CIA Annual Meeting

CIA Annual Meeting LOOKING BACK…focused on the future

Christian-Marc Panneton • e-mail: Christian-Marc.Panneton@inalco.com

Agenda • Why we need Stochastic Modeling • Importance of Parameters • LN Model and Parameters • Simulation - Correlation • RSLN Model and Parameters • Copulas

Why do we need stochastic Modeling? • If returns follow a normal distribution • Prices will follow a log-normal distribution Distribution of 1-year returns Distribution of value after 1 year

Guarantee • If a contract pays in one year • Max(Initial deposit, Current value) • Alternatively • Current value + Max(0, Initial deposit – Current value) Guarantee Pay-off Distribution • How to measure risk associated with such a pay-off?

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 Pay-off Distribution 2.39% 3.41% 11.42% • Expected pay-off • Zero payoffs weights => no tail information

CTE calculation • Mathematical definition • Payoff when • Involves an Integral Not easy to do!

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 • Solution: Stochastic Integration • Simple example: Calculate • From calculus, the exact solution is:

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 • More difficult example: Calculate CTE(80%) • 1-year guarantee • At maturity, no lapse, no death, no fees

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

If in regime 1 (low volatility regime) If in regime 2 (high volatility regime) • Before doing Stochastic modeling • Need a model • Log-normal Model • Regime Switching Model with two log-normal regimes • Once a model is selected • How to get model parameters? • Maximum Likelihood Estimation (MLE)

m s • Impact on CTE value - Log-Normal Model • CTE(80%), 10-year guarantee, TSX index High sensitivity of CTE to parameters To decrease CTE by 10% Increase m from 9.0% to 9.3% or Decrease s from 19.0% to 18.3% Higher s to reflect calibration

Increase lapse from 8.0% to 8.9% Increase m from 9.0% to 9.3% or Decrease s from 19.0% to 18.3% • 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 To decrease CTE by 10%

Stability of parameters • Log-normal calibrated model parameters for TSX • From January 1956 to ... CTE(80%) volatile: 85% to 147% s is stable: ± 0.4% m vary more: ±0.85%

Precision of parameters • Log-normal model calibrated parameters for TSX • From January 1956 to May 2005 m s 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.

p12 m2 m1 s1 p21 s2 • Impact on CTE value - RSLN Model • CTE(80%), 10-year guarantee, TSX index To decrease CTE by 10% Increase m2 by 1.3% or Decrease s2 by 2.3% Decrease p12 by 0.2% or Increase p21 by 0.8% Increase m1 by 0.5% or Decrease s1 by 1.5%

Stability of parameters • RSLN model parameters for TSX • From January 1956 to ... m1 : ± 0.7% p12 : ± 0.5% CTE(80%) volatile: 86% to 164%

Precision of parameters • RSLN model parameters for TSX • From January 1956 to May 2005 m1 s1 m2 s2 p12 p21 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.

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

Maximum Likelihood Estimation (MLE) • The likelihood function, L() is the joint density function of the observed data (xt) 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

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

Case Study #1 • Log-normal model, one variable: S&P/TSX TR • 2 parameters to estimate: m and s • Starting values for (monthly) parameters • m = 1% and s = 5% • Assuming m and s , find the density associated with each historical return • yFeb, 1956 = 3.84% => With Excel: NormDist(3.84%,1%,5%,False) = 6.793 Formula:

Use Excel Solver find m and s which will maximize the log-likelihood value • Constraint: Annual (d) Annualized • Case Study #1 • Take the log of the density Formula: • Sum all log density • Sum = 994.8 • Results:

With Excel, use matrix functions: -LN(2*PI()) - 0.5 * LN(MDeterm(W)) - 0.5 * SumProduct(MMult(y; MInverse(W)); y) = 4.183 • Then, Maximize with Excel Solver • Constraints: • 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

Agenda • Why we need Stochastic Modeling • Importance of Parameters • LN Model and Parameters • Simulation - Correlation • RSLN Model and Parameters • Sensitivity of CTE to Parameters

where subject to • Simulation in practice • How to generate correlated random variables? • Want • Can generate independent random variables • Solution: linear transformation

Simulation in practice • Solve • Solution: Constraint on r

Practical Issue: Constraints on correlations • Is enough? • Answer is No! • Look at the simulation process • 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

where • With 3 variables • Want: • Generate 3 independent random numbers: • Need linear combination: • Solve for • Solution

where • Easier with Matrix notation: • Mathematically, it means: • Square Root of Matrix by Cholesky Decomposition

2x2 correlation matrix • Determinant must be non-negative: • Determinant of all 2x2 sub-matrices must also be non-negative: • 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 • 3x3 correlation matrix: • Determinant must be non-negative:

Possible values for a 3x3 Matrix (–1, 1, 1) (1, 1, 1) (–1, 1,–1) (1, 1,–1) (–1,–1,–1) (1,–1,–1)

1 0.5 0 -1 -0.5 0 0.5 1 -0.5 -1 • Once one value is set (e.g.: ) • Possible values for

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 3x3 2x2

Possible values for a 4th index

1 0.5 0 -1 -0.5 0 0.5 1 -0.5 -1 • Once one value is set (e.g.: ) • Possible values for

Starting values for (monthly) parameters • 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:

Regime probabilities for next period • Regime probabilities in 2 periods • The stable distribution of the chain is given by • Case Study #5 • Initial probabilities to be in regime 1 or 2 • If start in regime 1: • Solve,

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

Take the log: • Incorporate the observed return information into regime probabilities • Add densities conditional on regime: • Continue with subsequent historical returns • Maximize with Excel Solver

Multi-variate RSLN • Each index has its own regime process • Nice in theory, but not in practice! • Simulate 10 indices • 2 m and s per index => 40 • 2 regime transition probabilities per index => 20 • 210 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 • 410 combinations => more than 1 million joint density to value

Multi-variate RSLN • Global Regime • Some limitations, but a practical solution! • Simulate 10 indices • 2 m and s 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

TSX - Uni-variate TSX - Multi-variate • Case Study #6 • RSLN model, 2 variables: TSX TR & S&P500 TR • 12 parameters to estimate • Same process except, use the joint density: • Parameter drift Can be explained by an higher probability to be in the high volatility regime

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.

TSX - Uni-variate High Vol. Regime Prob. S&P 500 - Uni-variate High Vol. Regime Prob. Topix - Uni-variate High Vol. Regime Prob. • 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

# regimes 2 3 # regimes 2 3 # corr. matrices 1 2 1 3 # corr. matrices 1 2 1 3 Log-lik 2362.9 2363.1 2366.2 2366.8 Log-lik 5197.9 5215.0 5251.1 5253.3 CTE(80%) 3.04% 3.34% 3.03% 3.50% CTE(80%) 0.42% 0.46% 0.59% 0.55% • Adding a Third Regime • TSX TR & S&P 500 TR • TSX TR, S&P 500 TR, CA-US, Topix

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