An Introduction to Stochastic Reserve Analysis

Gerald Kirschner, FCAS, MAAADeloitte Consulting

Casualty Loss Reserve Seminar

September 2004

Presentation Structure Background Chain-ladder simulation

methodology Bootstrapping simulation

methodology

Arguments against simulation Stochastic models do not work

very well when data is sparse or highly erratic.

Stochastic models overlook trends and patterns in the data that an actuary using traditional methods would be able to pick up and incorporate into the analysis.

Why use simulation in reserve analysis?

Provide more information than traditional point-estimate methods

More rigorous way to develop ranges around a best estimate

Allows the use of simulation-only methods such as bootstrapping

Simulating reserves stochastically using a chain-ladder method

Begin with a traditional loss triangle

Calculate link ratios

Calculate mean and standard deviation of the link ratios

Acc.Year 12 24 36 48

1 1,000 1,500 1,750 2,0002 1,200 2,000 2,3003 1,800 2,5004 2,100

Development Age

Acc.Year 12 - 24 24 - 36 36 - 48

1 1.500 1.167 1.1432 1.667 1.1503 1.389

Mean 1.500 1.157 1.143Std. Deviation 0.1179 0.0082 0

Link Ratios

Simulating reserves stochastically using a chain-ladder method Think of the observed link ratios for

each development period as coming from an underlying distribution with mean and standard deviation as calculated on the previous slide

Make an assumption about the shape of the underlying distribution – easiest assumptions are Lognormal or Normal

Simulating reserves stochastically using a chain-ladder method

For each link ratio that is needed to square the original triangle, pull a value at random from the distribution described by

1. Shape assumption (i.e. Lognormal or Normal)

2. Mean3. Standard deviation

Acc.Year 12 - 24 24 - 36 36 - 48

1 1.500 1.167 1.1432 1.667 1.1503 1.389

Mean 1.500 1.157 1.143Std. Deviation 0.1179 0.0082 0

Acc.Year 12 - 24 24 - 36 36 - 48

1 1.500 1.167 1.143

2 1.667 1.150 1.143

3 1.389 1.163 1.1434 1.419 1.145 1.143

Link Ratios

Link Ratios

Simulating reserves stochastically using a chain-ladder method

Simulated values are shown in red

Lognormal Distribution, mean 1.5, standard deviation 0.1179

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

1.168 1.249 1.330 1.411 1.492 1.573 1.654 1.735 1.816 1.897

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Random draw

Simulating reserves stochastically using a chain-ladder method Square the triangle using the

simulated link ratios to project one possible set of ultimate accident year values. Sum the accident year results to get a total reserve indication.

Repeat 1,000 or 5,000 or 10,000 times.

Result is a range of outcomes.

Enhancements to this methodology Options for enhancing this basic

approach Logarithmic transformation of link

ratios before fitting, as described in Feldblum et al 1999 paper

Inclusion of a parameter risk adjustment as described in Feldblum, based on Rodney Kreps 1997 paper “Parameter Uncertainty in (Log)Normal distributions”

Simulating reserves stochastically via bootstrapping

Bootstrapping is a different way of arriving at the same place

Bootstrapping does not care about the underlying distribution – instead bootstrapping assumes that the historical observations contain sufficient variability in their own right to help us predict the future

Actual Cumulative Historical Data

Acc.Year 12 24 36 48

1 1,000 1,500 1,750 2,0002 1,200 2,000 2,3003 1,800 2,5004 2,100

Ave Link Ratio 1.500 1.157 1.143

Recast Cumulative Historical Data

Acc.Year 12 24 36 48

1 1,008 1,512 1,750 2,0002 1,325 1,988 2,3003 1,667 2,5004 2,100

Development Age

Development Age

Simulating reserves stochastically via bootstrapping

1. Keep current diagonal intact

2. Apply average link ratios to “back-cast” a series of fitted historical payments

Ex: 1,988 =

2,300

Simulating reserves stochastically via bootstrapping

Actual Incremental Historical Data

Acc.Year 12 24 36 48

1 1,000 500 250 2502 1,200 800 3003 1,800 7004 2,100

Recast Cumulative Historical Data

Acc.Year 12 24 36 48

1 1,008 504 238 2502 1,325 663 3123 1,667 8334 2,100

Development Age

Development Age

3. Convert both actual and fitted triangles to incrementals

4. Look at difference between fitted and actual payments to develop a set of ResidualsResiduals

Acc.Year 12 24 36 48

1 (0.259) (0.183) 0.801 0.0002 (3.437) 5.340 (0.699)3 3.266 (4.619)4 0.000

Development Age

Simulating reserves stochastically via bootstrapping

pnn

5. Adjust the residuals to include the effect of the number of degrees of freedom.

6. DF adjustment =

where n = # data points and p = # parameters to be estimated

Residuals adjusted for # degrees of freedom= Residual * [n / (n-p) ]^0.5n = # data pointsp = # Parameters to be estimated = (2 * number of AY) - 1

