A s t m l{ulletin lo (1079) 274-282
H O W TO D E F I N E A BONUS-3,1AI.US SYSTEM W I T H AN
EXPONENTIAL UTII.ITY FUNCTION*
J E A N LEMAIRE
We c o m p u t e a m e r i t - r a t i n g aystem for a u t o m o b i l e t h i r d p a r t y liability i n s u r a n c e
b y t w o different ways, b o t h w i t h t h e help of a n e x p o n e n t i a l u t i l i t y f u n c t i o n
O) YVe a p p l y t h e principle of zero u t i l i t y to e x p o n e n t m l utilities
(n) \Ve b r e a k t h e s y m m e t r y b e t w e e n t h e o v e r c h a r g e s a n d t h e u n d e r c h a r g e s by
w m g h t m g t h e m d i f f e r e n t l y t h r o u g h t h e i n t r o d u c t i o n of a u t i l i t y function, in o r d e r
to p e n a h z e t h e overcharges.
"1.'he results are applied to t h e portfolio of a Belgmn c o m p a n y a n d c o m p a r e d to
t h e p r e m m m s y s t e m p r o v i d e d b y t h e e x p e c t e d v a l u e principle
l ) e u x m d t h o d e s dfff6rentus, bas6e~ sur l'emplol de fonction~ d ' u t f i l t 6 e x p o n e n tielles nous p e r m e t t e n t de d6fmir till s y s t h m e b o n u ~ - m a l u s en a s s u r a n c e a u t o m o b i l e "
(i) le p r i n c i p e de l'utfllt6 nulle:
(ii) la p d n a h s a t m n des i n j u s t m e s de la compagnie, o b t e n u e en p o n d 6 r a n t les
erreurs de p r i m e au m o y e n d ' u n e f o n c t i o n d ' u t i h t 6 (le m a n i & e ~. l m s e r la s v m 6 t r l e
e n t r e les p r i m e s t r o p 61ev6es et les 1)nines t r o p basses
l.es r 6 s u l t a t s th,6orlques s e n t a p p l i q u 6 s au portefeuille d ' u m , c o m p a g n m beige
et c o m p a r 6 s a u x p r i m e s f o u r m e s p a r le p r i n c i p e de l ' e s p 6 r a n c e m a t h & n a t i q u e .
I. TIlE EXPECTED VALUE PRINCIPLE
In automobile third p a r t y liability rate-making, the policy-holders are usually
differentiated by two kinds of discriminating variables: a set of a priori factors
(like power and use of the car, age and sex of the driver, territory . . . . ) and an
a postcriori classification scheme o1" merit-rating system.
Let us consider a given risk class, with all the a priori factors fixed and
suppose t h a t the introduction of more variables is either practically impossible
or would not improve the homogeneity of the risk class. The problem is then to
define an " o p t i m a l " bonus-lnalus system, where the meaning of the word
" o p t i m a l " is to be specified.
A first possibility is to charge each policy an a m o u n t proportional to its
expected n u m b e r of claims: this is the expected value principle, developed by
BOHLalANN (1964) and BICIISEL (1964) among other>. For our numerical
examples, we shall classically assume that the n u m b e r of claims of each policy
is Poisson distributed
,~(x)-
e - zkk
,!
(x>o),
with a F-structure function
~ae-.cXka-
dU(X) -
I'(a)
I
dX
(a, -~> o).
* A n earlier v e r s m n of fins p a p e r was p r e s e n t e d a t t h e z4th A.qTIN Colloquium,
" f a o r m m a , O c t o b e r 1978.
275
BONUS-MALUS SYSTEM
Ct
Ct
T h e F - d i s t r i b u t i o n has a m e a n m = - ,
a variance ~2=T
and a moment":'~
I t is well-known t h a t in this c a s e - - s e e for i n s t a n c e DERROX ( 1 9 6 2 ) - - t h e
d i s t r i b u t i o n of the n u m b e r of claims in the portfolio is a n e g a t i v e b i n o m i a l
~bt :
(]e+a--
I.)
