DEDUCTIBLES IN INDUSTRIAL FIRE INSURANCE

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D E D U C T I B L E S IN I N D U S T R I A L F I R E I N S U R A N C E ] URGEN STRAUSS

Municll I. INTRODUCTION

In recent times, the subject of deductibles in Industrial Fire insurance has gained significance to an increasing extent. In fact, up to a short time ago it was by no means common to apply deductibles in industrial Fire insurance in Europe. The situation is entirely different in the USA where deductibles are the usual thing and are even obligatory with individual risk categories (e.g. pe- troleum refineries), hazards (e.g. explosion), or special types of companies (e.g. f a c t o r y mutuals).

Today, however, there is a definite increase in the demand for deductibles in Industrial Fire insurance in Europe, too.

This trend may be explained by the fact that ill order to reor- ganize their business successfully, insurels have been forced to increase their l)remiums hy a considerable amount, i t seems to be an unwritten law that when t)remiums in general are increased considerably, the point will some time or other be reached when the insured do not accept any further premium increases. Thus, even if such increases are completely justified in a view of claims experience, the insured will demand that l)remiums are reduced either by the insurer granting rebates for appropriate loss prevention measures or by the introduction of deductibles. In the USA, for example, such a situation arose around z96o when considerable premium increases very quickly led to the introduction of substan- tial deductibles in Industrial business. In some European countries we are currently experiencing a similar development.

W h y are deductibles introduced and what purpose do they have ? When introducing or propagating deductibles, insurers m a y have various objectives. Basically, however, it is hoped that when a deductible is introduced the moral hazard involved will be improved since in such a case the insured will show greater interest in loss prevention and loss reduction. After all, whenever a loss does occur

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DEDUCTIBLES 1N INDUSTRIAL FIRE INSURANCE 379 u n d e r a policy with a deductible, tile insured will h a v e to bear p a r t of the financial b u r d e n himself.

A n o t h e r v e r y i m p o r t a n t point is t h a t when a d e d u c t i b l e is applied, tile insurer will not h a v e to settle small losses a n y m o r e - - w h i c h obviously relieves him of a considerable a m o u n t of work. One must therefore a p p r e c i a t e t h a t the " v a l u e " of a d e d u c t i b l e c a n n o t o n l y be m e a s u r e d in t e r m s of the a m o u n t involved. R a t h e r , one has to m a k e allowance for the s e t t l e m e n t e x p e n d i t u r e s a v e d b y the fact t h a t the insurer is not liable for m i n o r losses. A f u r t h e r a d v a n t a g e is t h a t if a d e d u c t i b l e is applied w i t h o u t a p r e m i u m r e d u c t i o n , it can be used to i m p r o v e the s t r u c t u r e of u n d e r - r a t e d risks, as in such cases the d e d u c t i b l e c o n s t i t u t e s an indirect p r e m i u m increase.

On the o t h e r hand, the a d v a n t a g e offered to the insured is tlhat the p r e m i u m he must p a y to o b t a i n i n s u r a n c e c o v e r a g e is reduced. T o g e t h e r with the policy of g r a n t i n g r e b a t e s to the insured w h e n e v e r he takes effective loss p r e v e n t i o n measures, deductibles are therefore suitable for helping an insurer keep his m a r k e t a n d p r e v e n t good risks front m o v i n g o v e r to outsider m a r k e t s from the local m a r k e t . Similarly, b y avoiding p r e m i u m increases t h r o u g h the i n t r o d u c t i o n of deductibles, the insurer can keep his business at- t r a c t i v e for the policyholder, thus p r e v e n t i n g a n y t r e n d s t o w a r d s self-insurance. Finally, it should also be considered t h a t w i t h o u t deductibles, it would be e n t i r e l y impossible in some cases to c o v e r severe risks.

2. D E F I N I T I O N OF A D E D U C T I B L E AND THE D I F F E R E N C E B E T W E E N DEDUCTIBLES AND E X C E S S OF LOSS INSURANCE

A d e d u c t i b l e can be defined as the p a r t i c i p a t i o n of t h e insured in a loss up to a c e r t a i n limit agreed on in advance.

