Reserving consecutive layers of inwards excess-of-loss reinsurance

ELSEVIER Insurance: Mathematics and Economics 20 (1997) 225-242

Reserving consecutive layers of inwards excess-of-loss reinsurance *

Greg Taylor 1 Tillinghast, GPO Box 3279, Sydney, NSW 2001, Australia

Received March 1996; received in revised form March 1997

Abstract

Consider an excess-of-loss reinsurance arranged in a number of layers. A loss reserve is required for each layer. There are two major reasons why the independent application of some conventional loss reserving technique to each layer is inappropriate. First, the experiences in different layers in respect of a particular treaty year will be linked; favourable or adverse experience in one layer is likely to be reflected in favourable or adverse experience in the next. Second, experience data will typically become sparse in the higher layers, rendering analysis in isolation from other layers relatively uninformative. The purpose of the present paper is to analyse the linkages between the loss experiences of different layers, and apply these to obtain linked loss reserves. A numerical example is provided. © 1997 Elsevier Science B.V.

Keywords: Reinsurance; Excess-of-loss; Reserving

1. Introduction

Consider the case of a reinsurer of two or more consecutive layers of an excess-of-loss (XoL) treaty. Suppose that the usual triangulation of incurred losses is available separately in respect of each layer, but no other claims information is available.

It would be possible, in reserving, to apply some particular type of analysis, e.g. chain ladder or some variant thereof, to each layer. To do this independently for each layer would, however, fail to recognise the coupling of consecutive layers' experiences.

For example, heavy experience in one layer will increase the frequency and/or the severity of hits on the next layer above. It is preferable, therefore, for each layer's experience to be modelled in a manner which takes due account of the modelled experience in the layer immediately below.

The following sections make some suggestions as to a very basic form of such modelling. Undoubtedly, more sophisticated techniques could be developed, as is briefly discussed in Section 8.

* Paper presented to the XXVIIth ASTIN Colloquium, Copenhagen, Denmark, September 1996. 1 Consultant, Tillinghast; and Professorial Associate, University of Melbourne.

0167-6687/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S0167-6687(97)00034-X

226 G. Taylor~Insurance: Mathematics and Economics 20 (1997) 225-242

2. Statement of the problem

The purpose of the present section is to describe in precise terms the problem discussed in Section 1. Consider K + 1 layers of insurance, labelled layers 0, 1, 2 . . . . . K. The labelling begins at 0 to accommodate a

ground-up layer which may not be reinsured, but for which experience is available. The use of a ground-up layer is not essential to the following analysis, however. The 0th layer may just as well be taken as a reinsured layer.

The layer number will be represented throughout by a bracketed superscript.

. (k) be the incurred loss in layer k deriving from treaty year i and as recorded at the end of development year Let Lij j ( = 1,2 . . . . ).

L (k) The data assumed available comprise an incurred loss triangulation for each layer, i.e. - i j " i = 1 . . . . . I ; j = 1,2 . . . . . l - i + l ; k = O , 1 . . . . K .

Let

dff) = upper limit of layer k,

Wi (k) : d f f ), k = O,

= d f f ) - d f f - ' , , k = 1,2 . . . . . K,

= height of layer k.

With this notation, layer k(> 0) reinsures W f f ) excess of dff-1). Incurred losses in a layer may be zero. The zero and non-zero cases are considered separately in each layer:

Pi(f. ) = Prob[Li (k) = 0], (2.1)

A(k) ~r--(k) r(k) ij : tZ tL i j IL i j ~ 0], (2.2)

for values of i, j representing the future, i.e. i + j > I + 1. It will be conveninent to represent the loss L!k. ) as a multiple of the relevant layer height. Thus, define tJ

and

Q(k) = L(k)/W(k) (2.3) ij ij - i

Ri(;) (k) (k) = a i j / W i •

Further, define

(2.4)

= t~(k) _ O(k) . (k) Wi(k), (2.5) AQi(~ ) ~i , j+l ~ i j = A L i j /

and ARi(; ) similarly. It is assumed that AL/(~ ) _> 0. A version of the chain ladder will be applied to the bottom layer. Age-to-age ratios will therefore be required. In

fact, it will be useful to define

(o) /L!O)I . (2.6) Zi j = l o g [ L i , j + 1. ,j J

3. General structure of the model

Since layer 0 is a ground-up layer, it is assumed here that

= o . (3.1)

G. Taylor~Insurance: Mathematics and Economics 20 (1997) 225-242 227

It is possible, of course, for the contrary to occur, i.e. for the line of business under consideration to have a non-zero probability of generating no claims at the primary level. It would be unusual to protect risks of this type with a sequence of XoL layers.

In any event, the possibility is ignored here. Its inclusion would not require any major structural change to the

It would be necessary simply to model the function p/(O). paper.

The functions P)~) are modelled for k > 0. In fact, the modelling is generally divided into three cases: l j

L!k ) = 0, (3.2) U

L(k) (k) , ij ~ O, A L l , j _ , = 0, (3.3) - ( k )

A L i , j _ l ~ O, (3.4)

where

- (k) (k) - (k) A L i , j _ 1 : Li j - - L i , j _ 1. (3.5)

_~ I(k) The model excludes the possibility o f Lit~ )_ 0 when ~i , j -1 > 0, and so the first and third of the enumerated cases are in fact mutually exclusive, as are other pairs of cases.

