ESTIMATION OF STOP LOSS PREMIUM IN F I R E INSURANCE
C. P. WELTEN
Amsterdam, Holland
The estimation of stop loss premiums can be based on some knowledge about the distribution function of the sum of all claims in a year (assuming t h a t the stop loss insurance relates to a period of one calender year). Generally speaking there are two methods to obtain this knowledge about the distribution function.
I. The first method is to construct a distribution function from data concerning:
a. the distribution function of the number of claims per year, taking into account the variability of the parameter(s) of this distribution function.
b. the distribution function of the insured sums. c. the distribution function of partial claims.
d. the correlation between the insured sum and the probability of occurring of a claim.
e. the probability of contagion.
2. The second method is to derive a distribution function from the year's totals of claims over a long series of years, expressed in e.g. units of the totals of insured sums in t h a t years.
In practice it is often difficult to find a useful basis to apply one of these methods. Data concerning the distribution function of the number of claims per year, of the insured sums, and of partial claims are mostly available, but often nothing is known about the correlation between the insured sum and the probability of occurring of a claim.
The second method is mostly not applicable because, if the year's totals of claims over a long series of, by preference recent, years are available, these data often t u r n out to be heterogeneous
ESTIMATION OF STOP LOSS PREMIUM IN FIRE INSURANCE 3 5 7
or to be c o r r e l a t e d w i t h time. If, in t h a t case, o n l y t h e d a t a of t h e m o s t r e c e n t y e a r s are used, t h e n u m b e r of these d a t a is o f t e n a too small basis for t h e c o n s t r u c t i o n of a d i s t r i b u t i o n function.
A special difficulty c o n c e r n i n g t h e calculation of stop loss p r e m i u m s is t h a t we h a v e to k n o w as m u c h as possible a b o u t e v e n t s t h a t h a p p e n o n l y seldom. O f t e n it is n o t e a s y t o o b t a i n t h a t knowledge.
I n t h e following we will describe a case of a stop loss c o v e r in fire insurance. In this case, c o n c e r n i n g a fire i n s u r a n c e c o m p a n y w i t h a r a t h e r small n u m b e r of i n s u r a n c e s w i t h r e l a t i v e high i n s u r e d sums, t h e values
t o t a l of claims, occurring in t h e y e a r j X lO 4 rj = t o t a l of i n s u r e d sums in t h e y e a r j
were available o v e r a period of 45 y e a r s (1916-196o). As well in t h e n u m e r a t o r as in t h e d e n o m i n a t o r t h e r e i n s u r e d p a r t s of claims resp. reinsured p a r t s of i n s u r e d sums are n o t included. T h e d a t a , n e c e s s a r y for a p p l i c a t i o n of t h e first m e t h o d , were o n l y p a r t l y available, a n d it was n o t possible to g a t h e r t h e m as yet.
T h e coefficient of c o r r e l a t i o n b e t w e e n r i a n d j p r o v e d to be - - o,o98. According t o Fisher's z ' - m e t h o d t h e d e v i a t i o n of this v a l u e f r o m zero is n o t significant, so t h a t t h e h y p o t h e s i s , t h a t t h e v a l u e s r i h a v e n o t a linear t r e n d , can n o t be rejected. F u r t h e r investiga- tions h a v e n o t given rise to t h e a s s u m p t i o n t h a t t h e r e is a non- linear t r e n d of some i m p o r t a n c e , a n d indications for lack of h o m o - g e n i t y are n o t found. F o r this r e a s o n s we a s s u m e d t h a t t h e 45 o b s e r v e d values r i can be considered as a r a n d o m sample f r o m a h o m o g e n e o u s p o p u l a t i o n .
