j. RANTALA 1. INTRODUCTION

Ast ln ]Bulletin 12 (1081) l -2 t

EVALUATION OF THE CAPACITY OF RISK CARRIERS BY MEANS OF STOCHASTIC-DYNAMIC PROGRAMMING

T. FENTIKAINEN and j. RANTALA

1. INTRODUCTION

The Ministry of Soc, al Affairs and Health, being the Supervising Office of Insurance m Finland, has established a special working group to investigate the problems involved with the solvency of insurers. A report will be compiled m a near future The capacity of risk carriers is one of the problems dealt with, and ~t will be preliminarily reviewed in this paper.

The problem was treated by the working group parallelly by means of

I. an empir ica l approach observing actual fluctuations in underwriting gains of insurers, and

2. a ~heorcLica! approach, constructing a stochastic-dynamic model and studying its behavlour, especmlly its sensitivity to numerous background factors.

First the methods of investigation are described and their application is then demonstrated using some numerical data. Because a comprehensive report will be published by the working group separately, only tile mare schedule is given. For the same reason the consideration is limited here to stochastic risks, omitting the fact that the solvency of an insurer is also jeo- pardized by numerous "non-stochastic" risks such as failure in investments, political interference of the authorities, mismanagement of the company, or misappropriation of its property.

2. STOCHASTIC-DYNAMIC MODEL

The state of an insurer is defined by means of state variables such as the volume and mix of the portfolio, reserves, etc. Then a number of transition equations are constructed to control the incoming and outgoing money flows, as shown in tlle attached schedule. The difference AU between these flows, the underwriting profit or loss, is accumulated into a risk reserve U. U is equivalent to the concept of tile solvency margin, if underevaluations of assets and overevaluations of liablltt~es (e.g. fluctuation reserves, catastrophe pro- visions, safety margins, etc. m underwrlting reserves) are included in it.

Numerous exogenous and endogenous factors can be taken into account, as referred to in the schedule

2 P E N T 1 K A I N E N A N D R A N T A L A

The model is dy+mm~c in tha t it cat] be made self-correcting ("adapt ive") . For exatnple, if the solvency rat io U/B is high, the level of the net re tent ions of reassurance can be increased, and vice versa If the prof i tabi l i ty is good, then more efforts can be al located for sales promotion. If tlle s tate of the insurer is becoming critical, then economizing m administrat ion, deduct ion of sales costs, etc. can be prog~ ammed, as can an increase in prenfium rates. Different kinds of business strategzes can be exper imented with, especially if the model is to be used to prognostmate the s ta te and future deve lopment of an insurer for the insurer 's own use. Such strategies could be aimed at increasing market shares by means of sales campaigns, by means of compeht ive reduct ions in premiums, etc. However , m the work of the I~innish s tudy group these features were not taken i n to account , instead a t tent ion was given more to finding general cond~tions on which the solvency may depend.

co E

E q)

(3_.

IMFLOW

Earned

rlek

premlums

Sarety bodlng

~adlng ¢or admlnlstratlon

Met Investment

earnlngs

RISK RESERVE U OUTFLObJ

t,U

i I

Clalmc paid and out=fondu

~I Re~'uran*el (net balance~

U1v1~enOc I

Variables end ossurnptlons

State varlobles ~ parameters

+ Por¢¢olio mlx, branches ]=1,2,

Clalm ~Ize dr

• Volume lndloators n=n1+r4+

(expected number or clolm¢)

+ Reserves

• Aesec$

Exogenout racier*

• chert period Fluctuation oF

the ba¢Ic proboblllCle*

• Long period bu~Ine#s cycles

+ InPlatlon

- steady race I x

- ~hock Impulsle$ ~1 x • Market oondltlon~

- real growth or the

per trollo g - #:ales response etc

• Rote or Interest i n

Bu~: lno~s s t roJ ;e91es

RQCe~ or premIultlc - c a r e t y ,iDad.lng~

¢ Reln~uronce - net recentlonc Mj

$ Sales errort*, admln1~tr

¢ 01vldend~

+ Sorely level (ruln probc~-

b i l l t y s and tlmQ ~pan T)

CAPACITY" OF RISK CARRIERS 3

The model is stochastic in that the claims (and possibly some other variables as well) are assumed to vary stochastically. For the purpose a random number generator was constructed to simulate the aggregate amount of claims. Four levels of stochasticity were assumed:

1. The m~mbar of claims varies at random (counting process). 2. The claim size ~ varies at random; distribution functions are given for each

portfolio section Sj(z). These functions also depend on the reassurance and its net retentions Mj, where j indicates the section (branch) of the portfolio.

3. Short term variatiom The expected nmnber of claims ~tj vary at random from year to year (being fixed reside each calendar year). Standard deviations ~/ and skewnesses "gj of the fluctuations in basic probabihties are given input parameters. One reason for this type of varmtion may be weather conditions.