The relevance of the case AL (k) = 0 is that it implies AL (k+l) = 0. The distribution of AL (k+l) will be continuous for strictly positive values but will have a discrete mass at zero. The separation of cases (3.3) and (3.4)

f - --~k) recognises this in the modelling of AL (k+l). Case (3.2) is a special case o AL = 0.

4 . M o d e l l i n g t h e u p p e r l a y e r s

4.1. Probabil i ty o f nil incurred

To calculate the array of values Pff), one must consider the evolution of each underwriting year, in particular (k) , , (k)

transitions of L i • ~ = ~ to Lij = [3. It is assumed ~aat the different underwriting years develop independently, and so any particular underwriting year

may be considered in isolation from the others. In this case, the suffix i is constant, and so is suppressed through the remainder of the present section unless the contrary is explicitly stated.

. (k) ( k - l) (k) The e v e n t L i j = Ot is abbreviated to {j, k, ~}. Similarly, the twofold e v e n t L i j = or, L i j = [3 is denoted by (k-l) {j, k, or, [3}. When A appears in place of a in this last expression, it d e n o t e s A i j .

The expression {j, k, or(> 0),/~} will denote {j, k, or, [3}, i.e., relating to a specific value of or, but subject to the (k-l) (k)

restriction that u > 0. In distinction from this, {j, k, ~ > 0, 13} will denote the e v e n t Li j > O, L i j : [3, i.e. unrelated to a specific value of a .

Transitions of the type mentioned above will be influenced by experience in the next layer below, specifically by

(h-l) (suffix i shown for consistency with the earlier notation). L}kj--_l~ and t i j It will be necessary, therefore, to consider transitions of pairs

L k- l ) L (k) 1 r (k-l) " (k)l i , j - - l ' i,j--lJ ~ [~ij ' L i j j ' (4.1)

or, in the abbreviated notation,

{ j - 1 ,k , ct, r } --+ { j , k , y, 6}. (4.2)


There are three distinct types of state to be recognised in such transitions. On the left-hand side, for example, of (4.2) they are

a = ~ = o , a # o , t~=o, a # o , ¢~#o,

respectively. The matrix of transitions between these states, written in the unabbreviated notation, appears as follows:

- - ( k - l ) - - 0 (k-l) (k-l) = ]/ ~ O, Li j = ~' St: L i j k = O, Li j . (k) = 0 L i j - - ~ 0

L ~ ) = 0 L i j . (k) _

- ( k ) . 0

L~kj_J~ = O, L i , j _ ' = 0 -* • •

L(k-l) (k) , L i , j - ~ i , j - I = O 1 5 0 , = 0 0 * :¢

. ( k - U # O, (k) . L i , j - i ~ i , j - I ~ a fl ~ 0 0 l

where the entries ~ are complementary to the remainder of the rows to give a unit row sum. Let this matrix be denoted by M} k). Appendices A and B develop the methodology for evaluating/k/~ k) for various

o - d

k(> O) and j . The following results are reproduced from there. The quantities found there to be required are:

Prob[{j, k, 0}l{j, k - 1, A], {j - 1, k, a, 0]],

Prob[{j, k - 1, y > 0}l{j - 1, k - 1, 0}],

Prob[{j - 1, k - 1, or, 0}].

Denote the three respective sets of these quantities by N~ k), R (k-l) ¢(k-1) "'j , ~ j - l • Note that the original target of the

investigation, Prob[{j, k, or}], may be expressed in terms of x! k) thus -j ,

Prob[{j, k, a}] = Prob[{j, k, 0, a}] + Prob[{j, k,/3 > 0, or}]. (4.3)

The SJ k) and M~ k) may be evaluated recursively with the following order of evaluation, and with J denoting I - i + 1, i.e. latest development year:

then

S~°)(all j ) , S(f)(all k) [given],

g (1), j = J + l , J + 2 . . . . . S (1), j = J + l , J + 2 . . . . . M(2),S (2) . . . .

Sj. f°) is treated as given since, by convention, {j, O, a, O} = {j, O, 0}, so that (3.1) gives Here

Prob[U, 0, 0H = 0.

Similarly M~ 1) is a special case of M~ k). Specifically, (A.6) gives (approximately)

Prob[{j, 1, y > 0, 8}l{j - 1, 1, or,/3}] = Prob[{j, 1, 8}l{j, 0, A}, {j - 1, 1, or, fl}], (4.4)

and this last quantity is dealt with below. The remainder of the recursion follows the schematic appearing hereunder. This is the same as the first schematic

of Appendix A, but rearranged to show the S! k) as the target of the recursion, in accordance with (4.3). J


/ s P -1 A j - I

V -1,

/ Mj(k-l)

R/(k-l)

"./-1

The equat ions required for the recursion, in order, are ( f rom Append ix A):

For SJ k),

Prob[{j , k, O, 0}] = Prob[{ j - 1, k, O, 0}] x Prob[{j , k, O, O}llj - 1, k, O, 011, (4.5)

Prob[{j , k, o~ > O, Oil = Prob[{ j - 1, k, O, 0}] x Prob[{j , k, ~ > O, Olllj - 1, k, O, 0}]