"c" /~:t
1
k
A p p l i e d to the following o b s e r v e d d i s t r i b u t i o n - - s e e LEnAIRE ( 1 9 7 7 ) - - t h i s
model p r o v i d e s a fairly g o o d fit, a c c e p t e d b y the Z°--test of g o o d n e s s of fit
TABLE
Absolute frequencies
Number
of claims
Observed
A dj ust:ed
96,978
9,24 °
704
43
9
o
106,974
96,895.5
9,222. 5
7 t 1.7
50-7
36
o
t o6,974
o
I
2
3
4
More than 4
Total
Mean = . 1Ol I Variance =
,
1
1o74, a = 1 6o49, "r = 15 8778`
S u p p o s e the risk class has been o b s e r v e d for t years, a n d let kl (l = 1. . . . .
be t h e n u m b e r of claims d e c l a r e d d u r i n g y e a r l. T h o s e hz's are realizations
r a n d o m variables K~, a s s u m e d to be i n d e p e n d e n t a n d e q u i d i s t r i b u t e d .
each set of o b s e r v a t i o n s (k~ . . . . , kt), we m u s t associate a p r e m i u m P t + l
Pt+l(l. .....
t)
of
To
=
le~).
[f P ( h , , . . . , k t [ X) d e n o t e s the p r o b a b i l i t y t h a t a p o l i c y - h o l d e r with given
p a r a m e t e r ?, will p r o d u c e a s e q u e n c e (kj . . . . . kt), the a posteriori d i s t r i b u t i o n
of X is
P(<,
du(x
I k, . . . . .
. . . , k, I X)dV(X)
k~) -
P(k~ . . . . .
k, I x)du(x)
o
T h e e x p e c t e d value principle defines t h e p r e m i u m Pg+~(k~ . . . . .
-Pt+, = K(~ + oO JrXclU(x I k~ . . . . .
o
w h e r e K is a c o n s t a n t a n d 0~ t h e s a f e t y loading.
let).
kt) b y
276
JEAN
LEMAIRE
I t is easier to define a bonus-naalus s y s t e m b y the relativitics
f XdU(X I k, .... , kt)
o
100
•
/XdU(Z)
o
i.e. the p r e m i u m the policy-holder has to t)ay if its initial p r e m i u m (l = o) is loo.
T h e negative binomial model has the interesting p r o p e r t y t h a t the a posteriori
distribution of the claim f r e q u e n c y X also a d m i t s a l'-distribution
7"0 t }(I t
du(x r kl,
...,
kt)
1 (j-'rtt
P(a')
-
dZ,
t
with p a r a m e t e r s a' = a + k
and v' = ,~+l, where k = Z kt is the total
I
1
n u m b e r of claims.
T h e relativities are in this case
a+
100
.
k
.r
-r+l
a"
Applied to our example, t h e y provide the following m e r i t - r a t i n g system.
TABLE. 2
k
t
0
1
2
3
4
o
1
2
3
4
152.69
144.15
136.51
211.3 °
199.48
188.92
179.41
269.92
254.82
241.32
229.18
5
6
I00
94.O7
88.81
84.1o
79.87
129.64
3 1 o . 16
365.51
293.73
278 95
346.14
328.73
398.55
378.5 °
In the following section we shall use two different apl)roaches to d e t e r m i n e
a b o n u s - m a l u s s y s t e m : the principle of zero utility a n d the penalization of
overcharges. T h e y h a v e one thing in c o m m o n : the use of utility functions.
As GERBER (1974a , 1974b ) has shown t h a t exponential utility functions, of
the f o r m
1
~,(x)
=
-
C
(1
-
e -c~)
(c>o)
possess v e r y desirable properties for the insurers, we shall only use this particular form in the sequel.