4 Bus*ross cycles. The basic probabihtics are also subject to long period variations. Business cycles are introduced into the model by means of autoregression rules or by deterministically or "half-deterministically" randomizing the phase of the cycle. Business cycles are caused by general economic cycles (booms, recessions), by cycles generating mechanisms in the insurance market, by mflatmn, etc.

Fig. I shows an example of the random flow (reahsatmn of the process or a "sample path") for a time span of 2 5 years.

15

~g

05

~0

Risk reserve/premium = solvency ratio u

T i m e

. . . . ~ . . . . 1; . . . . t ; . . . . . 2a . . . . . ~5 y e a r s

I"lg t A r c a h ~ a t m n of the bus ines s flo',~ p rocess

Instead of taking the absolute amount of the solvency margin U as the main indicator, it seems advisable to take the relative amount, the solvency ratio, denoted by u = U/B, where B is the premiunl income (cf. schedule).

4 P E N T I K , A I N E N ANT) R A N T A I . A

Relat ive variables of t h i s kind with dimension o in respect of mone ta ry unit are not direct ly affected by inflation as are the absolute amounts Hence they are s tatable variables for long-term prognoses where the value of money is not as~um('d to be constant

Following the idea of the Monte Carlo method the ~imulation is rcpcated numelous times. A bundle of sample paths i~ thu~ obtained, a~ ~hown in Fig. 2

15 S o l v e n c y r a t i o u

]0

P,,,

R 1

Ruin b a r r i e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~1~ i i i i i J i i i 1 ~ i i i i i i f i i

} 1; 1; 2'~ l"ime 2's lqg 2 A bund le~I~c ,ah~, t tmn~o[nn m~urer '~ lmmm,,~ flmv pr,~cess

A "sto¢'ha~t~c bundle" like that m Fig. z is an imlmrtant tool m analysing the solvency of an insurer "fhe ~hnpe and position of the bundle make it possible to draw conclusions on the solvency and other features of the p~ocess. If the b u n d l e i g g a f e l y o v e l the ruin banic t (eg the legal minimum amount of solvency margin) It indicates a solvent state.

A ~talyhc ;nethod

It is often possible as a short cut apploach to compute di lect ly the middle line of the stochastic bundle p lo t ted ill lrlg 2. The confines of the bundle are alto direct ly computable , when the 1)robal~ility i~ given, according to which the realizations will lie hetwe('n th(' c~mflne~ "l'he h leadth of " t h e stochast ic bundle" is denoted by tile range varml~les R,, and let a~ ~een in Fig 2 Due to the skewness of lhe claim~ pmce~s lhey may no! nece~amly be equal.

F rom tile middle line of tile stochastic bundle and frolrl the range~ R il i~

6 PENTII(AINEN AND RANTALA

1.1

t . o ~ ~ I O

O S

|SIp Irrll II+1 l i t ! 1170 1171 Ill A|

" 1 !

O,S

I11! IS90 1111 J$

I + t l

0 1 O S

l i sp 1171 11~1 1119 I|?G I I+I 11 -ns

I , s

o s

liSP 1190 1171 Ill

os

I S

t o , ~

g l

111? I170 I191 11|! 1190 | l i e I I 12

I 1

o s

1117 I170 II11 13

i I [ ..~--"~'-- I , I

1111 1170 l l ; l f lCt 1170 lIFe e* 15

0 S " - - ' ' ] "

1111 fiFO I |71 ml

I s i s

os _ '" " ~ ' ' , o s ' : .r

I l l2 tt?O 117| 1112 liFO l l t l ¢+ ¢t

II1! t17o 119! c:l

I S

O S

I J f I liFO IITJ (*

I I

| 0

[ . . . . . . . . nls~ 119o 117e

cs

F~g 4 An example of the bus tness flow of 17 F m m s h m s u r a n c e c o m p a n l e s Sohd line = f l uc tua t i on r e s e r v e / p r e m m m s ( m d t c a t m g here so lvency ra t io ~l) and do t t ed hne = cla ims ra t io A'/.B The d a t a were recezved from the S u p e r v m n g OMce The so lvency marg ins are no t p u b h s h e d and therefore , for t he sake of a n o n y m i t y , on ly the codes oI the corn-

p a i n e | are g~ven

CAPACITY OF RISK CARRIERS 7

B- EMPIRICAL APPROACII

To test the goodness of ht of the model and to cal ibrate its parameters , a large amount of da ta from the actual flow of business of all Finnish non-hfe insurance companies was collected. An example of the da ta is given in Fig 4. The do t ted line repreqents the clmms ratio, i.e. the aggregate claims/earned premmms on the company ' s own retent ion The solid line is in tended to indicate the flow of the soh, ency ratio. The total a inounts of the solvency margins were not available, and ~o instead the so-called f luctuat ion (equalization) reserve, which is a special ty prescribed m the Finnish Insurance Company Act, was used. The accumula ted profi t from risk business has to be t ransferred to this reserve, and the loss from underwri t ing business sub t rac ted from it. Hence the reserve's flow mdmates the variat ions of the acc,~mulated underwri t ing p roh t (_+) quite well Only m cases where the equahza tmn reserve was occasionally exhaus ted to zero does this idea not functmn. In theqe cases the solvency margin was entirely composed of the company ' s own booked capital, to which are added hidden reserves like underevaluat lon of assets, etc

Fig. 4 m a y be umque in tha t the unsmoothed result is genmnely given. I t m ay deviate from convent ional published flow charts in tha t no significant smoothmgs (or manipulat ions) are present.