+ Prob[{ j - 1, k, y > 0, 0}1 x Prob[{j , k, ct > 0, 0}l lJ - 1, k, A, 0I]. (4.6)

r,(k) For ~j ,

Prob[{j , k, y > 0}[{j - 1, k, 0}]

= {Prob[{j, k, y > 0}l{j - 1, k, 0, 0}] × Prob[{j - 1, k, 0, 0]]

+ Prob[{j , k, y > 0}[{j - 1, k, A, 0}] × Prob[{ j - 1, k, ce > 0, 0}]}/Prob[{j - 1, k, 0}]. (4.7)

Note the special case for k = 1, in which

Prob[{ j - 1, 1, 0, 0}] = 0, (4.8)

because of (3.1). For M) k),

Prob[lj, k , y > O , 8 } l { j - l , k , o ~ ( > O ) , ~ } ] = P r o b [ { j , k , S I l { j , k - l , A } , l j - l , k , ot(>O),~}], (4.9)

Prob[{j , k, y > 0, 8}l{j - 1, k, 0, 0}]

=Prob[{ j , k , 3 } l { j , k - l , A } , { j - l , k , O , O } ] x P r o b [ { j , k - l , y > O } l { j - l , k - l , O } ] . (4.10)

The condit ional probability which appears on the right-hand sides of (4.4), (4.9) and (4.10), respectively, is discussed in Appendix B. There the following approximation is adopted:

, (k-l)1 1~,~17,0}] {1 exp t r (k- 1), Aq~k/__l?]}- I (4.11) Prob[{j , k, 0}l{j, k - 1, "ij ,, {j - 1, k, , _ = + Jlqi j


with

] fit[q~(1 +q)llog(fllp), q >_ 1, f ( q , p ) I - o o , q < 1, (4.12)

for constants rio, flJ > 0. Here q and Aq are defined in terms of l in the same way as Q and A Q are defined in terms of L (see (2.3) and

(2.5)). At the end of the recursion illustrated by the earlier schematic, all S) k) (and other quantities) will be available.

These may be used as inputs to (4.3) to yield Prob[{j, k, 0}], the probability of nil incurred.

4.2. Size of non-zero incurred

Section 4.1 calculates probabilities of zero incurred and of non-zero incurred. In the event that incurred is non-zero, it will be necessary to estimate its expected size.

Amounts of incurred losses are expressed in terms of R~; ) (see (2.4)), whose evolution with development year is now examined.

If movement occurs between ~ijo(k-1) and ~i,j+l't~(k-1) i.e. AQI~ -1) > 0, it may generate movement between ~ij O(k) . ~ ( k ) . A,,~(k) • ~ ( k - l )

a n a ~i,j+l' 1.e . ~'~i.j-I > 0. Even if there is no movement in layer k - 1, i.e. zx~dij = 0, movement may still occur in layer k.

This may reasonably be modelled as

An(k) ~(k) ̂ o(k-l) R[;) g(k)(j), (4.13) l~ij = u z-xtxij -k-

for constant a (k) > 0 and suitable function g(k)(.). Note that, in the absence of movements in layer k - 1, g(k)(j) acts as an age-to-age factor, and must therefore

converge to 0 sufficiently rapidly with increasing j if Ri~; ) is to be asymptotically constant. A simple possibility is

g(k) (j) = b(k) j -s , (4.14)

~(k) to converge with increasing j . where b(k)(> 0) is a constant, and s would need to exceed 1 for Kij Estimates h (k), ~(k)(j) of a (k), g(k)(j) are made on the basis of observed Ri(; ) and the relation (4.13). This is

done by means of regression in the present paper.

Relation (4.13)then enables R~,~]+ l tO be predicted recursively on the basis of A R ~ - l ) and R~; ). This means that

future values -(k) o f l~ij , j = I - i + 2, I - i + 3 . . . . may be predicted on the basis o f R~k)l_i+ 1 provided that future ~: o ( k - 1 ) values t , l ~ij are available.

5. M o d e l l i n g t h e b o t t o m l a y e r

By (3.1), it is assumed here that

~.(0) = 0. (5.1) J

With this assumption, age-to-age factors are meaningful. The bottom layer is modelled here by means of a regression- based variation of the chain ladder.


Consider the variable Zij , defined by (2.6). It is assumed that

Zij + K ~ log N(Izij , t72), (5.2)

for a suitable constant K, and with IZij, t72. specific parametric forms depending on i, j . In fact, it will be assumed here that

uij = h i ( i ) + h2(j), (5.3)

t72 = ~r2 / h3( j ) , (5.4)

for suitable functional forms hi (.), h2(.) and h3(.), and for constant cr 2 > 0. Values of ~2 and of the parameters involved in these functional forms are estimated (by regression techniques in

the present paper), leading to fitted values Yij corresponding to observations log(Zij -~- K) . The model values of ElL (o) /L(O)l the age-to-age factors • i,j+l I ij j are calculated as

exp{exp[l~ij + ½~r2/h3(j)] - K}, (5.5)

where 62 is the estimate of tr 2.

6. Assembling the predictions

Sections 4.1, 4.2 and 5 deal with three sets of predictions. Each individual prediction is dependent on data and/or one or more other predictions. They therefore need to be evaluated in a specific order, as set out below.

