The statistical distribution of incurred losses
and its evolution over time
III: dynamic models
Greg Taylor
November 2000
Casualty Actuarial Society
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Table of Contents
1. Introduction .............................................................................................................. 1
2. The parametric framework ..................................................................................... 3
3. Kalman filter............................................................................................................. 7
4. Evolutionary model with unknown second moments ........................................... 9
5. Estimation of second moments .............................................................................. 11
6. Prediction and prediction error ............................................................................ 17
7. Numerical example................................................................................................. 19
8. Acknowledgements ................................................................................................. 35
9. References ............................................................................................................... 36
Appendices
A
Covariance of prediction errors
B
Covariance of squared prediction errors
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1.
1
Introduction
This paper is written at the request of, and is partly funded by, the Casualty
Actuarial Society’s Committee on Theory of Risk. It is the third of a trio of
papers whose purpose is to answer the following question, posed by the
Committee:
Assume you know the aggregate loss distribution at policy inception
and you have expected patterns of claims reporting, losses
emerging and losses paid and other pertinent information, how do
you modify the distribution as the policy matures and more
information becomes available? Actuaries have historically dealt
with the problem of modifying the expectation conditional on
emerged information. This expands the problem to continuously
modifying the whole distribution from inception until it decays to a
point. One might expect that there are at least two separate states
that are important. There is the exposure state. It is during this
period that claims can attach to the policy. Once this period is over
no new claims can attach. The second state is the discovery or
development state. In this state claims that already attached to the
policy can become known and their value can begin developing.
These two states may have to be treated separately.
In general terms, this brief requires the extension of conventional point
estimation of incurred losses to their companion distributions. Specifically, the
evolution of this distribution over time is required as the relevant period of
origin matures.
Expressed in this way, the problem takes on a natural Bayesian form. For any
particular year of origin (the generic name for an accident year, underwriting
year, etc), one begins with a prior distribution of incurred losses, which applies
in advance of data collection. As the period of origin develops, loss data
accumulate, and may be used for progressive Bayesian revision of the prior.
When the period of origin is fully mature, the amount of incurred losses is
known with certainty. The Bayesian revision of the prior is then a single point
distribution. The present paper addresses the question of how the Bayesian
revision of the prior evolves over time from the prior itself to the final
degenerate distribution.
This evolution can take two distinct forms. On the one hand, one may impose no
restrictions on the posterior distributions arising from the Bayesian revisions.
These posterior distributions will depend on the empirical distributions of
certain observations. Such models are non-parametric.
Alternatively, the posterior distributions may be assumed to come from some
defined family. For example, it may be assumed that the posterior-to-data
distribution of incurred losses, as assessed at a particular point of development
of the period of origin, is log normal. Any estimation questions must relate to
the parameters which define the distribution within the chosen family.
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These are parametric models. They are, in certain respects, more flexible than
non-parametric, but lead to quite different estimation procedures.
The first paper (Taylor 1999a) dealt with non-parametric models only. The
second (Taylor 1999b) dealt with parametric models. The present paper
addresses the case of dynamic models, in which parameters are allowed to
evolve from one period of origin to the next.
Familiarity with the earlier papers will be assumed here. In particular, the
Bayesian background introduced and described there will be assumed.
As far as possible, the notation used here will be common with the earlier
papers.
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2.
The parametric framework
2.1
Basic framework
Let X(i, j) denote some real valued stochastic variable that is indexed by year of
origin i and development year j, i  0, 0  j  J for fixed J > 0.
Note that i + j labels a particular year of calendar time (experience year), and
the data for the year k = i + j constitute the vector:
 X  k , 0  , X  k  1,1 ,..., X  0, k   ,
T
(2.1)
which will be denoted X(k).
For brevity, the whole discussion here is formulated in terms of years. However,
any other period, consistently replacing years, would be equally satisfactory.
Let X   h  denote the data triangle:
X   h    X  i, j  : i  0,0  j  J ,0  i  j  h

h
X  k .
(2.2)
k 0
Generally, the use of the prime, as in (2.2), will indicate the incorporation of all
past experience years in the associated quantity. Equation (2.2) shows that the
data triangle grows over time by the addition of diagonals.
Denote
  i, j   E X  i , j  ,
(2.3)
and suppose the X  i, j  |   i, j  form a mutually stochastically independent set.
In this set-up, X  i, j  corresponds to one component of the vector Y  i, j  , and
  i, j  to one component of the vector   i, j  , in the second of the earlier
papers.
Just as the X  i, j  were represented in diagonals (see (2.1)), so may the   i, j 
be represented. Thus
  k     k , 0  ,   k  1,1 ,...,   0, k   .
T
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(2.4)
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2.2
4
Parameters linking development years
Section 8 of the paper on parametric models represented parameters   j  ( 
depended only j there) in terms of basis functions, weighted by a smaller set of
underlying parameters, as follows:
Q
  j     qq  j  ,
(2.5)
q 1
where q   were the pre-defined basis functions, and the  q constants
requiring estimation.
More precisely,   j  in the earlier paper was a vector, each component having
the representation (2.5).
This representation is extended in the present paper to the   i, j  , which depend
on both i and j:
Q
  i, j     q  i   q  j  .
(2.6)
q 1
Note that the basis functions have not been changed, and still depend only on j.
However, their coefficients now vary with i, creating variation in  with i.
It is convenient to express (2.6) and related formulas in matrix form:
  i, j   U T  j 
1x Q
 i 
(2.7)
Q x1
where matrix and vector dimensions are optionally written below their
associated symbols, and
U T  j   1  j  ,..., Q  j  
(2.8)
T  i   1  i  ,...,  Q  i   .
(2.9)
It is now possible to represent the vector E  X  k   in the form:
  k   E  X  k    C  k 
  k  ,
 k 1 x  k 1Q  k 1Q x1
where
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(2.10)
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C  k   diag U T  0  ,U T 1 ,...U T  k  
U T  0 
 1x Q

U T 1

1x Q










T
U  k 

1x Q
  k   T  k  , T  k  1 ,..., T  0   .
T
2.3
(2.11)
(2.12)
Evolution of parameters
In the previous two papers, all parameters such as   i  have been assumed
constant over time. For the dynamic models that form the subject of the present
paper, these parameters are assumed to evolve from one year of origin to the
next, as follows:
  i   G  i   i  1  v i  ,
(2.13)
where G  i  is a deterministic Q x Q matrix and v  i  is a stochastic Q-vector
with
E  v  i    0
(2.14)
V v  i    V  i  .
(2.15)
From (2.12) and (2.13),   k  evolves over k as follows:
  k   G  k 
 k 1Q x1
  k  1 v  k 
 k 1Q x kQ
kQ x1
 k 1Q x1
(2.16)
with
G  k 
 1

