stochastic reserving in general insurance peter england, phd emb younger members’ convention 03...
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Stochastic Reserving in General Insurance
Peter England, PhDEMB
Younger Members’ Convention
03 December 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
P re d ic tive D istrib u tion
V a ria b ility(P re d ic tio n E rro r)
R e se rve e stim a te(M e a su re o f lo ca tio n)
Example
357848 766940 610542 482940 527326 574398 146342 139950 227229 67948 0352118 884021 933894 1183289 445745 320996 527804 266172 425046 94,634 290507 1001799 926219 1016654 750816 146923 495992 280405 469,511 310608 1108250 776189 1562400 272482 352053 206286 709,638 443160 693190 991983 769488 504851 470639 984,889 396132 937085 847498 805037 705960 1,419,459 440832 847631 1131398 1063269 2,177,641 359480 1061648 1443370 3,920,301 376686 986608 4,278,972 344014 4,625,811
18,680,856 3.491 1.747 1.457 1.174 1.104 1.086 1.054 1.077 1.018 1.000
Prediction Errors
Mack's Over-Distribution dispersed Negative
Year Free Poisson Bootstrap Binomial Gamma Log-Normal2 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
10000 14000 18000 22000 26000 30000 34000
Total Reserves
Figure 1. Predictive Aggregate Distribution of Total Reserves
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:
The cumulative claims are defined by:
and the development factors of the chain-ladder technique are denoted by
1
: 1, , 1; 1, ,
: 2, ,
ij
j
ij ikk
j
C j n i i n
D C
j n
Basic Chain-ladder
1
11
, 11
, 2 2 , 1
, , 1
ˆˆ
ˆˆ ˆ 3, ,
n j
iji
j n j
i ji
i n i n j i n i
i j j i j
D
D
D D
D D j n i n
Over-Dispersed Poisson
~ ( )
log
log log
ij ij
ij ij ij ij
ij ij
ij ij ij
C IPoi
C
Var C o C
likelihood C
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
1 2
log
( ) (log )
plus many others
ij i j
i i
i i
η c a b
(t) c a b.t d (t)
(t) c a s t s (t)
(Chain ladder type)
(Hoerl curve)
(Smoother)
Chain-ladder
ijijij
ijij
jiij
Clikelihood
ba
bacη
loglog
log00
1
1
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
In general, remembering that
121
321
n
n
bbb
bbbb
n eee
eeee
01 b
Variability in Claims Reserves
Variability of a forecast Includes estimation variance and process
variance
Problem reduces to estimating the two components
21
variance)estimation variance(processerror prediction
Prediction Variance
22
2
2 2
2 2
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ ˆ2
ˆ ˆ
E y y E y E y y E y
E y E y y E y
E y E y E y E y y E y E y E y
E y E y E y E y
Prediction variance=process variance + estimation variance
Prediction Variance (ODP)
ikijikij
ijijij
ijijij
Cov
VarMSE
VarMSE
),(2
)(
)(
2
2
Individual cell
Row/Overall total
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. Fit chain ladder model2. Obtain Pearson residuals3. Resample residuals4. Obtain pseudo data, given
5. Use chain ladder to re-fit model, and estimate future incremental payments
C
rP
,*Pr
**PrC
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
22
221
2
ˆ)ˆ(ˆ
)ˆˆexp(ˆ
)(
),(~log
ijij
ijijij
ijij
ijij
ijij
Var
m
mC
INC
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 non-constant process variances
Prediction Variance
1)ˆ,ˆ(expˆˆ2
1)ˆexp(ˆ
1)ˆexp(ˆ)(
22
22
ikijikij
ijij
ijijij
Covmm
mMSE
mCMSE
Individual cell
Row/Overall total
Over-Dispersed Negative Binomial
1,j
1,
1 variance
and 1mean
withbinomial, negative ~
jij
jij
ij
D
D
C
Over-Dispersed Negative Binomial
, 1
j , 1
~ negative binomial, with
mean and
variance 1
ij
j i j
j i j
D
D
D
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
, 1
, 1
~ normal, with
mean and
variance
ij
j i j
j i j
D
D
D
Joint modelling
1. Fit 1st stage model to the mean, using arbitrary scale parameters (e.g. =1)
2. Calculate (Pearson) residuals3. Use squared residuals as the response in a
2nd stage model4. Update scale parameters in 1st stage model,
using fitted values from stage 3, and refit5. (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
, 1
2, 1
Specifies first two moments only
has mean and
variance
ij j i j
j i j
D D
D
Mack’s Model
2
1
11
1
, 1, 1
Provides estimators for and
ˆ
and
j j
n j
ij iji
j n j
iji
ijij i j ij
i j
w f
w
Dw D f
D
Mack’s Model
1 2
2
1
212 1
21 1
1
1 ˆˆ
ˆ 1 1ˆ ˆˆ ˆ
n j
j ij ij ji
nk
i in n kk n i ikk
qkq
w fn j
MSEP R DD D
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:
1Estimated Ultimate 1
f
The Bornhuetter-Ferguson Method
Let the initial estimate of ultimate claims for accident year i be
The estimate of outstanding claims for accident year i is
nininiM
32
11
11
3232
nininninin
iM
iM
Comparison with Chain-ladder
replaces the latest cumulative claims for accident year i, to which the usual chain-ladder parameters are applied to obtain the estimate of outstanding claims. For the chain-ladder technique, the estimate of outstanding claims is
nininiM
32
1
1321, ninininiD
Multiplicative Model for Chain-Ladder
1
~ ( )
( )
with 1
is the expected ultimate for origin year
is the proportion paid in development year
ij ij
ij ij ij
n
ij i j kk
i
j
C IPoi
C
E C x y y
x i
y 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:
iiix ,t independen~
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!
“I believe that stochastic modelling is fundamental to our profession. How else can we seriously advise our clients and our wider public on the consequences of managing uncertainty in the different areas in which we work?”
- Chris Daykin, Government Actuary, 1995
“Stochastic models are fundamental to regulatory reform”
- Paul Sharma, FSA, 2002
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.
G e n e r a l I n s u r a n c e A c t u a r i e s & C o n s u l t a n t s