One pronounced general feature can be found from the p lo t ted curves. There are periods when the solvency ratios go ei ther up or down m several consecutive years, i e. some kind of business cycles appear, the shape of wlfich are very

1.5

1.0

0.5

1962

0 0 o

0 0 0 0 ~0 oo°o

O O

o O O

• • o ° • o • e e

o • e • O • ) *

• "" " " ' ' " 0 0 ~ ' . , . o ~ t 5 • e e 0 O• 0

• "o 0 . d ~ ~ o

• ". 0 0 0 0 . "V ° ! "-. . . . . . . . , / . . .y

1970 1978

l:~g 5 The flow of the risk xcscrve U/l? of six large non-hfe compames

1.2

1.0

0.@

PENTIKAINEN AND RANTALA

(Trends adjusted)

M a r i n e lilt

0,~ AccKlent ,n~ A Underwt rang

0,1 ~ / ~ . . . ~ . ~ L o ~ s ~ at,o Gain

/--

-0.i ~ I n d u s t r l a l Production (Devlal;lonlrrom normal level~,~

-0.2 Fig 6a The joint marine and employment accident business of all Fmmsh insurance c o m p a m e s D e v m t m n from the ave rage levels Note the clear cor re la t ion wi th the general

economm cycles, whmh are ind ica ted by the indus t rml p roduc t ion index

8

O3

0.6

0.4 I • J _ _ ,

1960 1965 19'70 1975 Fig 6b Loss ratio of the joint business of all Flllnish insurance compamcs coneermng motor tlurd party habfllty The actual and smoothed data The smoothed data can be interpreted to represent the long period varmtmns and the devmhon of the actual data

from tile smoothed data mainly the short period varlatmn

similar to the well-known business or growth cycles known m the national economy and to many industrial or commercial enterprises. To show this more clearly the flow of the solvency ratio u of six large companies is included in the same figure (Fig. 5). The business cycles of tile companies are obviously synchronized m time.

]rigs. 6a and 6b show the joint business of all Finnish insurance companies Due to the large volume of material the ordinary Poisson fluctuation is neg- ligible in size and the variations from year to year are caused by long-period business cycles and by short-term variation of the basic probabilities. In Fig. 6b the smoothed flow describes long-period business cycles and it can be assumed that deviations in the actual values from it are caused mainly by short-term variations

The mechanism behind business growth is obviously quite a complicated one and as yet not well known. One of the reasons for this phenome~aon is clearly the reflection of the normal growth cycles in natmnal economies. A boom gives rise to increased loss ratios for many non-life branches owing to the increased activity in industry and other sectors. A recession can have the opposite effect. TOe influence of business cycles varies from sector to sector. Another reason is a general mechanism characteristic of free markets generally, and is by no means confined to insurance markets. Good profitability stimulates competition, new enterprises appear and the market shows clear signs of com- petitive premium reductions and increased sales promotion expenditure. So this favourable market is soon "spoiled" and downswing can be expected Due to the reluctance of the market mechanism this swing will continue for several years until poor results again compel the insurers to increase rates and reduce competition, thus making the market ready for a new upswing

Obviously other background factors also exist, and these may differ in time and space.

A considerable amount of literature has been published concerning the general econometric models, growth cycles, etc. It is astonishing that, the corresponding phenomena for insurance markets have received little attention. However, some recent notable works by HELTEN, I(ARTEN and BECKER Call be referred to (cf. references).

The purpose of the empirical approach was, of course, to find guidance to assist with the theoretical approach and also to calibrate the model parameters so as to get a model capable of realistically simulating real world phenomena and of explaining the actual business flow behaviour and fore- casting the range of fluctuations. The importance of the business cycles was stressed, as was the necessity of incorporating them into the model.

I 0 PENT1K.~INEN AND RANTALA

'4 APPLICATIONS

4.1. Some general comments, a standardized insm'cr

The Finnish working group has produced a great collection of applications aimed at monitoring the complicated problem of solvency. It is not possible to review this side of the work here. The apphcatlons given in the following are intended only to demonstrate how the model drafted above can be applied.

The number of variables and assumptions involved in the model is quite large. This makes it difficult to get an), idea of how the numerous variables affect the solvency and other behaviour of the model, and especially difficult to determine their interdepcndences. One solution to tiffs "communication difficulty" was to construct a special "standard insurer", which corresponds to a typical, average insurance company The basic data and the solvency in- dicators were calculated for this particular case Then one or more of the background variables, such as the size of the insurance company, the level of reassurance, inflatmn and numerous others were taken, in turn, as moving variables, whereas all the others were fixed. In this way it was possible to get a concept of the sensmvlty of the model to various background factors.