1
G  k   






,


1 
(2.17)
each submatrix here of order Q x Q and
v  k   vT  k  , 0 .
T
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(2.18)
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Substitution of (2.14) and (2.15) into (2.18) yields
E v  k    0
(2.19)
V  k  0
V v  k   
,
0
 0
(2.20)
denoted V   k  .
Now recall (2.10), but expressed in the form:
X k   C k 
 k 1 x1
 k 1 x  k 1Q
  k   w  k 
 k 1Q x1
 k 1 x1
(2.21)
with
E  w  k    0.
(2.22)
In addition, denote
W  k   V  w  k    V  X  k  |   k   .
(2.23)
The v  i  and w  k  are assumed mutually stochastically independent.
The structure described by (2.16) and (2.21) may be recognised as the Kalman
filter structure (Kalman, 1960).
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3.
7
Kalman filter
Consider the forecast of a new diagonal X  k  1 from data X   k  . Let
Xˆ  k  1 denote the forecast, and suppose that it must satisfy two conditions:
(i)
(ii)
it must be linear in the data; and
it must be least squares in the sense that
2
  E  Xˆ  k  1  X  k  1
is a minimum within the admissible forecasts, where the expectation is
unconditional.
These two conditions correspond to (4.8) and (4.9) in the previous paper (Taylor
1999b).
The forecasts Xˆ  k  1 , k = 0, 1, 2, etc may be calculated according to the
Kalman filter recursion (Kalman, 1960), as follows:
ˆ   h  1| h   G  h  1 ˆ   h | h 
(3.1)
  h  1| h   V   h  1  G  h  1   h | h  GT  h  1
(3.2)
Xˆ  h  1| h   C  h  1 ˆ   h  1| h 
(3.3)
L  h  1| h   W  h  1  C  h  1   h  1| h  CT  h  1
(3.4)
K  h  1    h  1| h  CT  h  1 L1  h  1| h 
(3.5)
ˆ   h  1| h  1  ˆ   h  1| h   K  h  1  X  h  1  Xˆ  h  1| h  
(3.6)
  h  1| h  1  1  K  h  1 C  h  1    h  1| h  .
(3.7)
The recursion is initiated at h = -1 in (3.3) and (3.4) with given values ˆ   0 | 1
and   0 | 1 , which are the prior-to-data estimates of   0  and its uncertainty
V v  0   . The recursion then proceeds through (3.3) – (3.7), and then cycles
through (3.1) – (3.7) with increasing h.
Throughout this recursive scheme, a symbol of the form Q  h2 | h1  represents an
estimate of quantity Q  h2  at time h2 on the basis of data up to and including
time h1 . Thus, (3.3) with h = k is the estimator sought in the present case, and
(3.4) its covariance matrix.
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Equation (3.6) indicates how the parameter estimate ˆ  is affected by new data
X. It is seen that the previous estimate ˆ  is adjusted by a multiple (in fact, a
matrix multiple) of its associated prediction error.
Thus, as data emerge and reveal errors in earlier predictions, the filter corrects
for them in its subsequent predictions. The matrix K is called the Kalman gain
matrix.
A very useful dissertation on the Kalman filter is given by Neuhaus (1989),
where it is noted that
L  h  1| h   V  X  h  1  Xˆ  h  1| h  
(3.8)
  h2 | h1   V   h2   ˆ   h2 | h1   ,
(3.9)
for h2  h1 or h1  1 .
Note also that, by (3.1),
E ˆ   h  1| h   G  h  1 E ˆ   h | h 
 G  h  1   h 
(3.10)
if ˆ   h | h  is an unbiased estimator of   h  . Then, by (2.14) and (2.16),
E ˆ   h  1| h    E   h  1  .
(3.11)
Thus, unbiasedness of ˆ   h | h  as an estimator of   h  implies unbiasedness
of ˆ   h  1| h  as an estimator of   h  1 .
Similarly, by (3.3), one can show that unbiasedness of ˆ   h  1| h  implies
unbiasedness of Xˆ  h  1| h  as an estimator of X  h  1 . Induction on h leads
to the following result.
Proposition 3.1
Suppose that ˆ   0 | 1 is an unbiased estimator of   0  . Then, for each h = 0,
1, etc,
ˆ   h  1| h  is an unbiased estimator of   h  1
Xˆ  h  1| h  is an unbiased estimator of X  h  1
ˆ   h  1| h  1 is an unbiased estimator of   h  1 .
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4.
9
Evolutionary model with unknown second moments
Section 2.3 described a model which incorporated parameter evolution.
introduced the two sets of second order quantities V(i) and W(k).
It
These were regarded as known quantities. Consider, however, the case where
they are unknown. In the simplest such case, they will involve just one
unknown parameter, thus:
V  i   V *  i 
(4.1)
QxQ
W  k   W *  k  ,
(4.2)
 k 1 x  k 1
where  is the unknown scalar parameter, and V *  i  and W *  k  are known
matrices.
Generally, a starred symbol will represent the corresponding unstarred symbol
after removal of the multiplier  .
Suppose in addition that the initial value   0  of the Kalman filter (Section 3) is
subject to the uncertainty:
  0 | 1   *  0 | 1 ,
(4.3)
where  *  0 | 1 is a known matrix, independent of  .
QxQ
Proposition 4.1
When (4.1) – (4.3) hold, the quantities   h2 | h1  ,
h2  h1 or h1  1, and
L  h  1| h  in the Kalman filter are proportional to  . Moreover, the Kalman
gain matrices K(h) are independent of  .
This result is simply proved by reference to (3.2), (3.4), (3.5) and (3.7), and
induction on h.
As a consequence of the proposition, one may write
  h2 | h1    *  h2 | h1  ,
h2  h1 or h1  1
L  h  1| h   L *  h  1| h  ,
(4.4)
(4.5)
where the starred quantities here may be obtained from a “starred recursion”
corresponding to (3.1) – (3.7), but with (3.2), (3.4) and (3.7) replaced as follows:
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 *  h  1| h   V  *  h  1  G  h  1  *  h | h  GT  h  1
(3.2a)
L *  h  1| h   W *  h  1  C  h  1  *  h  1| h  CT  h  1
(3.4a)
 *  h  1| h  1  1  K  h  1 C  h  1   *  h  1| h 
(3.7a)
where
V *  h  0
V  * h  
.
0
 h 1Q x  h 1Q  0
(2.20a)
This recursion is initiated with given values of ˆ   0 | 1 and  *  0 | 1 .
By (3.8) and (4.5),
V  X  h  1  Xˆ  h  1| h    L *  h  1| h  ,
(4.6)
where L *  h  1| h  is a known matrix given by (3.4a).
Write (4.6) as
V   h  1| h    L *  h  1| h 
(4.7)
with   h  1| h  denoting the prediction error
  h  1| h   X  h  1  Xˆ  h  1| h  .
(4.8)
Then
TrV   h  1| h     Tr L *  h  1| h  .
(4.9)
Recall from Proposition 3.1 that
E   h  1| h   0.
(4.10)
Hence  T is an unbiased estimator of V    . Then, by (4.9), an unbiased
estimator of  is
ˆ  h  1  Tr   h  1| h  T  h  1| h   / Tr L *  h  1| h 
 T  h  1| h    h  1| h  / Tr L *  h  1| h  .
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(4.11)
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5.
Estimation of second moments
5.1
Unbiased estimation
When data X  h  1 are available, (4.11) provides h + 2 unbiased estimators of
 , viz ˆ  0  , ˆ 1 ,..., ˆ  h  1 . Then any convex combination of these estimators
is itself an unbiased estimator.
Proposition 5.1. Let
ˆ   h  1  ˆ  0  ,..., ˆ  h  1 
T
(5.1)
and
R  h  1  V ˆ   h  1  .
(5.2)
Consider estimators of  of the form
ˆ  h  1  aT  h  1 ˆ   h  1 ,

