error, the copound negative binoial first order approxiation is slightly better than the other two, whereas for sup-error, the copound binoial zeroth order is slightly better than the other two first order approxiations. Table 2 shows that these general conclusions do not necessarily hold for each k-value. We copare our approxiations with Hipp s first order approxiation h (1), which is considered for this Gerber portfolio in Hipp (1986). We recall fro 3.1 that h (1) is intrinsically ore coplicated than our rather siple first order approxiation, in that h (1) requires y i *2 for each i =1,,. As ight be expected, h (1) is a uch ore accurate appproxiation, with -error and sup-error The Gerber 100 portfolio is obtained fro the above portfolio by ultiplying all the nubers in the body of Table 1 by 100, resulting in a portfolio of 3100 policies (see Kuon et al. (1993)). The errors for the various approxiations for this portfolio are given in Table 4. We see again that the first order approxiations are iproveents on the zeroth order ones. For this exaple, the zeroth order copound binoial approxiation is better than all others in the table for -error, but for sup-error, both the first order copound Poisson and negative binoial approxiations iprove on the zeroth order copound binoial. Hence, as a general conclusion fro these two exaples, perhaps it is worth considering (a) copound binoial approxiations, and (b) the siple first order approxiations for the copound Poisson and the copound negative binoial. For the Gerber 100 portfolio, we see again, as expected, that the Hipp first order approxiation does very uch better, with -error and sup-error TABLE 4 GERBER 100 PORTFOLIO: ERRORS FOR THE ZEROTH AND FIRST ORDER COMPOUND POISSON, COMPOUND BINOMIAL AND COMPOUND NEGATIVE BINOMIAL APPROXIMATIONS Poisson Binoial Negative Binoial zeroth first zeroth/first zeroth first -error sup-error

14 392 SUSAN M. PITTS TABLE 5 EXAMPLE FROM KLUGMAN ET AL. Age group No. of policies q i b i Rolski et al. (page 144) copare the (zeroth order) copound Poisson, copound binoial and copound negative binoial approxiations given in (1.1), (1.2) and (1.3) respectively for a different sall portfolio of 29 policies, and they find that, in ters of relative error, the copound binoial is better than the copound Poisson, which is in turn better than the copound negative binoial. This is in line with the results for the zeroth order approxiations for the two exaples above. They also find that Hipp s first and second order approxiations give uch saller relative errors. For an exaple involving continuous rando variables, we consider an exaple fro Klugan et al. (1998), page 373, of a portfolio of 900 policies which divide into three groups, where for an individual in the ith group, the clai aount distribution is x i =(1 q i )d 0 + q i y i, where y i corresponds to an exponential distribution with ean b i, see Table 5. The true total clai aount distribution has a ass at zero and a density over (0, ). The densities for the true total clai aount distribution, the zeroth order noral approxiation, and the first order noral approxiation in (3.8) are shown in Figure 1. The first order approxiation does a better job than the zeroth order approxiation of capturing the shape of the true density. For the zeroth order approxiation, the and sup-errors are and respectively, and the corresponding errors for the first order approxiation are and STOP-LOSS PREMIUMS In considering copound distribution approxiations for the individual odel, several authors also copare the resulting approxiations for stop-loss preius (for exaple, Hipp (1986), Kuon et al. (1993), De Pril and Dhaene (1992), Dhaene and Sundt (1997)). In this section, we extend our functional approach to stop-loss preius, and we obtain new first order approxiations that are refineents of the existing approxiations in the literature. In this section we restrict for siplicity to non-negative rando variables and we consider eleents of A that are concentrated on [0, ). For a non-negative total clai aount S with distribution function F S we suppose (S)<. Then the stop-loss preiu with retention y(> 0) is 3 3 # S # + y y _ ^S - yh i = ^x - yhf ( dx) = ^1 - F ( x) hdx, S

15 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 393 FIGURE 1. True (solid line), zeroth order noral approxiation (dotted line) and first order noral approxiation (broken line) for the exaple fro Klugan et al. where x + = ax{x,0}. This last expression otivates our choice of functional, but first we define a relevant space. Let A 1 ={a A : a is concentrated on [0, ), # t a (dt)< }, where a is as 3 in 2. For a in A 1, put a 1 = #( 1+t) a (dt)= a + # a ((t, )) dt. It is readily 0 checked that (A 1, 1 ) is a Banach space, and that for a,b A 1, their convolution product a * b is also in A 1, with a * b 1 a 1 b 1. For (a 1,,a ) in A 1, we have that F(a 1,,a )=a 1 * * a is in A 1. For a in A 1 and y > 0, define T y (a) by 3 T () a = # a^] t, 3gh dt. y y The condition that a is in A 1 ensures that T y (a) <. When a is a probability easure, then T y (a) is the stop-loss preiu with retention y. Thus we have the stop-loss functional, T y F : A 1, where # ^ a1,..., ah 7 a1 *... * a^] x, 3gh dx. y 3 When x i =(1 q i )d 0 + q i y i as in 3, we have that T y F(x 1,,x ) is ((S y) + ) where S is the total clai aount for the individual odel. When F(a 1,,a ) is one of the approxiations to this total clai aount distribution given in