One critical question concerning the standard insurer approach is whether

U / B

1 . 5

1 . 0

0 . 5

0 . 0 , , I . , , , , I

100 1000 B

Fig . 7 M u u m u m s o l v c n c y r a t i o as a f l ,nct~on of p r e m m m i n c o m e B ( s t a n d a r d in su re r ) S o h d h n e ' N e t ~e ten t~on c o n s t a n t l ) o t t e d l ine : A c c o r d i n g to a spec i f i c scale , n e t r e t e n t i o n i n c r e a s e s x~]th pre[ll lutn IIICOIIIC ./3 (In / l l l lhollS of i i~onetarv uni t) U p p e r c u r v e s , b u s m e s q cyc le Hlch tded L o w e r c u r v e s b u s i n e s s cyc le e x c l u d e d R u i n b a r r i e r is o I B a n d t t m c

s p a n io y e a r s

C A P A C I T Y O F R I S K C A R R I E R S 1 1

or not the actual c ircumstances of insurers of different sizes and types are too far from each other to be described by one s tandard case only. For example, if the portfolio is large, then the risk f luctuat ions are expected to be smoothed well, but for a small portfolio they can predominate This can be seen in Fig. 7.

The min imum solvency margin, as i l lustrated m Fig. 3, was calculated for the s tandard insurer. However , the volume of p remmm income B on the company ' s own re ten tmn (including safety loadings and loading for adminis- t ra t ive expenses) was changed. As the sohd hnes show, the min imum solvency margin depends great ly on the size of the company This is especially t rue in the case where the business cycles were not assumed (lower line). However , in pract ice the level of net re tent ions in reassurance is adjus ted according to the size of the insurer. Normally, large compames have considerably larger net re tent ions than small ones. For this reason a special scale was const ructed according to which the net re ten tmn was dependent on the size of the company. An impor tan t observat ion was that the slze of the company no longer had any sigmflcant influence (dot ted line). This observat ion justified the use of one standard insurer as "a ya rds t i ck" even if all results are to be tes ted separate ly and the s tandard insurer me thod can give only prel iminary hints of solvency structures .

4.2. I~tflation Another example of an application considered here is the influence of inflation. I t is advisable to discuss separa te ly the cases where the ra te of inflation iz is assumed to be s teady, 1.e. the same from year to year, and where the rate varies from year to year.

1 . 0

U

/0.5 B

0.0

Y l e l O o i i --: . . . . .-.' •

interest loading

I k I , I , ! , ]

20 40 60 80 i00

Equilibrmm

]rig 8 Influence of a steady mflatlon and other factors

12 PENTIK.AINEN A N D RANTALA

The influence of a steady inflation rate and some other most important background factors to the solvency ratio is shown in Fig. 8, which was generated by means of the Monte Carlo method.

The real growth in the portfolio is measured by the growth rate t 0, which is defined as the real growth m premium volume/3. Hence the nominal increase in B is composed of both the inflation and real growth increments

(4.2.1) s ( t + = +

where t is the time in years. Another background factor is the safety loading X. It xs composed of the

conventional safety loa&ng in premiums added by the yield of interest for the underwriting reserve, which is available to reinforce the total underwriting gain of the insurer.

In addition, the yield of interest, rate in, added to the solvency margin was also taken into account.

The configuration shown in Fig. 8 gives rise to some observations of interest. The stochastic bundle of realizations is essentially different from that which is customary in conventional risk theory Normally, the standard deviations and hence the breadth of the bundle continuously increase as time t increases. It is also well known that the final (for a infinite time span) probability of ruin can be less than 1 only if the bundle, i.e. solvency ratio u, tends to infimty Here the bundle quite obviously has a finite asymptotic range and a certain equihb- r~um level. This can be explained by means of the background factors mentioned above as follows:

As shown in Fig. 8 there are actually four principal forces in action: In- flation continuously reduces the solvency ratio, because it causes an increase in the denomlnator of u = U/B, i.e. B is nominally growing. For the same reason real growth also reduces the solvency margin and forces the solvency ratio down. On the other hand, the yield of interest continuously increases the solvency ratio, as does the safety loading X (if it is positive, as of course it must be for any sound business in the long run) The combined effect of inflation, real growth and interest is proportional to the actual size of u, whereas safety loa&ng is proportional to the business volume B. Hence in the upper sector of the figure the former forces are strong and in the lower sector weak, whereas the safety loading effect is the same throughout. If the multlplicative joint effect of inflation and real growth is larger than that of the yield of interest, i.e.

(4.2.2) (l + , = ) . (l +Tg) > 1 + ,,~,

then thele is always a certain equilibrium level where these forces are equal If the actual size of u. is above this eqmlibrium level, then tile forces pressing down are stronger. Tile reverse is true if it is below the equilibrium level.