Although this definition is v e r y closely r e l a t e d to the definition of Excess of Loss insurance, the basic difference between the two is t h a t Excess of Loss insurance is usually c o n c l u d e d on a first loss basis where the sum insured does not fully c o r r e s p o n d to the value of the risk, whereas in the case of a d e d u c t i b l e the t r a d i t i o n a l c o n c e p t of full value insurance with its u n d e r - i n s u r a n c e clauses r e m a i n s unaffected.

Speaking of deductibles as such, we m a y basically distinguish between a m o u n t deductibles a n d t i m e deductibles. Referring to

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D E D U C T I B L E S IN I N D U S T R I A L F I R E I N S U R A N C E

a m o u n t deductibles first of all, the most i m p o r t a n t c a t e g o r y we h a v e h e r e is t h a t of tile so-called " p u r e " deductible where the insurer does not p r o v i d e a n y i n d e m n i f i c a t i o n at all for losses below the a m o u n t agreed on, while when i n d e m n i f y i n g losses exceeding t h a t a m o u n t , he will be responsible for the claim minus tlm deductible a m o u n t . Vv'hen a p p l y i n g a n o t h e r kind of deductible, the so- called franchise, the insurer is required to i n d e m n i f y a n y losses exceeding the agreed limit in full (i.e. he c a n n o t d e d u c t the insured's share), while the insured is responsible for a n y losses lower t h a n the a m o u n t agreed on. T h e reason w h y such franchises are not as common as " p u r e " deductibles is most p r o b a b l y because policyholders always feel r a t h e r a n n o y e d when t h e y h a v e to p a y losses just below the fixed a m o u n t . T h e situation is similar when one applies the so- called d i s a p p e a r i n g d e d u c t i b l e which is a kind of c o m b i n a t i o n of a " p u r e " d e d u c t i b l e and a franchise. T h e special c h a r a c t e r i s t i c of this d i s a p p e a r i n g franchise is t h a t here the insured is liable for all losses up to the a m o u n t agreed on (which makes this the same as a " p u r e " deductible), while the insured's share is r e d u c e d as the a m o u n t of the claim increases so t h a t when a c e r t a i n limit is reached, the i n s u r e d does not h a v e to p a y a n y t h i n g at all. It can be seen, therefore, t h a t of t h e various kinds of a m o u n t deductibles, the deductible as such is the easiest to use. C o m p a r e d with a franchise, it offers the a d v a n t a g e of being n o n - m a n i p u l a t a b l e , so t h a t indemnification does not d e p e n d on w h e t h e r a loss has exceeded the a m o u n t fixed or not.

W h e n a p p l y i n g a time deductible, the d e d u c t i b l e is defined in units of time. I t is obvious t h e r e f o r e t h a t such deductibles are only possible with an insurance where a loss occurs over a certain period of time and is not an i n s t a n t a n e o u s event. Thus, a time d e d u c t i b l e is quite suitable in Fire Loss of Profits insurance. When a p p l y i n g a time deductible, we m u s t again distinguish between two different t y p e s : F i r s t of all there is the " p u r e " time d e d u c t i b l e as such w h e r e the insured is responsible for t h a t share of a loss c o n s t i t u t e d b y the period agreed on, while secondly we also h a v e p r o p o r t i o n a t e time deductibles where the insured pays a certain p e r c e n t a g e in the overall loss resulting from the ratio between the time of the deductible a n d the d u r a t i o n of the loss as a whole. As p r o p o r t i o n a t e t i m e deductibles c a n n o t be m a n i p u l a t e d in a n y way, t h e y a p p e a r

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DEDUCTIBLES IN INDUSTRIAL FIRE INSURANCE

381

preferable to standard time deductibles where the insured has the possibility in some cases of limiting the amount of a claim for, say, the first three days, thus increasing it for the following period and manipulating loss development.

As there are still considerable difficulties in calculating rebates for time deductibles, we shall confine ourselves to " p u r e " amount deductibles in the following.

3. REBATE CALCULATION WITH DEDUCTIBLES

Assuming that the basic t)reinium provided for in the tariff is correct, the question of calculating the rebate is of first and foremost importance when introducing a deductible. With this in mind, it should therefore be noted that the gross premium charged by the insurer is made up of the following components:

a) the risk premium required for covering claims expenditure on the basis of the equivalence principle;

b) the cost surcharges (and here it is sufficient in the present context to distinguish between the cost of claims settlement and the cost items which are dependent and independent of the premium); and

c) the profit margin.

