2007 Annual Meeting ● Assemblée annuelle 2007 Vancouver

1. Canadian Institute of Actuaries L’Institut canadien des actuaires 2007 Annual Meeting ●Assemblée annuelle 2007 Vancouver

2. IP-41 Stochastic Modeling for Insurance Products Stochastic Scenario Development for Inexperienced Users Julia Wirch-Viinikka Investment Products Pricing 2007 Annual Meeting Assemblée annuelle 2007

3. Objectives and Constraints Scenario Generators Start Simple: Fixed Income Scenarios Equity Scenarios Picking a model: Key Considerations Calibration What can go wrong? 2007 Annual Meeting Assemblée annuelle 2007 Stochastic Tools

4. What is your objective? Appropriate Pricing (Products/Derivs) Variability and Risk Management Duration, Convexity, Hedging Aggregation across products Reserve and capital calculation (tail results) What are your constraints? Time, computation power Analysis of results Expertise What will you do with the results? 2007 Annual Meeting Assemblée annuelle 2007 Is stochastic modeling going to give you a better answer?

5. Models are dependent on Highly variable random variables Complex correlations/ dependency structures Models are highly sensitive to assumptions and RVs Asymmetry of Results Optionality / one extreme tail Low Frequency, High Severity Fat tailed distribution Systematic/Non-diversifiable Risk 2007 Annual Meeting Assemblée annuelle 2007 When is Stochastic Important?

6. P: Real World Probability Often based on economic theories and statistical data Historical data provides long-term averages Full distribution of possible outcomes Used for cash flow projection, tail events Q: Risk Neutral Probability Replicates current market prices & implied volatilities No Arbitrage “free lunch” Only the mean has significance Used to price financial instruments, where investors hedge risk and require no risk premium 2007 Annual Meeting Assemblée annuelle 2007 P’s and Q’s: Stochastic Etiquette

7. Tail risk: Use real-world valuation to measure tail risk Average cost: Use “real world” inputs when you are willing to accept the “average” result with a high amount of variability Use risk neutral when you want results (e.g. a price or a profit measure) which you can be very confident can be realized (through hedging) EXAMPLES: Hedging (Financial Engineering) Market-consistent pricing - RN Risk Management, Valuation and Pricing (Actuarial Modeling) Tail exposures – RW Volatility - RW Averages – RW/RN Static Hedging - RN Dynamic Hedging – RW/RN 2007 Annual Meeting Assemblée annuelle 2007 RN vs RW?

8. REGIME 1 Low Volatility s1 High Mean m1 Correlation r1 2007 Annual Meeting Assemblée annuelle 2007 Traditional Actuarial Stochastics • Wilkie (Cascade Structure) • Factor Models More recent addition: • RSLN-2: REGIME 2 High Volatility s2 Low Mean m2 High Correlation r2

9. Yield Curve Spot / Yield / Forward curves Starting yield curve How many key rates do you need? How to fill in the rest of the yield curve? General Characteristics Initial yield curve should match actual Mean Reversion (to a non-fixed target rate) Long Periods of Relative Stability Range of Shapes (Normal, Inverted, Humped) Correlation between maturities and with Inflation Non-negative No exploding rates 2007 Annual Meeting Assemblée annuelle 2007 Interest Rate Models

10. Mean Reversion DRIFT: (Long-term mean – Current rate) dt Hull & White (Vasicek with time dep params): dr=theta(a(t)-r)dt+v(t)dz Non-Negative RANDOM TERM: volatility * sqrt(current rate) dz CIR: dr=theta(a-r)dt+v sqrt(r) dz As interest rates go toward zero, the random term diminishes and the interest rate is pulled up toward the long term average Easier Calibration BDT: mean reverting with lognormal distribution (mutually dependent mean reversion and volatility terms): d(ln r)=(a(t)+(v’(t)/v(t))ln r)dt + v(t) dz Black-Karasinski: Lognormal – indep parameters (like BDT) HJM: generalization - any volatility function Easier to parameterize when vol is indep of the forward rate Market Models More flexible: separation of risk between volatility and correlation 2007 Annual Meeting Assemblée annuelle 2007 A Few Models

11. Are they related? Direct relation shows zero correlation However… Bond Funds and Equity Indices show 30%-60% correlation Duration analysis can explain 90%+ of bond fund returns: an( int – int-1) = Bond Fund Return (t-1,t) 2007 Annual Meeting Assemblée annuelle 2007 Yield Curve vs. Equity

12. Random Number Generators: Are they really random? Do they generate enough variables before repeating? Do you have Enough Scenarios? Representative Sampling Variance Reduction Techniques only helps you get a quicker estimate of the mean. Test robustness on much larger sample size Especially important for tail measures Are your Parameters Right? Validation and Recalibration 2007 Annual Meeting Assemblée annuelle 2007 What to watch out for:

13. 2007 Annual Meeting Assemblée annuelle 2007 Enough Scenarios? • Convergence / Sampling error • Variance Reduction Techniques may help • Many techniques work for averages not tails

14. + Maximum 75th Percentile Median 25th Percentile Minimum Outliers 2007 Annual Meeting Assemblée annuelle 2007 + + + Histograms & Box Plots Stochastic Results

15. Model risk:the possibility of loss or error resulting from the use of models. Model misspecification Assumption misspecification Inappropriate use or application Inadequate testing, validation, and documentation Lack of knowledge or understanding, user and/or management Error and negligence Beware: Often there is a false sense of precision 2007 Annual Meeting Assemblée annuelle 2007 The Small Print

16. 2007 Annual Meeting Assemblée annuelle 2007 Questions?