Hence there is a compressing drift against tlus equilibrium level, which ex- plains the general behaviour of the process.

A rule of thumb is that the real growth of non-life business is about 1.5 x the growth rate of GNP. Hence it generally varies in range 4-8% per annum (the Fmmsh figure for the past 17 years is 6%) The rate of inflation usually varies in different countries between 5 and 15% Normally the nominal actual average ymld bf interest does not come up to the level of the multiplicatlve joint effect of inflation and real growth, i.e. the conchtion (4.2.2) may be vahd, at least in the long run as far as we can see. If it is not, then the structure of the process is essentially different from that shown in Fig 8, and the bundle is moving to infimty [

It can be shown that the eqmhbrium level is X

1 - - rrn

where X = the safety loading (see p. Iz) and

1 --bl n ~ r ' ~ = (l+,g) (~+,~)

= the relative interest factor relative to the nominal growth of the premium income.

Using the average figures of the Finnish companies for the 3,ears 1964-1979 we get the eqmhbrmm level 66%.

\ \qthout going into the matter more deeply we may conclude that the four background factors shown in Fig. 8, inflation included, are significant for any solvency consideration

The actual rate of inflation vm ies from year to year. Hence the assumption of its constant value was a simphflcatmn This assumptmn is relaxed in Fig. 9.

Introductmn of a shock inflation into the model produced drastic effects, as can be seen. The effcct of varying inflation depends, anaong other things, on the assumptions of how claims and premmms will react to ~t. If an unexpected shock inflation appears, the insurers are not sufficiently prepared to change premium rates immediately. This effect is escalated by the fact that the premmms are normally collected at the beginning of the insurance period and correspond (m the best case) to the expected level of inflatmn during the forthcoming period If the actual rate of inflation exceeds the expected (if any) amount, claims and expenses increase almost immedmtely, though premiums can only be corrected after some tmae lag. In the example given in 1Fig. 9 this time lag was assumed to be two years Fig 9 demonstrates the use of the stochastic-dynamm model as a "sensitivity analyser" Only one realization was generated. The rate of inflation in two particular years was varied and the

14 PENTIKAINEN AND RANTALA

15

l g

Q5

I ................................................................. ut

L t I i i • i i i i ~ i ~ i i I i J J ; i ; J J ] " l

5 1o 15 20 25

Fig 9 Sohd hnc wt th circles Only a n teady inflation rate of 8 % / a n n u m assumed Weaker hne in addi t ion a shock inflation rate of 2 o % / a n n u m m years 3 and 4 a.ssumed

The shaded area marks the difference between the original and changed flows

whole process was then regenerated using exactly the same random numbers as before.

4 . 3 . B u s i n e s s c y c l e s

Empirical data already showed that long-term variations in risk exposure may have a very significant influence on the solvency ratio. This observation was reinforced by the theoretical model when a variation of the basic probabilities was introduced. The influence of the variations was assumed to meet the expected value of claims, which is one of the basic variables in the conventional risk theoretical formulae as follows:

(4.3.1) n(t) = n(o). (l +~g)t. (1 +z,{t))

Here n(t) is the expected number of claims in year t, ig is the real growth of the business (for the sake of simplicity it was assumed to be constant from year to year) and z,(t) is an auxiliary "cycle variable" which indicates the deviations of the average risk exposure from its normal value. The scale was defined so as to give a long-term mean value of zero for zs.

As mentioned above, the cycle variable zs can be introduced into the model in different ways. The s:mplest way is to assume it to be deterministic, perhaps following the sine form

(4.3.2) z~(t) = z ~ • sin (cot + v)


where zm is the ampl i tude of the wave and the coefficient

(4-3 3) o~ = 27r/T~

is the f requency factor. Here Ts is the wavelength and v is a phase varmble. More soplust icated models are achieved if several "sine formed" waves are composed together and a noise term is added; i.e. the methods of tm~c series are applied

15

10

0 5

00

CI

Fig lo

I t can be seen from Fig. 10 how dramat ica l ly Fig. 2 changes when the business cycle is assumed. The ampl i tude (15% of the normal amount of claims) and wavelength (12 years) are no more than their empirical values.

Because our purpose was only to demons t ra te tile model, wc are not going to discuss whether or not the results are reahst lc and what kind of conclusions can be dra,,~,n. The model can be developed by incorporat ing into it buil t- in dynamics This could s imulate the rational behawour of the managemen t when an unfavourable change in the solvency rat io is imminent , or the kind of action to be expected if the solvency ratio gets ve ry high.

We will present another generalisation of the business cycle assumption.

16 P E N T I K A I N E N AND RANTALA

The phase variable v in equat ion (4.3.2) can be randomized, tile result being shown in Fig. 11.