(5.3)
where a(h + 1) is a non-stochastic (h + 2)-vector. The MVU estimator is given
by
a  h  1  R 1  h  11/ 1T R 1  h  11 ,
(5.4)
where 1 denotes a vector with all entries equal to unity.
Moreover, for this a ( h + 1),
ˆ  h  1   1T R 1  h  11 1 .
V 

 
(5.5)
Proof
The proof is an elementary and well-known exercise in constrained optimisation.
Application of (5.3) and (5.4) requires a knowledge of R(h + 1). One requires
quantities of the form
Cov ˆ  f  1 , ˆ  g  1   Cov T  f  1| f    f  1| f  , T  g  1| g    g  1| g   /
Tr L *  f  1| f  Tr L *  g  1| g   .
(5.6)
This last quantity involves covariances of second moments, i.e. involves fourth
moments, of the observations. A practical estimate requires parametric
assumptions, which enable the expression of fourth moments in terms of second.
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Appendices A and B calculate the covariance in (5.6) on the assumption that the
observations X(i, j) are normally distributed. The relevant results obtained from
there are as follows:
Cov T  f  1| f    f  1| f  , T  g  1| g    g  1| g  
 2 2 Tr  M *  f  1, g  1 M *T  f  1, g  1 
[from (B.18)]
(5.7)
where, by (A.25) and the comment at the end of Appendix A,
M *  h, g   C  h  N *  h, g  CT  g   C  h  Q *  h, g   hgW *  h 
(5.8)
and N *  h, g  and Q *  h, g  are calculated by means of the following
recursions (see (A.20) and (A.14)):
N *  h, g   P  h  1 N *  h  1, g 
 gh V   h  1  G  h  K  h  1W  h  1 K T  h  1 GT  h  for h  g
Q *  h, g   P  h 1 Q *  h 1, g 
for h  g  1
(5.9)
(5.10)
with
P  h   G  h  1 1  K  h  C  h  
[by (A.11)].
(5.11)
The recursion for Q is initiated (see A.18)) by:
Q *  g  1, g   G  g  1 K  g W *  g  .
(5.12)
Also
Q *  g , g   0.
(5.13)
The recursion for N* is initiated by N *  g , g  1 , which is itself obtained from
another recursion (see A.22) – (A.24)):
N *T  g , g  1  P  g   P  g  1 N *  g  1, g   V  *  g  1
G  g  K  g  1W *  g  1 K T  g  1 GT  g  
(5.14)
N *  0,1   *  0 | 1 PT  0  .
(5.15)
N *  0,0   *  0 | 1 [by (A.30)].
(5.16)
By definition of M(f,g) in Appendix B,
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L  f  1| f   M  f  1, f  1
and so
L *  f  1| f   M *  f  1, f  1 .
(5.17)
Substitute (5.7) and (5.17) into (5.6):
Cov ˆ  f  , ˆ  g   22Tr  M *  f , g  M *T  f , g  / Tr M *  f , f  Tr M *  g , g 
(5.18)
for f, g = 0, 1, 2, …, h + 1.
By definition (5.2), R(h + 1) has elements given by (5.18). They each involve
the unknown factor 2 2 , but note that this cancels in (5.4). Therefore, (5.4) is
equivalent to:
1
1
a  h  1   R *  h  1  1/ 1T  R *  h  1  1


(5.19)
with R *  h  1 having (f,g) element
Tr  M *  f , g  M *T  f , g   / Tr M *  f , f  Tr M *  g , g   .
(5.20)
ˆ  h  1 is defined by (5.3) with a (h + 1) given by (5.19).
Then the estimator 
By (5.5), (5.18) and the definition of R *  h  1 ,
1
ˆ  h  1   22 1T  R *  h  1  1 1 .
V 
 

 
5.2
(5.21)
Credibility estimation
Now assume that the parameter  is a latent parameter, i.e. it is a sampling from
some prior distribution with d.f. F  , mean  and variance 2 .
According to the credibility of the earlier papers, specifically Taylor (1999b,
ˆ  h  is
Section 4), the least squares estimate of  that is linear in 
ˆ h
  h   1  z  h     z  h  
(5.22)
with
z  h   1  S  h  
1
ˆ  h |    / V E 
ˆ

S  h   E V 
    h |  .
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(5.23)
(5.24)
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14
By (5.21),
ˆ  h |     2 E 2  1T  R *  h   1 1
E V 
 
 
 

1
1
1
 2  2   2  1T  R *  h   1 .


(5.25)
ˆ  h  is an unbiased estimator of  ,
Since 
ˆ  h |     V     2 .
V E 


(5.26)
Substitute (5.25) and (5.26) in (5.24):
1
1
S  h   2 1   2 / 2  1T  R *  h   1 .


