1. 2006 Seminar for the Appointed Actuary Colloque pour l’actuaire désigné 2006 2006 Seminar for the Appointed Actuary Colloque pour l’actuaire désigné 2006 Canadian Institute of Actuaries Canadian Institute of Actuaries L’Institut canadien des actuaires L’Institut canadien des actuaires

2. Stochastic interest rate models Implications for the valuation of policy liabilities September 21, 2006 Geoffrey H. Hancock, FSA, FCIA Director Mercer Oliver Wyman

3. 2 Seminar for the Appointed Actuary, September 2006 PD-4 Introduction to Stochastic Modeling Uses for interest rate models Real world vs. risk neutral Stylized facts Model forms Random number generators Sampling error

4. 3 Seminar for the Appointed Actuary, September 2006 Stochastic valuation approaches, including stochastic interest rate models, are increasingly necessary to value complex financial instruments Increased computing power, combined with distributed processing More complex guarantees incorporated into product designs Deterministic models cannot adequately capture interest rate and other market risks Increased use of stochastic models Development of more sophisticated risk measurement and management techniques Increased disclosure requirements from markets and regulators

5. 4 Seminar for the Appointed Actuary, September 2006 Not a necessary “evil”, stochastic economic scenarios are in fact a valuable element of an insurance company’s risk measurement toolkit with several business applications Strategy Formulation Trading / Optimization Cost Effectiveness Strategy Formulation Trading / Optimization Cost Effectiveness Risk Management Embedded Options Competitive Features Sensitivities Embedded Options Competitive Features Sensitivities Product Design & Pricing Statutory  GAAP Economic  Fair Value Embedded Value Statutory  GAAP Economic  Fair Value Embedded Value Financial Reporting Capital Management Business Plan ‘What if’ Scenarios Capital Management Business Plan ‘What if’ Scenarios Forecasting Economic Scenarios

6. 5 Seminar for the Appointed Actuary, September 2006 Both ‘real world’ and ‘risk neutral’ models serve a purpose, and sometimes you actually need both! Time-varying parameters to match market prices and term structure Focus: asset pricing (today) “Implied volatility” used as fitting parameter in market calibration Recalibrate if conditions change Model Structure Probability Measure Constant parameters to model realistic behaviour through time Focus: plausible future outcomes Model is calibrated based on history and future expectations Recalibrate if dynamics change Concept of replication & the “law of one price”  No risk free profits Only the mean has significance  No information possible outcomes Used to price financial instruments Investors hedge risk & do not require a premium  E[R] = risk free rate “Implied volatility” used to calibrate to observed market prices Can be theory or data based  Economic and statistical Provides full distribution of possible outcomes in the “real world” Used for cashflow projections Investors require a premium for bearing risk  E[R] is a subjective element of the model Usually calibrated to historic data Arbitrage FreeEquilibrium Risk Neutral (Q measure) Real World (P measure)

7. 6 Seminar for the Appointed Actuary, September 2006 Interest RatesInflation  Bounded nominal rates  Mean reversion, but non-fixed target rate  Rate volatility decreases with maturity  Heightened volatility when rates are high  High correlation between maturities  Variety of yield curve shapes  Non-persistence of extremes (high or low)  Realized = expected + random exogenous shock  Mean reversion, but non-fixed target rate  Volatility “clustering”  Various forms  Relationship to other economic factors Real World Models Historic Observations & “Stylized Facts”

8. 7 Seminar for the Appointed Actuary, September 2006 CLIFR set up a working group to develop guidance for the Calibration of Interest Rate Models Have a set of calibration standards that can be applied consistently to a wide range of interest-sensitive insurance and investment products, including both long and short term products Investigate and develop methodologies and standards for the calibration of interest rate models for determining reserves to be held by life insurance companies ObjectiveMandate To get “quick wins”, limit the scope to a few specific products for which complete calibration criteria can be more readily determined Leave other products to second phase or later working group Scope Original timeline called for preliminary calibration criteria to be developed by now, but turnover in working group has slowed progress DecJanFebMarAprMayJunJulAugSepOctNov

9. 8 Seminar for the Appointed Actuary, September 2006 The working group developed some general principles, … 1.Be sufficiently robust to narrow the range of practice, but allow the actuary to apply reasonable judgement to specific circumstances; 2.Be applied to the set of scenarios produced, not to the model parameters or inputs; 3.Be applied to not only the near term, but also the steady state portions of the scenarios produced; 4.Be applied to more than one point on the yield curve including a mix of short, medium, and long-term points; 5.Promote the development of scenario sets that measure exposure to yield curve shocks as well as long-term paths of declining as well as rising interest rates, consistent with history; 6.Look at average rate distributions corresponding to extended periods of time as well as rate distributions at selected points in time.