15

05

cr

0 ~

Fig II The phase of the business cycle is randonnsed * = the insurers went brokcl

4.4. Solvency profile Vinally, ye t ano ther way to benefit from the model is presented. For any combinat ion of the given values of the background factors the minimuln solvency can be compu ted by tile me thod ment ioned in connection with Fig 3- The rnmimum solvency ratio is first computed for each combinat ion of background factors. The solvency rat io can be p lo t ted as a horizontal column. In this way it is possible to show in one picture in a very concen t ra ted shape how solvency depends on different value combinat ions of tile background factors.

Fig. 12 is in tended to i l lustrate the influence of the basic assumptions of the model

First only the number of clauns was assumed to be a random variable, whereas claim size and all other aspects were constant . As expected, the necessary m i n m m m solvency ra tm can be quite small, in our example 9% of the earned p remmms on the company ' s own retention.


M i n i m u m solvemy ratio u = U/B in par cant

Pure Poxsson

varlet[on (nr of claims)

+

c l a i m S I Z e

va I" l a ~ i O l l

+

s h o t E - Eer r , i

[ IUC ~Ua [ lOllS

÷

l o n g - C . e r m

[ ]UC EUa t l O l l S

( b u s i n e s s cyc l e s )

"shock" xnf let. ton (20 % In the f i r s t yea r )

9.2

9 2

2 2 9

IllIIIlIlllIIlllll11111111111111

- - ] = f O r o n e } e a r

~IIHHm = 'for l0 year s

R u l n p r o b d b x l l t } : i

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Illlllll]llllllllllllIIIIIIllllllIll lllIl lllIIll]l HtlIIIlll 11 o I I

50 % 188 °/o u/B

Fig 12 H o ~ does the m i n i m u m soh, cncy ra t io del)cnd on the d i f f e ren t bas ic a s s u m p - t ions ) ( R u m ba r r i e r = o I 13) P u r e PoJsson = on ly the ~tumber of c la ims v a n e s a t r a n d o m , no o t h e r e l e m e n t of the p roce s s Claun size = m a d d t t m n , t he raze of ilxdlvidual c la ims also f l u c t u a t e s S h o r t - t e r m f l u c t u a t i o n : in ad(h t ion , t h e bas ic p r o b a b i h t i e s are also s u b j e c t to s h o r t - t e r m f l u c t u a t t o n s B us i ne s s cycles = l o n g - t e r m cycles m the baste probal ) f l l tms are i n t r o d u c e d Shock m f l a t m n = finally a o n e - y e a r shock in f la t ion w a s

a s s u m e d

The next step was to randomlse the claim size, following which the short-span variations of basic probabilities were also introduced. Then, long-term business cycles were assumed, and finally a shock inflation impulse was added to the model (standard inflation 9 0 , shock inflation rate 20% and lasting only the first year).

One-year and l o-year rum probabilities are computed alternately. The figure again shows how significant the assumpUon concerning business

cycles is. The rate of inflation also affects the solvency conchtion greatly. The computation was performed for a "standard insurer" which in size,

portfolio mix, and otherwise corresponds to an average insurer irl Finland. In Fig 13 numerous other background factors were experimented with by

applying the same technique It ts possible in this way to create a "solvency profile", which provides in one single picture at least a general concept of the degree of influence of many background factors.

1 8 P E N T I K A I N E N A N D R A N T A L A

Solvency profile

"g t l .nda ld ' insurer

[3 : 5 8 5

B : 2 6 5

B : 52

M = 1

r l = 5

V i : l O

V l r l a t l o n of the i n s u r e r ' s need Io r ~olvcnc~ raargln

~ P | I I l I i l l l r t i l l l lTl l i l t~ 9~ "I. ~ l t h ~ l t :-~.:;~ " ~ ~ -

%ze of the i n s u r e r ~1 m 2 ~ 1

~]illJlllll I l l i l i l l f l l l l l | ; ' ~ "/"

$~=e & ne t r e t e n t i o n

S = s e s , m = 1 0 ~ H I I I I : I I ! I I I ~

B : S2 , ~ : t r[~-IIIl;:)llIll:lll'

h l t h c l J t s h ~ k in [ l~ tkur l hra~Lh

~ l t h s h r m c h 2

brunch .3

hrdnch 4

NIX 62 18 1 14

~5 38 l 15

T~me-sp'm

7 : 1 ) e a r

T = 5

T : 2 S ~ "

el

IB = ]15/.tak)

~ S y .

I@1 "/

Be "/.

n o v .

S h o r t - t e r r a fluc tuat ton

O h = 0 0 6 , lib : e 25 I~4 ",L

O~ -" O 10 , ,in = e 25 I 1 2 4 "].