(5.27)
In the case h = -1 (no data),   h  is given by its prior:
  1  .
5.3
(5.28)
Summary of algorithm
The complete algorithm incorporating the Kalman filter of Section 3, as
modified by Section 4, and the present section’s estimation of second moments,
is as follows.
ˆ   h  1| h   G  h  1 ˆ   h | h 
(3.1)
 *  h  1| h   V  *  h  1  G  h  1  *  h | h  GT  h  1
(3.2a)
  h  1| h     h   *  h  1| h 
(4.4)
Xˆ  h  1| h   C  h  1 ˆ   h  1| h 
(3.3)
L *  h  1| h   W *  h  1  C  h  1  *  h  1| h  CT  h  1
(3.4a)
ˆ  h  1   X  h  1  Xˆ  h  1| h   X  h  1  Xˆ  h  1| h  / Tr L *  h  1| h 
(4.11)
T
At this point, there follows an inner recursion to calculate the  h  2 x  h  2
matrix R(h + 1) (see below)
1
1
a  h  1   R *  h  1  1/ 1T  R *  h  1  1


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(5.19)
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ˆ   h  1  ˆ  0  ,..., ˆ  h  1 
T
(5.1)
ˆ  h  1  aT  h  1 ˆ   h  1

1
S  h  1  2 1   2 / 2  1T  R *  h  1  1


z  h  1  1  S  h  1 
(5.3)
1
(5.27)
1
(5.23)
ˆ  h  1
  h  1  1  z  h  1    z  h  1 
(5.22)
L  h  1| h     h  1 L *  h  1| h 
K  h  1   *  h  1| h  C T  h  1  L *  h  1| h 
(4.5)
1
ˆ   h  1| h  1  ˆ   h  1| h   K  h  1  X  h  1  Xˆ  h  1| h  
 *  h  1| h  1  1  K  h  1 C  h  1   *  h  1| h 
  h  1| h  1    h  1  *  h  1| h  1 .
(3.5)
(3.6)
(3.7a)
(4.4)
The recursion is initiated, as in Section 4, at h = -1 with given values of
ˆ   0 | 1 and  *  0 | 1 and with
  1  .
(5.28)
It begins at (3.3), runs through to (4.4) with h = -1, then cycles through (3.1) –
(4.4) with h = 0, 1, etc.
The inner recursion, between steps (4.11) and (5.19), is as follows, where
throughout
P  h   G  h  1 1  K  h  C  h   .
(5.11)
For each value of h in the outer recursion, calculate:
Q *  h  1, g   P  h  Q *  h, g  g  0,1,..., h 1
(5.10)
Q *  h  1, h   G  h  1 K  h W *  h 
(5.12)
Q *  h  1, h  1  0
(5.13)
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N *  h  1, g   P  h  N *  h, g 
h1, g V   h   G  h  1 K  h W  h  K T  h  GT  h  1 g  0,1,..., h  1, h  0
(5.9)
where, in the case g = h + 1, N *  h, h  1 , is given by
N *T  h, h  1  P  h   P  h  1  N *  h, h  1   V  *  h  1
T
G  h  K  h  1W *  h  1 K T  h  1 GT  h  
(5.14)
M *  h  1, g   C  h  1 N *  h  1, g  C T  g 
C  h  1 Q *  h  1, g    h 1, gW *  h  1
g = 0, 1, …, h + 1
(5.8)
Calculate R* (h + 1) as the symmetric matrix with (f, g) element
Tr  M *  f , g  M *T  f , g   / Tr M *  f , f  Tr M *  g , g   .
This
inner
recursion
calculates
values
of
(5.20)
Q *  h  1, g  , N *  h  1, g  ,
g  0,1,..., h  1 in terms of Q *  h, g  , N *  h, g  , g  0,1,..., h , all of which are
calculated at previous iterations of the outer recursion.
The first loop of the outer recursion, h = -1, requires the value of R(0), hence
N *  0,0  and Q *  0,0 . These are given by
Q *  0,0  0
(5.13)
N *  0,0   *  0 | 1 .
(5.16)
The second loop of the outer recursion, h = 0, requires the value of R(1), hence
N * 1,0  . This is given by
N * 1,0  N *T  0,1
[by (A.6) and (A.28)]
 P  0   0 | 1 ,
(5.29)
by (5.15).
Note that the inner recursion calculating N *  h  1, g  and Q *  h  1, g  for
g = 0, 1, …, h + 1 calls on just N *  h, g  and Q *  h, g  for g = 0, 1, …, h.
Once the N *  h  1, g  and Q *  h  1, g  have been calculated, and are ready for
use in the next iteration of recursion, the N *  h, g  and Q *  h, g  may be
discarded.
Note also that, at each iteration of the outer recursion, R(h + 1) is constructed by
the augmentation of R(h) with an additional row and column.
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6.
Prediction and prediction error
6.1
Prediction
Predictors are given by Neuhaus (1989, Section 2.3). Define
ˆ   h | k   G  h  ˆ   h  1| k 
(6.1)
whence
ˆ   h | k   G  h, k  2  ˆ   k  1| k 
for h  k  2
(6.2)
with
G  h, k  2  G  h  G  h 1 ...G  k  2 .
(6.3)
Also define
Xˆ  h | k   C  h  ˆ   h | k  .
(6.4)
Then Xˆ  h | k  is the Kalman filter’s unbiased estimate of X(h) based on data up
to and including experience year k.
6.2
Prediction error
The same section of Neuhaus (1989) gives:
  h  1| k   V   h  1  G  h  1   h | k  GT  h  1
(6.5)
where   h | k  denotes MSEP ˆ   h | k   .
This provides a recursion by which   k  2 | k  ,   k  3| k  , etc can be
calculated starting from   k  1| k  given by (4.4).
Also
L  h | k   W  h   C  h    h | k  CT  h 
(6.6)
where L  h | k  denotes MSEP  Xˆ  h | k   .
Equation (6.6) gives MSEP for each future experience year h. It is also possible
to obtain prediction covariances across different future years from Neuhaus
(1989, Section 2.2). Define