By introducing a deductible, three factors in this calculation model are influenced : the risk premiunl, the costs for settling claims, and the costs which are a function of the premium.

In the following, I would therefore like to examine these three factors ill greater detail.

3.I. Risk premium

3.1.1. Calculation of the loss elinaination ratio

The degree to which the risk premium call be reduced when a deductible is introduced depends basically on the so-called loss elimination ratio. This is the factor which indicates which percen- tage in claims expenditure--in terms of overall claims v o l u m e - - t h e insurer is able to save by introducing the deductible. The loss

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D E D U C T I B L E S I N I N D U S T R I A L F I R E I N S U R A N C E

elimination ratio, which shall here be referred to as a, can be determined as soon as the d i s t r i b u t i o n of claim a m o u n t is k n o w n '

..I

J" xf(x) dx + A J" f(x) dx

o A

( )

a = I

f x f (x) dx

o N o t e : a = loss elimination r a t i o A = d e d u c t i b l e

f(x)

= d i s t r i b u t i o n of claim anaount X

I t is assumed in this f o r m u l a (I) t h a t the a m o u n t of a n y individual loss is i n d e p e n d e n t of claim f r e q u e n c y (i.e. the n u m b e r of claims r e l a t e d to the n u m b e r of risks o b s e r v e d during a given period).

Up to a short time ago, it was h a r d l y possible in G e r m a n y to obtain a n y r e p r e s e n t a t i v e statistics showing the distribution of claim a m o u n t s . It m u s t therefore be a p p r e c i a t e d t h a t a n u m b e r of m a j o r I n d u s t r i a l Fire insurers in G e r m a n y h a v e p a r t i c i p a t e d in a special statistical s u r v e y on the basis of which all risks a n d claims prevailing in some selected fields of i n d u s t r y h a v e been compiled o v e r a n u m b e r of years. In so doing, i n f o r m a t i o n has been p r o v i d e d on the sums insured,

PM~Ls,

and p r e m i u m s as far as the r e l e v a n t risks are concerned, while at the same time code figures have been deter- m i n e d in a c c o r d a n c e with t h e b r e a k d o w n of the G e r m a n Fire tariff into classes of risks. T h e overall objective in this v e n t u r e was to l)rovide a c o m p l e t e s u r v e y of the each individual risk a n d not just a s t u d y of overall treaties. Losses were then registered s e p a r a t e l y on the basis of the claim a m o u n t s for each individual risk. This statistical s u r v e y has been p r o v i d e d for Fire and Fire Loss of Profits s e p a r a t e l y .

W h e n selecting the individual fields of i n d u s t r y to be included in this investigation, it was decided to a t t a c h special i m p o r t a n c e to those risk categories which, according to G e r m a n statistics, are c h a r a c t e r i z e d b y considerable differences in t e r m s of claim fre- q u e n c y and average claim a m o u n t s . This was done because it can be a s s u m e d t h a t the loss elimination ratio depends to a considerable e x t e n t on these two factors of claim f r e q u e n c y a n d average claim a m o u n t s . A first analysis of the m a t e r i a l o b t a i n e d has indicated as

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D E D U C T I B L E S IN I N D U S T R I A l , F I R E I N S U R A N C E 38,.3

an initial result that the distribution of clailns may be approximated by using a log-normal distribution. This applies particularly to low claim amounts. On the basis of a logarithmic probability paper, Fig. I provides an example of such an approximation by showing the values obtained for Fire and Fire Loss of Profits for all of the statistical material (i.e. not broken down according to risk categories). 9 9 9 8 99.95 9 9 9 99.8 99.5 99 98 97

j J

1 60 ,o 7 . 3 0 , ' J F i r e I I I I I 10 25 50 100 250 500 1000 Claim Amount F i g . t. D i s t r i b u t i o n of c l a i m a m o u n t s in I n d u s t r i a l F i r e a n d F L O P .