D l s t r l N l t l o n o f t i l e l z e o f one c i a ~ (N-]OPO t)

branches ? ~ q 7"6 % D of the s t z e of on c l a m

b ~ a ~ ~ ~ ' ~ , ' ~ " ~ ~a "/" branches 2 1 1 - t 1 4 ~ : ~ lO~

• 1 Al l the o t h e r p.arameter~ a r c kep t c o n s t a n t , the branches have the saa~ f l u c t u a t ~ Y s . s a f e r ) ~ a r g l r ~ e tc

kmt= 0

k u = t = S q ,

kta t : 6 "~

z= = 0

Z . : O 8 2 5

Z . = 0 2 2 5

Z . : 8 3 ¢ 8

I Conpogl l tOn of the p in t f o l i o

1 (illi[bililk(L~i~@itLli,iiOiUTll~k ]05 ./. ~mii~i,.,~ "2" /" l hm~liglllltllUilllillllllllill:t!ll¢ , f,lll,@ll~ ~ ~ -~ hlUtlIIIIIIIUIIIIIIIUItillilIi!IUIitlTZT'~ 123 -/.

IIIIIllllllllflllllllllllllllllilllllU 91 ./. JlIIt[IIIIIIIIIIIIIIIIIIIIIIIIU ~ "/-

Tota l s a f e t y l oad ing

- - 3 13o -I. 121 ./.

- i 1~2 ./. lO'q %

a m p h t u d c of the f ] u c t u a t ion

~ q2 -/. ~ r ~ t t ~ 6 5 -/.

4 #rb~rg:i't~r'-';~'~l . ~q

I~ Mi!~iiii.,~1;,;,d': :

Real growth

% = 8 "/°~II1111UIIIIIJlI[IIIlHJ1011111@[I

ig = 4"/.hlIHUHHIHIIIIIIIUIIIIIilIII 1°= vs. Nl l l~111~1111111111~lllTilT[ll t o = o hlllllllllUIIIIIIIIlIIIIIllllll

I n f l a t i o n

i - - )

8 6 "/.

78 %

l o n g - t e r ~

-/°

~ ' r ' ~ = - - - f l 12~ -/. : i ~ ) 4 L ~ I I 152 "/.

)Ill ioI -/o

8 6 " / -

8 0 q .

75 q-

69 "/.

S'/. 7 9 " / o

7-/. I I ~s- I . 11 "1. ~ 102 "/.

i3 q. I i 1 ] ] - / .

el l i ~ m , , ~ . E ' i i - . i : ' - i i 7) ' " .... ~t 17] ./.

(lh ~. ,E4.1~'N~/~:--~;] . -=-- 'T~LX~ ] 2 7 - / .

d i l l ~ , ~ . . . ~ 1 ~ , L - 2 _ ~ 7 " - ' ~ ! i 115-1 .

I11 = I n f l a t l o n sho,.k 2 y e a r s , 20 ; , ]ag Z year~ [

CHI = I n f l a t i o n sh~:k 3 y e a r s , 20 I , no lag ( I l l ) : I n f I o t l o n shock I ~ ' ] r . 230 I,. no lag

I R u m probabi h ty

£ : • [ H l l l l I I I [ I H I L [ 72 m~. 1@ /. . . . . . . . ]]~,Hm~ 79 "/.

e 5 "/. UJJ' ,H' , ' , ' , ' ,H[I I [ I . IJJ I

c I "/o I I I I H I H I I I I I ' , ' , I ' , " , H I ' , ,:',', n O "/.

f l=~g 13 A s o l v e n c y p l o f f l l e d e m o n s t r a t i n g h o w t h e s o l v e n c y o f a n i n s u r e r d c p e n d s o n

v a r i o u s b a c k g r o u n d f a c t o r s T m ~ e s p a n i o y e a r s R u m b a r r m r zs o . ] B


Monetary values such as pre lmums B and net re tent ion M are given in millions of Fmk.

The portfoho mix was cons t ruc ted by making use of four different typical sections and combining them in the proport ions shown in the figure. Section 1 comprised a motor car, family proper ty , etc., branches where the individual claim sizes are typica l ly small and the risk exposure is not very sensit ive to seasonal or o ther varlatmns. Section 2 comprised industrial and marine insurance, etc., branches where the indwldual claim sizes can be large. Those branches where the risk exposure ~s very sensitive to shor t - te rm fluctuataons, like forest insurance and cledi t insurance, were placed in section 3. Section 4 comprxsed internat ional reassurance

This showed tha t differences m portfolio mix do not grea t ly affect solvency ratios. This as easily unders tood since normal reinsurance cuts off the risk tops. F rom the point of view of solvency the remaining distr ibutions of the risks on the insurer 's own retent ion are fairly sunila~

Fur the r details concerning this profile idea can be found in the for thcoming research report .

4.5. Safety loa&ng The evaluat ion of sufficmnt safety loading can be based on the equil ibrium level and on the width of the stochastic but-idle The equihbr ium level must be so hagh tha t the &stance from the equilibrium level to the acceptable min imum level ( ~ r u i n barrier) is suffmient; a.e. it must be at least a half of the width of the stochast ic bundle Tills approach is in fact some kind of var iant of tlle well-known s tandard deviat ion principle on the company level taking into account the business cycles and the other va tmt ions and factors inflation, growth, etc If the NP-approxmaat ion is used m calculat ing the limits of the bundle then also the skewness of the distr ibution of the total amount of claims influences on the safety loading. (NP-approximat lon is described in the book by BEARD et. al (1977). An extension of the me thod to the short tall of the distr ibution of the total clann amount will be presented m a for thcoming paper by T. PENTIKAINEN.)