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  h1 , h2 | k   E   h1   ˆ   h1 | k    h2   ˆ   h2 | k 
18
T
(6.7)
L  h1 , h2 | k   E  X  h1   Xˆ  h1 | k   X  h2   Xˆ  h2 | k  .
T
(6.8)
Note that
  h, h | k     h | k 
(6.9)
L  h, h | k   L  h | k  .
(6.10)
From Neuhaus,
  h1 , h2 | k   G  h1 , h2  1   h2 | k 
(6.11)
L  h1 , h2 | k   C  h1    h1 , h2 | k  C T  h2 
(6.12)
for h1  h2  k .
Note that   h1 , h2 | k  and L  h1 , h2 | k  are given for the case h2  h1  k by
  h1 , h2 | k   T  h2 , h1 | k 
(6.13)
L  h1 , h2 | k   LT  h2 , h1 | k  .
(6.14)
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7.
19
Numerical example
The present section will illustrate the results of Section 5 by reference to the
same real data set as used in the earlier papers. The data appeared in the form of
incurred losses, adjusted to constant dollar values for inflation, in Table 7.1 of
Paper II. They were converted to logged age-to-age factors in Table 7.2 of the
same paper.
These tables are repeated as Tables 7.1 and 7.2 below.
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Table 7.1
20
Incurred Losses
Period
Incurred losses to end of development year n=
of origin
0
$000
1
$000
2
$000
3
$000
4
$000
5
$000
6
$000
7
$000
8
$000
9
$000
10
$000
11
$000
12
$000
13
$000
14
$000
15
$000
16
$000
17
$000
1978
9,268
18,263 20,182 22,383 22,782 26,348 26,172 26,184 25,455 25,740 25,711 25,452 25,460 25,422 25,386 25,520 25,646 25,469
1979
9,848
16,123 17,099 18,544 20,534 21,554 23,219 22,381 21,584 21,408 20,857 21,163 20,482 19,971 19,958 19,947 19,991
1980
13,990 22,484 24,950 33,255 33,295 34,308 34,022 34,023 33,842 33,933 33,570 31,881 32,203 32,345 32,250 32,168
1981
16,550 28,056 39,995 42,459 42,797 42,755 42,435 42,302 42,095 41,606 40,440 40,432 40,326 40,337 40,096
1982
11,100 31,620 40,852 38,831 39,516 39,870 40,358 40,355 40,116 39,888 39,898 40,147 39,827 40,200
1983
15,677 33,074 35,592 35,721 38,652 39,418 39,223 39,696 37,769 37,894 37,369 37,345 37,075
1984
20,375 33,555 41,756 45,125 47,284 51,710 52,147 51,187 51,950 50,967 51,461 51,382
1985
9,800
1986
11,380 26,843 34,931 37,805 41,277 44,901 45,867 45,404 45,347 44,383
1987
10,226 20,511 26,882 32,326 35,257 40,557 43,753 44,609 44,196
1988
8,170
1989
10,433 19,484 32,103 38,936 45,851 45,133 45,501
1990
9,661
1991
14,275 25,551 33,754 38,674 41,132
1992
13,245 29,206 36,987 44,075
1993
14,711 27,082 34,230
1994
12,476 23,126
1995
9,715
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24,663 36,061 37,927 40,042 40,562 40,362 40,884 40,597 41,304 42,378
18,567 26,472 33,002 36,321 37,047 39,675 40,398
23,808 32,966 42,907 46,930 49,300
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Table 7.2
21
Logged incurred loss age to age factors
Period
of origin
0
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
0.678
0.493
0.474
0.528
1.047
0.747
0.499
0.923
0.858
0.696
0.821
0.625
0.902
0.582
0.791
0.610
0.617
1
0.100
0.059
0.104
0.355
0.256
0.073
0.219
0.380
0.263
0.270
0.355
0.499
0.325
0.278
0.236
0.234
2
0.104
0.081
0.287
0.060
-0.051
0.004
0.078
0.050
0.079
0.184
0.220
0.193
0.264
0.136
0.175
Logged age to age factor from development year n to n+1
development year n=
4
5
6
7
8
9
10
11
3
0.018
0.102
0.001
0.008
0.017
0.079
0.047
0.054
0.088
0.087
0.096
0.163
0.090
0.062
0.145
0.048
0.030
-0.001
0.009
0.020
0.089
0.013
0.084
0.140
0.020
-0.016
0.049
-0.007
0.074
-0.008
-0.008
0.012
-0.005
0.008
-0.005
0.021
0.076
0.069
0.008
0.000
-0.037
0.000
-0.003
-0.000
0.012
-0.019
0.013
-0.010
0.019
0.018
-0.028
-0.036
-0.005
-0.005
-0.006
-0.050
0.015
-0.007
-0.001
-0.009
0.011
-0.008
0.003
-0.012
-0.006
0.003
-0.019
0.017
-0.022
-0.001
-0.026
-0.011
-0.028
0.000
-0.014
0.010
0.026
-0.010
0.015
-0.052
-0.000
0.006
-0.001
-0.002
0.000
-0.033
0.010
-0.003
-0.008
-0.007
12
-0.001
-0.025
0.004
0.000
0.009
13
14
15
16
-0.001 0.005 0.005 -0.007
-0.001 -0.001 0.002
-0.003 -0.003
-0.006
Average 0.699 0.250 0.124 0.065
0.049 0.020 -0.001 -0.013 -0.004 -0.006 -0.006 -0.007 -0.003 -0.003
0.001
0.004 -0.007
Standard
deviation 0.169 0.121 0.095 0.045
0.052 0.033 0.017
0.004
0.002
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0.019 0.013
0.018
0.021
0.014
0.013
0.002
Casualty Actuarial Society
22
As in the example of Section 8.3 of Paper II (see (8.38)), it is assumed that (2.6)
takes the form:
  i, j   1  i  j  1   2  i  j  1
2
3
(7.1)
ie in (2.11),
2
3
U T  j    j  1 ,  j  1  .


(7.2)
Consistently with (8.39) in Paper II, it is assumed that
V  X  i, j  |   i    x  0.8
2j
(7.3)
ie in (2.23)
2
2k
W  k    diag 1,  0.8 ,...,  0.8 


(7.4)
ie
2
2k
W *  k   diag 1,  0.8 ,...,  0.8  .