Bottom layer (Section 5). For each treaty year, develop the most recent incurred loss figure through future years by means of the chain ladder factors (5.5).

size of non-zero incurred (Section 4.2). Predict future values of R~ }) on the basis of the Upper layers predicted I

• (2) from A 0) and so on. R~ °). Hence develop future values of Ai~l ). Then develop future Aij " - i j

Upper layers - probability o f nil incurred (Section 4.1). The precedence of the quantities required for prediction J

of the P.~(')tj was examined in detail in Section 4.1. The schematic there indicates that the future S) 1) can be evaluated

recursively over increasing j on the basis of the 3) 0) (which are degenerate because of (3.1)). The S) l) then

yield predictions of the p(l)ij" The p/~2) are then predicted on the basis of the S) 1), the Pi(~ ) from the S) 2), and so on.

Upper layers - incurred losses. These are predicted by means of the simple formula

(k) (k) (k) Pij ]Ai j" E[Lij ] = [1 - (6.1)

7. Numerical example

The following example uses a real data set which has been disguised by adjustment. The adjustment has been made carefully, however, in such a way as not to disturb the main features of the data.

Because the recommended treatment of upper layers differs from that of the bottom layer, the example deals with the bottom layer and one upper layer. Because it is desirable to illustrate the methodology with a sparse data triangle, the upper layer chosen has k > 1. For the sake of the example, suppose k = 3.

Although details of layers other than k = 0, 3 are not given here, they have in fact been used in the estimation of layer-independent parameters, such as in (4.12).

232

Table 1 Projected claim costs

G. Taylor~Insurance: Mathematics and Economics 20 (1997) 225-242

Treaty year Incurred losses Layer 0 Layer 3 To 1995 Projected ultimate To 1995 ($M) ($M) ($M)

Projected ultimate ($M)

1983 29.4 29.9 1984 168.1 174.3 11.6 12.2 1985 77.9 83.4 0 1.0 1986 129.6 142.4 1.8 2.7 1987 125.6 142.2 0 2.5 1988 180.5 212.7 45.0 48.8 1989 91.0 119.4 0 3.6 1990 127.2 182.9 93.7 102.8 1991 49.6 102.1 7.7 12.1 1992 55.3 157.3 10.0 17.4 1993 44.6 209.7 0 21.1 1994 7.8 122.0 10.0 41.8

The incurred loss data for the example are set out in Appendix C. Those of layer 3 (Table 2) are seen to be quite sparse and generally unsuitable in isolation for application of conventional projection techniques. It is here that the activity in the lower layers, e.g. layers 0 and 2 in Table 2, becomes useful.

The schema described in Sections 4--6 has been applied and selected numerical details given in Appendix D. Appendix D. 1 deals with layer 0 and Appendix D.2 with layer 3. The results obtained are shown in Table 1.

In practice, the projected results in respect of the more recent treaty years might require reconsideration due to their relative unreliability, but this has not been done here.

8. D i s c u s s i o n a n d f u r t h e r r e s e a r c h

All of the above developments have been carried out on the basis of aggregate data. In practice, it may be possible to obtain individual loss data, particularly in those layers where simple triangulation provides little information.

Some additional data would enable some exploration of individual claim size distribution, which could then supplement the above models of loss development. The precise form of the additional analysis would depend on the form of additional data available. For example, a complete history of development of each claim might be available; or alternatively, individual estimated loss sizes might be available only at the latest date of experience tabulation.

The working above makes clear that, without this individual claim information, one is compelled to model aggregate data with some sort of "development year effect" (e.g. (4.13), (5.3) and (5.4)). Care is needed that this does not lead to over-parametrisation of the model.

In this respect, Appendix D. 1 shows that seven parameters are used to model the bottom layer, and Appendix D.2 uses between six and eight to relate each upper layer to the one below it. These parameters are estimated from (see Appendix C): - 125 data points in the bottom layer; - 77 data points in each upper layer (26 of them non-zero in the layer modelled in Appendix D.2).

This seems reasonable parsimony of parameters, especially if one recalls that a simple chain ladder model applied to each layer would involve 30 parameters in the bottom layer, 22 in each upper layer, and still produce inconsistencies between layers by omitting the linkage between them modelled in Section 4.


Appendix A. Probabilistic relations between layers

A.1. An approximation

Lemma. Let X be a random variable, and A, B specific events. Then, as a first-order approximation in X,

Prob[X > 0, AIBI ~. Prob[AIX = E[XIX > 0, B], B] × Prob[X > 01B]. (A.1)

Proof. Let p denote a genetic pdf. Then

> O, AIB] = f p (X = x, a lB) dx = f p(AIX = x, B )p (X = xLB) Prob[X dx. (A.2) o , t /

x>0 x>0

NOW expand P(AIX = x, B) in a Taylor series about x = E[XbX > 0, B] = / z say. Then

p(AIX = x, B) = p(AIX = tz, B) + (x - / z ) ~-~Px (AIX = x, B)lx=u + O(x -/z) 2. (A.3)

Substitute (A.3) in (A.2) to obtain as an approximation

> 0, ALB] = p(AIX = Iz, B) f p (X = xLB) + ~ r p ( A I X = x, B)lx=u f (x - Iz)p(X = xIB). Prob[X

x>0 x>0 (A.4)

By definition of/~, the second integral in (A.4) is zero. Then (A.4) is equivalent to the result sought. []

