- i -
Abstract
Let
x.J.
be the total claim amount of an insurance policy
in calendar year
i.
We assume that the
x.J. 's
are con-
ditionally independent given an unknown random parameter
e, and that
a..m(8)
J.
+[3.
J.
i.
J.
=
<P·
J.
V(m(8)) = 1
E(m(8)) ::: 0
for all
EV(x.l 8)
In the present paper it is under these assump-
tions shown how to calculate the credibility estimator of
m(8)
by recursive updating.
the unknown parameters
folio data.
described.
a..,
J.
We also give estimators for
s.,
J.
and
<P·J.
based on port-
Some generalizations of the model will be
Finally we mention some related models.
- 1 -
1.
Introduction
In credibility models of insurance experience rating it is
usually assumed that for a given policy the risk characteristics are generated by an unknown random parameter
8 descri-
bing how this policy may differ from other similar policies
in the portfolio.
In the simplest case we assume that the
total claim amounts from different years are conditionally
independent and
variance
identica~ly
s 2 (e), given
e.
distributed with mean
m(8)
and
The assumption of identical dis-
tribution is in many cases rather unrealistic.
important reason is inflation.
One very
And factors influencing the
risk may change; for instance, motor insurance claim amounts
may be influenced by improved roads and increased traffic.
In the present paper we shall modify the model by assuming
that the total claim amount from calendar year
a. m (e)+ B.
~
~
s~(e)
and variance
propose estimators for
~
~
has mean
given e , and we shall
a., 8. , and
~
~
!P.
~
= E(s~(e))
~
based
on portfolio data.
The model will be generalized into two directions:
1) to the case of estimating loss ratios and 2) to the case
when
2.
m(8)
develops randomly over time.
Preliminaries
2 A.
With a few minor exceptions we use the notation of
Sundt (1979a).
2 B.
Let
m
All displayed moments are assumed to exist.
be an unknown random variable.
(1)
We shall say
that an estimator
m
is a better estimator of
another estimator
m( 2 )
if
that is, we use quadratic loss.
m
than
- 2 -
Let
0
call an estimator
on
m
of
""
m
if
where
We shall
be observable random variables.
g 0 ,g 1 , ••• ,gn
m
m (based
may be written
are non-random numbers.
estimator of
m
m
x 1 , ••• ,xn).
(based on
a linear estimator of
will be called the
The best linear
~redibility
estimator of
A model with identical policies
3.
3 A.
We consider an insurance policy that has been in force
since calendar year
c
inclusive.
To get convenient nota-
tion we shall assume that one insurance year covers one
calendar year.
Let
policy in year
i.
x.
~
be the total claim amount of the
We shall assume that
xc,xc+ 1 , •••
are
conditionally independent given an unknown random parameter
e,
and that for all
i
E(x.j8)=a.m(8)+j3..
~
~
(1)
~
For the present we shall assume that the
are non-random numbers.
X.
~
1
S
Without loss of generality we let
<P. = EV(x. I e).
~
~
(2)
It is assumed that different
a.
are positively correlated, that is, that
tive for all
8.~ 's
and
~
V(m(8)) = 1.
E(m(8)) = 0
~ve introduce
a.'s
1.
is posi-
i.
Formula (1) says that the policy has a risk element
m(e)
that remains unchanged as time passes, and that the conditional means of the claim amounts are linear transformations of
m(e).
The coefficients and constant terms are the same for
all policies in the portfolio and can be estimated from
portfolio data.
An assumption like
E(x.j 8)
~
= a.m(8)
~
would
be natural to take care of inflation, and in (1) we have
- 3 -
added a constant term
3 B.
$.
1.
that gives further flesibility.
In this subsection we are going to describe how to
calculate
and
and
based on
xt, the credibility estimators of
xc, xc+ 1 , ... ,xt_ 1 .
m( e)
We are going to give
the formulae on recursive form as described in Sundt (1980a).
Let
yi
= (xi-8i)/ai.
given
e'
and
Then
yc,yc+ 1 , ...
are
in~ependent
E(y.!e) = m(8)
1.
