PANJER vs KORNYA vs DE PRIL : A C O M P A R I S O N FROM A P R A C T I C A L
P O I N T OF VIEW
BY S KUON, A. REICH AND L. REIMERS
The Cologne Re, Cologne
ABSTRACT
We compare three modern methods for calculating the aggregate claims distribution with respect to their computation amount and accuracy" Pan ler's algorithm,
the approximation method of Kornya and the most recent, exact procedure of
De Pril. They are compared numerically in the case of actual Life portfolios. The
computauon amount of De Prll's method is much greater than that of the two
others, which do not differ substantially in this respect. The accuracy of Kornya's
and Panjer's methods is remarkably high in the examples considered. However,
as the accuracy of Kornya's approximation method can be determined easily
m advance, this procedure turns out to be the most useful one for the problems
arising from practical work.
KEYWORDS
Aggregate claims distribution; computation amount; individual model; collective
model.
1. INTRODUCTION
In the years since 1980 a remarkably large number of methods have been made
available to insurance mathematicians for calculating the aggregate claims
distribution of a portfolio numerically. In the monograph of GERBER (1979) one
can find those methods which were used up to 1980 and above all the models to
be examined in pracuce and theory, namely the so-called individual and (due to
LUNDBERG (1909)) the collective model of risk theory. A m o n g the methods
developed after 1980, of which there is no summarizing description in the internauonal literature, we would like to quote as new ideas or applications the recurswe
formula of PANJER (1981), the method of Fast Fourier Transform (FFT) of
BERTRAM (1981), the approximation method of KORNYA (1983) and the method
of DE PRIL (1986). For practitioners and theoreticians, the quesuon arises which
of all these methods should be given preference in situations relevant in practice.
With regard to BUHLMANN (1984), who compared the algorithm of Panjer
with the FFT method, it is only necessary to analyse one of these two methods
m more detail. We have chosen the Panjer algorithm, because our experience
proves that most practitioners choose the Panjer method. Therefore, the aim
ASFIN BULLETIN Vol 17, No 2
184
KUON, REICH AND REIMERS
o f thts p a p e r Is to c o m p a r e the m e t h o d s o f P a n j e r , K o r n y a and De Pril from a
practmal point o f view.
Such a c o m p a r i s o n ~s feasible only m the case o f Life p o r t f o l i o s ; i.e. m the
l a n g u a g e o f risk t h e o r y in such an individual model, where the claim a m o u n t
d i s t r i b u t i o n s o f the individual policies are (individual) two point d i s t r i b u t i o n s
throughout:
(i) T h e m e t h o d o f P a n j e r first o f all exactly d e t e r m i n e s under certain c o n d i t i o n s
the aggregate claims d i s t r i b u t i o n m a collective model. F o r a calculation o f
an individual m o d e l , one has to t r a n s f o r m this situation into a statable collective model (due to GERBER (1984) and HIPP (1985) there is an estimate
for the e r r o r which arises from this transition). The Pan.ler a l g o r i t h m ,
however, then enables the calculation (up to a c h e c k a b l e error term) o f the
aggregate claims d i s t r i b u t i o n m any individual model.
(i0 The m e t h o d o f K o r n y a evaluates up to a prescribed a c c u r a c y the aggregate
claims d i s t r i b u t i o n in an individual model. The original m e t h o d d e v e l o p e d
by KORNYA (1983) took o n l y Life p o r t f o h o s into a c c o u n t , but it can be
generalized to an a r b i t r a r y individual m o d e l , as was shown by H I e p (1986).
(ii0 The m e t h o d o f De Pril m a k e s possible the exact calculation o f the aggregate
claims d i s t r i b u t i o n m the i n d i v i d u a l model, but only in the case o f Life portfolios. At present, a g e n e r a l i z a t i o n to m o r e general situations is not within
sight.
It is t h e r e f o r e precisely in the case o f Life p o r t f o l i o s that all three m e t h o d s can
be c o m p a r e d wlth one a n o t h e r T h e fact that in this case the m e t h o d s o f P a n j e r
and K o r n y a only lead to an a p p r o x i m a t i o n is no d i s a d v a n t a g e from a practical
and theoretical p o i n t o f view p r o v i d e d that error estimates for the inaccuracies
are sufficiently small.