A.2. Transition probabilities

Consider the matrix M} k) set out in Section 4.1. It contains three elements (denoted by *) requiting calculation. Each is of the form

Prob[{j, k, y > 0, 8}l{j - 1, k, a,/~}]. (A.5)

By the lemma, this may be approximated by

Prob[{j, k, ~}l{J, k - 1, A}, {j - 1, k, or, 13}] x Prob[l j , k - 1, y > 0}l{j - 1, k, or, fl}], (A.6)

A(k-1) where the context makes clear that A denotes --ij "

Consider the specific cases of or, fl that arise in the final member of (A.6). These are enumerated in Section 4.1 • (k)

just prior to setting out Mj . Evidently,

P r o b [ { j , k - l , y > 0 } l { j - l , k , o t , / ~ } ] = l f o r d > 0 . (A.7)

This relates to the second and third rows of M} k). J

In this case (A.6) may be reduced to give

Prob[{j, k, y > 0, 6}l{J - 1, k, a,/~}]

= Prob[{j, k, 8}l{j, k - 1, A}, {j - 1, k, ~,/~}] f o r d > 0. (A.8)

Consider the alternative cases in which a = 0, i.e. the (1, 2) and (1, 3) elements of M~ k). Here 13 = 0 necessarily. Then

Prob[{j, k - 1, y > 0}[{j - 1, k, 0, 0}] = Prob[{j, k - 1, y > 0}l{j - 1, k - 1, 0}]. (A.9)


From (A.6), (A.8) and (A.9), all transition probabilities may be calculated from probabilities of two forms, viz:

Prob[{j, k, ~}[{j, k - 1, A}{j - 1, k, or, fl}], (A.IO)

and

Prob[{j, k, y > O}l{j - 1, k, 0}]. (A.11)

A.3. The probability (A. 11)

Recall the general result for conditional probabilities,

Prob[AIB1 o r . . . or Bn] = Y~n=l Prob[AIBm] Prob[Bm] )--~m = 1 Prob[Bm ] ' (A. 12)

provided that the events B1 . . . . . Bn are mutually exclusive. In the present context, this gives

Prob[{j, k, y > 0}[{j - 1, k, 0}]

{Prob[{j, k, y > 0}l{j - 1, k, 0, 0} x Prob[{j - 1, k, 0, 0}]

+fProb[{ j , k , i ,>O} l { j - l , k , ot(>O),O}]xProb[{j-l,k, ol(>O),O}]}/Prob[{j-l,k,O}]

(A.13)

where the variable of integration is or. The first probability within the integral may be approximated by the replacement of ot with its expected value,

conditional on positivity (in much the same way as in the lemma), reducing it to

Prob[{j, k, y, 0}l{j - 1, k, A, 0}]. (A.14)

Substitution of (A. 14) in (A. 13) then gives

Prob[{j, k, 7, > 0}l{j - 1, k, 0}]

= {Prob[{j, k, 9/ > 0}l{j - 1, k, 0, 0} x Prob[{j - 1, k, 0, 0}]

+ Prob[{j, k, y > 0}[{j - 1, k, A, 0}] x Prob[{j - 1, k, ot > 0, 0}]}/Prob[{j - 1, k, 0}] (A.15)

The conditional probabilities appearing here are in fact the (1, 3) and (2, 3) elements of Mj! k). Their multipliers a r e

unconditional probabilities relating to the end of development year j - 1.

A.4. Unconditional probabilities

Consider the unconditional probabilities appearing in (A. 15), viz. Prob{j - 1, k, or, 0}. These may be calculated the usual forward recursion involving the transition matrix M! k). by J

G. Taylor/Insurance: Mathematics and Economics 20 (1997) 225-242 235

Prob[(j - 1, k, 0, O)] = Prob[(j - 2, k, 0,O)l x Prob[Ij - 1, k, 0,O)lI.j - 2, k O,O)l, (A. 16)

Prob[{j - 1, k,a! > O,O}] = Prob[{j -2,k,O,Ol x Prob{j - 1, k,cr > 0,OllIj -2,k,O,O)l

+Prob[(j -2,k, y > 0,O)l x Prob[{j - 1, k,a! > O,O]lIj -Zk,A,Oll,

(A.17)

where the second of the two members on the right-hand side is an approximation obtained by application of the

lemma to a more precise expression. Thus, unconditional probabilities at time j - 1 are expressed in terms of unconditional probabilities at time j - 2

and the transition matrix M?) J-1’

AS. Summary of the recursion

By Appendices A.2-A.4 (in particular, (A.lO), (A.1 1) and (A.15)-(A.17)), the quantities required for the com-

putation of M:k’ are:

Prob[Ij, k, Vl(j, k - 1, A), Ij - 1, k, a, B)l,

Prob[(j,k-l,y>O)](j-l,k-l,O]],

Prob[{j - 1, k - l,cz,O]].

(A.18)

(A.19)

(A.20)

Denote the respective sets of quantities (A.18), (A. 19) and (A.20) by NY’, R,(k-‘) and Sy_y’). Estimation of NY’ is the subject of Appendix B. With this notation, the recursion developed in Appendices A.2-A.4 may be represented

diagrammatically as follows:

/‘“,

N!c’ J

R,Ck-‘) J

AFk\

M,Ck- 1) (k-1) J A,-1

rf’\

s’i”-‘1 i2

AI-‘) J2

&If-” J 1

An abbreviated form of this diagram which recognises just the nodes involving M and S is as follows:

236

/ .~j(t-l)


•(t-l) -2 j,,lx.(k-i)

- !