As the
mt
yi's are linear transformations of the
must be the credibility estimator of
m(8)
X.' S,
1.
based on
yc, ... ,yt_ 1 , and formula (11) in Sundt (1980a) gives
q>t-1
wt-1 -2 at-1
,..,
mt =
q>t-1
2
a .L.. 1
Wt-1
y
+
<tJ t-1 t- 1
(f.'lt-1
Wt-1 +-2tVt-1 +--;at ... 1
at-1
.....
m = 0
c
we
We rewrite
wt =
mt =
(3)
( 3)
""'
(4)
mt-1
= 1•
and (4) as
Wt-1 t.pt- 1
(5)
2
at-1 Wt-1+tpt-1
2
a t-1 l/Jt-1
2
at-1 wt-1+qJt-1
xt-1- 8t-1
at-1
+
q>t-1
a2 1/J
+tp
t-1 t-1 t-1
mt-1
(6)
- 4 "'
mt'
we can easily find
When we have
by
( 7)
3 C.
The
~.'s
a.'s,
1
. B.'s,and
1
are supposed to be unknown
1
and therefore have to be estimated from portfolio data.
We
assume that we have a portfolio of independent policies that
satisfy the conditions of subsection 3 A and have the same
(a.,s.,~.)'s.
Suppose that klN
in both years
k
1
1
1
and
1~
claim amount of policy
and let
1
-
We introduce
klxj = klN
policies have been in force
in year
-1
klxij
j
denote the total
(i=1 , ... ,k 1 N;j=k,l).
kl
Li= 1 klxij"
The obvious estimator of
Bk
1s
Let
klN
= kl;r-1
i~ 1
<klxik-klxk><klxil-klxl).
We easily see that
k=l
k:t:l •
As
a2 =
k
a
rs
r<s<k, we estimate
for
A
ak
a rkask
=
ak
by
'
,/ir<s<k kwrsarkask
;
V Lr<s<k kwrs~rs
where the
W
k rs
instance choose
's
are non-random weights.
proportional to
rs
One could for
N.
~
5 -
can now be estimated by
....
"'
!Ck = akk -
3 D.
3 A-B.
"'
ak ·
Let us now return to the situation of subsections
vJe see that at the end of year
estimates of
( (l • ,
l
for i=c, ... , t-1
13 l• ' 'P.l )
described in the previous subsection.
~
mt
t-1
we can get
by the method
Hence we may estimate
by putting these estimates into (5) and (6).
we also need estimates
by (7) we see that to estimate
of
Unfortunatel~
and
However,
these quantities cannot be
estimated from the available data unless we introduce some
more structure.
The author believes that because of the
uncertainty by the choice of such structure it should be
used to construct estimators
s;
and
to be used only
in formula (7), but that in recursion (5)-(6) we should use
A
estimators
a., "'s., and
l
l
(p.
l
as developed in subsection 3 C.
The choice of additional structure seems to depend very
much on the actual situation, and we shall therefore restrict
ourselves to some vague general suggestions.
We shall for the rest of subsection 3 D assume that the
ai's and Si's are random variables independent of the
of the portfolio.
8's
Then all expectations and covariances
introduced in subsections 3 A-C becomes the analogous conditional quantitites given the
a l. ' s and Bl• ' s .
One possibility is to assume that
known parametric forms
a(i;y)
,...
and
E(a.) and
l
b( i; y).
E(B.) have
l
Then we may
A
find an estimator y
of the unknown parameter vector
y
"'
"'
based on the available
A
8t
by
a~= a(t;,r>
a.'s and B.'s and estimate
l
l
at
and
A
and 8~ = b(t;r).
Because of the appro-
ximative nature of the assumption of parametric forms
.,.. 6 -
a(i;y)
and
f"'<J
recent
-
we ought to give more weight to the most
b(i;y)
......
a.'s
and
~
"
than to the older ones when construe-
f3.~ ' s
....
ting the estimator y
.
f"'<J
The special case where the (a.,f3.)'s
are independent and
~
l
identically distributed, is closely related to the model
described in Sundt (1979b).
In this case we may estimate
and
"
B~ = Li<t twiei ,
where the
and
are non-random weights.
Another approach is to make some martingale assumption.
We shall give a few cases.
Suppose that {a.} is a martingale.
Then
~
.t =
and
~
This
t-1 would be a natural estimator of
solution seems intuitively very sound; as we have no data for
a.
a.'~
the next year, the best we can to is to use what we have found
for the present year.
estimate
et .
Nmv- let
gale.