In the following section we present the three m e t h o d s m a u n i f o r m way. In Secn o n 3 we then c o m p a r e these m e t h o d s by m e a n s o f actual p o r t f o l i o s and portfolios derived f r o m these m o r d e r to answer the question o f which m e t h o d is the
best one for numerical c o m p u t a t i o n s . As p r a c t i t i o n e r s , there is no need for us to
look at the p o r t f o l i o s given h~therto in a c t u a r i a l literature, which m general are
both theoretical and small.
2. UNIFORM DESCRIPTION O F ' T H k METHODS
For the individual m o d e l o f a Life p o r t f o l i o , c h o o s e finitely m a n y m o r t a h t y rates
q~ . . . . . q j and (up to a fixed m o n e t a r y umt) fimtely m a n y risk sums i = 1. . . . . I,
such that each policy o f the p o r t f o l i o has a m o r t a h t y rate qj,,, Jo ~ J, and a risk
sum t0, to .~ I. D e n o t e by n,j the n u m b e r o f all policies with m o r t a l i t y rate qj and
risk sum t. T h e d l s m b u t i o n function o f a p o h c y Y,//in a cell O , J ) , l = 1, . ,n,j, is
Ftj(x)=
l - qj,
O~x<t
1,
x l> I.
PANJER vs KORNYA vs DE PRIL A COMPARISON
185
The &strlbutlon function
F(x) = prob(S ~< x)
o f the aggregate claml S can be expressed on the (usual) assumption o f the
independence o f all the pohcles by the convolution formula
F(x)=
,~- F,*"',(x)
t,J
As ~s well known, for large portfolios it is impossible to carry out these convoluuons numerically. Because the support o f S ~s contained m No, ~t ~s sufficient
for a c o m p u t a t i o n o f F to calculate
f ( x ) = prob(S = x ) ,
(x = 0, 1.... ).
(0 The method o f Panjer. To apply this method to the
must first o f all assign a collecnve model to the individual
done in various ways, but we have chosen the most standard
chosen a specml c o m p o u n d Polsson distribution, which
equations
/
given s~tuanon one
model. This can be
procedure. We have
is defined by the
J
k= E
Z qjn,j
t=l 3=[
1
G(x) =
i
J
Z
Z
t=l
j=[
where
IO,
0~<x<t
F'~°)(x) = 1,
x i> t
is independent o f j. The density function g o f the probablhty d~strlbunon G is
given by
g(t) = ~
(t = 1,2 . . . . . I ) ,
where
J
X, = ~
q~n,j.
3=1
The aggregate claims S in the individual model are therefore approximated by an
assocmted collective model
N
g=Zx,
t=!
where the "claim a m o u n t s " X, are independent and identically distributed with
distribution function G. The X, are also assumed to be independent o f the "claim
n u m b e r " N, which follows a Poisson distribuuon with parameter X. The density
f ( x ) = prob(S = x)
186
KUON, REICH AND REIMERS
o f the distribution function of ,S can now be calculated (see PANJER, 1981):
f(0) = e x p ( - k),
and then recursively by
](x)
= x_ ....
X
vg(v)f(x-
v)
( x = 1,2 .... ).
v=l
In case of large X, f(0) may numerically equal zero. For example, PANJER and
WILLMOT (1986) give two different methods (decornposmon o f the portfolio and
exponential scaling) to handle this problem.
For the error
sup l F ( x ) - F'(x)[ ,
where F, F denote the corresponding distribution functions, GERBER (1984) and
HIPp (1985) have given error estimates.
(n) The m e t h o d o f Kornya. We assume qj ~< 1[3, j = l ..... J, and define for
any K ~ N
&(K)
-
-
K + I ,=l j=l
rl U
In view of
A(K) --+ 0
for K--* c~
one can find to any given accuracy e > O.a K E N satisfying
e x p ( A ( K ) ) - 1 ~< e.
For such a fixed K, consider the special polynomial
--(K)
Om Um ,
=
m
=
0
where
b~K)= ~.a (-l)/' k ~ ,/,g(lq~sqs) ~
k=l
k
~'=1 J = l
and
b~ff') =
/~
( - 1)~
I,&lm
k
g=l
n,,,/,,a
denote the non-vanishing Taylor coefficients o f Q(K).
