Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid

and Incurred Claims

Huijuan Liu

Cass Business SchoolLloyd’s of London

10/07/2007

IBNR – Incurred But Not Reported claims. It is the aggregate claims, denoted as .

IBNER – Incurred But Not Enough Reported claims. It is the developed amount from the existing claims which have already been reported. It is regarded as the incremental old claims, denoted as .

Schnieper(1991) introduces a model that is designed for excess of loss cover in reinsurance business using IBNR claims.

Main approach – separating the aggregate IBNR run-off triangle into two more detailed run-off triangles according to the claims reported time, i.e. the new claims and the old claims.

• The new claims, denoted as , are the claims that first reported in development year j, unknown in development year j-1.

• The old claims (decrease), denoted as , are defined as the decrease in all claims that occurred between development year j and j-1, known in development year j-1.

ijN

ijD

Background

ijX

ijD

Outline

• The Schnieper Separation Approach for the best estimates

• Theoretical Approach to the approximation of the Mean Squared Error of Prediction (MSEP) for the Schnieper model

• Bootstrap and Simulation Approach for estimating the MSEP and the predictive distributions

• A Numerical Example

• Discussion and Further work

The Schnieper’s Separation Approach

Incremental IBNR

Incremental True IBNR

Incremental Decrease IBNER

+

According to when the claim is reported, we can separate the IBNR into the True IBNR and IBNER

New ClaimsDecrease from

Old Claims

1, jiij XX

ijN ijD

Development year j

Accident

year i

Schnieper’s Model Assumptions

, 1ij i j ij ijX X D N

1, jiij XX

ijDijN

jijiij EXNE 1, jjijiij XXDE 1,1,

, , , 1 , 11i j t ij i j t i j t ij j t i j t ij i j tE X X E E X X X E X X E

2

1, 1 1, 1 1

1

ˆ ,..., jj j n j j n j

ii

Var X XE

2

1, 1 1, 1 1

, 11

ˆ ,..., jj j n j j n j

i ji

Var X XX

21, jijiij EXNVar 2

1,1, jjijiij XXDVar Independence between accident years

Motivation for Estimating the MSEP

‘Level 2’Prediction Variance / Variability

‘Level 1’Best Estimate / Mean

‘Level 3’Predictive Distribution

Prediction Variance

Prediction Variance = Process Variance + Estimation Variance

• Process Variance – the variability of the random

variables

• Estimation Variance – the variability of the

fitted model

Process Variance – the squared error of the modelling process. It is straightforward to estimate, given the random variables are identically independent distributed. It is the sum of the process variances of each individual random variable from the underlying distribution.

Estimation Variance – the squared error of the fitted underlying model. It is relatively more complicated to estimate, due to the same parameters involved for the loss claim predictions.

Recursive Approach – to obtain the prediction variances of row totals (or ultimate losses) from different accident years. In the other word, the estimated process variances of the row totals are, recursively, calculated from the latest observed claims data (the leading diagonal) in the run-off triangle.

Process / Estimation Variances of the Row Totals

1 1

ˆn n

in ik in iki i

Var X X Var X

1

1 1 1

ˆ ˆ ˆ2 ,n n n n

in ik in ik sn sp tn tqi i t s t

Var X X Var X Cov X X

where , and . 1k n i 1p n s 1q n t

Therefore, the prediction variance of overall total loss claim is the approximate sum of process variance and estimation variance, which is written as follows,

The above equation is expanded when considering the row totals, i.e. the prediction variance is approximated by the sum over process variances, estimation variances of row totals and all the covariance between any two of them. So we have