S)l), j = J + 1, J + 2, M(2), .~(2) • ° - ~ ~ j , . . -

This indicates that the recursion, over j and k, must be carried out in the following order:

S)°)(all j ) , sJk)(all k) [given].

Then

MO), j = J + l , J + 2 . . . . .

where J denotes I - i + 1. Note that the values of M~ l) require various quantities related to the 0th layer, e.g.

R (°) = Prob[L~ °) > 01L~°_) 1 ---- 0].

The modelling of such quantities is treated separately in Section 5.

Appendix B. Main model for transitions between layers

Appendix A.5 identifies

Prob[{j, k, 8}]{j, k - 1, A}, {j - 1, k, c~, fl}], (B.1)

as a quantity required for the calculation of M (k). One interpretation of this conditional probability is that the probability of incurred losses in layer k flipping

from zero to non-zero in development year j depends on incurred losses in layer k - 1 in year j and year j - 1; equivalently, on the level of incurred losses in layer in year and their change in year k.

(k l) Thus the probability (B. 1) may be expressed in terms of Lij- and AL~k/._I~; or, by (2,5), equivalently in terms

of Qi(k-l) and AO (k-l) ~..i,j-I • That is, for example,

Prob[{j, k, 0/l{j, k - 1, l~kj-I)}, {j - 1, k, l{k/_l~, 0}]

= P r ° b [ L i ( k ) = o i L } k - l ) = l } k-l),l(k-l)~i,j-I =/(k-1)i,j-l'Li,j-i(k) " =0 ]

= fn[q{] -1), Aq~k,j-_ll)], (B.2)

where q, Aq are defined in terms o f / i n the same way as Q, AQ are defined in terms of L (see (2.3) and (2.5)).


Since Q expresses losses L as a multiple of layer width, it is reasonable to adopt the same function for all layers on the right-hand side of (B.2). For convenience this function is written in the logistic form

Prob[Li(~ ) = 0 L (k-l) l(k-l) L(k-l) ~(k-l) L(k) = O] i j = ' i j ' i , j - I = t i , j - I ' i , j - I

{1 + exp f [ q ~ - l ) , A~(k-1)l/-l , (B.3) - - t t i , j _ 1 JJ

where the function f is still to be specified. Note that f (q, p) will usually be - c o if q < 1 (no exhaustions of layer k - 1). Subject to this,

{ + o o as p ~ oo (all q), (B.4) f (q , P) - o o as p ~ 0 (allq),

and it may further be reasonably assumed that

Of/Sp, ~f/Oq > 0 forq > 0 and all p. (B.5)

Note that f (q , p) is bounded above for fixed p and q ~ co; otherwise, the probability in (B.3) would approach 0 by virtue of large q~- I even if A k- l ' qi,j- l were small.

With this last observation taken into account, together with (B.4) and (B.5), a reasonable form to assume for fitting of f to data is

[ to[q~(1 +q)]log(fllp), q > 1, (B.6) f (q , P)

I - c o , q < l ,

with riO, fll positive constants.

Appendix C. Data for numerical example

C.1. Incurred losses

The example is concerned with the modelling of layers k = 0. The experience in layer 3 is, however, dependent on that in layer 2.

The triangulations of incurred losses for layers k = 0, 2, 3 are shown in Table 2. The bold heading of the final column of the table draws attention to the fact that this a 19-month experience

period. All other experience periods are annual. Note that the data given for layer 0 in Table 2 satisfy assumption (3.1) except in development year 1. This is

sufficient for present purposes. The coverage provided by layers 2 and 3 is shown in Table 3.

Appendix D. Numerical results

D. 1. Bottom layer

The model is set out in Section 5. The particular forms chosen for hi (.), h2(.) and h3(.) were:

hi (i) = kl max(0, i - 1981) + k2 max(0, i - 1983) + k3 max(0, i - 1989) + k4l(i = 1990),

h2(j) = k5 + k6 log j + k7j, ½ + (j - 6)2/36 for j > 2,

h 3 ( j ) : 2 5 1 ½ + ( J - 6 ) 2 / 3 6 ] f o r j < 2 ,

(D.1)

(D.2)

(D.3)

238

Table 2


Treaty Incu~edlosses(US $000)to end of year 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 LayerO 1980 3175 1981 8255 1982 898 1983 2055 1984 281 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Layer 2 1984 0 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Layer 3 1984 0 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

2880 8380

878 20070 24521

0

8750 0

8750 0

10 177 10 177 13 522 13 522 13 393 13 393 13 393 13 393 12289 12289 14380 14380 14630 16491 17426 19426 13191 21016 42871 21103

878 688 688 688 688 688 688 688 688 688 23 114 26 852 31 541 29915 26234 27770 29011 28940 29626 29428 48521 97 925 101 255 115 101 110344 140672 143 979 163 217 167623 168077 20772 49438 86483 91 173 85040 84653 83 897 84755 82909 77 860

0 10275 71040 99494 113 687 121 807 112025 121 764 129391 129629 0 29780 56 137 72902 89 189 98 204 113639 120470 125616

0 15 000 15 328 54685 123488 149594 176693 180549 0 0 1250 21 231 48 202 75 254 90957