277
BONUS-MALUS SYSTEM
2. THE P I ( i N C I P L E OF ZERO U T I L I T Y
T h i s is G e r b e r ' s w o r k , a n d all we h a v e to d o is to a p p l y f o r m u l a (19) in GERBER
(19742) to a d e g e n e r a t e d i s t r i b u t i o n (since a m e r i t - r a t i n g s y s t e m is b a s e d
s o l e l y on t h e n u m b e r of c l a i m s a n d n o t on t h e i r a m o u n t ) . \Ve o l ) t a i n
Pt+l(kl,
....
let) =
l< " - - c
l Log
l
).+,r"
I
[c < Log (-~ + ~)3
T h i s c r e d i b i h t y p r e n a i u m is a n o n - d e c r e a s i n g c o n t i n u o u s f u n c t i o n of c.
A c h o i c e a c = .4 y i e l d s a r e a s o n a b l e i n i t i a l p r e m i u m P t = . 1262; s i n c e t h e p u r e
p r e m i u m is . l o t t, it c o r r e s p o n d s t o a s a f e t y l o a d i n g of a b o u t 2 5 % .
T h e t a b l e of r e l a t i v i t i e s differ d e c e p t i v e l y l i t t l e fl'om t h e p r e c e d i n g one.
TA 13L 15. 3
t
o
t
2
3
4
o
I
2
3
4
5
6
352 55
143 90
~36.17
)29 23
211 it
190.14
188.45
178.85
269.67
254.38
240.72
228.50
309.62
293 oo
278.07
364.86
345.28
327.68
397.55
377.3 °
IOO
93.99
86.66
83.90
79 62
T h e d i f f e r e n c e s a r e s m a l l , c v e u for v e r y u n r e a s o n a b l e v a l u e s of c. T i l e foll o w i n g t a b l e w a s c o m p u t e d w i t h c = 1.65, a v a l u e t h a t t r e b l e s t h e i n i t i a l p r e mium.
TABLE 4
t
o
o
t
2
3
4
5
6
IOO
i
93.13
~51.t7
209.20
26723
2
3
87.16
81-9 °
z41.46
132.94
295.77
183.97
250.08
235.O1
304.38
286.04
337.07
388,11
4
77 25
325 39
~73.52
221.66
269.79
337 92
366.06
O n e n o t i c e s t h a t in b o t h c a s e s all t h e p r c m m n ] s a r e b e l o w t h e c o r r e s p o n d i n g
figures of t a b l e 2. T h i s n a t u r a l l y i m p l i e s a h i g h e r i n i t i a l p r e m i u m P1.
278
JEAN LEMAIRE
..3' PENALIZATION OF OVERCHARGES
In this section we develop an idea of FEI~.REIRA (a977).
If we represent the a posteriori distributions ot the claim frequencies [like
in -fig. I -for t = 3, h = o (group l) and t = 3, k = 2 (group 2)], we observe t h a t
those distributions substantially overlap, All the t)olicy-holders of the second
group must pay a premium 2.24 times higher than the members of group l,
and yet, m a n y of them have an actual claim frequency (see shaded area in
fig. l) t h a t place them below the average of group I. Those people are thus
strongly overcharged: t h e y p a y more than twice their fair premium. The
problem is that, since no distinction amoung group 2 is available, we do not
know which of the group 2 policies are those with the lowest claim frequencies.
The problem increases with the wtlue of k since the injustices due to overcharges grow in a m o u n t and do not become so scarce in q u a n t i t y since the
variance of dU(X] k~ . . . . . let) grows (linearly in the negative 1)mornial case)
with the n u m b e r of accidents, for given t.
dU(~
8
r--,x\
6
5
4
3
2
01
02
0.3
04
05
lqg. i
The rates o b t a i n e d by the expected value 1)rlnclple possess interesting
properties, (see BC'HL~IANN (1964)), t h e y minimize the sum of the squared
'errors' (overcharges and undercharges) taken over all the policies of a given
group, they cn~ure financial stability in the sense t h a t tile prenliums will
c o m p e n s a t e for the expected losses for every value of t. H o ~ e v e r , it might
seem unfair, from the policy-holder's point of view, that the1,' treat overcharges
and undercharges s y m m e t r i c a l l y : ' p a y i n g too much' and 'not paying enough'
13ON U S - M A L U S
279
SYSTEM
ale valued the s a m e w a y ; one imght argue t h a t the error which consists of
asking too m u c h to a policy-holder is more severe t h a n the opposite one. Some
sense of equity c o m m a n d s us to distinguish t h e m ; one should weight differently
both kinds of errors, penalizing the overcharges.