10. 9 Seminar for the Appointed Actuary, September 2006 … and outlined a set of quantitative and qualitative criteria that would need to be developed. QualitativeQuantitative  Robustness of the Scenario Generator  Use of multiple interest rate models  Integration with Equity Market Risk  Characteristics not specifically mentioned  Initial State & Steady State Yield Distributions  Left & Right Tail Thickness  Shape Distribution & Inversion Frequency  Yield Curve Change Characteristics  Average Yield Distribution Characteristics  Correlation Characteristics  Sampling Error Tolerance  Characteristics not specifically mentioned The AAA also has a working group developing guidance for interest rate models. It would make sense for these two groups to work more closely together and reach the same or similar end point.

11. 10 Seminar for the Appointed Actuary, September 2006 Some examples of calibration criteria envisioned by the AAA Economic Scenario Subgroup, but still a work in progress… QualitativeQuantitative  Adherence to stylized facts, and/or commentary on desirable attributes  Initial state & steady state yield distributions for short and long rate maturities  Left and right tail measures as a function of median or mean at various time horizons  Shape distribution & inversion frequency (long – short spread)  Correlations among maturities

12. 11 Seminar for the Appointed Actuary, September 2006 Several interest rate models have been proposed in the literature and by practitioners. Example 1:  ~ Short Real Rate of Interest:  ~ Long – Short Bond Spread:  ~ Expected Inflation:  ~ Realized Inflation: where: Waiting Time between Jumps ~ Poisson( ) Key State Variables for the Real World Model:

13. 12 Seminar for the Appointed Actuary, September 2006 Yield Curve Interpolation Intermediate term yield of maturity m given by: where: Remainder of curve fit by Nelson-Siegel (maturity w in years): Simple model, but explains over 90% of yield curve movements!  ~ deterministic function (depends on shape of yield curve) Example 1 continued

14. 13 Seminar for the Appointed Actuary, September 2006 Example 2: “Multiplicative shock” model The 3-month, 5-year and 10-year Treasury yields each evolve according to the following process: where m denotes the maturity index, e.g. 1 = 10-year (primary rate) 2 = 3-month (secondary rate) 3 = 5-year (mid-term rate) The “pre-shock” rates are bounded  extreme “post-shock” rates are possible due to random shock, but do not tend to persist:

15. 14 Seminar for the Appointed Actuary, September 2006 Commentary on the “Multiplicative shock” model Primary (long) rate slowly mean-reverts with weight  1 to a long-term fixed target  1 Secondary (short) rate mean-reverts with weight  2 to the long rate less a constant spread  2 The mid-term rate mean-reverts to a simple function of the short and long yields, with the relationship varying slightly depending on whether the curve is inverted Extreme rates are avoided through the use of bounds m L and m U, but applying them before the stochastic (random) innovation avoids “flatlining” and produces much more realistic paths The remainder of the yield curve can be obtained by interpolation or extrapolation using the relationships from the “closest” historic curve (e.g. using sum of weighted squared difference for 3 key rates) Example 2 continued

16. 15 Seminar for the Appointed Actuary, September 2006 Example 3: “Additive shock” model, e.g. the Cox-Ingersoll-Ross variety The 3-month, 5-year and 10-year Treasury yields each evolve according to the following process: … where each of the are as with the Multiplicative shock model Negative rates are possible, although unlikely due to the Other models: Hull & White Heath-Jarrow-Morton Stochastic volatility (e.g., AAA C-3 Phase I Risk-Based Capital generator)

17. 16 Seminar for the Appointed Actuary, September 2006 That is, bond return is a function of – the level of interest rates (interest earned) – the change in the level of interest rates (duration effect) But there are other factors which affect bond fund returns (e.g. credit losses, liquidity preferences, etc.) which are more difficult to model – Add a random component to allow for these additional factors: where is correlated to the random shocks for equity returns, thereby increasing the bond-equity correlation While interest rate changes and equity returns may not be strongly correlated, bond and equity returns do have significant correlation Intuitively, bond returns can be a simple function of interest rates: Is the interest rate of maturity m at time t

18. 17 Seminar for the Appointed Actuary, September 2006 Reproducible  Seed value to ‘prime’ algorithm  Robustness of generator should be independent of seed Long Periodicity Independence Statistical Routines Random Number Generator is always critical, but often overlooked! Desirable Characteristics  Ability to produce large number of values before repeating  Should be >> required number of samples  Free from bias, trends and autocorrelation  Pass statistical tests for ‘randomness’ (e.g., DIEHARD)  Samples are often from Uniform(0,1) distribution  That is, cumulative density ~ U( x ) = F( x )  Need inverse of F( x )  Require fast, accurate statistical routines ~ e.g.,   1 ( x )

