1. CIA Annual Meeting LOOKING BACK…focused on the future

2. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Christian-Marc Panneton e-mail: Christian-Marc.Panneton@inalco.com Christian-Marc Panneton e-mail: Christian-Marc.Panneton@inalco.com

3. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Agenda Why we need Stochastic Modeling Importance of Parameters LN Model and Parameters Simulation - Correlation RSLN Model and Parameters Copulas

4. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Why do we need stochastic Modeling? If returns follow a normal distribution Distribution of 1-year returns Distribution of value after 1 year Prices will follow a log-normal distribution

5. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future If a contract pays in one year Max(Initial deposit, Current value) Guarantee Pay-off Distribution How to measure risk associated with such a pay-off?Guarantee Alternatively Current value + Max(0, Initial deposit – Current value)

6. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Expected pay-off Zero payoffs weights => no tail information Pay-off Distribution 3.41% 11.42% 2.39% Pay-off at a specified probability Equivalent to VaR measure Limited tail information Expected pay-off with a specified probability CTE measure More tail information

7. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future CTE calculation Mathematical definition Payoff when Involves an Integral Not easy to do!

8. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Solution: Stochastic Integration Simple example: Calculate Stochastic integration Generate a uniform random number between 0 and 5 Calculate Repeat n times Calculate the average of all samples Multiply by the width of the interval: 5 From calculus, the exact solution is:

9. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future More difficult example: Calculate CTE(80%) 1-year guarantee At maturity, no lapse, no death, no fees Exact Solution: Hardy, NAAJ April 2001 Stochastic integration Generate a standard normal random number Calculate Payoff: Repeat n times Sort and calculate the average of the 20% highest payoff

10. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Stochastic integration is easier to do No complex integrals to calculate Need only to simulate market returns and determine pay-off according to each path Drawback: Computer intensive Only 20% of random paths are used to calculate CTE(80%) Aggregation: the worst paths for a specific contract are not necessarily the same for another contract Some articles explore these topics 2003 Stochastic Modeling Symposium

11. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Agenda Why we need Stochastic Modeling Importance of Parameters LN Model and Parameters Simulation - Correlation RSLN Model and Parameters Copulas

12. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Before doing Stochastic modeling Need a model Log-normal Model Regime Switching Model with two log-normal regimes If in regime 1 (low volatility regime) If in regime 2 (high volatility regime) Once a model is selected How to get model parameters? Maximum Likelihood Estimation (MLE)

13. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Impact on CTE value - Log-Normal Model CTE(80%), 10-year guarantee, TSX index   To decrease CTE by 10% Increase  from 9.0% to 9.3% or Decrease  from 19.0% to 18.3% High sensitivity of CTE to parameters Higher  to reflect calibration

14. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Comparison with Lapse Assumption CTE(80%), 10-year guarantee, TSX index No mortality 8% per year lapse assumption Expect high sensitivity because SegFund guarantee is a lapse supported product Increase  from 9.0% to 9.3% or Decrease  from 19.0% to 18.3% To decrease CTE by 10% Increase lapse from 8.0% to 8.9%

15. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Stability of parameters Log-normal calibrated model parameters for TSX From January 1956 to...  is stable: ± 0.4%  vary more: ±0.85% CTE(80%) volatile: 85% to 147%

16. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Precision of parameters Log-normal model calibrated parameters for TSX From January 1956 to May 2005  MLE 8.61 % 18.71 % s.e. 2.23 % 0.45 % CTE(80%) partial derivative – 30.3 13.3 Lower CTE(80%) by 10% + 0.1 s.e. – 1.7 s.e.

17. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Impact on CTE value - RSLN Model CTE(80%), 10-year guarantee, TSX index To decrease CTE by 10%     p  p  Increase   by 0.5% or Decrease   by 1.5% Increase   by 1.3% or Decrease   by 2.3% Decrease p  by 0.2% or Increase p  by 0.8%

18. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Stability of parameters RSLN model parameters for TSX From January 1956 to...   : ± 0.7% p 12 : ± 0.5% CTE(80%) volatile: 86% to 164%

19. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Precision of parameters RSLN model parameters for TSX From January 1956 to May 2005         p 12 p 21 MLE 15.4 % 11.8 % -19.1 % 25.2 % 4.3 % 19.4 % s.e. 2.3 % 0.6 % 11.1 % 0.6 % 1.9 % 6.7 % CTE(80%) partial derivative -20.2 5.9 -7.5 3.6 42.8 -12.2 Lower CTE(80%) by 10% + 0.2 s.e. – 3.1 s.e. + 0.1 s.e. – 4.9 s.e. – 0.1 s.e. + 0.1 s.e.

20. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Agenda Why we need Stochastic Modeling Importance of Parameters LN Model and Parameters Simulation - Correlation RSLN Model and Parameters Copulas

21. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Maximum Likelihood Estimation (MLE) Given a particular set of observed data, what set of parameters gives the highest probability of observing the data? The likelihood function is proportional to the probability of actually observing the data, given the assumed model and a set of parameters (  ) Maximizing the likelihood function is equivalent to maximizing the probability of observing the data

22. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Maximum Likelihood Estimation (MLE) The likelihood function, L(  ) is the joint density function of the observed data (x t ) given the parameters in  If the returns in successive periods are independent, then this density is the product of all the individual density functions More convenient to work with the log-likelihood

23. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study Monthly Data (January 1956 to May 2005) S&P/TSX Total Return Index S&P 500 Total Return Index CA-US Exchange Rate Topix Index (Japan) First, convert index values to returns

24. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #1 Log-normal model, one variable: S&P/TSX TR 2 parameters to estimate:  and  Starting values for (monthly) parameters   = 1% and  = 5% Assuming  and , find the density associated with each historical return y Feb, 1956 = 3.84%=> With Excel: NormDist(3.84%,1%,5%,False) = 6.793 Formula:

25. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #1 Take the log of the density Sum all log density Sum = 994.8 Use Excel Solver find  and  which will maximize the log-likelihood value Constraint:Formula: Results: Annual (  ) Annualized

26. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #2 LN model, 2 variables: TSX TR and S&P 500 TR 5 parameters to estimate: Same process except, use the joint density = 4.183 -LN(2*PI()) - 0.5 * LN(MDeterm(  )) - 0.5 * SumProduct(MMult(y; MInverse(  )); y) = 4.183 With Excel, use matrix functions: Then, Maximize with Excel Solver Constraints:

27. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Agenda Why we need Stochastic Modeling Importance of Parameters LN Model and Parameters Simulation - Correlation RSLN Model and Parameters Sensitivity of CTE to Parameters

28. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future where Can generate independent random variables Simulation in practice How to generate correlated random variables? Solution: linear transformation subject to Want

29. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Simulation in practice Solve Solution: Constraint on 

30. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Cases Study #3 and #4 LN model, 3 var.: TSX, S&P 500 and CA-US 9 parameters to estimate LN model, 4 var.: TSX, S&P 500, CA-US and Topix 14 parameters to estimate Practical Issue: Constraints on correlations Is enough? Answer is No! Look at the simulation process

31. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future With 3 variables Want: Generate 3 independent random numbers: where Need linear combination: Solve for Solution

32. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Easier with Matrix notation: Mathematically, it means: where Square Root of Matrix by Cholesky Decomposition

33. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Restrictions on correlation values Correlation matrix must be Semi-Definite Positive All eigenvalues are non-negative The product of the eigenvalues of a matrix equals its determinant 2x2 correlation matrix Determinant must be non-negative: 3x3 correlation matrix: Determinant must be non-negative: Determinant of all 2x2 sub-matrices must also be non- negative:

34. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Possible values for a 3x3 Matrix (1, 1, 1) (–1, 1, 1) (–1, 1,–1) (–1,–1,–1)(1,–1,–1) (1, 1,–1)

35. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Once one value is set (e.g.: ) Possible values for -0.5 0 0.5 1 -0.500.51

36. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future More and more restrictions as the size is increased For a 4x4 correlation matrix: Determinant must be non-negative Determinant of all sub-matrices must be non-negative 3x32x2

37. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Possible values for a 4 th index

38. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Once one value is set (e.g.: ) Possible values for -0.5 0 0.5 1 -0.500.51

39. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Agenda Why we need Stochastic Modeling Importance of Parameters LN Model and Parameters Simulation - Correlation RSLN Model and Parameters Copulas

40. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #5 RSLN model, one variable: S&P/TSX TR 6 parameters to estimate: Initial probabilities to be in regime 1 or 2? Define the Transition Matrix: Starting values for (monthly) parameters

41. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #5 Initial probabilities to be in regime 1 or 2 If start in regime 1: Regime probabilities for next period Regime probabilities in 2 periods The stable distribution of the chain is given by Solve,

42. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #5 MLE Calculate densities: When in regime 1 When in regime 2 With regime switching process Starts in regime 1 and stays in regime 1 Starts in regime 2 and switch to regime 1 Starts in regime 1 and switch to regime 2 Starts in regime 2 and stays in regime 2

43. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Add densities conditional on regime: Take the log: Incorporate the observed return information into regime probabilities Continue with subsequent historical returns Maximize with Excel Solver

44. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Multi-variate RSLN Each index has its own regime process Nice in theory, but not in practice! Simulate 10 indices 2  and  per index => 40 2 regime transition probabilities per index => 20 2 10 correlation matrices (45 correlations per matrix) => 46,080 Assuming 40 years of monthly historical data only 4,800 data points for the 10 indices! The density calculation will involve all possible regime combinations 4 10 combinations => more than 1 million joint density to value

45. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Multi-variate RSLN Global Regime Some limitations, but a practical solution! Simulate 10 indices 2  and  per index => 40 2 global regime transition probabilities => 2 2 correlation matrices (45 correlations per matrix) => 90 The density calculation will involve only 4 regime combinations

46. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Case Study #6 RSLN model, 2 variables: TSX TR & S&P500 TR 12 parameters to estimate Parameter drift TSX - Uni-variate TSX - Multi-variate Can be explained by an higher probability to be in the high volatility regime Same process except, use the joint density:

47. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Limitation of Global Regime Probability of being in either regime is modified TSX - Uni-variate High Vol. Regime Prob. TSX - Change in High Vol. Regime Prob.

48. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Limitation of Global Regime OK if indices are in the same regime at the same time If not ! Fix parameters to their univariate estimates for the most significant exposure (e.g. TSX), then estimate the remaining parameters Add more global regimes Add some local regimes TSX - Uni-variate High Vol. Regime Prob. S&P 500 - Uni-variate High Vol. Regime Prob. Topix - Uni-variate High Vol. Regime Prob.

49. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Adding a Third Regime TSX TR & S&P 500 TR TSX TR, S&P 500 TR, CA-US, Topix # corr. matrices 1 2 1 3 Log-lik 2362.9 2363.1 2366.2 2366.8 CTE(80%) 3.04% 3.34% 3.03% 3.50% # regimes 2 3 # corr. matrices 1 2 1 3 Log-lik 5197.9 5215.0 5251.1 5253.3 CTE(80%) 0.42% 0.46% 0.59% 0.55% # regimes 2 3

50. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Agenda Why we need Stochastic Modeling Importance of Parameters LN Model and Parameters Simulation - Correlation RSLN Model and Parameters Copulas

51. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Copulas A function that links univariate marginals to their full multivariate distribution Sklar theorem: Provide a unifying and flexible way to study multivariate distributions A lot of interest and research in the context of credit derivatives: tail events

52. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future Copula test TSX TR & S&P 500 TR # corr. matrices 1 2 1 3 Log-lik 2362.9 2363.1 2366.2 2366.8 CTE(80%) 3.04% 3.34% 3.03% 3.50% # regimes 2 3 Gaussian Log-lik 2363.8 2364.0 2372.4 2373.4 CTE(80%) 3.06% 3.27% 3.72% 4.06% Student The choice of copula can materially affect the CTE value!

53. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future RSLN, 2 regimes, Student copula, 1 correlation matrix

54. CIA Annual Meeting Session 2403: Stochastic Modeling LOOKING BACK…focused on the future RSLN, Student copula 2 regimes, 1 corr. matrix3 regimes, 3 corr. matrices

55. CIA Annual Meeting LOOKING BACK…focused on the future