Since all the m e m b e r s of g r o u p 2 m u s t p a y the s a m e a m o u n t , this practically
m e a n s that this p r e m i u m m u s t be reduced. C o n s e q u e n t l y the p r e m i u m for
the first g r o u p m u s t be raised m order to restore the financial balance. However, as the highest risk classes arc nearly always the least populated, the
increase of the t ) r e m m m of g r o u p I will usually be quite small.
The fact t h a t the p o p u l a u o n of the sub-groups decreases with k can be
illustrated t)3, the results of a simulation performed on the portfolio described
in section ,. CORLIER, LEMAIRE a n d i~'iUI[OKOLO (1979) o b t a i n e d the following
populatmns.
"1A BLF. 5
h
1
o
O
I0,O00
1
~,o59
877
58
6
2
8,297
7,584
6,091
1472
197
3 I
1947
2238
381
600
73
13o
3
,I
l
2
3
4
5
6
o
o
o
2
!
o
12
29
2
8
1
4
Consequently it is only necessary to raise the p r c m i m n of g r o u p J by I B.F.
in order to dinaimsh the c o n t r i b u t i o n s of g r o u p z by 2o B.F.
One way to t r e a t the p r o b l e m a s y m m e t r i c a l l y is to index one's preferences
over all o v e r c h a r g i n g and undercharging possibilities by a utility function,
and to m a x i m i z e the weighted e x p e c t e d utihty, n a t u r a l l y with the condition
that the s y s t e m will be financially stable.
For a given value of t, let us denote b\"
- m + 1 the n u m b e r of groups (m is the m a x i n m m value of k),
- A:~ the popuhttion of those groups,
m
-
N
=
E
I
-
p~-
=
N~.,
o
Pt~-~(k,, .....
-- all(},
I~)
- x =
-~-
.(xuu(x).
u
dU(X[/~',
k~),
.....
I~t) ,
alld
280
JEAN LEMAIRE
Using exponential utility functions, with arguments equal to tim difference
between the premium p~ and the real value X, we shall maximize
Z=N
t
E N
c1 It -
e
o
ctU(x I ?e)
e-~(x-~,,)]
o
subject to the condition
m
'E N ep e
N
X,
or minimize the Lagrangian function
+=cN
k
,0
(I
I
(2)
~p~ = o ~ N k
.u
~',~-~,dU(XI~)
= ~
....
o
If we denote by Me(x) = i c zx dU(X I k) the. momentgenerating function of
0
the a posteriori distribution of X, equations (2) simplify to
or
l
-
(3)
Pk = c
i
log0~
-
-- l o g M k (
-- c)
c
k
=
o ....
m.
1
Let 13 = - L o g ~. ~3 can be obtained by multiplying (3) 1)3' Nk, sumnlirlg
C
up over all the values of k and dividing 1)5, N. We get
1
N
N•
2 Nepe N12 Nt~ N112
c
k,,o
k,,o
k
Log M~.( - c)
o
and, 1)5, (1)
m
1 E
~3 = ~, + z~c
k,
NA-LogMk(-c).
o
B()NUS-MAI.US
.qVS'I'Ir..M
28 1
Finally
Pk
=
Pt+~(k~ ....
kt,) =
'['2
~, +
N ~ l,og/14 d - c )
-
'
C
N
- LogM-k(-c)
I ,u
]
The value of c can be chosen in order to reflect one's preferences over the
a s y m m e t r y of the errors. If we set c = i i.5, it means that it is necessary
to undercharge two policies by .o3 each in order to compensate for a single
overcharge of .o4 A choice of c = 17. 5 summarizes a sense of fairness that
requires two undercharges of .o4 to balance one overcharge off .o4.