,
m ~< IK,
PANJER
vs K O R N Y A
vs D E P R I L
A COMPARISON
187
The simple recursmn f o r m u l a
a~ A') = exp b~ h')
a~A,~= - 1 ~
n
mahdi,, h(K)
~ ....
n i> 1,
Ill = [
then leads to the coefficients o f the p o w e r series o f exp Q(K)(u), i.e.
exp Q(K)(u)= Z
a~K)u".
n=O
Finally, ~f one defines
F(x)(X) = ~
I aU~l,
i/=0
then the result o f KORNYA (1983) states that
[ F(x) - F(X~(x)[ ~<e x p ( A ( K ) ) - 1 ~< e
holds for all x ~< Z m,j.
010 The method o f De Prll P u t t i n g
A,k=(-l)~+J t
± n,j
( t = l . . . . . l,k~> 1),
J=l
the result o f DE PRIL (1986) states that
I
f ( 0 ) = 1-~
t=l
f(x)
1
=X
J
~
(I - qj)"',
J=l
..... ~
t,/,I
z.~
~
t=l
k=l
A,k f ( x -
kt)
holds for all x ~< ~ m,s.
3. NUMERICAL COMPARISON
As all o f the m e t h o d s will turn o u t to be sufficiently precise, we only need to look
at the c o m p u t a t i o n a m o u n t as an o b v i o u s measure o f their usefulness. The
a m o u n t o f c o m p u t a t i o n can be quantified as the n u m b e r o f floating point o p e r a t~ons, c o u n t e d s e p a r a t e l y as b a r o p e r a t m n s ( a d d i t i o n s and substract~ons) and d o t
o p e r a t i o n s (mult~phcations a n d dwls~ons), o r as the c o m p u t a t i o n t~me ( C P U
time). We must e m p h a s i z e that C P U time d e p e n d s heavily on c o m p u t e r type, prog r a m m m g language and style, and that the t u r n a r o u n d u m e m a y be m a n y t~mes
longer. T h e following m e a s u r e m e n t s have been c a r n e d out on p r o g r a m s w r m e n
m VS A P L on an IBM 4381 under V M / C M S .
Crucial for the c o m p u t a t i o n a m o u n t ~s not only the size o f the p o r t f o h o but
also the n u m b e r o f values o f the aggregate clmms d i s t r i b u t i o n to be c o m p u t e d .
G w e n a certain step width ,6 (e.g. a fixed n u m b e r o f m o n e t a r y units) the values
188
KUON, REICH AND REIMERS
o f F(0), F ( A ) . . . . . F(L • A ) are to be c o m p u t e d for a certain n a t u r a l n u m b e r L.
In the following examples we have chosen L so that L • A is a b o u t the size o f
# + 3 • o, where # and o denote the mean and s t a n d a r d d e v i a t i o n o f the aggregate
claims d i s t r i b u t i o n , which can easily be c o m p u t e d b e f o r e h a n d .
In the a l g o r i t h m s o f P a n j e r and K o r n y a , the c o m p u t a t i o n a m o u n t d e p e n d s
on I, J, K ( K o r n y a only) and L, but not on the p o r t f o l i o size. The n u m b e r o f
b a r and dot o p e r a t i o n s (BO resp. DO) can be expressed explicitly. F o r large L
we have, in the case o f P a n j e r ' s a l g o r i t h m , a p p r o x i m a t e l y BO --- DO = I . L;
in the case o f K o r n y a ' s a l g o r i t h m , a p p r o x i m a t e l y B O - ~ DO = K . I - L .
De P r l l ' s a l g o r i t h m needs a b o u t BO = L • H i • (0.5 • L + J - 1) + J and
DO --- L • H i . (0.5 • L + J + 1 . 5 ) + L • ( J + 1 ) + M - 1, where
I
J
M=Z
Z
t=l J=l
is the n u m b e r o f policies and
HI=~
1-=< 1 + I n I
t=l
I
is the tth h a r m o n i c n u m b e r .