19. 18 Seminar for the Appointed Actuary, September 2006 Despite a good quality RNG, sampling error inherent in Monte Carlo simulations can be significant, especially for tail measures The CTE Estimator Bias is small (i.e., expected value of CTE estimator  true CTE) Sampling error can be very significant (especially if N  1000) – Variance of CTE estimator is larger for fat-tailed loss distributions See “Variance of the CTE Estimator”, Manistre & Hancock (NAAJ, Sep 2003): Stratifica- tion  Seems to work well, but often based solely on the scenarios (model inputs, not results)  Need to test for effectiveness before putting into production Other Techniques  Variance reduction  biased sampling (put more results in the tail)  Use of “control variate” (alternatively, a “reference portfolio”)  Fit empirical results to known loss distribution for analytic tractability

20. 19 Seminar for the Appointed Actuary, September 2006 Several approaches are used to reduce sampling error. Example 1: use a ‘representative sample’ of stochastic scenarios Significance Method / stratified sampling: 1. Define a significance measure, e.g.  2. Calculate S for each of N stochastic scenarios 3. Sort the paths in ascending order based on S 4. Choose n representative scenarios by selecting every [(N/n)*k – (N/2n)] th scenario, each equally likely Other approaches involve selecting n scenarios out of N using a distance measure (think of it as the difference between two significance measures), then assigning each of the original N scenarios to the n selected ones using the distance measure, with the number of scenarios being assigned to a given selected scenario determining the relative likelihood of that selected scenario. See Efficient Stochastic Modeling for Large and Consolidated Insurance Business: Interest Rate Sampling Algorithms, by Yvonne Chueh, NAAJ, Volume 6, Number 3 (July 2002) These methods are based on scenarios only. They do not require a valuation using all N scenarios.

21. 20 Seminar for the Appointed Actuary, September 2006 Example 2: a ‘control variate’ approach identifies any valuation bias introduced by the sampling of the n scenarios High precision of true distribution for control variate may be obtainable in closed form Use the results of prior steps to adjust preliminary values for actual portfolio Can test several control portfolios for improvement Estimate Actual Values Used as a “reference” for the actual portfolio Must be highly positively correlated to actual portfolio Design Control Variate Obtain “true” distribution for desired metrics (outputs) Must be able to establish measures to high precision Evaluate Control Run ”control” & “actual” over the valuation scenarios Valuation scenarios are a manageable (small) set Run Valuation Scenarios

22. 21 Seminar for the Appointed Actuary, September 2006 Example 3: previous two approaches can be combined to produce very accurate results Select and Evaluate Control Variate Needs to be highly correlated to actual portfolio If portfolio to be valued is complex, could use a small cross-section of actual inforce portfolio Perform stochastic valuation of control variate using large N (e.g. 20,000+) scenarios  This is deemed to be the “correct” answer. Select Valuation Scenarios Sort N valuation results Choose n (<<N) scenarios by stratifying the results, and picking a representative scenario for each stratum (e.g. the mid-point result) Difference relative to example 1 is that sorting and selection is based on valuation results rather than on scenario significance measures Determine Adjustment Factor Perform stochastic valuation of control variate using n scenarios  Adjustment Factor Value Full Portfolio Perform stochastic valuation of Full Portfolio using n scenarios  Adjust result for bias identified in prior step:

23. 22 Seminar for the Appointed Actuary, September 2006 Sampling Error of AAR at CTE90 Random Sampling (1) vs. Representative Scenarios (2)

24. 23 Seminar for the Appointed Actuary, September 2006 Reducing Sampling Error The Control Variate Approach: Results

25. 24 Seminar for the Appointed Actuary, September 2006 CIA prescribed scenarios can produce more extreme valuation results than guideline CTE60-80 range from a stochastic valuation Prescribed scenarios: 1.Grade from current to 5% flat after 20 years 2.Grade from current to 12% flat after 20 years 3.(i) Grade long rate from current to 12%, then to 5%, etc (ii) Short rate = 60% of long 4.(i) Grade long rate from current to 5%, then to 12%, etc (ii) Short rate = 60% of long 5.Same as #3 except short rate moves to 120% of long rate, then to 40%, etc 6.Same as #4 except short rate moves to 40% of long rate, then to 120%, etc 7.Implied forward curve At year-end 2005, due to very low interest rate environment, scenario #7 was the most severe for most insurers June 2006 Exposure Draft for section 2330 modifies and adds to these prescribed scenarios.