As is to t)e expected, the Z a test indicates that the hypothesis that the claim distribution is a log-normal distribution m a y not be rejected with a statistical reliability of 9o0/0 . Also, this result ties up very weU with the observations made by G. Ferrara (see Refer- ences [i]) who has shown oll the basis of material compiled in I t a l y that claims distribution can be approximated quite effectively by way of a log-normal distribution. (It should nevertheless be noted

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384 D E D U C T I B L E S I N I N D U S T R I A L F I R E I N S U R A N C E

at this point t h a t the material compiled by G. F e r r a r a is repre- sentative of claim a m o u n t s far lower t h a n the claim a m o u n t s in the material compiled in Germany.)

If, therefore, we state on the basis of equation (I) log c ~ ( l o g :z:-~) e

2cr2

f ( x ) - 1 / ~ x e (2)

with the expected value E(log X) = ~ and the variance De(log X) = c~, the following two integrals must be solved:

log e ,: (log z--:)~

I~(G) -- i / ~ S e 2°~" dx

Iz(A) ,4 log e f I aog2~..z-~)______~

= - e d x

x

B y way of some substitutions, it can be shown t h a t ~2('n '°)0~ +2"~ In '° (log ~ - - ~ )

It(G) = e 2 • • -- e In lO

and

I~_(A) A ( I @ ~ )

Here, • is tile distribution function of the (o.z) normal distribution :

t 2 I ~ - . ~

¢ , ( x ) -

1/~

-~S e " (tl

We then obtain the following equation for the loss elimination ratio a :

a(A)=. (

oln o)+

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It is obvious of course t h a t the loss elimination ratio determined b y way of these equations c a n n o t be applied to all risk categories. In fact, even within one and the same class of risk, we must distinguish in terms of rebate calculations between large and small risks considering t h a t if one applies a deductible of DM I o o , o o o . - - for a small object insured, most losses will not have to be passed on to the

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DEDUC'FIBI.I~S IN INDUSTRIAL 17IRE INSURANCE ~ 8 5

insurer, whereas if the same deductible is applied to a large object in the same class of risk, it might only cover small partial losses. In order to calculate a suitable deductible rebate, the insurer must determine the loss elimination ratio on the basis of homogeneous classes of risk--homogeneous not only in terms of the type of risk invoh, ed, but also in terms of risk magnitude. This is precisely where the problem arises in Industrial Fire insurance, where it is extremely difficult to form homogeneous risk classes in the way just mentioned, particularly considering that the risk classes obtained must be sufficiently reliable statistically in spite of the considerable fluctuations to be expected within the framework of overall claims experience. In the following it shall therefore be a t t e m p t e d to show which individual factors in the material available were found to definitely have an influence on the loss elimination ratio, using statistical fitting methods.

3.I.2 Dependence of the loss elimination ratio on the absolute a m o u n t of the deductible.

Fig. 2 shows how tile loss elimination ratio increases in Industrial Fire and F L o P as tile deductible (expressed in DM) of tile overall material compiled increases.

Fife FLoP / I I I I I I ~ Deductible i n D M 1 0 0 0 25 .50 100 2 5 0 5 0 0 1.000

Fig. 2. Dependcnce of the loss e l i m i n a t i o n r a t i o on the a b s o l u t e a m o u n t of

the clcductibte.

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386 D E D U C T I B L E S i N I N D U S T R I A L F I R E I N S U R A N C E

This g r a p h shows t h a t in FLOP, tile r e b a t e g r a n t e d to the insured in r e t u r n for the same deductible a m o u n t must be much lower t h a n in Fire. This situation is easy to explain:

In FLOP, the a v e r a g e claim is m u c h g r e a t e r t h a n in Fire so t h a t most F L o P claims exceed the d e d u c t i b l e agreed upon quite considerably. T h e result of this n a t u r a l l y is t h a t the d e d u c t i b l e o n l y suffices for a small p e r c e n t a g e of claims.