0 16288 78577 68 328 121 708 127 232 0 0 8175 23 873 49615

0 7500 16254 55 280 0 17 295 44586

0 7750

9688 9892 10072 10075 10075 10075 10075 10076 10076 10076 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7500 7500 8657 8792 7500

0 0 0 0 11 444 11 444 11 885 8771 8771 0 0 0 0 1080 9 116 15000 22500

0 0 0 0 0 2851 1 129 0 0 5 000 10 000 30 000 50 000

0 0 0 10000 10000 0 0 0 10 847

0 0 5 000 0 5 000

9688 9892 11 620 11 624 11 624 11 624 11 624 11 625 11 625 11 625 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2632 1810 1810 1810

0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 000 27 232 45 000

0 0 0 0 0 0 0 0 0 10 000 20 000 52 000 93 650

0 0 0 5191 7740 0 0 0 10000

0 0 0 0 10000

Table 3 Layer limits Year Layer 2 Layer 3

Lower limit ($M) Upper limit ($M) Lower limit ($M) Upper limit ($M) 1984-1988 26.25 33.75 33.75 48.75 1989-1994 17.5 22.5 22.5 32.5


where I (.) is an indicator function which takes the value 1 when the condition occurring as argument is satisfied, and 0 otherwise.

The function hi(.) defines a piecewise linear function of treaty year with changes of slope at 1981, 1983 and 1989, and with treaty year 1990 regarded as exceptional.

The function h3 (j) is chosen to produce rough homoscedasticity. The data set out in Appendix C were analysed by the GLIM package. Outliers, defined as observations producing

standardised residuals numerically greater than 3, were given no weight in the regression. This yielded the following parameter estimates:

Parameter Estimate kl --0.3603 k2 +0.6045 k3 -0.3796 k4 -0.4908

k5 + 1.970 k6 -3 .074 k7 +0.1637

a2 1.505

3 . 3 0 0 3 . 0 0 0 2 . 7 0 0 2 . 4 0 0 2 . 1 0 0 1 . 8 0 0 1 . 5 0 0 1 . 2 0 0 0 . 9 0 0 0 . £ 0 0 0 . 3 0 0 0 . 0 0 0

- 0 . ' 1 0 0 - 0 . 1 ; 0 0 - O . g O 0 - 1 . 2 0 0 - 1 . S O 0 - 1 . 0 0 0 - 2 . 1 0 0 - 2 . 4 0 0 - 2 . 7 0 0

R R

2 R R R 2 R R 2 R 2 R R R R 3 R R 2 R R

R R 2 R R 2 R 2 R R R R

3 2 R R 2 4 2 3 R R R R R 4 3 R R R R

R R 3 R R R

R R R

197e~oo 19e1.oo 1.4~oo 1907~oo 199oloo 1993~oo 1996~o

Fig. 1. Standardised residuals by treaty year.

3 . 3 0 0 3 . 0 0 0 2 . ' 7 0 0 R R 2 . 4 0 0 2 . 2 0 0 1 . 0 0 0 1 . $ 0 0 R R R R R 1 . 2 0 0 2 R R R R 0. 900 R 2 R R R 0.600 3 R 2 R R R R 0 . 3 0 0 3 2 R 2 R 0 . 0 0 0 2 2 R R 2

- 0 . 3 0 0 R 2 2 2 R 2 R R R R - 0 . 6 0 0 R R 2 R R R 2 R - 0 . 9 0 0 R R R R R 2 2 R R - 1 . 2 0 0 R R R R R R - 1 . 5 0 0 R R - 1 . 8 0 0 R - 2 . 1 0 0 R - 2 . 4 0 0 R - 2 . 700

. . . . . . . . : . . . . . . . . . : . . . . . . . . . : . . . . . . . . . : . . . . . . . . . : . . . . . . . . . : . . . . . . . . . :

o . 0 0 3 . 0 0 1; . 0 o 9 . 0 0 12 . o 0 1 5 . 0 0 1 8 . 0 o

Fig. 2. Standardised residuals by development year.

24O

Table 4 Incurred losses


Treaty year To 1995 ($000) Projected ultimate ($000) 1983 29428 29930 1984 168077 174306 1985 77 860 83364 1986 129629 142355 1987 125 616 142212 1988 180549 212745 1989 90957 119402 1990 127232 182 878 1991 49615 102 113 1992 55 280 157285 1993 44586 209722 1994 7750 122022

The value K in (5.2) was set to zero, and cases of Zij < 0 were excluded from the regression. Note that h2(j) does not converge to zero for increasing j . It does however provide reasonable values over the range of j (1-16) dealt with here. Incurred loss development is ignored beyond development year 16.

The residual plots shown in Figs. 1 and 2 indicate reasonable effectiveness of h l (.) in tracking variations related to treaty year; and of h3 (.) in removing heteroscedasticity.

Model age-to-age factors were produced for this model in accordance with (5.5). These were chained, then applied to the 1995 incurred losses, to give the results in Table 4. Further development of treaty years earlier than 1983 has been ignored.

The reliability of the estimates decreases with increasing treaty years.

