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