16 394 SUSAN M. PITTS TABLE 6 GERBER PORTFOLIO: ERRORS FOR THE STOP-LOSS APPROXIMATIONS Poisson Binoial Negative Binoial zeroth first zeroth/first zeroth first ax abs error , then the corresponding stop-loss preiu T y F(a 1,,a ) is the resulting zeroth order approxiation to the true stop-loss preiu. For a first order approxiation, we need the Fréchet derivative of T y. It is easy to check that T y is linear and bounded, and that hence T y is Fréchet differentiable at s in A 1, with derivative T y,s = T y at s. Using the chain rule for Fréchet derivatives (Cartan (1971), Theore in Chapter 1), we see that the stop-loss functional is Fréchet differentiable at a =(a 1,,a ) with derivative given by (T y F) a (h) =T y F a (h) where h =(h 1,,h ). This leads to first order approxiations for the stop-loss preiu with retention y which are obtained by applying T y to the previous first order approxiations for F(x 1,,x ). We apply this for the Gerber portfolio of 31 policies. Table 6 gives the axiu absolute error for the stop-loss approxiations for retention values y =0,, MISCELLANEOUS REMARKS 1. The first order approxiations (3.1) and (3.3) ay be relevant for other areas of applied probability where sus of independent rando variables are of interest. In the insurance context, the functional approach is not liited to the specific approxiations given in 3. For exaple, an even sipler and cruder approxiation is obtained if we rescale the clai sizes so that one unit is (S)/ and so (S) =. Taking a i = d 1, i =1,,, gives a zeroth order approxiation d for the distribution of S. Fro (3.3), the first order approxiation is s. d xin * d ] -1g d 2. Error bounds are an iportant aspect of the literature on approxiations for the individual risk odel. The functional approach does indeed give rise to error bounds on the zeroth order approxiations via the Mean Value Theore (see Proposition of Chapter 1 in Cartan (1971)). Suppose that a =(a 1,,a ) and x =(x 1,,x ) are in A, and that the a i s and the x i s are probability.

17 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 395 distributions (as they are in our applications). Then the Mean Value Theore error bound for using F(a) =a 1 * * a as an approxiation for F(x) =x 1 * * x iplies F() x - F() a # xi - ai, (6.1) so that the bound is not new, but recovers an existing known bound, see Feller (1971), page 286, and also Dhaene and Sundt (1997). 3. Higher order derivatives give rise to higher order approxiations. Consider a general ap C : U B 1 B 2 where U is an open subset of B 1 and (B i, i ),,2, are Banach spaces. The second derivative of C at a is a bounded bilinear ap C a (, ) fro B 1 B 1 to B 2, and is identifiable with the derivative of the ap a 7 C a, U L (B 1,B 2 ), where L (B 1,B 2 ) denotes the set of bounded linear operators fro B 1 to B 2 ( 5.1 in Chapter 1 of Cartan (1971)). For our convolution product ap F, this leads to a second order approxiation F(x) =F(a)+F a (x a)+ 2 1 F a (x a, x a), see Theore in Cartan (1971). Following a siilar approach to that of Proposition 2, we find that F a (x a, x a) = ( x - a ) * (x j a j ) * % a, i j i i k k i, j for a =(a 1,,a ) and x =(x 1,,x ) in A. For the stop-loss functional, we use the fact that (T y F) a (x) =T y (F a (x)) (see 5), and that T y is linear and bounded, to see that (T y F) a (x a, x a) =T y e ^ xi - aih *_ x j - a j i * % ako. i j k i, j Even when a 1 = =a, both these second derivatives still involve the su x x i j i * j. For calculation via fast Fourier transfor, in general this gives no siplification over direct calculation of x 1* * x. For higher order approxiations, the coputational burden increases. Thus the ain practical gain of the present approach is achieved by the first order approxiations arising fro the first derivative, and there sees little further benefit fro going on to higher order approxiations beyond the first. 4. The second derivative also provides an upper bound on the -error of the first order approxiations. This bound is given by F(x) F(a) F a (x a) sup F 2 1 x a 2, ( 1 - t) a + tx 0 # t # 1