The same approach could be used to
n i -- api
-
a~-'i-1
and assume that
{n·}
l
is a martin-
Then
( 8)
A
and
A
nt*
= 8t-1 - 8t-2 would be a reasonable estimator of
"
As st = nt + 8t-1 ' we estimate Bt by B~ = nt + 1\_ 1 =
"
= 2Bt_
1 -st_ 2 . The present martingale assumption can be
nt.
A
interpreted as a very weak assumption of linear trend in the
f3.~ 1 s.
An analogous approach could of course be used in the
estimation of
Now suppose that
gale.
Then
{o .}
~
given by
o.
~
=a./a. 1
~
~-
is a martin-
- 7 -
and
o~
= ~t-l/~t- 2
would be a reasonable estimator of
ot .
As
The quantity
oi
could 1JB thought of as a rate of inflation,
and the martingale assumption would then say that the expected
inflation of next year is equal to the inflation of the present
year.
If
8.l
is interpreted as rate of inflation, it would be
natural to assume that it is also related to the
e.'s.
l
Let
( 9)
Then
Bt
could be estimated by
A
A
The assumption (9) says, roughly speaking, that the expected
claim amounts of next year are equal to the expected claim
amountsof the present year increased by a multiplicative
inflation, and in addition we get an additive element, ~hich
according to our present knowledge has expectation zero.
As an intermediate case between (8) and (9) we could
assume that
Then
St
could be estimated by
- 8 -
3 E.
St
He have now proposed several estimators of
based on claim data from before year
t.
at
and
However, the
insurance company may also possess additional information
that ought to be incorporated into the estimators
S~.
a*
t
and
For instance, in motor insurance one ought to use greater
estimated values of
and
st
than indicated by the avail-
able data if it 1s known that the speed limits are to be in-.
creased in year
t.
And the company ought to incorporate
available prognoses about inflation.
3 F.
As we have seen in the two previous subsections,
there are several approaches that can be used to find estia~
mators
Bt•
and
Experience and knowledge would probably
give the actuary some idea that some of the approaches are
better than others in his actual situation.
the claim data from year
t
However, when
are available, one ought to
examine different choices of
a*
t
and
S~
approaches seem to be better than others.
and see if some
It
s~ems
that the
function
where the sum is taken over all policies that have been in
force in year
a~
S~
and
t, is useful in this connection5 the estimators
that minimize
Qt
would be preferable.
If this
analysis indicates that one approach of finding estimators
a~
8~
and
is better than the others, it would be natural
to use this approach for the estimators
Instead of
A
(a -a*)
t
t
Qt
and
one could of course minimize the functions
A
2
and
the estimated
minimize
Qt.
cst-8~) 2 , but as we essentially want to fit
Xt
Is
to the
xt's, it seems more natural to
- 9 -
4.
Credibility for loss ratios
4 A.
We shall now modify the model of Section 3 to credi-
bility estimation of loss ratios.
Our approach is a genera-
lization of a model by Blihlmann and Straub (Buhlmann &
Straub (1970); Blihlmann (1971)).
We consider an insurance portfolio that has been ceded
since calendar year
c
inclusive.
It is assumed that one
reinsurance year covers one calendar year.
direct insurance risk premium of year
reinsurance claims of the same year.
ratio of year
1
is
x. = s./p ..
l
l
l
1
Let
and
p.l
s.l
be the
the total
Then the observed loss
It is assumed that the xi's
are conditionally independent given an unknown random parameter
e, and that assumptions (1) and
are satisfied with positive
and
Let
and
based on
a.'s.
l
(2) of subsection 3 A
We further assume that
be the credibility estimators of
m( e)
xc, .•. ,xt_ 1 , and let
The situation is obviously the same as in subsection 3 B,
and we get
xt-1- 13 t-1
cp t-1
+
a
p
a 2 ,,,
t-1
t-1 t-1 '~"t-1
=0
=1
+~
¥t-1
- 10 -
4 B.
The difference from the model of Section 3 appears
when we are going to estimate the
(a·,"·,
]. JJJ. 1.0·)
. ]. 's
by data
from a portfolio of ceded portfolios as the different ceded
portfolios have different amounts of direct insured premiums.
We assume that we have a portfolio of independent ceded
portfolios that satisfy the conditions given in subsection 4A
I
and have the same
(a.]. ,e.]. ,tp.)
