More on Stochastic Reserving in General Insurance GIRO Convention, Killarney, October 2004 Peter England and Richard Verrall

More on Stochastic Reserving in General Insurance

GIRO Convention, Killarney, October 2004

Peter England and Richard Verrall

Overview

The story so far

Bootstrapping recursive models (including Mack's model)

Working with incurred data

Bayesian recursive models

A comparison between bootstrapping and Bayesian methods

Bootstrapping and the Bornhuetter-Ferguson Technique

Including curve fitting for estimating tail factors

Developments in Stochastic Reserving

Background

England, P and Verrall, R (1999), Analytic and bootstrap estimates of prediction errors in claims reserving, Insurance: Mathematics and Economics 25, pp281-293.

England, P (2002), Addendum to “Analytic and bootstrap estimates of prediction errors in claims reserving”, Insurance: Mathematics and Economics 31, pp461-466.

England, PD and Verrall, RJ (2002), Stochastic Claims Reserving in General Insurance, British Actuarial Journal 8, III, pp443-544.

+ many other papers


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 – “Chain Ladder” Structure

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


Over-Dispersed Poisson

)(E)(

)(

log

)(~

ijij

ijij

ijij

ijij

CoCVar

C

IPoiC

jiCij year t developmen and year origin in claims lIncrementa


Example Predictor Structures

)(log)(

log

21 (t)stsac(t)

(t)db.tac(t)

bacη

ii

ii

jiij

Chain Ladder

Hoerl Curve

Smoother


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

ikijikij

ijijij

ijijij

Cov

VarMSE

VarMSE

)(2

)(

)(

2

2

Individual cell

Row/Overall total


EMBLEM DemoODP Chain Ladder with constant scale parameter


Mack’s Model

jiDij year t developmen and year origin in claims Cumulative


Mack, T (1993), Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bulletin, 22, 93-109

1,2

1,

jijij

jijij

DDV

DDE

Specifies first two moments only:

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


ResQ Demo – Mack’s ModelDevelopments in

Stochastic Reserving

Parameter Uncertainty - Bootstrapping

> Bootstrapping is a simple but effective way of obtaining a distribution of parameters

> The method involves creating many new data sets from which the parameters are estimated

> The new data sets are created by sampling with replacement from the observed data (or residuals)

> The model is re-fitted to each new data set

> Results in a (“simulated”) distribution of the parameters


Reserving and Bootstrapping

O b ta in fo re ca st, in clu d ing p roce ss e rro r

O b ta in re sid ua ls a n d p se ud o d a taR e -fit s ta tis t ica l m o de l to p seu d o d a ta

D e fine a n d f it sta tis tica l m o d e l

Any model that can be clearly defined can be bootstrapped

(see the England and Verrall papers for bootstrapping the ODP)


Bootstrapping Mack’s Model



1,

2

1,

1,

ji

j

ji

ij

jji

ij

DD

DV

D

DE

Bootstrapping Mack’s ModelDevelopments in


factorst developmen-pseudo ˆ

residual edstandardis ˆ

residual ˆ

,~

**

2

j

ij

jijij

j

ijijij

ij

ijijijij

ij

jjij

w

rf

fwr

fwr

wNf

Recursive Models: ForecastingDevelopments in


> With recursive models, forecasting proceeds one-step at a time:

Move one-step ahead by multiplying the previous cumulative claims by the appropriate bootstrapped development factor

Include the process error by sampling a single observation from the underlying process distribution, conditional on the mean given by the previous step

Move to the next step

Note that the process error is included at each step before proceeding

Igloo Demo 1 – BootstrappingChain Ladder Model Only


ODP – with constant scale parameter

Bootstrapping Mack’s Model

ODP – with non-constant scale parameters

Negative Binomial Recursive Model

1,

1,

)1(

jijjjij

jijij

DDV

DDE


This is a recursive equivalent to the ODP model


Igloo Demo 2 – BootstrappingNegative Binomial – Chain Ladder Model only


Negative Binomial – with constant scale parameter

Negative Binomial – with non-constant scale parameters

Compare results with ODP and Mack shown earlier:

> ODP and Negative Binomial are very close

> Results with non-constant scale parameters are close to Mack’s method

Bootstrapping Recursive Models: Advantages

> Consistent with traditional deterministic actuarial techniques

Individual points can be weighted out for n-year volume weighted averages, exclude high/low etc

Curve fitting can be incorporated

> Bootstrap version of Mack’s model can be used where negative incrementals are encountered

For example: Incurred claims

> Bootstrapping incurred claims:

Gives distribution of Ultimates and IBNR

Can be combined with Paid to Date to give distribution of Outstanding claims

Must be combined with (simulated) Paid to Incurred ratios to give distributions of payment cash flows


Reserving and Bayesian Methods

O b ta in fo re ca st, in clu d ing p roce ss e rro r

O b ta in d is trib u tion o f p ara m e te rsu s in g G ib bs sa m p ling

D e fine a n d f it sta tis tica l m o d e l

Any model that can be clearly defined can be fitted as a Bayesian model


Excel Demo – Gibbs SamplingODP - Chain Ladder Model Only


Parameter Number

Name ValueStandard

ErrorExpected

ValueStandard

Error

1 Mean 12.506 0.173 12.500 0.165 - Row (1)2 Row (2) 0.331 0.153 0.335 0.1503 Row (3) 0.321 0.158 0.322 0.1554 Row (4) 0.306 0.161 0.307 0.1575 Row (5) 0.219 0.168 0.220 0.1646 Row (6) 0.270 0.171 0.266 0.1707 Row (7) 0.372 0.174 0.369 0.1708 Row (8) 0.553 0.186 0.549 0.1839 Row (9) 0.369 0.239 0.352 0.239

10 Row (10) 0.242 0.428 0.172 0.434 - Col (1)

11 Col (2) 0.913 0.149 0.910 0.15712 Col (3) 0.959 0.153 0.956 0.15913 Col (4) 1.026 0.157 1.024 0.16714 Col (5) 0.435 0.184 0.427 0.19115 Col (6) 0.080 0.214 0.065 0.22216 Col (7) -0.006 0.238 -0.029 0.24417 Col (8) -0.395 0.310 -0.434 0.31618 Col (9) 0.009 0.320 -0.045 0.32819 Col (10) -1.380 0.897 -1.827 1.016

EMBLEM GIBBS

Igloo Demo 3 – Bayesian MethodsODP, Negative Binomial and Mack’s modelComparison with bootstrapping


Bayesian Stochastic Reserving: Advantages

> Overcomes some practical difficulties with bootstrapping

> Sets of pseudo-data are not required, therefore far less RAM hungry when simulating

> Arguably more statistically rigorous and theoretically appealing

> Flexible approach

Informative priors

Bayesian Bornhuetter-Ferguson Method (see latest NAAJ)

Model uncertainty

Individual claims and additional covariates


The Bornhuetter-Ferguson Method and Bootstrapping

> Pseudo-development factors give simulated proportion of ultimate to emerge in each development year

> BF prior loss ratio gives prior Ultimate

> Adjust pseudo-data to take account of BF prior Ultimate and simulated proportions, before forecasting

> Forecast based on adjusted pseudo-data

> BUT, simulate the BF prior ultimate making assumptions about precision of prior

> Add simulated forecasts to historic Paid-to-Date to give distribution of Ultimate.


A Bayesian Bornhuetter-Ferguson Method

> A Bayesian framework is a natural candidate for a stochastic BF method

> Bayesian recursive models offer the best way forward

> Work in progress!

Verrall, RJ (2004), A Bayesian Generalised Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving, NAAJ, July 2004


Bootstrapping and Curve FittingDevelopments in


Igloo Demo 4 – Bootstrapping

Curve fitting and Tail Factors


Summary of Developments

> ODP with non-constant scale parameters

> Bootstrap version of Mack’s model

> Recursive version of ODP: Negative Binomial model

Recursive models allow weighting out of points (exclude Max/Min, n-year volume weighted averages etc)

> Bootstrap version of the Negative Binomial model

> Curve fitting, tail factors and bootstrapping

> Bayesian stochastic reserving

Any clearly defined model (ODP, Mack, NB, curve fitting etc)

> A stochastic Bornhuetter-Ferguson method


Documents

More on Stochastic Reserving in General Insurance GIRO Convention, Killarney, October 2004 Peter England and Richard Verrall