Transcript Stochastic Reserving Methods

Stochastic Reserving in
General Insurance
Peter England, PhD
EMB
GIRO 2002
Aims
• To provide an overview of stochastic
reserving models, using England and
Verrall (2002, BAJ) as a basis.
• To demonstrate some of the models
in practice, and discuss practical
issues
Why Stochastic Reserving?
• Computer power and statistical methodology
make it possible
• Provides measures of variability as well as
location (changes emphasis on best estimate)
• Can provide a predictive distribution
• Allows diagnostic checks (residual plots etc)
• Useful in DFA analysis
• Useful in satisfying FSA Financial Strength
proposals
Actuarial Certification
• An actuary is required to sign that the
reserves are “at least as large as those
implied by a ‘best estimate’ basis
without precautionary margins”
• The term ‘best estimate’ is intended to
represent “the expected value of the
distribution of possible outcomes of the
unpaid liabilities”
Conceptual Framework
Reserve estimate
(Measure of location)
Variability
(Prediction Error)
Predictive Distribution
Example
357848 766940 610542
352118 884021 933894
290507 1001799 926219
310608 1108250 776189
443160 693190 991983
396132 937085 847498
440832 847631 1131398
359480 1061648 1443370
376686 986608
344014
3.491
1.747
1.457
482940
1183289
1016654
1562400
769488
805037
1063269
527326
445745
750816
272482
504851
705960
574398
320996
146923
352053
470639
146342
527804
495992
206286
139950
266172
280405
227229
425046
67948
1.174
1.104
1.086
1.054
1.077
1.018
1.000
0
94,634
469,511
709,638
984,889
1,419,459
2,177,641
3,920,301
4,278,972
4,625,811
18,680,856
Prediction Errors
Mack's
OverDistribution dispersed
Negative
Year
Free
Poisson Bootstrap Binomial Gamma Log-Normal
2
80
116
117
116
48
54
3
26
46
46
46
36
39
4
19
37
36
36
29
32
5
27
31
31
30
26
28
6
29
26
26
26
24
26
7
26
23
23
22
24
26
8
22
20
20
19
26
28
9
23
24
24
23
29
31
10
29
43
43
41
37
41
Total
13
16
16
15
15
16
Figure 1. Predictive Aggregate Distribution of Total Reserves
10000
14000
18000
22000
26000
Total Reserves
30000
34000
Stochastic Reserving Model Types
• “Non-recursive”
– Over-dispersed Poisson
– Log-normal
– Gamma
• “Recursive”
– Negative Binomial
– Normal approximation to Negative
Binomial
– Mack’s model
Stochastic Reserving Model Types
• Chain ladder “type”
– Models which reproduce the chain ladder results
exactly
– Models which have a similar structure, but do not
give exactly the same results
• Extensions to the chain ladder
– Extrapolation into the tail
– Smoothing
– Calendar year/inflation effects
• Models which reproduce chain ladder results
are a good place to start
Definitions
Assume that the data consist of a triangle of incremental
claims:
C
ij
: j  1,
, n  i  1; i  1,
, n
The cumulative claims are defined by:
j
Dij   Cik
k 1
and the development factors of the chain-ladder technique are
denoted by

j
: j  2,
, n
Basic Chain-ladder
n  j 1
j 
D
i 1
n  j 1
ij
D
i 1
i , j 1
Dˆ i ,n i  2  ˆn  j  2 Di ,n i 1
Dˆ i , j  ˆ j Dˆ i , j 1
 j  n  i  3,
, n
Over-Dispersed Poisson
Cij ~ IPoi ( ij )
log ij  ij
 Cij   ij
Var Cij   o  Cij 
log likelihood    Cij log ij  ij 
What does Over-Dispersed
Poisson mean?
• Relax strict assumption that variance=mean
• Key assumption is variance is proportional
to the mean
• Data do not have to be positive integers
• Quasi-likelihood has same form as Poisson
likelihood up to multiplicative constant
Predictor Structures
ηij  c  ai  b j
(Chain ladder type)
i (t)  c  ai  b.t  d log(t)
(Hoerl curve)
i (t)  c  ai  s1 (t )  s2 (log(t))
(Smoother)
plus many others
Chain-ladder
ηij  c  ai  b j
a1  0
b1  0
log  ij  ij
log likelihood  Cij log  ij   ij
Other constraints are possible, but this is usually the easiest.
This model gives exactly the same reserve estimates as the
chain ladder technique.
Excel
•
•
•
•
•
Input data
Create parameters with initial values
Calculate Linear Predictor
Calculate mean
Calculate log-likelihood for each point in the
triangle
• Add up to get log-likelihood
• Maximise using Solver Add-in
Recovering the link ratios
Incrementals
e
e
c
c  a2
e
e
c  a3
c  a4
e
e
c  b2
c  a2  b2
e
c  a3  b2
c  b3
e
e
c  a 2  b3
e
c  b4
Recovering the link ratios
Calculate ratios of cumulatives, which are the
same for each row. Eg row 2:
Column 2 to Column 1:
e
c  a2
c  a2  b2
e
c  a2
e
1 e