T h e first e x a m p l e derives from a real Life insurance p o r t f o h o . It consists o f
104,652 risks aged from 15 to 74 with risk sums from DM 2,000 to DM 200,000
in steps o f D M 2,000 (that is I = 100, J = 60). The expected aggregate claim
a m o u n t s to p . = D M
4 5 3 7 mflhon, the s t a n d a r d d e v i a t i o n is a = D M
0.511 million. T h e aggregate claims d i s t r i b u t i o n is to be c o m p u t e d for a value o f
L = 3,000 which c o r r e s p o n d s to an a m o u n t o f D M 6 m d h o n or a p p r o x i m a t e l y
# + 3 • o and c o m p r i s e s 99.6% o f its mass.
Algorttlun
Panjer
Kornya ( K = 5)
De Prd
BO
DO
CPU seconds
Error estimate
298,049
1,530,560
24,121,798
307,150
1,378,250
24,443,487
4 200
9 748
~ 500
2 10 4
3 3 10 8
0
Owing to lack o f c o m p u t e r capacity, we had to omit the actual c o m p u t a t i o n o f
/~ with De P r i l ' s a l g o r i t h m and could only count the n u m b e r o f o p e r a t i o n s .
To e s t i m a t e the a c c u r a c y o f K o r n y a ' s a l g o r i t h m , we used the f o r m u l a from
Section 2; to estimate the error o f the d i s t r i b u t i o n c o m p u t e d a c c o r d i n g to P a n j e r ,
we a d d e d the distance between the K o r n y a and P a n j e r d i s t r i b u t i o n to the K o r n y a
b o u n d , a p p l y i n g the triangle inequality. A c c o r d i n g to Section 2 it is possible to
estImate the e r r o r o f the collective model following H w P (1985), but for this
a n o t h e r a p p l i c a t i o n o f P a n j e r ' s a l g o r i t h m is needed and the c o m p u t a t I o n a m o u n t
increases c o n s i d e r a b l y . ( W h a t is more, H i p p ' s estimate is rather pessimistic,
especmlly in the case o f large p o r t f o l i o s . )
189
PANJER vs KORNYA vs DE PRIL A COMPARISON
This e x a m p l e d e m o n s t r a t e s the usefulness o f the a b o v e given f o r m u l a e for the
n u m b e r o f F L O P s o f K o r n y a ' s , P a n j e r ' s and De Prll's al g o r i t h m s. O w i n g to different p r o g r a m structures, the n u m b e r o f F L O P s per second differs s o m e w h a t for
the a l g o r i t h m s
A c o n s i d e r a t i o n c o n c e r n i n g the choice o f step width .6: halving ,5 doubles both
L and I. F r o m the f o r m u l a e given above, it follows as a rule o f t h u m b that all
the a l g o r i t h m s need four times the previous c o m p u t a t i o n a m o u n t .
We will now e x a m i n e the p e r f o r m a n c e o f D e P r d ' s a l g o r i t h m in c o m p a r i s o n
with P a n j e r ' s and K o r n y a ' s if applied to smaller p o r t f o l i o s T o achieve this, we
take s u b p o r t f o h o s from the e x a m p l e a b o v e , consisting o f the I youngest age
classes and the J smallest sum classes. For K o r n y a ' s a l g o r i t h m we c h o o s e K = 5;
L is again d e t e r m i n e d from p. and a.
CASE I / = J = 10, 10,953 rtsks, /~=DM 137,346, o=DM 45,861, L=250, 99 99% of mass
Algornhm
Panjer
Kornya (K=5)
De Prll
BO
DO
CPU seconds
Error es~Hnate
2,304
13,060
97,334
3,065
11,525
112,744
0 231
0327
4 250
I 05 10-~
5 6 10 ~5
0
CA',~II / = J = 15, 27,687 risks, #=DM 467,757, o=DM 99,046, L=400, 99 890//0 of mass
Algorithm
Panjer
Kornya (K= 5)
De Prll
BO
DO
CPU seconds
Error estimate
5,719
31,215
281.928
6,935
27,625
319,060
0386
0549
9 779
1 10 10 -4
I 3 10 - i a
0
CASF. Ill / = J = 20, 46,698 risks, p. = DM 945,215, o = DM 156,920, L = 700, 99 68% of mass
AIgornhm
Panjer
Kornya (K=5)
De Pill
BO
DO
CPU seconds
Error estimate
13,509
72,120
923,999
15,630
65,750
991 ,I 18
0750
I 030
22 878
1 14 I0 -a
2 5 I0 14
0
O u r last e x a m p l e is a Life p o r t f o h o g e n e r a te d from scratch. It consists o f 1,019
risks with sums ranging from D M 2,000 to DM 50,000 (step width is 2,000, i.e.