An obvious m e t h o d to find a useful d i s t r i b u t i o n f u n c t i o n f r o m t h e values r i is b a s e d on t h e a s s u m p t i o n t h a t this d i s t r i b u t i o n f u n c t i o n belongs to t h e w e l l k n o w n class of P e a r s o n - c u r v e s . T o a p p l y this m e t h o d t h e c e n t r a l m o m e n t s of t h e sample (the values r x
. . . r45 ) are c a l c u l a t e d first. These m o m e n t s t u r n o u t to be :
r n 1 = 2 , 2 5 6 m3 = 9,176
358 ESTIMATION OF STOP LOSS PREMIUM IN FIRE INSURANCE
F r o m these s a m p l e - m o m e n t s u n b i a s e d e s t i m a t e s of t h e c e n t r a l m o m e n t s of t h e p o p u l a t i o n can be d e r i v e d (see e.g. Cram6r, Mathe- m a t i c a l m e t h o d s of Statistics, p. 35I). T h e s e e s t i m a t e s are in t h e p r e s e n t case: S (Wl) = 2,256 S (~,) = 4,3Ol S (W3) = 9, 821 S (~4) = 51,554 T h e P e a r s o n - c u r v e t h a t has t h e s e c e n t r a l m o m e n t s , p r o v e s to b e a B - f u n c t i o n :
(r
- -f(r) = ( s - - p)ra+n-1 . B ( m . n ) ( ° ' 7 5 8 < r < 6 ' 6 3 8 ) w h e r e p = 0,758 s = 6,638 m = o , I 3 4 n = 0,392 A first c o n s i d e r a t i o n showed a l r e a d y t h a t IO o u t of 45 v a l u e s of t h e sample are s i t u a t e d below, a n d one a b o v e t h e i n t e r v a l of t h e B - f u n c t i o n . Classification of t h e 45 v a l u e s in t h e five i n t e r v a l s o - I ; I - 2 ; 2 - 3 ; 3 - 5 a n d > 5, a n d t e s t i n g w i t h t h e " e x p e c t e d " n u m b e r s in these i n t e r v a l s on t h e basis of t h e a b o v e m e n t i o n e d B - f u n c t i o n gives a v a l u e X 2 = 8 , I 6 I , w i t h four degrees of f r e e d o m ; this does n o t lead t o r e j e c t i o n of t h e c o n s i d e r e d f r e q u e n c y - f u n c t i o n . N e v e r t h e l e s s it is r e m a r k a b l e t h a t according to t h e B - f u n c t i o n I I o u t of 45 values h a v e to fall in t h e i n t e r v a l > 5 ; in fact 7 values fall in this interval. I t is t h e r e f o r e v e r y d o u b t f u l , w h e t h e r t h e a b o v e m e n t i o n e d B - f u n c t i o n gives a usefull r e p r e s e n t a t i o n of t h e " t a i l " of t h e a c t u a l f r e q u e n c y - f u n c t i o n . F i n a l l y we m e n t i o n a b o u t t h e B - f u n c t i o n , t h a t this f u n c t i o n leads t o a r i s k - p r e m i u m for t h e stop-loss-cover (without safety-loading) a d I , I 6 at a stop-loss-limit of 2,5, or a p r e m i u m ad 0,79 a t a limit of 3,5 (per IO.OOO units of t h e t o t a l of i n s u r e d sums).T h e i n a c c e p t a b l e results of application of t h e B - d i s t r i b u t i o n h a v e g i v e n rise to a t t e m p t a n o t h e r f r e q u e n c y - f u n c t i o n n a m e l y t h e l o g n o r m a l f r e q u e n c y - f u n c t i o n .
E S T I M A T I O N O F S T O P LOSS P R E M I U M I N F I R E I N S U R A N C E 359 I I - - (?. ,__-IL)_~ - - e ~ o t
f(r) dr
,/_---- V z ~ r r where dts Z l n r i ~ t - -- 0,394 45 #6 Z (ln ri - - V.)2 62 ~_f-t 45 = 0,945Classification of the observed values r i in the same intervals as mentioned in the description of the B-distribution leads to 7. * = 4,I48 with four degrees of freedom; this figure does not lead to rejection of the lognormal frequency-function. The " e x p e c t e d " number of values in the interval > 5 is 4,4; the observed number is 7. The deviation is indeed not significant, but nevertheless there is some suspicion against the lognormal frequency-function t h a t it underestimates the " t a i l " of the actual frequency-function of r. Finally we mention about the lognormal frequency-function, t h a t on the basis of this function the stop-loss-premium amounts to 0,72 at a stop-loss-limit of 2,5 and to 0,48 at a limit of 3,5 (per IO.OOO units of the total of insured sums).