A m o n g the statistical m a t e r i a l available, there were relatively few large losses with a claim a m o u n t of more than DM z million, even t h o u g h experience shows t h a t in G e r m a n y such losses are b y no rneans i n f r e q u e n t a n d in fact c o n s t i t u t e the m a j o r p r o p o r t i o n of the total claims e x p e n d i t u r e incurred by G e r m a n I n d u s t r i a l Fire insurers. This o b v i o u s l y m e a n s t h a t as soon as one is c o n f r o n t e d with losses of such i m p o r t a n t m a g n i t u d e , the statistical reliability of the material a n a l y z e d is not v e r y good. F o r this reason it m a y be t h a t the log-normal distribution derived front the statistical m a t e r i a l will decrease to zero too q u i c k l y for high claim a m o u n t s , the result of this being t h a t the insurer will u n d e r e s t i m a t e the p r o b a b i l i t y of large losses. This d r a w b a c k is a l r e a d y made e v i d e n t b y Fig. ~ where with the higher claim a m o u n t s in F L o P the empirically d e t e r m i n e d figures c a n n o t be a p p r o x i n m t e d so a c c u r a t e l y with the log-normal distribution. W h e n calculating rebates to be g r a n t e d to the insured in r e t u r n for deductibles, this would m e a n t h a t on the basis of the log-normal distribution the loss elimination ratio would be over- e s t i m a t e d , the r e b a t e g r a n t e d thus being too high. I t is a well- known fact t h a t as soon as large losses are considered, the P a r e t o distribution in Fire insurance provides a m u c h b e t t e r a p p r o x i m a t i o n of the statistical material (see [o] and [3]), whereas of course the P a r e t o distribution is not suitable for describing the claim distribution for small claims this being the claim c a t e g o r y which is also i m p o r t a n t when a p p l y i n g a deductible. I t can be shown t h a t as far as large claims are concerned, a good a p p r o x i m a t i o n can be reached b y a p p l y i n g the P a r e t o distribution. W i t h this in mind, it is p e r h a p s a d v a n t a g e o u s to write the claim distribution

f(x)

for the purpose of calculating the loss elimination ratio in the following m a n n e r :

c log e (log ~ - ~ ) e gel 2

c for :v < B

f(x)

= l / ~ a:~ - - (4)

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D E D U C T I B L E S IN I N D U S T R I A L F I R E I N S U R A N C E 387

By suitable selection of the constants to be applied in this equation, we can reach a standardization so that jr f ( x ) d x = z. I n

o

addition, the distribution density should be continuous at point B. By introducing the claim distribution according to (4), the formula for the loss elimination ratio becomes somewhat more complicated.

Assuming for example that B > A and ~ > 2, we then ol)tain = /~(l'(l°g~ - ~ crlnlo).-I- c A ( q , ( l ° g f f - - ~ ' ) - - ( l ) ( l ° g ~ - - ~ ' ) ) + _{- - 1 3 - ~ + l}A d

)

_ _ _ _ / ~ - ¢ t + 2 with jt~ ~ C e ~'ln 104~-' ¢~CZ(ln lO)u

If B is mucll larger than the deductible, it is quite sufficient when calculating the rebate to estimate the risk premium share for losses larger than B. In the rebate calculation, the loss elimination ratio is then corrected accordingly by applying it not to the entire risk premium, but rather to a risk premimn which has been reduced correspondingly. This procedure would be aPl)roximately in line with the concept developed by Mr. Thonaazin (see [7]), who splits up the risk premium into a basic portion for covering small and medium-sized losses while applying a loading for large losses, such loadings increasing with the size of the risk. When introducing deductibles, ~'[r. Thomazin then only wants to grant a premium rebate for the basic 1)remium and not for the loading mentioned. 3.r.3 Dependence of the loss elimination ratio on tile individual branches of industry

The studies specified in the foregoing have been repeated for the individual branches of industry. In so doing it was found that when applying a fixed deductible, tile loss elimination ratio obtained differs to a certain extent from one branch to another. In other words, the claim distribution starts to take effect at different points.

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388 DEDUCTIBLES IN INDUSTRIAL FIRE INSURANCE

i t is nevertheless advisable to a r r a n g e these individual branches of i n d u s t r y in groups where the effect is more or less similar, simply because otherwise the scales for rebates would b e c o m e too c o m p l i c a t e d and coml)rehensive. [n the case of the risks studied a n d e v a l u a t e d , it was possible to form one g r o u p c o n t a i n i n g the steel and iron i n d u s t r y , the a u t o m o b i l e i n d u s t r y , the electrical i n d u s t r y and the chemical i n d u s t r y . A n o t h e r g r o u p is c o n s t i t u t e d b y oil refineries and the c h i p b o a r d i n d u s t r y where a p a r t i c u l a r l y large n u m b e r of m a j o r and total losses m a y be e x p e c t e d and where partial d a m a g e is of s e c o n d a r y significance only. Applying the same deductible, the loss elimination l atio to be o b t a i n e d here is only a b o u t one-third to one- half of the ratio o b t a i n a b l e in the a b o v e - m e n t i o n e d risk categories. This shows quite clearly t h a t when g r a n t i n g rebates, we must distinguish carefully between risks suseei)tible to large losses to a considerable e x t e n t , and risks where the chance of a large loss is not so great.