1
b2
Column 3 to Column 2:
e
c  a2
c  a2  b2
c  a2  b3
e
e
c  a2
c  a2  b2
e
e
1 e  e

b2
1 e
b2
b3
Recovering the link ratios
In general, remembering that
b1  0
e  e  e  e
n  b1 b2
bn1
e  e  e
b1
b2
b3
bn
Variability in Claims Reserves
• Variability of a forecast
• Includes estimation variance and process
variance
prediction error  (process variance  estimation variance)
• Problem reduces to estimating the two
components
1
2
Prediction Variance


2
2



E  y  yˆ   E  y  E  y    yˆ  E  y  






2

 E  y  E  y    yˆ  E  yˆ  


2
2





ˆ
ˆ
ˆ
ˆ
 E  y  E  y    2 E  y  E  y   y  E  y    E  y  E  y  




2
2



 E  y  E  y    E  yˆ  E  yˆ  




Prediction variance=process variance + estimation variance
Prediction Variance (ODP)
Individual cell
MSE  ij   ij Var (ij )
2
Row/Overall total
MSE   ij    ij Var (ij )
2
 2 Cov(ij ,ik )  ij  ik
Bootstrapping
• Used where standard errors are difficult
to obtain analytically
• Can be implemented in a spreadsheet
• England & Verrall (BAJ, 2002) method
gives results analogous to ODP
• When supplemented by simulating
process variance, gives full distribution
Bootstrapping - Method
• Re-sampling (with replacement) from
data to create new sample
• Calculate measure of interest
• Repeat a large number of times
• Take standard deviation of results
• Common to bootstrap residuals in
regression type models
Bootstrapping the Chain Ladder
(simplified)
1.
2.
3.
4.
Fit chain ladder model
Obtain Pearson residuals
Resample residuals
Obtain pseudo data, given
C r
*
*
P
rP 
C

r ,
*
P
 
5. Use chain ladder to re-fit model, and
estimate future incremental payments
Bootstrapping the Chain Ladder
6. Simulate observation from process
distribution assuming mean is
incremental value obtained at Step 5
7. Repeat many times, storing the
reserve estimates, giving a predictive
distribution
8. Prediction error is then standard
deviation of results
Log Normal Models
• Log the incremental claims and use a
normal distribution
• Easy to do, as long as incrementals are
positive
• Deriving fitted values, predictions, etc is
not as straightforward as ODP
Log Normal Models
log Cij ~ IN (  ij ,  )
2
 ij  ij
(Cij )  mij
ˆ ij  exp( ˆ ij  ˆ )
m
2
ij
1
2
ˆ  Var (ˆij )  ˆ
2
ij
2
Log Normal Models
• Same range of predictor structures
available as before
• Note component of variance in the
mean on the untransformed scale
• Can be generalised to include nonconstant process variances
Prediction Variance
Individual cell