(7.5)
The evolution of parameters (see (2.13), (2.16) and (2.20)) will be assumed
subject to the following
G i   1
(7.6)
2x 2
V  i   12  1
(7.7)
2x 2
i.e.
V * i  
1
2
1.
(7.8)
2x 2
The following parameters are assumed for  :
  0.5 x  0.19  ,
2
2   0.01 .
2
(7.9)
These parameters, together with (6.4), set the prior for W(k) as representing half
the parameter variances assumed in the example of Paper II (see just after
(8.55)). This allows for the parameter variance of the earlier paper to be only
partly attributable to uncertainty in  in the present case, and also partly
attributable to evolutionary variation.
In common with the same example in the earlier paper, the Kalman filter is
initiated with
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23
 0.97 
  0 | 1  
.
 0.37 
(7.10)
It is assumed that
 *  0 | 1  1 .
(7.11)
2x 2
Table 7.3 gives estimates Xˆ  h  1| h  from (3.3). These are used to develop
estimates Xˆ  h | k  , h > k, from (6.4). Table 7.4 gives the associated standard
2
errors  L jj  h  1| h   , where the subscript indicates the element of the
covariance matrix.
1
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Table 7.3
Filtered logged age to age factors
Period of
Origin
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
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0
1
2
3
4
0.60000
0.65215
0.54568
0.49364
0.50990
0.85428
0.78785
0.59788
0.79870
0.84538
0.75509
0.79913
0.70136
0.84473
0.68892
0.75663
0.66637
0.63755
0.20603
0.18491
0.17317
0.17419
0.24854
0.23444
0.18884
0.21997
0.23262
0.21308
0.22108
0.21059
0.24790
0.22840
0.24864
0.22985
0.22212
0.09327
0.08576
0.08302
0.10139
0.11281
0.09710
0.09156
0.10559
0.10268
0.09980
0.10450
0.11139
0.11775
0.11679
0.11569
0.10953
0.05341
0.04960
0.05477
0.05784
0.05977
0.05438
0.05423
0.05866
0.05867
0.05960
0.06487
0.06727
0.07192
0.06715
0.06673
0.03453
0.03503
0.03530
0.03542
0.03725
0.03586
0.03433
0.03775
0.03932
0.04069
0.04417
0.04578
0.04633
0.04325
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Logged age to age factor from development year n to n+1, filtered on data up
to preceding experience year, for n =
5
6
7
8
9
10
11
12
0.02552
0.02444
0.02375
0.02383
0.02625
0.02454
0.02393
0.02714
0.02843
0.02993
0.03197
0.03175
0.03199
0.01877
0.01760
0.01689
0.01769
0.01898
0.01792
0.01813
0.02032
0.02182
0.02271
0.02340
0.02302
0.01417
0.01308
0.01300
0.01332
0.01437
0.01401
0.01391
0.01611
0.01702
0.01724
0.01759
0.01096
0.01034
0.01011
0.01038
0.01149
0.01090
0.01133
0.01285
0.01328
0.01329
0.00889
0.00826
0.00806
0.00848
0.00907
0.00901
0.00920
0.01026
0.01045
0.00728
0.00670
0.00670
0.00674
0.00758
0.00743
0.00746
0.00824
0.00604
0.00567
0.00536
0.00569
0.00635
0.00610
0.00605
0.00516
0.00462
0.00456
0.00482
0.00528
0.00500
13
0.00433
0.00392
0.00391
0.00405
0.00438
14
15
0.00374 0.00328
0.00340 0.00294
0.00333 0.00279
0.00337
16
17
0.00288 0.00248
0.00250
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Table 7.4
Standard errors of filtered logged age to age factors
Period of
Origin
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
0
1
2
3
0.22159
0.20329
0.19124
0.17740
0.25712
0.23260
0.24359
0.24657
0.23627
0.22825
0.21964
0.21787
0.22633
0.22605
0.22224
0.21743
0.21162
0.21162
0.10199
0.09691
0.09022
0.13110
0.11886
0.12473
0.12647
0.12136
0.11739
0.11307
0.11226
0.11669
0.11661
0.11469
0.11224
0.10927
0.10929
0.07608
0.07074
0.10269
0.09301
0.09752
0.09881
0.09476
0.09160
0.08820
0.08753
0.09096
0.09088
0.08937
0.08745
0.08512
0.08513
0.05635
0.08174
0.07401
0.07756
0.07856
0.07531
0.07279
0.07007
0.06952
0.07224
0.07217
0.07096
0.06943
0.06758
0.06759
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Standard error of filtered logged age to age factor from development year n to n+1,
filtered on data up to preceding experience year
4
5
6
7
8
9
10
11
12
0.06527
0.05908
0.06190
0.06269
0.06009
0.05807
0.05590
0.05546
0.05762
0.05756
0.05660
0.05538
0.05390
0.05390
0.04721
0.04947
0.05009
0.04801
0.04640
0.04466
0.04431
0.04603
0.04598
0.04521
0.04424
0.04306
0.04306
0.03955
0.04004
0.03838
0.03709
0.03570
0.03542
0.03680
0.03676
0.03614
0.03536
0.03442
0.03442
0.03202
0.03069
0.02966
0.02855
0.02832
0.02943
0.02939
0.02890
0.02828
0.02752
0.02752
0.02454
0.02372
0.02283
0.02265
0.02353
0.02351
0.02312
0.02262
0.02201
0.02201
0.01897
0.01826
0.01812
0.01882
0.01880
0.01849
0.01809
0.01761
0.01761
0.01460
0.01449
0.01506
0.01504
0.01479
0.01447
0.01409
0.01409
0.01159
0.01205
0.01203
0.01183
0.01158
0.01127
0.01127
0.00963
0.00963
0.00947
0.00926
0.00902
0.00902
13
0.00770
0.00757
0.00741
0.00722
0.00722
14
15
16
17
0.00606 0.00474
0.00593 0.00462
0.00577 0.00462
0.00578
0.00370
0.00370
0.00296
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Figure 7.1 illustrates the credibility smoothing of estimated  , displaying the
raw estimates from (5.3) and their smoothed versions according to (5.22).
Figure 7.1
Credibility smoothing of theta
0.08
0.07
0.06
0.05
Theta
0.04
0.03
0.02
0.01
0.00
-5
0
5
10
15
20
Experience year
Unsmoothed
Smoothed
Let Y(i) denote the vector of quantities X(i, j) relating to accident year i, ie
Y T  i    X  i, 0  , X  i,1 ,..., X  i, J   .
(7.12)
Define an unbiased estimator Yˆ  i | k  of Y  i  based on data X   k  :
 Xˆ 0  i | i 



 Xˆ 1  i  1| i  1 





Yˆ  i | k    Xˆ k i  k | k 


 Xˆ k i 1  k  1| k  




 Xˆ  i  J | k  
 J

(7.13)
where Xˆ j  h | k  denotes the j-th component of vector Xˆ  h | k  , j = 0, 1, etc.
Note that in (7.13) the first k – i + 2 components are obtained from the Kalman
filter, whereas the last J – k + i – 1 are obtained from (6.2) and (6.4). The first
k  i  1 components are not used in the following.
Adopt the convention in (7.13) that
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27
 Xˆ 0  i | i  1 