1 1

ˆn n

in ik in iki i

Var X X Var X

nX1

Process / Estimation Variances Approach

2nX2, 1nX

1nX 2nX nnX

2

, 1

2

, 1

ˆˆ

ˆ ˆ1

k ti k t ik

k t i k t ik

E X Var

E Var X

, 1ˆ ˆk t i k t ikVar Var X

2 ˆi k tE Var

,ˆi k t ikVar X

,, i k t iki k t ikV Var X X

2

, 11 k t i k t ikV

2 2, 1k t i k ti k t ikX E

Process Variance

Estimation Variance

nX1

2, 1nX

1nX

2ˆnX

2ˆnX ˆ

nnX

Covariance Approach

Correlation = 0

Calculate correlation between estimates

Calculate correlation using

previous correlation

ˆ ˆ,sj sp tj tqCov X X , 1 , 1

, 1 , 1

ˆ ˆ,ˆ

ˆ ˆ

s j sp t j tq

j

s j sp t j tq

Cov X XVar

E X E X

1n s j n

2

, 1 , 1ˆ ˆ ˆ1 ,

ˆ

j s j sp t j tq

s t j

E Cov X X

E EVar

Covariance between Row Totals (i.e. the ultimate losses) – is caused by the same parameter estimates in the row total predictions. It is also estimated recursively.

nX1

1nX ˆnnX2

ˆnX 3

ˆnX , 1

ˆn nX

2ˆnX2, 1nX

3, 2nX 3, 1ˆ

nX 3ˆnX

4, 3nX 4, 2ˆ

nX 4, 1ˆ

nX 4ˆnX

Results

22

, 1

21, 1

ˆ ˆˆ ˆ ˆ ˆ1

ˆ ˆˆ ˆ ˆ ˆ

n n nin ik i n ik

in i ni n tk

X Var Var X

Var Var X E Var

1 , 1 , 1 , 1 , 1

21 1

, 1 , 1

ˆ ˆˆ ˆ ˆ ˆ ˆ,2

ˆ ˆ ˆˆ ˆ ˆ1 ,

n n n s n sp t n tq s n sp t n tq

t s tn s t ns n sp t n tq

Var Cov X X X X

Cov X X E EVar

The prediction variance is estimated as follows,

This looks complicated but is simple to implement in a spreadsheet, due to the recursive approach.

Original Data with size m

Draw randomly with replacement, repeat

n times

Simulate with mean equal to

corresponding Pseudo Data

Pseudo Data with size m

Simulated Data with size m

Estimation Variance

Prediction Variance

Bootstrap

X triangle 1 2 3 4 5 6 7 exposure

1 7.5 28.9 52.6 84.5 80.1 76.9 79.5 10224

2 1.6 14.8 32.1 39.6 55 60 12752

3 13.8 42.4 36.3 53.3 96.5 14875

4 2.9 14 32.5 46.9 17365

5 2.9 9.8 52.7 19410

6 1.9 29.4 17617

7 19.1 18129

Example Schnieper Data

N triangle 1 2 3 4 5 6 7

1 7.5 18.3 28.5 23.4 18.6 0.7 5.1

2 1.6 12.6 18.2 16.1 14 10.6

3 13.8 22.7 4 12.4 12.1

4 2.9 9.7 16.4 11.6

5 2.9 6.9 37.1

6 1.9 27.5

7 19.1

D Triangle 2 3 4 5 6 7

2 -3.1 4.8 -8.5 23 3.9 2.5

3 -0.6 0.9 8.6 -1.4 5.6

4 -5.9 10.1 -4.6 -31.1

5 -1.4 -2.1 -2.8

6 0 -5.8

7 0

Analytical & Bootstrap

Reserves Estimates Prediction errors Prediction errors %

Analytical Bootstrap Analytical Bootstrap Analytical Bootstrap

2 4.4 4.4 9.5 8.8 215% 200%

3 4.8 4.8 14.3 13.5 298% 283%

4 32.5 32.6 29.8 30.8 91% 94%

5 61.6 61.1 41.5 41.6 69% 68%

6 78.6 78.1 44.9 44.0 58% 56%

7 105.4 104.6 51.5 49.8 49% 48%

Total 287.3 285.7 111.1 114.5 39% 40%

-100 0 100 200 300 400 500 600 700

0.001

0.002

0.003

0.004Density

Svar1 N(s=112)

-50 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700

25

50

75

100

125 Svar1

Fig. 1 Empirical Predictive Distribution of Overall Reserves

Empirical Prediction Distribution

• Apply the idea of mixture modelling to other situation, such as paid and incurred data, which may have some practical appeal.

• Bayesian approach can be extended from here.

• To drop the exposure requirement, we can change the Bornheutter-Ferguson model for new claims to a chain-ladder model type.

Further Work

Thank You

For Your Attention!