ˆ exp(ˆ )  1
MSE (Cij )  m
2
ij
2
ij
Row/Overall total


ˆ ij2 exp(ˆ ij2 )  1
MSE   m


ˆ ij m
ˆ ik exp Cov(ˆij ,ˆik )   1
 2 m
Over-Dispersed Negative
Binomial
Cij ~ negative binomial, with
mean  j  1Di , j 1 and
variance j  j  1Di , j 1
Over-Dispersed Negative
Binomial
Dij ~ negative binomial, with
mean  j Di , j 1 and
variance  j   j  1 Di , j 1
Derivation of Negative
Binomial Model from ODP
• See Verrall (IME, 2000)
• Estimate Row Parameters first
• Reformulate the ODP model, allowing
for fact that Row Parameters have been
estimated
• This gives the Negative Binomial model,
where the Row Parameters no longer
appear
Prediction Errors
Prediction variance = process variance +
estimation variance
Estimation variance is larger for ODP than NB
but
Process variance is larger for NB than ODP
End result is the same
Estimation variance and
process variance
• This is now formulated as a recursive
model
• We require recursive procedures to
obtain the estimation variance and
process variance
• See Appendices 1&2 of England and
Verrall (BAJ, 2002) for details
Normal Approximation to
Negative Binomial
Dij ~ normal, with
mean  j Di , j 1 and
variance  j Di , j 1
Joint modelling
1. Fit 1st stage model to the mean, using
arbitrary scale parameters (e.g. =1)
2. Calculate (Pearson) residuals
3. Use squared residuals as the response in a
2nd stage model
4. Update scale parameters in 1st stage model,
using fitted values from stage 3, and refit
5. (Iterate for non-Normal error distributions)
Estimation variance and
process variance
• This is also formulated as a recursive
method
• We require recursive procedures to
obtain the estimation variance and
process variance
• See Appendices 1&2 of England and
Verrall (BAJ, 2002) for details
Mack’s Model
Specifies first two moments only
Dij has mean  j Di , j 1 and
variance  Di , j 1
2
j
Mack’s Model
Provides estimators for  j and  2j
n  j 1
ˆ j 
w
i 1
n  j 1
f
ij ij
w
i 1
ij
wij  Di , j 1 and f ij 
Dij
Di , j 1
Mack’s Model
n  j 1
1
2
ˆ
ˆ
j 
w
f



ij
ij
j
n  j i 1


2

2 
n 1
ˆ k 1  1

1
2
ˆ
ˆ
MSEP  Ri   Din  2
 nk
ˆ
ˆ
k  n i 1 k 1  Dik
Dqk


q 1







Comparison
• The Over-dispersed Poisson and
Negative Binomial models are different
representations of the same thing
• The Normal approximation to the
Negative Binomial and Mack’s model
are essentially the same
The Bornhuetter-Ferguson Method
• Useful when the data are unstable
• First get an initial estimate of ultimate
• Estimate chain-ladder development
factors
• Apply these to the initial estimate of
ultimate to get an estimate of
outstanding claims
Estimates of outstanding
claims
To estimate ultimate claims using the chain ladder technique, you
would multiply the latest cumulative claims in each row by f, a
product of development factors .
Hence, an estimate of what the latest cumulative claims should be is
obtained by dividing the estimate of ultimate by f. Subtracting this
from the estimate of ultimate gives an estimate of outstanding
claims:

1
Estimated Ultimate  1  
f 

The Bornhuetter-Ferguson Method
Let the initial estimate of ultimate claims for
accident year i be M i
The estimate of outstanding claims for accident
year i is


1

M i 1 
n i  2 n i 3  n 

 Mi
1
n  i  2 n  i  3  n
ni  2ni 3  n  1
Comparison with Chain-ladder
Mi
1
n  i  2 n  i  3  n
replaces the latest cumulative
claims for accident year i, to which the usual chainladder parameters are applied to obtain the estimate of
outstanding claims. For the chain-ladder technique, the
estimate of outstanding claims is
Di ,n i 1 n i  2 n i 3 n  1
Multiplicative Model for
Chain-Ladder
Cij ~ IPoi ( ij )
(Cij )  ij  ij
E  Cij   xi y j
n
with
y
k 1
k
1
xi is the expected ultimate for origin year i
y j is the proportion paid in development year j
BF as a Bayesian Model
Put a prior distribution on the row parameters.
The Bornhuetter-Ferguson method assumes there
is prior knowledge about these parameters, and
therefore uses a Bayesian approach. The prior
information could be summarised as the following
prior distributions for the row parameters:
xi ~ independen t i ,  i 
BF as a Bayesian Model
• Using a perfect prior (very small
variance) gives results analogous to the
BF method
• Using a vague prior (very large
variance) gives results analogous to the
standard chain ladder model
• In a Bayesian context, uncertainty
associated with a BF prior can be
incorporated
Stochastic Reserving and
Bayesian Modelling
• Other reserving models can be fitted in a
Bayesian framework
• When fitted using simulation methods, a
predictive distribution of reserves is
automatically obtained, taking account of
process and estimation error
• This is very powerful, and obviates the need
to calculate prediction errors analytically
Limitations
• Like traditional methods, different stochastic
methods will give different results
• Stochastic models will not be suitable for all
data sets
• The model results rely on underlying
assumptions
• If a considerable level of judgement is
required, stochastic methods are unlikely to
be suitable
• All models are wrong, but some are useful!
References
England, PD and Verrall, RJ (2002) Stochastic Claims
Reserving in General Insurance, British Actuarial Journal
Volume 8 Part II (to appear).
Verrall, RJ (2000) An investigation into stochastic claims
reserving models and the chain ladder technique,
Insurance: Mathematics and Economics, 26, 91-99.
Also see list of references in the first paper.