D.2. Upper layers

D.2.1. Probability of nil incurred The model summarised by (4.11) and (4.12) was fitted to data comprising all transitions from zero incurred at

the beginning of a development year to zero or non-zero incurred, respectively, at the end of that year. Note that/50, 151 do not depend on layer in (4.12), and so (4.11) was fitted for all k = 1-5 simultaneously.

The results were as follows:

Parameter Estimate 130 0.8236 /~l 3.6529

D.2.2. Size of non-zero incurred The model summarised by (4.13) and (4.14) was fitted to data comprising all non-zero incurred losses from

Table 2 for layers k = 1-5. After a small amount of experimentation, s = 3 was chosen in (4.14) and b <k) = b, independently of k. In

addition, data became so sparse in the upper layers that a (3), a (4) and a (5) were assumed equal. The resulting version of (4.13) was fitted to the data using weighted least squares with weight j4/(i + j)2

ai (k) to achieve rough homoscedasticity with respect to development year j and associated wi th observation A experience year i + j (here i = 1 corresponds to treaty year 1984, the earliest for which upper layer experience is available).

This weighting factor indicates: - the strong increase in reliability of the model with increasing development year; - the fact that variability of experience appears to have increased over past experience years.


The r e s u l t s w e r e as fol lows:

Parameter Est imate

a C1) 0.1428

a (2) 0.3591

a (3) (= a (4) = a (5)) 0.6952

b 8.265

D.2.3. Numerical projections The models descr ibed in Appendices D.2.1 and D.2.2 are applied to produce Tables 5 and 6, respectively,

appl icable to layer 3.

As indicated by (4.11) and (4.13), these results will depend on year-by-year activity in layers 0 - 2 which is not

shown here.

C o m m e n t is required on the bold figures in Table 6. These are assumed values o f Ai(~ ) for 1995 in those cases

where LI 3) = 0. These values are required by (4.13) for the project ion o f future values o f "(3) l~t i j . ~(3)

These artificial values of Ai(~ )~ have been obtained by averaging the values o r t~ij observed for 1995 in respect

o f certain treaty years i.

With a couple o f exceptions, the "cer ta in" treaty years are those preceding the entry under considerat ion.

Table 5 Probability of nil incurred losses in layer 3 Treaty Projected probability (%) of nil incurred losses to end of year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 1994 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1993 100 73 47 31 21 15 12 10 9 8 7 6 5 1992 0 0 0 0 0 0 0 0 0 0 0 0 1991 0 0 0 0 0 0 0 0 0 0 0 1990 0 0 0 0 0 0 0 0 0 0 1989 100 95 91 88 85 82 80 78 76 1988 0 0 0 0 0 0 0 0 1987 100 93 87 81 77 73 70 1986 0 0 0 0 0 0 1985 100 96 93 89 87 1984 0 0 0 0

Table 6 Expected value of non-zero incurred in layer 3 Treaty Model expected value of incurred losses (US $000) (provided non-zero) to end of year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 1994 10000 24864 28661 31 652 34060 35966 37421 1993 11339 13 102 15 130 16888 18326 19411 20062 1992 10000 11 264 12754 13998 14958 15520 15970 1991 7740 8547 9485 10222 10647 10990 11 275 1990 93 650 95674 97310 98587 99622 100476 101 190 1989 12986 13462 13 815 14096 14330 14529 14701 1988 45000 45856 46586 47187 47692 48123 48496 1987 6717 7021 7300 7539 7747 7930 8100 1986 1810 2025 2236 2421 2586 2740 1985 6717 6892 7054 7195 7322 1984 11625 11837 12041 12227

38432 39242 39913 40486 20582 21 017 21 393 21 722 16347 16671 16953 17 203 11518 11728 11913 12083

101795 102312 102766 14852 14993 48 831

40980 41415 41815 22016 22295 17436


Table 7 Expected value of incurred (zero or non-zero) in layer 3 Treaty Model expected value of incurred losses (US $000) to end of year 1995 1996 1997 1 9 9 8 1 9 9 9 2000 2001 2002 2003 2004 2005 2006 2007 2008 1994 10000 24864 28661 31652 34060 35966 37421 38432 39242 39913 40486 40980 41415 41810 1993 0 3503 7988 11 679 14476 16446 17573 18439 19 143 19737 20245 20689 21 101 1992 10000 11264 12754 13998 14958 15520 15970 16347 16671 16953 17203 17436 1991 7740 8547 9485 10222 10647 10990 11275 11518 11728 11913 12083 1990 93650 95674 97310 98587 99622 100476 101190 101795 102312 102766 1989 0 648 1 1 9 7 1 6 8 7 2135 2550 2934 3294 3646 1988 45000 45856 46586 47187 47692 48123 48496 48831 1987 0 491 968 1 3 9 5 1779 2129 2458 1986 1810 2025 2236 2421 2586 2740 1985 0 261 524 757 972 1984 11625 11837 12041 12227

Thus, for example, that for i = 1987 is taken as the average over i = 1984, 1986. The same average is taken for

i = 1985. The average for i = 1993 omits the exper ience o f / = 1990 since it appears abnormal. It is calculated as

Average (10 000, 7740, 45 000 x 2, 1810 x 2, 11 625 × 2), where the factors o f ] reflect the change o f layer l imits

be tween 1988 and 1989.

The results o f Tables 5 and 6 are combined by means o f (6.1) to produce Table 7.

Documents

Reserving consecutive layers of inwards excess-of-loss reinsurance