's.
].
Suppose that
folios have been ceded both calendar years
let
port-
and 1 , and
denote the observed loss ratio of Portfolio
klxij
J.n year
k
klN
(i = 1, ... 'kl N; j = k,l)
j
Let
klN
klxj = li=1 kl aij klxij '
where the
kl aij' s are non -random weights.
Then
k =1
:k=l=l
with
Let
where the constants
klbi
are chosen so as to satisfy
klN
Li=1 kl bi kl ci = 1 '
Then we have
k=l
k
*1
.
].
- 11 -
with
and
ak
may be estimated by
~
Ct
rs
are non-random weights, e.g. proportional
k wrs 's
I rs p r rs p s , tJhere
.,
where the
to
j
~
(j)k =
.
can now be estimated by
(j)k
~
= r,s
~2
Ctkk - Ctk
kc
•
As choice of
klaij
and
klbi
we propose
with
(cf. Sundt (1980b), subsection 3B).
As
kkN
v
E
i=1
~= kkN
E
i=1
is the best linear unbiased estimator of
available claim amounts (see Sundt (1978)).
based on the
we propose to
~
estimate
sk
kkN
~
13 k
12 -
by
kkpik
r
~z
~
kkxl.k
i=1 kkpik ak +r.pk
N
kk
kkpik
i~1 kkPik a~+ 0k
=
For estimation of
year
st
by claim data from before
t , we refer to subsections 3D-F .
.5. Estimation when
SA.
and
~
a
varies with time
In subsection 3A we assumed that the claim amounts
xc,xc+ 1 , ...
of an insurance policy depended on an unknown
random parameter
e •
a
(ac,ac+ 1 , ... ) of unknown random parameters
is a sequence
and that
x.
l
Now we are going to assume that this
depends on
a only through
e.l ,
that is, we
allow the individual risk characteristics of the policy to
change as time passes.
This is a very natural assumption;
e.g., in motor lnsurance a car owner's driving abilities are
not constant.
We shall assume that
xc,xc+ 1 , ...
are inde-
pendent glven
e , and replace assumptions (1) and (2) by
E(x.!a> =a. m(a.)+B.
l
l
l
l
E(m(e.))
l
= o
C(m(a.),m(a.))
l
J
= p I i-J· I •
(10)
Assumptions similar to (10) have been studied by Sundt (1980a).
It is assumed that the
a.'s
l
are positive, and that pE<0,1] .
- 13 -
In the same way as 1n subsection 3B we find
=0
SB.
= 1
We are now go1ng to develop estimators of the ai's,
e.'s,
1
~.'s,
1
and
p
•
Assume that we have a portfolio of
independent policies that satisfy the conditions given in the
previous subsection and have the same
-
Let
~
(a 1. , B1• ,<.p.1 ) ' s and
and let
k
=1
k
=I=
1 .
~
Bk
As for all
a
a
__r_-_3~'~r_-_1___r_-_2-L,r_
a r-3,r-2 a r-1,r
by
P~
I fr
ak ·
r
= a r-3,r
a
r-2,r-1
a r-3,r-2 a r-1,r
we suggest to estimate
p
a
where the non-random weights
proportional to
r- 3 ,r N •
= p2,
by
wr <a r-3,r-1 ar-2 ,r +a r-3 ,r
-; - 2 E w
a
r r r-3,r-2 r-1,r
_
•
be defined as in subsection 3C,
klN' klxj' Bk' and
We estimate
p
wr
a
r-2,r-1
).
'
could e.g. be chosen
-- 14
~
We also have
r
ak-3 k-2
= ak-1,k/~-3,k-~ ak-2,k-1
and may estimate
=~
ak-l,k
~k
I
ak
by
a
t
k-3,k-2
ak-3,k-1 ak-2,k-1 •
can now be estimated by
For the estimation of
by claim data from
and
before year t , we refer to subsections 3D-F.
SC.
As we in Section 4 nodified the model of Section 3
to estimation of loss ratios, we can do a similar extension
of the present model.
SA we then replace
In the model assumptions of subsection
EV(xi!a>
= mi
by
EV(xila>
= ~i/Pi
and
get
,,,'~'c = 1
=0
,....
X
t
We shall not go any further into this model.