ˆ  i  1| i  1 
X

1
Yˆ  i | i  1  
.


 Xˆ  i  J | i  1 


Write   i, k  for the (J + 1)-vector:
1,...,1
 0,..., 0,

T  i, k   

 k  i  1 terms J  k  i terms 
(7.14)
and define
R  i | k   T  i, k  Yˆ  i | k  .
(7.15)
In the present example, this is the logged age-to-ultimate factor for accident year
i when data are available up to and including experience year k.
The values of R(i|k) are displayed in Table 7.5.
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Table 7.5
Filtered logged age to ultimate factors
Period of
Origin
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
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0
1
2
3
4
1.09826
1.11100
0.99314
0.96688
1.08274
1.38609
1.26782
1.14576
1.36463
1.39586
1.33365
1.37866
1.33344
1.44233
1.30705
1.34438
1.25411
1.20673
0.51657
0.47102
0.44436
0.44450
0.61052
0.57923
0.47346
0.53236
0.56284
0.51918
0.53572
0.52152
0.61032
0.57903
0.62910
0.58775
0.56918
0.29701
0.27516
0.26663
0.32224
0.35335
0.30508
0.28672
0.32759
0.31830
0.31014
0.32678
0.35062
0.37271
0.37340
0.36949
0.35129
0.20026
0.18680
0.20632
0.21654
0.22097
0.20090
0.20114
0.21573
0.21567
0.22054
0.24068
0.25138
0.26897
0.25240
0.25041
0.14528
0.14799
0.14904
0.14857
0.15467
0.14940
0.14318
0.15650
0.16346
0.16991
0.18496
0.19243
0.19451
0.18218
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Filtered logged age to ultimate factor from development year n,
filtered on data up to that experience year, for
5
6
7
8
9
10
11
0.11636
0.11171
0.10835
0.10816
0.11847
0.11089
0.10827
0.12248
0.12851
0.13585
0.14534
0.14450
0.14544
0.09004
0.08452
0.08100
0.08457
0.09025
0.08532
0.08652
0.09676
0.10414
0.10864
0.11197
0.11021
0.06973
0.06441
0.06401
0.06537
0.07023
0.06863
0.06823
0.07892
0.08351
0.08469
0.08635
0.05407
0.05105
0.04990
0.05111
0.05643
0.05361
0.05579
0.06323
0.06538
0.06546
0.04300
0.03996
0.03901
0.04095
0.04373
0.04347
0.04446
0.04950
0.05043
0.03369
0.03102
0.03102
0.03113
0.03497
0.03432
0.03451
0.03803
0.02599
0.02441
0.02306
0.02446
0.02728
0.02623
0.02602
12
13
14
15
16
17
0.01997
0.01786
0.01763
0.01865
0.02038
0.01931
0.01438
0.01304
0.01300
0.01345
0.01454
0.00994
0.00906
0.00885
0.00898
0.00619
0.00554
0.00527
0.00288
0.00250
0.00000
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The MSEP of the R(i|k) may also be estimated. By (7.15),
MSEP  R  i | k    T  i, k  MSEP Yˆ  i | k    i, k  .
(7.16)
Because of the positioning of the zeros in   i, k  , only the bottom right
 J  k  i x  J  k  i
sub-matrix of MSEP Yˆ  i | k  is required. A typical
element of this sub-matrix is
E  X  i, h1  i   Xˆ h1 i  h1 | k   X  i, h2  i   Xˆ h2 i  h2 | k  ,
h1 , h2  k  1, k  2,..., i  J
(7.17)
which is equal to the  h1  i, h2  i  element of L  h1 , h2 | k  from (6.8). That is,
MSEPh1 i ,h2 i Yˆ  i | k   Lh1 i ,h2 i  h1 , h2 | k 
(7.18)
for h1 , h2 = k + 1, …, i + J.
Equivalently,
MSEPj1 j2 Yˆ  i | k    L j1 j2  i  j1 , i  j2 | k 
(7.19)
for j1 , j2  k  i  1, k  i  2,..., J .
After substitution of (7.19), equation (7.16) generates a triangle of quantities
MSEP  R  i | k   , whose square roots are set out in Table 7.6.
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Table 7.6
MSEP of filtered logged age to ultimate factors
Period of
Origin
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
d:\219548251.doc
0
1
2
3
4
0.06855
0.09930
0.09608
0.09559
0.12478
0.13698
0.13364
0.13698
0.13197
0.12425
0.11816
0.11534
0.11803
0.11971
0.11710
0.11322
0.10926
0.10750
0.03226
0.03355
0.03683
0.04421
0.04529
0.04537
0.04547
0.04385
0.04206
0.04091
0.04071
0.04134
0.04133
0.04038
0.03926
0.03834
0.03802
0.02027
0.02241
0.02695
0.02661
0.02688
0.02666
0.02553
0.02451
0.02389
0.02387
0.02424
0.02407
0.02346
0.02281
0.02232
0.02220
0.01407
0.01695
0.01646
0.01668
0.01647
0.01571
0.01510
0.01472
0.01473
0.01496
0.01481
0.01442
0.01402
0.01372
0.01367
0.01074
0.01033
0.01050
0.01034
0.00985
0.00946
0.00923
0.00925
0.00939
0.00928
0.00903
0.00878
0.00860
0.00857
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MSEP of filtered logged age to ultimate factor from devleopment year n,
filtered on data up to that experience year, for
5
6
7
8
9
10
11
0.00652
0.00664
0.00653
0.00621
0.00597
0.00583
0.00584
0.00594
0.00586
0.00570
0.00554
0.00543
0.00541
0.00421
0.00413
0.00393
0.00378
0.00369
0.00370
0.00376
0.00371
0.00361
0.00351
0.00344
0.00343
0.00262
0.00249
0.00239
0.00234
0.00235
0.00239
0.00235
0.00229
0.00223
0.00218
0.00218
0.00157
0.00151
0.00148
0.00149
0.00151
0.00149
0.00145
0.00141
0.00138
0.00138
0.00095
0.00093
0.00094
0.00096
0.00094
0.00092
0.00089
0.00087
0.00087
0.00058
0.00059
0.00060
0.00059
0.00057
0.00056
0.00055
0.00055
0.00037
0.00037
0.00037
0.00036
0.00035
0.00034
0.00034
12
13
14
15
0.00023
0.00022
0.00022
0.00021
0.00021
0.00021
0.00013
0.00013
0.00013
0.00012
0.00012
0.00007
0.00007
0.00007
0.00007
0.00004
0.00004
0.00004
16
17
0.00001 0.00000
0.00001
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Tables 7.5 and 7.6 lead to estimates of (unlogged) ultimate incurred losses
through the log normal formula:
incurred losses of accident year i as at end of experience year k
x