I = 25) and ages ranging from 15 to 64 years ( J = 50). Th e m o r t a h t y is 50°7o o f
the G e r m a n m o r t a l i t y table A D S t 60/62 m o d . T o avoid a v o l u m i n o u s table, the
class sizes n,j are defined by the s o m e w h a t artificial f o r m u l a
n u = [0.5 + max[O, 1.7 • e x p ( - 0 . 0 0 0 5 . ( 4 l 2 +,22)) + 0 . 5 . c o s ( t + , j ) ] ]
1 ~< i~< 25, 1 ~<j ~ 50.
( I x ] denotes the largest integer less than or equal to x). T h e aggregate claims
distribution was c o m p u t e d for L = 100 ( a p p r o x i m a t e l y # + 3 • o, 99.5O7o o f mass).
190
Algoruhm
Panjer
Kornya (K=3)
Kornya (h = 5)
Dc Prfl
KUON, REICH AND REIMERS
BO
DO
3,349
11,375
16,525
36,504
3,675
4,825
5,150
43,470
CPU seconds
0
0
0
3
104
158
174
180
Error esumate
6 7 10 '~
2 3 10-6
3 4 10- io
0
4. CONCLUSIONS
De P1fl's a l g o r i t h m , w h i c h gives exact results, is a r e m a r k a b l e p r o g r e s s in t h e o r y ,
but i n v o l v e s m u c h g r e a t e r c o m p u t a u o n a m o u n t t h a n the two o t h e r m e t h o d s .
P a n j e r ' s a l g o r i t h m gives an a p p r o x i m a t i o n to the a g g r e g a t e c l a i m s d l s t r l b u u o n
m m i n i m a l time. Its a c c u r a c y is sufficient for m o s t p r a c t i c a l p u r p o s e s , but it can
o n l y be e s t i m a t e d at s o m e a d d ~ u o n a l e x p e n s e .
K o r n y a ' s a l g o r i t h m needs m o r e c o m p u t a t i o n s t h a n P a n j e r ' s , b u t it a l l o w s us,
g w e n K, to e s t i m a t e its a c c u r a c y , or, ~f we ask for a specific a c c u r a c y , to c o m p u t e
K. In each e x a m p l e the a c t u a l C P U u m e was o n l y slightly h i g h e r t h a n that for
P a n j e r ' s a l g o r i t h m . T h e a c c u r a c y o f K o r n y a ' s a l g o r i t h m , ff a p p h e d to small p o r t f o h o s , is so high that it c o m p u t e s p r a c t i c a l l y the e x a c t d i s t r i b u t i o n .
K o r n y a ' s a l g o r t t h m t u r n s o u t to be a m e t h o d for c o m p u t i n g a g g r e g a t e clam~s
d l S t r l b u u o n s w h i c h is well stuted for b o t h s m a l l a n d large L i f e p o r t f o h o s .
AUTHORS' NOTE
M. V a n d e b r o e k a n d N. De Prll r e f o r m e d us at p r o o f - s t a g e that t h e y h a v e d r a w n
a suntlar c o n c l u s t o n , w h c r e the e x a c t m e t h o d o f De Pril is t a k e n as a basis for
a new a p p r o x i m a t i o n p r o c e d u r e . T h i s p a p e r will be p u b l i s h e d in a specml issue
o f the B u l l e t i n o f the R o y a l A s s o c t a t t o n o f Belgtan A c t u a t t e s , d e d i c a t e d to the
80th b i r t h d a y o f P r o f e s s o r E. F r a n c k x .
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S. KUON, A . REICH, L. REIMERS
The Cologne Re, Department for Research and Development, Theodor-HeussRing 11, D-5000 Koln ], Federal Republic' of Germany.