Acc.Year 12 24 36 48

1 (0.473) (0.335) 1.462 0.0002 (6.275) 9.749 (1.275)3 5.963 (8.433)4 0.000

n 10p 7DF 1.82574

Development Age

Simulating reserves stochastically via bootstrapping

7. Create a “false history” by making random draws, with replacement, from the triangle of adjusted residuals. Combine the random draws with the recast historical data to come up with the “false history”.

Random Draw from Residuals

Acc.Year 12 24 36 48

1 1.462 (0.335) 5.963 1.4622 9.749 (8.433) (0.473)3 (1.275) (6.275)4 9.749

False History= [residual * (fitted incremental ^ 0.5)] + fitted incremental

Acc.Year 12 24 36 48

1 1,055 497 330 2732 1,680 445 3043 1,615 6524 2,547

Development Age

Development Age

Simulating reserves stochastically via bootstrapping

8. Calculate link ratios from the data in the cumulated false history triangle

9. Use the link ratios to square the false history data triangle

Cumulated False History

Acc.Year 12 24 36 48

1 1,055 1,551 1,881 2,1542 1,680 2,125 2,4293 1,615 2,2674 2,547

Ave Link Ratio 1.367 1.172 1.145

Squaring of the Cumulated False History

Acc.Year 12 24 36 48

1 1,055 1,551 1,881 2,1542 1,680 2,125 2,429 2,7823 1,615 1,615 1,893 2,1684 2,547 3,480 4,080 4,673

Development Age

Development Age

Simulating reserves stochastically via bootstrapping Could stop here – this would give N

different possible reserve indications. Could then calculate the standard

deviation of these observations to see how variable they are – BUT this would only reflect estimation variance, not process variance.

Need a few more steps to finish incorporating process variance into the analysis.

Simulating reserves stochastically via bootstrapping

10. Calculate the scale parameter Φ.

Incorporate Process Variance in the modelCalculate scale parameter Φ = Pearson chi-squared statistic / # degrees of freedom

Pearson χ2 = sum of the squares of the unscaled Pearson residualsDF = # data points / # parameters to be estimated

Acc.Year 12 24 36 48

1 0.067 0.034 0.641 0.0002 11.811 28.514 0.4883 10.667 21.3334 0.000

Φ = 40.2879

Development Age

Simulating reserves stochastically via bootstrapping

11. Draw a random observation from the underlying process distribution, conditional on the bootstrapped values that were just calculated.

12. Reserve = sum of the random draws

Calculate Incremental Future PaymentsAcc.Year 12 24 36 48

12 3533 278 2754 934 600 592

Pull random draws from a series of Gamma distributionsmean = incremental future payment from the previous stepvariance = Φ * mean

Acc.Year 12 24 36 48

12 3133 213 2804 1,047 597 501

RESERVE = sum of random draws = 2,951

Development Age

Development Age

Pros / Cons of each methodChain-ladder Pros More flexible - not

limited by observed data

Chain-ladder Cons

More assumptions Potential

problems with negative values

Bootstrap Pros Do not need to

make assumptions about underlying distribution

Bootstrap Cons Variability limited

to that which is in the historical data

Selected References for Additional Reading England, P.D. & Verrall, R.J. (1999). Analytic and bootstrap

estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, 25, pp. 281-293.

England, P.D. (2001). Addendum to ‘Analytic and bootstrap estimates of prediction errors in claims reserving’. Actuarial Research Paper # 138, Department of Actuarial Science and Statistics, City University, London EC1V 0HB.

Feldblum, S., Hodes, D.M., & Blumsohn, G. (1999). Workers’ compensation reserve uncertainty. Proceedings of the Casualty Actuarial Society, Volume LXXXVI, pp. 263-392.

Renshaw, A.E. & Verrall, R.J. (1998). A stochastic model underlying the chain-ladder technique. B.A.J., 4, pp. 903-923.