18 396 SUSAN M. PITTS where a =(a 1,,a ) and x =(x 1,,x ) are in A (Theore in Chapter 1 of Cartan (1971)). If the a i s and x i s are all probability distributions, then sup 0 t 1 F (1 t)a + tx 1, and so we obtain the following bound on the error for the new first order approxiation to F(x) F(x) F(a) F a (x a) 1 x a 2 = x a d i - i n (6.2.) For the stop-loss functional, we find T y F(x) T y F(a) (T y F) a (x a) 1 Ty x a 2 x a, e i - i o because T y The error bounds in Rearks 2 and 4 above are not necessarily the best bounds that can be achieved in particular cases. Together, (6.1) and (6.2) tell us what we ight have expected: if x i is sufficiently close to a i for each i, then the zeroth order approxiation is close to the true distribution, and the first order approxiation is even closer. However, these bounds do not preclude the possibility of good zeroth and first order approxiations even when the righthand sides of (6.1) and (6.2) are not sall. Different approaches fro the functional approach considered here ay well give rise to better error bounds. Nevertheless, the nuerical exaples in 4 and 5 deonstrate that the functional approach can provide useful and relatively easily calculated approxiations that are iproveents upon the known zeroth order approxiations in the copound Poisson, the copound negative binoial, and in the noral case. 2 2 ACKNOWLEDGEMENTS I would like to thank the referees for their helpful coents, which have led to iproveents in both the content and presentation of the paper. REFERENCES CARTAN, H. (1971) Differential Calculus. Herann, Paris and Kershaw, London. CEKANAVICIUS, V., and WANG, Y.H. (2003) Copound Poisson approxiation for sus of discrete rando variables. Adv. Appl. Prob. 35, CHARLES, D. (2002) A coparison of possible ethods for approxiating aggregate clais probabilities of the individual risk odel. Unpublished MPhil thesis, University of Cabridge. CRAMÉR. H. (1930) On the atheatical theory of risk. Forsäkringtiebolaget Skandia , Parts I and II, Stockhol, 1930, 7-84, In Harald Craér: Collected Works, Vol I, ed: A. Martin-Löf, Springer-Verlag, DE PRIL, N. (1986) On the exact coputation of the aggregate clais distribution in the individual life odel. ASTIN Bulletin 16, DE PRIL, N. (1989) The aggregate clais distribution in the individual odel with arbitrary positive clais. ASTIN Bulletin 19, 9-24.

19 AN APPROACH TO APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 397 DE PRIL, N., and DHAENE, J. (1992) Error bounds for copound Poisson approxiations of the individual risk odel. ASTIN Bulletin 22, DHAENE, J., and SUNDT, B. (1997) On error bounds for approxiations to aggregate clais distributions. ASTIN Bulletin 27, DUNFORD, N. and SHWARTZ, J.T. (1958) Linear Operators: Part I, General Theory. Interscience, New York. EMBRECHTS, P., GRÜBEL, R. and PITTS, S.M. (1993) Soe applications of the fast Fourier transfor algorith in insurance atheatics. Statistica Neerlandica 47, FELLER, W. (1971) An Introduction to Probability Theory and Its Applications. Vol. 2, 2nd Ed. Wiley, New York. GERBER, H.U. (1979) An Introduction to Matheatical Risk Theory. Huebner Foundation for Insurance Education, Philadelphia. GRÜBEL, R. (1989) Stochastic odels as functionals: soe rearks on the renewal case. J. Appl. Prob. 26, GRÜBEL, R. and PITTS, S.M. (1992) A functional approach to the stationary waiting tie and idle period distributions of the GI/G/1 queue. Ann. Probab. 20, HIPP, C. (1986) Iproved approxiations for the aggregate clais distribution in the individual odel. ASTIN Bulletin 16, KLUGMAN, S.A, PANJER, H.H., and WILLMOT, G.E. (1998) Loss Models: fro Data to Decisions. Wiley, New York. KORNYA, P.S. (1983) Distribution of aggregate clais in the individual risk theory odel. Transactions of the Society of Actuaries 35, KUON, S., RADTKE, M., and REICH, A. (1993) An appropriate way to switch fro the individual risk odel to the collective one. ASTIN Bulletin 23, PANJER, H.H. (1981) Recursive evaluation of a faily of copound distributions. ASTIN Bulletin 12, PITTS, S.M. (1998) A functional approach to approxiations for the observed open ties in single ion channels. Journal of Matheatical Analysis and Applications 219, PITTS, S.M. (1999) Approxiations for sei-markov single ion channel odels. In Sei-Markov Models and Applications. eds J. Janssen and N. Linios. Kluwer, Dordrecht POLITIS, K. and PITTS, S.M. (1998) Approxiations for the solutions of renewal-type equations. Stochastic Processes and their Applications 78, ROLSKI,T.,SCHMIDLI, H., SCHMIDT, V., and TEUGELS, J. (1999) Stochastic Processes for Insurance and Finance. Wiley, Chichester. RUDIN, W. (1986) Real and Coplex Analysis. (3rd ed.) McGraw-Hill, New York. SUSAN M. PITTS University of Cabridge