It must doubtlessly be possible to obtain a frequency-function, which fits the observed data of the sample reasonably well, also in the interval > 5. B u t then too it is not at all certain t h a t such a frequency-function gives a good representation of the " t a i l " of the actual frequency-function; only if the sample should contain some hundreds of observed values it could be possible to make sure t h a t the " t a i l " is well represented.
For t h a t reason it is perhaps useful to look for an estimate of the stop-loss-premium, which is not based on a n y assumption about the analytical shape of the underlying distribution function. This can be done as follows.
If the r a n d o m variable r (being lO 4 × the ratio between the total of claims in a year and the total of insured sums in t h a t year) 24
360 E S T I M A T I O N OF STOP LOSS P R E M I U M I N F I R E I N S U R A N C E h a s t h e f r e q u e n c y - f u n c t i o n f(r), a r a n d o m v a r i a b l e h c a n be defined w i t h t h e f r e q u e n c y - f u n c t i o n F ( h ) : F(h) = o if h < o 6 F(o) = f(r)dr/ o F(h) = f ( h + G) i f h > o
T h e n t h e m e a n P of h is t h e risk p r e m i u m for t h e s t o p loss c o v e r a t t h e limit G ; , P , beeing t h e s t a n d a r d d e v i a t i o n of h, is t h e s t a n d a r d - d e v i a t i o n of t h e y e a r l y stop-loss-risk. T h e 45 o b s e r v e d v a l u e s h i c a n b e d e r i v e d f r o m t h e v a l u e s rj a c c o r d i n g t o : h i = o if r i ~ G h i - ~ r i - - G i f r i > G T h e v a l u e s h c a n be c o n s i d e r e d as o b s e r v e d v a l u e s of t h e r a n d o m v a r i a b l e h .=. F(h). F r o m t h i s follows"
i~ hj
I. t h a t P - - ~-1 is an u n b i a s e d e s t i m a t e of t h e s t o p loss p r e m i u m P. 45 45(hi- p)~
2 - - 1 2. t h a t S~ = ~- is a n u n b i a s e d e s t i m a t e of __44 ap' 45 45 3. t h a t ~ ' , b e e i n g t h e e x p e c t e d v a l u e of ( P - - P ) * , is 45 T h e r e f o r e t h e e s t i m a t e of t h e s t a n d a r d d e v i a t i o n of t h e e s t i m a t e c a n be c a l c u l a t e d as ~ - - . S~ 44 2 I n t h e p r e s e n t case t h e e s t i m a t e s P a n d ~ t u r n o u t t o b e : a t a limit G = 2,5; P = o,77; % = o,22a t a limit G = 3,5; P = o,51; ~ = o,15 I n p r a c t i c e t h e s t o p loss p r e m i u m c a n be c a l c u l a t e d as
ESTIMATION OF STOP LOSS PREMIUM IN F I R E INSURANCE 361
In this formula the value of P + ~ can be considered as the upper limit of a confidence interval, which contains the stop loss premium P with a probability, for which Gauss'inequality gives a low estimate. The value ~r~ is a security-loading.
CONCLUSION
The abovementioned case demonstrates that in calculating stop-loss-premiums a cautious choise of the underlying distribution function is necessary, and that it is perhaps better to make no assumption concerning the shape of that function. It p r o v e s - - a t least in some cases--to be possible to obtain a "quantified uncer- t a i n t y " about the stop loss premium without such assumption.
A P P E N D I X j ,~ j r, I 0,66 24 0,58 2 6,47 25 0,26 3 0,45 26 1,36 4 4,51 27 1,98 5 6,48 28 4,90 6 2,03 29 3,36 7 4,69 30 1,90 8 O, 26 31 0,97 9 0,67 3 z 1,40 10 3, 88 33 6,O9 I I 2,50 34 o,81 12 0,30 35 o,77 13 0,95 36 o,49 14 I , I 9 37 2,07 15 2,78 38 o,76 16 o,5I 39 5,37 17 1,28 4 ° 0,79 18 7,26 41 1,7o 19 0,44 42 o,93 2o I,OI 43 5, I I 21 1,59 44 I , I I 22 o,88 45 1,61 23 6,42
