1. A Stochastic Framework for Incremental Average Reserve Models Presented by Roger M. Hayne, PhD., FCAS, MAAA Casualty Loss Reserve Seminar 18-19 September 2008 Washington, DC Page based on Title Slide from Slide Layout palette. Design is 2_Title with graphic. Title text for Title or Divider pages should be 36 pt titles/28 pt for subtitles. PRESENTER box text should be 22pt. DATE text box is not on master and can be deleted. The date should always be 18 pts.

2. 2February 18, 2016 Reserves in a Stochastic World  At a point in time (valuation date) there is a range of possible outcomes for a book of (insurance) liabilities. Some possible outcomes may be more likely than others  Range of possible outcomes along with their corresponding probabilities are the distribution of outcomes for the book of liabilities – i.e. reserves are a distribution  The distribution of outcomes may be complex and not completely understood  Uncertainty in predicting outcomes comes from – Process (pure randomness) – Parameters (model parameters uncertain) – Model (selected model is not perfectly correct) Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

3. 3February 18, 2016 Stochastic Models  In the actuarial context a stochastic could be considered as a mathematical simplification of an underlying loss process with an explicit statement of underlying probabilities  Two main features – Simplified Statement – Explicit probabilistic statement  In terms of sources of uncertainty two of three sources may be addressed – Process – Parameter  Within a single model, the third source (model uncertainty) usually not explicitly addressed Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

4. 4February 18, 2016 Basic Traditional Actuarial Methods  Traditional actuarial methods are simplifications of reality – Chain ladder – Bornhuetter-Ferguson – Berquist-Sherman Incremental Average – Others  Usually quite simple thereby “easy” to explain  Traditional reserve approaches rely on a number of methods  Practitioner “selects” an “estimate” based on results of several traditional methods  No explicit probabilistic component Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

5. 5February 18, 2016 Traditional Chain Ladder  If C ij denotes incremental amount (payment) for exposure year i at development age j  Deterministic chain ladder Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.  Parameters f j usually estimated from historical data, looking at link ratios (cumulative paid at one age divided by amount at prior age)  Forecast for an exposure year completely dependent on amount to date for that year so notoriously volatile for least mature exposure period

6. 6February 18, 2016 Traditional Bornhuetter-Ferguson  Attempts to overcome volatility by considering an additive model  Deterministic Bornhuetter-Ferguson Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.  Parameters f j usually estimated from historical data, looking at link ratios  Parameters e i, expected losses, usually determined externally from development data but “Cape Cod” (Stanard/Buhlmann) variant estimates these from data  Exposure year amount not completely dependent on to-date number

7. 7February 18, 2016 Traditional Berquist-Sherman Incremental  Attempts to overcome volatility by considering an additive model  Deterministic Berquist-Sherman incremental severity Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.  Parameters E i exposure measure, often forecast ultimate claims or earned exposures  Parameters α j and τ j usually estimated from historical data, looking at incremental averages  Berquist & Sherman has several means to derive those estimates  Often simplified to have all τ j equal

8. 8February 18, 2016 A Stochastic Incremental Model  Instead of incremental paid, consider incremental average A ij = C ij /E ij  First step translating to stochastic, have expected values agree with simplified Berquist-Sherman incremental average Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.  Observation – the amounts are averages of a (large?) sample, assumed from the same population  Law of large numbers would imply, if variance is finite, that distribution of the average is asymptotically normal  Thus assume the averages have Gaussian distributions (next step in stochastic framework)

9. 9February 18, 2016 A Stochastic Incremental Model – Cont.  Now that we have an assumption about the distribution (Gaussian) and expected value all needed to specify the model is the variance in each cell  In stochastic chain ladder frameworks the variance is assumed to be a fixed (known) power of the mean Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.  We will follow this general structure, however allowing the averages to be negative and the power to be a parameter fit from the data, reflecting the sample size for the various sums

10. 10February 18, 2016 Parameter Estimation  Number of approaches possible  If we have an a-priori estimate of the distribution of the parameters we could use Bayes Theorem to refine that estimates given the data  Maximum likelihood is another approach  In this case the negative log likelihood function of the observations given a set of parameters is given by Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

11. 11February 18, 2016 Distribution of Outcomes Under Model  Since we assume incremental averages are independent once we have the parameter estimates we have estimate of the distribution of outcomes given the parameters Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.  This is the estimate for the average future forecast payment per unit of exposure, multiplying by exposures and adding by exposure year gives a distribution of aggregate future payments  This assumes parameter estimates are correct – does not account for parameter uncertainty

12. 12February 18, 2016 Parameter Uncertainty  Some properties of maximum likelihood estimators – Asymptotically unbiased – Asymptotically efficient – Asymptotically normal  We implicitly used the first property in the distribution of future payments under the model  Define the Fisher information matrix as the expected value of the Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function  The variance-covariance matrix of the limiting Gaussian distribution is the inverse of the Fisher information matrix typically evaluated at the parameter estimates Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

13. 13February 18, 2016 Incorporating Parameter Uncertainty  If we assume – The parameters have a multi-variate Gaussian distribution with mean equal to the maximum likelihood estimators and variance- covariance matrix equal to the inverse of the Fisher information matrix – For a fixed parameters the losses have a Gaussian distribution with the mean and variance the given functions of the parameters  The posterior distribution of outcomes is rather complex  Can be easily simulated: – First randomly select parameters from a multi-variate Gaussian Distribution – For these parameters simulate losses from the appropriate Gaussian distributions Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