§.
Conclusion. Related models
6A.
The methods treated in Sections 3 - 5 may seem a bit
inconsequent; at the end of year
of
Bi's
a.
and
1
B.1
t-1
we have estimators
assuming no connection between
a.'s and
1
from different years, but then for the estimation of
and
Bt
we suddenly introduce some structure.
The
- 15 -
reason for the introduction of this structure is, as argued
in subsection 3D, the need of additional assumptions to be
able to estimate
But as we do not feel too
and
confident about these assumptions, we are willing to use them
only when strictly necessary.
6B.
1
Alternatively, we may find it reasonable that for all
E(x.! 8) =
J.
X·'
b(8) ,
J. -
design vector, and
.
where
b(8)
~
y.
~J.
is a known non-random
is a vector function of
e .
Such
models were introduced in credibility theory by Taylor (1975)
and Hachemeister (1975).
Of later contributions to the theory
we mention Jewell (1975), Taylor (1977), De Vylder (1977,1978),
and Norberg (1980).
6C.
These regression models assume that different policies
are independent, and that time-heterogeneity occurs in accordance with known design vectors.
assume that to each calendar year
An opposite approach is to
J.
there is connected an
unknown random parameter
n.J.
folio in that year.
ni's are assumed to be independent
The
that influences the whole port-
and identically distributed, and for a policy with random
risk parameter
e and the
function
year.
e
n.J. 's
F<·l •,•)
the conditional distribution of
is of the form
J.S
F<·le,nt),
given
where the
independent of the policy and the
Such models may describe cases where purely random
elements influence the whole portfolio; e.g., in motor insurance a winter with extremely icy roads may lead to many
accidents.
Models of this sort have been treated by Welten
(1968) and Sundt (1979b).
- 16 -
fl.cknowledgement
The present research was supported by Association of
Norwegian Insurance Companies and the Norwegian Research
Council for Science and the Humanities.
References
Buhlmann, H . . (1971).
Credibility procedures,
Proceedings
of the 6th Berkeley symposium on mathematical statistics
and probability, Vol.1, pp. 515-525.
University of
California Press, Berkeley and Los Angeles.
Buhlmann, H. & Straub, E.
(1970).
Glaubwurdigkeit fur
Schadensatze. Mitteilungen der Vereinigung schweizerischgr
Versicherungsmathematike, 2Q, 111-133.
De Vylder, F. (1977).
distributionfree
Optimal parameter estimation in semicredibility theory.
Paper presented to
the 13th ASTIN colloquium in Washington D.C.
De Vylder, F. (1978).
theory.
Parameter estimation in credibility
ASTIN Bulletin 10, 98-112.
Hachemeister, C.A. (1975).
Credibility for regression models
with application to trend.
In Credibility: Theory and
applications (ed. P.M. Kahn), pp. 129-163.
Academic Press,
New York.
Jewell, W.S.
(1975).
Bayesion regression and credibility
theory.
RM-75-63.
International Institute of Applied
Systems Analysis, Laxenburg, Austria.
Norberg, R. (1980).
Empirical Bayes credibility.
Submitted
for publication 1n Scand. Acturial J.
Sundt, B. (1978).
On models and methods of credibility.
Statistical Research Report 1978-7.
Institute of Mathe-
matics, University of Oslo.
Sundt, B. (1979a).
model.
A hierarchical credibility regression
Scand. Actuarial J. 107-114.
- 17 -
Sundt, B. (1979b).
An insurance model with collective
seasonal random factors, Mitteilungen der Vereinigung
schweizerischer Versicherungsmathematiker 79, 57-64.
Sundt, B. (1980a).
Recursive credibility estimation.
Submitted for publication in Scand. Actuarial J.
Sundt, B. (1980b).
models.
Parameter estimation in some credibility
Statistical Research Report 1980-6.
Institute of Mathematics, University of Oslo.
Taylor, G.C. (1975).
loss ratios.
Credibility for time-heterogeneous
In Credibility: Theory and applications
(ed. P.M. Kahn), pp. 363-389.
Taylor, G.C. (1977),
Academic Press, New York.
Abstract credibility.
Scand.
arial J. 149-168.
Welten, C.P. (1968).
ASTIN
Bulletin~'
The unearned no claim bonus.
25-32.
Actu~·