In the following tables there are given some few examples of appropr ia te safe ty loadangs These loadmgs are proport ional to tile gross premium on the insurer 's own retention. Figures are calculated for a typical Finnish insurance company at 1 °/o safety level and with minimum level as zero. As said above also the business-cycles are taken into account The notat ions are as follow: '

n = the expected number of claims/year m the first year ; M = the net re tent ion in millions of F m k , B = the gross premium on the own retent ion in millions of F m k ,

Eu = the equilibrium level.

20 P E N T I K A I N E N AND RANTALA

TAI~LE I

S a f e t y I o a ( h n g a n d t h e raze a n d t h e n e t r e t e n t i o n of t he m s m e r , w h e n ~x = o o 9 , zn = 0 0 8 5 ,

'tg ~ 0 0 6

n M 13 X E u

5000 o 5 12 o 049 o 81 l o o o o o 7 25 o 044 o 72 20000 I o 52 o 0 4 ° o 66 40000 2 o I15 o 039 o 64 8oooo 5 o 265 o 039 o 63

1Ooooo xo o 586 o 037 o 61

The net retention is adjusted according to the size ance with general practice. Then the safety loadmgs Another example is given in table 2 fixing the sine of the rate of mflatmn and also the rate of interest vary.

of the insurer in accord- do not vary very much. the company but letting

TAI3LI; 2

S a f e t y l o a ( h n g a n d m f l a t t o n , x~hen ~t ~ 40000, ./14 = 2 a n d 1 3 = zl 5

z n t o * z X Eu

o o 8 5 o o 6 o o 5 o o 1 8 o 73 o 085 o 06 o 07 o 030 o 68

o 085 o 06 O 09 o 039 0.64 o o 8 5 o o 6 o II o o 4 7 o 6 o

o 085 o 06 o t3 o 054 o 57 o l o o o 6 o 13 . ° ° 4 q o 5 9 o oO o o6 o or) o 049 o 60

If the average mflatmn increases from 5% to 13% and the rate of interest doesn't increase then the safety loading and the premmm level must be raised by 3.6% umts m order to keep the same safety level. In addition one must do of course also the usual index-corrections. It ~s seen that although the safety loading increases the equilibrium level decreases.

REFERENCES

]_3EARD ]~ J~, PI:NFI|xA1NEN T a n d E PESONEN (1977) Risk Theory, C h a p m a n a n d Ha l l , L o n d o n

BEcl ;ER, F (1o79) Analyse 7ran Zettrethen der A~aftfahrtvevs~cherung und Feue~Jer- stcherung 7n .4bhang~ghett 1,on gesamtwt)tschafthchen G)ossen, V e r e m zur F o r d e r u n g det V e r s ) c l ) e r u a g s w ~ s s e n s c h a f t an de r U m v e r s ) t a t Mannl )em~ e V

HILLrL;N, E ( t977) 13us*ness cycles and l~lsuJa~tce, T h e G e n e v a P a p e r s on lRmk a n d l n s m a n c e ' Essay ' s m the E c o n o m i c T h e o r y of l { l sk a n d I n s u r a n c e


I(ARaEN, W. (i973) Zu Inhaltt und Abgrenzung der Ruckstelhmg fur drohendc Verluste aus schwebcnden Geschaftcn ill \rer~tchcrungsbllanzen, l"ers~cherlr~rgsrv, rlschaft 23, 73

I~^RTEN, k¥ (1975) Zur Begr .ndung e~ner sachgerechten Schwanh*~ngsrl~ckstellu~rg, "Sorgen, versorgen, verslchern", Festschrlft fur Heinz Gebhardt, Hans Kalwar, l(a~lsruhe

I(ARTE.N, "VV (1980) 7he 1few "Schwankungsrttckslellultg" *n a~lnual statentenls of Get.man ~mm'ers A n appllcatzon of the theory of risks, The Gcneva Papers on ]~isk and Insurance

.PI2NTIKAINEN, T (197 o) The fluctuatton reserve as stipulated in the ~-lnnlsIl insurance company act of 1953. B,dlettlz of the Fr,~msh hzs,t~.alzce lnformatton Ce~ttre.

]9ENTIKAINEN, "1" (1975) A model of stochastlc-dytmmlc prognosis, Scaltdt~$av~an Ac- tuavzal Journal

.PENrlKAINEN. T (1978a). A solvency testing model bulldtng approach for business planning. Scand~*mwaT~ .4clual tal Jolcv~tal

'PEN rlKAINEN. '1" (1978b), Dynamic programming, an approach for analysing competition strategies, Ast tn Bulletin

Tra~zsactw~ts, 2 tst Inter'national Congress of .4 or.as zes. Zu rmh & Lausanne, 198o

Documents

j. RANTALA 1. INTRODUCTION