14. 14February 18, 2016 Berquist-Sherman Average Paid Data Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo. AccidentMonths of DevelopmentForecast Year1224364860728496Counts 1969178.73361.03283.69264.00137.9461.4915.478.827,822 1970196.56393.24314.62266.89132.4649.5733.668,674 1971194.77425.13342.91269.45131.6666.739,950 1972226.11509.39403.20289.89158.939,690 1973263.09559.85422.42347.769,590 1974286.81633.67586.687,810 1975329.96804.758,092 1976368.847,594 Estimates α1α1 α2α2 α3α3 α4α4 α5α5 α6α6 α7α7 α8α8 Parameter143.78316.77251.78197.68102.5346.2321.367.36 Std. Error6.211.549.167.625.253.753.072.41 κτp Parameter8.58711.12650.5782 Std. Error0.23210.00770.0303

15. 15February 18, 2016 Forecast Average Expected Values Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo. AccidentMonths of Development Year24364860728496Total 1969 19709.34 197130.5410.5241.06 197274.4334.4011.85120.68 1973185.9683.8438.7513.34321.9 1974403.89209.4894.4543.6515.03766.5 1975579.48454.96235.97106.3949.1716.931,442.91 1976821.26652.77512.5265.81119.8455.3919.072,446.64

16. 16February 18, 2016 Forecast Average Variances Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo. AccidentMonths of Development Year24364860728496Total 1969 19708.19 197128.108.1936.29 197280.8433.119.65123.60 1973235.5193..7438.4011.19378.84 1974709.12331.88132.1051.1115.771,242.97 19751,039.02785.45367.61146.3259.9317.472,415.80 19761,657.071,270.62960.54449.55178.9373.2921.364,611.37

17. 17February 18, 2016 Example Accident Year Results Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo. Process OnlyIncluding Parameter Uncertainty AccidentStandard Percentile YearMeanDeviationMeanDeviation5%95% 1969000000 197080,98126,50380,55136,44224,148144,035 1971408,50063,754407,01982,070274,928545,616 19721,169,365106,4481,169,765137,850945,6621,399,015 19733,087,023172,0603,086,394233,7092,702,4573,476,160 19745,986,335216,2255,984,922344,2125,425,0056,551,203 197511,676,044307,38011,671,230549,68510,783,70512,583,860 197618,579,788375,62618,581,701808,46517,258,89819,916,569 Total40,988,036572,74240,981,5811,513,55738,528,69643,485,373

18. 18February 18, 2016 Example Next Year Results Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo. Process OnlyIncluding Parameter Uncertainty AccidentStandard Percentile YearMeanDeviationMeanDeviation5%95% 1969000000 197080,98124,81780,55136,44224,148144,035 1971303,85952,742302,55368,934192,431418,164 1972721,23087,122721,793105,826551,032898,662 19731,783,372147,1711,783,236172,9671,502,2862,075,631 19743,154,365207,9743,154,597240,8342,764,6843,559,245 19754,689,180260,8364,686,348309,9094,179,6445,204,351 19766,236,615309,1306,236,267372,6675,629,2616,854,599 Total16,969,602489,38416,965,345652,96815,893,88918,045,385

19. 19February 18, 2016 Distribution of Outcomes from Model Page based on Title Only from Slide Layout palette. Design is 01_Title with photo. Subtitles are Part of Title Field, then Modified Manually (see next page)

20. 20February 18, 2016 Some Observations  The data imply that the variance for payments in a cell are roughly proportional to the square root of the mean in this case, much lower than the powers of 1 and 2 usually used in stochastic chain ladder models  The variance implied by the estimators for the aggregate future payment forecast is 573K  Incorporating parameter risk gives a total variance of outcomes within this model is 1,514K  Obviously process uncertainty is much less important than parameter  MODEL UNCERTAINTY IS NOT ADDRESSED HERE AT ALL Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

21. 21February 18, 2016 More Observations  We chose a relatively simple model for the expected value  Nothing in this approach makes special use of the structure of the model  Model does not need to be linear nor does it need to be transformed to linear by a function with particular properties  Variance structure is selected to parallel stochastic chain ladder approaches (overdispersed Poisson, etc.) and allow the data to select the power  The general approach is also applicable to a wide range of models  This allows us to consider a richer collection of models than simply those that are linear or linearizable Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.

22. 22February 18, 2016 Some Cautions  MODEL UNCERATINTY IS NOT CONSIDERED thus distributions are distributions of outcomes under a specific model and must not be confused with the actual distribution of outcomes for the loss process  An evolutionary Bayesian approach can help address model uncertainty – Apply a collection of models and judgmentally weight (a subjective prior) – Observe results for next year and reweight using Bayes Theorem  We are using asymptotic properties, no guarantee we are far enough in the limit to assure these are close enough  Actuarial “experiments” not repeatable so frequentist approach (MLE) may not be appropriate Subtitles are Part of Title Field, then Modified Manually (see next page) Page based on Title and Text from Slide Layout palette. Design is 01_Title with photo.