An Introduction to Stochastic Reserve Analysis Gerald Kirschner, FCAS, MAAA Deloitte 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

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. Mean 3. Standard deviation

Simulating reserves stochastically using a chain-ladder method Simulated values are shown in red 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

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 3. Convert both actual and fitted triangles to incrementals 4. Look at difference between fitted and actual payments to develop a set of Residuals

Simulating reserves stochastically via bootstrapping 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

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”.

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

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.

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

Pros / Cons of each method Chain-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 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 Renshaw, A.E. & Verrall, R.J. (1998). A stochastic model underlying the chain-ladder technique. B.A.J., 4, pp