exp R  i | k   12 MSEP  R  i | k   .
These estimates appear in Table 7.7.
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(7.20)
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Table 7.7
Estimates of ultimate claims incurred
Period of
Origin
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
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0
1
2
3
4
28,765
31,435
39,629
45,654
34,886
67,141
77,396
33,003
47,581
43,943
32,890
43,872
38,883
64,118
51,894
59,716
46,182
34,265
31,111
26,259
35,716
44,738
59,559
60,380
55,112
42,931
48,128
35,185
32,378
33,508
44,747
46,520
55,875
49,688
41,643
27,438
22,768
33,015
55,941
58,953
48,936
56,335
50,657
48,599
37,097
37,151
46,137
48,420
49,596
54,120
49,179
27,539
22,543
41,213
53,166
48,834
44,014
55,596
47,407
47,251
40,605
42,294
50,426
56,543
50,120
57,005
26,486
23,933
38,850
49,909
46,354
45,093
54,815
47,043
48,835
41,981
43,898
55,824
57,253
49,563
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Ultimate incurred claims, as estimated at end of development year n,
filtered on data up to that experience year, for n=
5
6
7
8
9
10
11
29,697
24,182
38,359
47,787
45,019
44,169
57,791
45,984
51,208
46,591
42,961
52,291
57,173
28,698
25,320
36,965
46,267
44,251
42,796
56,967
44,545
50,993
48,859
44,452
50,890
28,112
23,900
36,316
45,212
43,342
42,567
54,866
44,292
49,414
48,604
44,090
26,890
22,731
35,600
44,335
42,477
39,878
54,970
43,278
48,444
47,218
26,884
22,291
35,299
43,365
41,690
39,595
53,308
43,419
46,699
26,599
21,521
34,638
41,731
41,330
38,684
53,283
44,033
26,127
21,690
32,631
41,440
41,264
38,343
52,746
12
13
14
15
16
17
25,977
20,853
32,779
41,090
40,651
37,801
25,792
20,234
32,771
40,886
40,791
25,640
20,141
32,538
40,459
25,679
20,058
32,338
25,720
20,041
25,469
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Figure 7.2 gives a plot of the evolving distribution of estimated ultimate incurred
losses for accident year 1980. The illustrated distributions are based on the
parameters in Tables 7.5 and 7.6. The figure corresponds to Tables 7.1 and 8.1
in Taylor (1999b).
Figure 7.2
Fig 7.2
Accident year 1980
Probability density
10
8
At
At
At
At
At
6
4
2
end
end
end
end
end
1980
1981
1983
1988
1995
0
10
20
30
40
50
60
70
80
Incurred losses ($000)
Thousands
The different distributions can be identified by their increasing concentration
(and therefore increasing peak height) with increasing development. Thus, the
distribution at the end of 1995, with only two years of development remaining, is
highly concentrated.
Figure 7.2 differs little visually from Figure 8.1 of the previous paper. However,
Figure 7.3 provides a little more detail of the comparison of results derived from
the Kalman filter (present paper) and from the credibility approach (previous
paper).
Figure 7.3
Acc yr 1980
(previous)
$60,000
$55,000
Acc yr 1986
(previous)
$50,000
Acc yr 1990
(previous)
$45,000
$40,000
Acc yr 1980
(present)
$35,000
Acc yr 1986
(present)
Development year
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14
12
10
8
6
4
2
$30,000
0
Predicted ultimate incurred
('000)
Comparison of development with previous paper
Acc yr 1990
(present)
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The graph provides comparisons in respect of accident years 1980, 1986 and
1990. In the earliest of these years, the two approaches produce very similar
results, the Kalman filter being slightly lower at early development but slightly
higher at late development.
By accident year 1986, the Kalman filter is producing materially higher results.
By accident year 1990, the gap has widened further.
These differences result from the dynamic nature of the filter. It is seen from
Table 7.2 that age-to-age factors at all development years up to say 9 show an
increasing trend over most of the experience periods.
While the experience rating nature of the credibility method in the previous
paper causes it to follow the trend, the model is static in its parameters. The
Kalman filter, on the other hand, contains dynamic parameters, and therefore
adapts more rapidly to the changing claims experience.
The Kalman filter is also somewhat tighter in its predictions, as Table 7.8
indicates. Here the coefficients of variation of the final diagonal of incurred
losses (Table 7.7) are calculated from Table 7.6 and compared with the
corresponding c.v.’s from Table 8.4 of the previous paper.
By the log normal assumption in use here, the c.v. associated with any forecast is
estimated as:
c.v.  exp  MSEP   1 2 .
1
(7.21)
Table 7.8
Predictive coefficients of variation
Accident year
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
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Estimated predictive coefficient of variation of ultimate
incurred losses
Kalman filter
Credibility
%
%
0
0
0.3
0.4
0.6
0.7
0.8
1.0
1.1
1.3
1.4
1.7
1.8
2.2
2.3
2.8
3.0
3.6
3.7
4.5
4.7
5.6
5.9
7.0
7.4
8.8
9.3
11.0
11.7
13.9
18.4
17.4
19.7
33.7
Casualty Actuarial Society
8.
35
Acknowledgements
I am grateful to my colleagues Steven Lim and Graham Taylor, who
programmed calculations for the numerical example in Section 7.
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Casualty Actuarial Society
9.
36
References
Kalman, R.E. (1960). A new approach to linear filtering and prediction
problems. Journal of Basic Engineering, 82, 340-345.
Neuhaus, W. (1989). Dynamic linear models and the discrete Kalman filter.
Statistical Memoir No. 1/89 from the University of Oslo.
Taylor, G.C. (1999a). The statistical distribution of incurred losses and its
evolution over time I: non-parametric models. Research Paper No. 72, Centre
for Actuarial Studies, University of Melbourne.
Taylor, G.C. (1999b). The statistical distribution of incurred losses and its
evolution over time II: parametric models. Research Paper No. 73, Centre for
Actuarial Studies, University of Melbourne.
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