Using the P-structure function and the pOl)ulations of table 5, we obtain
the followingrelativitics f o r e = l l . 5
TABLE 0
. . . .
~_2
--_
k
1
o
0
IOO
95.48
I
2
3
4
l
91.58
87.73
84 26
and f o r c =
2
3
140 17
184.62
229.55
134 28
128 68
123.52
177.o2
169.63
i62 79
219.74
21o 48
202.05
o
0
100
l
2
3
4
5
6
262 45
251.43
24].32
305 17
292 38
280-58
333 23
3i9.84
17. 5.
"I'AB L I';
l
4
95.93
92.39
88 91
85 60
7
1
2
3
4
5
6
136.14
13o.97
125 98
121.39
176.36
169.54
163 06
157 08
216 56
208 13
2oo. t 4
192.77
246.69
237.21
228 46
285 28
274.29
264 i 5
3 i t 36
299.86
4. CONCLUSION
Comparing the differeut tables, we notice that the merit-rating systems defined
by tables 2 to 4 require very high surcharges for the bad risks. Although perfectly justified, it seems very difficult to enforce them practically, for
political and colnmercial reasons. In fact, no system in operation in the world
present such severe maluses. A consequence is that in m a n y countrms the
borms-ruaiu> systems arc oul of balance financially, in the scnsu that lilt'
maluses are too small to c o m p e n s a t e for the bonuses.
2t~2
JEAN LEMAIRE
O n the o t h e r h a n d , the s y s t e m s c o n s t r u c t e d b y the t e c h n i q u e d e v e l o p e d in
3 are less severe (and the h i g h e r the a v e r s i o n to i n j u s t i c e c, tim ~maller the
r a t m b e t w e e n t h e e x t r e m e premiunas). If it is i m p o s s i b l e for p r a c t i c a l r e a s o n s
to c h a r g e the n e c e s s a r y m a l u s c s , o n e c o u l d t h i n k of a p p l y i n g this t e c h m q u e ,
s m c c it p r o d u c e s s m a l l e r m a l u s e s a n d v e t preserve~ f i n a n c i a l s t a b i l i t y .
(:niversild Lzbrc dc 13ru.xelles
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I IQ-12Q
I3UHLI~tANS, H (1964). Optimale I~ramienstufen.~y~temc, ,lhttezlunfien der I'ereznigung
Schwezzerzscher [;erstckerungsmatkemaDker, 64, i93-213.
('orLl ER, [:, I.E.XLXH~E,J. and I) 31Um~KOLO.(1O7~)) Smaulat~on of an automobde portf~dm,
The Geneva papers on Hsk and inmwance, 12, 4Oi4(9
I)ERRO,X, M (1962) Mathematlsche Problelne der Automobil~erslchcrung: M2ttedun,een
der Vere~mgung Schwetzertscher I'ersLcherungsmalhematzker, 62, lo 3-123.
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Alternative to the Classical Approach, presented at the X X I l l t h international
.Meeting of the Inat~tute of ,Management Sciences Athens.
(;ERBER, H (1974 a) ()n addltixc premmm calculation prmeq~les, ANT"IN l~zdlehn, 7,
2 15-222
((]ERBER, H. (1~)74b) ()n iteratlve l)renuum calculation principles: Mtltedungen der
I:eretnigung Schwezzeriscker I'ersicherungsmathemaltker, 74, 163-172
IASMAIRE, J . (1977) Selection procedures of rcgre~mn analysts apphed to automobile
insurance; 3,1711edungen tier Irereimgu~zg Schwetzertscher I:erstckerungsmalkemattker.
77, 65-7e.