3.1. 4 D e p e n d e n c e of the loss elimination ratio on the P M L

It has a l r e a d y been p o i n t e d out t h a t the loss elimination ratio does not only d e p e n d on the absolute a m o u n t of the deductible g r a n t e d , but also on the size of the risk. T h e most o b j e c t i v e criterion for assessing the size of a risk is of course the sum insured of the i n d i v i d u a l complex. H o w e v e r , as it was not possible to the full e x t e n t with. the statistical material studied to split up the figures shown in each t r e a t y according to the individual complex, it was decided to a p p l y the P M L for the purpose of d e t e r m i n i n g the "size of a risk", even t h o u g h we were n a t u r a l l y aware of the possil)ility t h a t the P M L is not fixed properly. We felt t h a t the possibility of error resulting f r o m this would be within a c c e p t a b l e limits, p r o v i d e d risks with a relatively wide P M L span would be p u t into one group. I t was then f o u n d t h a t if the deductible stays the same, the loss elimination ratio will decrease when the P M L increases (see Fig. 3 where the s t e p - f u n c t i o n of the loss elimination ratio we o b t a i n e d first has been f i t t e d into a curve). [ t is nevertheless i n t e r e s t i n g to note t h a t this d e p e n d e n c e of the loss elimination ratio on the P M L was only found in risk groups with a c o m p a r a b l e claim ratio ( = claims in per cent of the p r e m i u m ) and thus having similar claim experience. First of all, this d e p e n d e n c e on the P M L had been

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D E D U C T I B L E S IN I N D U S ' F I / I A L F I R E I N S U R A N C E 389 c o v e r e d up c o m p l e t e l y b y the opposite d e p e n d e n c e of the loss elilnination ratio on claim experiellce (see 3.I.5).

b loss eliminaxion ratio

deductible DM 100.000 deduct;hie DM 25,000 I I I Ii 10 50 100 PML in mio. DM Vig. 3. D e p e n d e n c e o f t h e l o s s e l i m i n a t i o n r a t i o o n t h e P M L .

One m a y say t h a t a p p r o x i m a t e l y the loss elimination ratio cor- responding to a d e d u c t i b l e of the a m o u n t A is a function of the size of a risk, as is shown b y the following e q u a t i o n :

a ( A ) = g(X) • (log P M L ) -f~, ~ > o

Here, g(A) is a f a c t o r which depends on the general level of the loss elimination ratio of the d e d u c t i b l e A.

3.I.5 D e p e n d e n c e of the loss elimination ratio on claim e x p e r i e n c e We have f o u n d t h a t the loss elimination ratio depends basically on the claim ratio, i.e. on claim experience. The loss elimination ratio o b t a i n e d in treaties with f a v o u r a b l e claim ratios is m u c h higher t h a n t h a t o b t a i n e d in treaties with u n f a v o u r a b l e claim ratios. On the o t h e r hand, however, it must be realized t h a t if treaties from the i n d i v i d u a l b r a n c h e s of i n d u s t r y were only to be g r o u p e d on the basis of the claim ratio criterion for each individual t r e a t y , this would c o n s t i t u t e an e x t r e m e act of selection as in this case a t r e a t y will v e r y quickly b e c o m e a " b a d " t r e a t y as soon as losses s t a r t to occur. T h e same c h a r a c t e r i s t i c as just described can h o w e v e r also be o b s e r v e d if the criterion of " g o o d " a n d " b a d " claim e x p e r i e n c e is related not to individual risks, but to individual b r a n c h e s of

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390 D E D U C T I B L E S IN I N D U S T R I A L F I R E I N S U R A N C E

industry. This can be explained by tile fact that in treaties and branches of industry with good claim experience and thus with a low claim ratio there will still be a number of losses, but the size of those losses and thus the average loss degree has proved to be low. This in turn means t h a t in the case of such risks the deductible can absorb a larger proportion of claims than in the case of risks with a high claim degree.

