General Insurance Spring SeminarFriday 21 May 2004, Staple Inn Hall

Reserve Ranges & Stochastic Reserving

1. Background2. Thomas Mack Method3. Multi-Line4. Alternative “Practical” Approach

Ranges in Practice

Percentages Varying Parameters Stochastic

Stochastic Methods: Problems

To match deterministic model BF methodology Incurred claims: redundancy Spurious data Tail Multi-lines/correlation Large losses Latent claims

Stochastic v Deterministic

Not same mid-point:

Use range unchanged Move range Pro-rata adjustment

Thomas Mack Method

Assumptions:

Independence of accident years E(Ci,k+1|Ci1,…Cik) = Cikfk

Var(Ci,k+1|Ci1,…Cik) = Cikk2

Mack Adjustments

Alternative development factors Omit data cells By year to give correct overall result BF Tail

Formulae

Standard error by accident year given by:

kI

jjk

ki

I

iIk k

kiIi

CCfCXes

1

1

12

222 11

)).(.(

2

1

1,2

1

1

kI

jk

jk

kjjkk fC

CC

kI

Formulae: last s.e. Given by:

22

232

3

422

1 ,,min III

II

Formula: Overall Standard Error Standard error overall and adjustment given by:

I

i

I

iIkkI

nnk

kkI

ijjIiIi

C

fCCXesXes

2

1

1

1

22

1

22 /2)).(.()).(.(

I

iiX

Xf

1

2

Formula: Tail Factor Adjustment

Tail factor adjustment given by:

222 )).(.()()).(.( iii YesCRes

BF A Priori Loss Ratio Variability

Choices:

Time Series Rate/Inflation Adjusted

BF as Credibility Formula

Formula for standard error:

))((2)(1

22oommmo

n

mmm spspcsps

Multi-Line

Correlation between classes Use @Risk and Correlation MatrixPaper gives simple alternative: basic copula

Chain-Ladder Practical Alternative Follow deterministic methodStandard errors based on selected/alternative development factorsUse @Risk to complete triangleAllow for correlation where appropriatePart-year triangles/large losses/omit cells etc.BF adjustment as above

Graph of Results

Distribution of Reserves

0

50

100

150

200

35 60 85 110

Amount Millions

Nu

mb

ers