3.I.6 The problem of anti-selection

The situation just described now brings us to one of the most i m p o r t a n t points in this discussion on deductibles, namely to the problem of anti-selection. If deductibles are granted without proper supervision in return for a rebate, there may be a danger of anti- selection by the policyholders which is basically due to the heterogeneity of risks within the same risk and rating group. Of course, it is quite obvious that in practice this heterogeneity can never be avoided completely. Therefore there will be a danger that policyholders who generally have a bad claim record will not be interested in obtaining a deductible while policyholders with a good claim record will naturally want to have a deductible. In an extreme case, this would mean that the insurer would be deprived of a considerable a m o u n t of the premium by granting deductibles for treaties that would hardly involve any claim expenditure anyway, while on the other hand he would have to indemnify just about the same losses in the case of bad risks where the insured is not interested in obtaining a deductible. This shows quite clearly that the rebates actually granted must be lower than the rebates originally cal- culated on the basis of the claim distribution observed in a heterogeneous class of risk.

As the problem of anti-selection results mainly from the fact that the same average premium has to be paid by both "good" and " b a d " risks within the same class of risks, the danger of anti- selection will of course be much less if an experience rating is provided for in the underlying tariff as in this way the risks involved can be artificially made a little more homogeneous.

When calculating rebates, it must therefore be considered in this c o n t e x t whether and to what extent the rating applied is already based on experience rating. In addition, problems relating to the

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D E D U C T I B L E S IN I N D U S T R I A L F I R E I N S U R A N C E 1391 credibility t h e o r y must also be considered, i.e. ill how far the claim experience of one individual t r e a t y can be r e g a r d e d as statistically reliable within a g r o u p of heterogeneous risks. In t e r m s of absolute numbers, it is nevertheless difficult to cope with the problem of anti-selection, p a r t i c u l a r l y considering t h a t this p r o b l e m relates to the question of m o r a l hazard. It is obvious, therefore, t h a t careful u n d e r w r i t i n g can c o n t r i b u t e c o n s i d e r a b l y to solving this problem. I n so doing, the claim experience so far a n d a b o v e ,all claim fre- q u e n c y a n d the a v e r a g e claim a m o u n t m a y p r o v i d e v e r y i m p o r t a n t assistance in a p p r e c i a t i n g the clanger of anti-selection. F o r the reasons a l r e a d y m e n t i o n e d it is therefore essential not to specify rigid r e b a t e rates, b u t r a t h e r to use r e b a t e b a n d - w i d t h s for under- writing purposes.

M a t h e m a t i c a l model investigations can of course be v e r y helpful in d e t e r m i n i n g these r e b a t e band-widths. In this c o n t e x t , [ would like to refer to the work done b y E. N e u b u r g e r (see [4]), who as far as I know was the first to develop a m a t h e m a t i c a l model with the help of which the effects of anti-selection can be assessed. In his work, N e u b u r g e r assumes t h a t the portfolio i n v o l v e d is heterogeneous a n d can be split up into two more h o m o g e n e o u s classes of risk; h o m o g e n e o u s in t h a t both of these classes have a different a v e r a g e claim degree, i.e. the a v e r a g e claim a m o u n t p e t loss occur- rence r e l a t e d to the sum insured (in practice, treaties with an a v e r a g e claim degree of < d are g r o u p e d u n d e r class I, while treaties with an a v e r a g e claim degree of > d come u n d e r class 2). In addition, it is also necessary to know, or at least estimate, the d i s t r i b u t i o n of claim degrees in the two classes of risk. 13y proceeding on the basis of different a s s u m p t i o n s a b o u t the n u m b e r of treaties in the two classes of risk and the n u m b e r of treaties in each class which take a deductible, one can s t u d y the effects of anti-selection. If all the insured in the class with the lower average claim degree then take treaties with a d e d u c t i b l e while all the insured in the class with the higher claim degree h a v e treaties w i t h o u t a deductible, we h a v e the case of excessive anti-selection.

3.2. The influence of the deduclible on costs

W h e n calculating the loss elimination ratio, we know to w h a t e x t e n t the risk p r e m i u m m a y be r e d u c e d b y a p p l y i n g a deductible.