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

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

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

5. Product Design & Pricing
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
Risk Management
Embedded Options
Competitive Features
Sensitivities
Product Design & Pricing
Statutory  GAAP
Economic  Fair Value
Embedded Value
Financial Reporting
Capital Management
Business Plan
‘What if’ Scenarios
Forecasting
Economic Scenarios

6. Risk Neutral (Q measure)
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
Arbitrage Free
Equilibrium
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
Structure
Model
Risk Neutral (Q measure)
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
Real World (P measure)
Probability
Measure

7. Real World Models Historic Observations & “Stylized Facts”
Interest Rates
Inflation
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

8. Leave other products to second phase or later working group
CLIFR set up a working group to develop guidance for the Calibration of Interest Rate Models
Objective
Mandate
Scope
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
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
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
Original timeline called for preliminary calibration criteria to be developed by now, but turnover in working group has slowed progress

9. The working group developed some general principles, …
Be sufficiently robust to narrow the range of practice, but allow the actuary to apply reasonable judgement to specific circumstances;
Be applied to the set of scenarios produced, not to the model parameters or inputs;
Be applied to not only the near term, but also the steady state portions of the scenarios produced;
Be applied to more than one point on the yield curve including a mix of short, medium, and long-term points;
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;
Look at average rate distributions corresponding to extended periods of time as well as rate distributions at selected points in time.

10. Initial State & Steady State Yield Distributions
… and outlined a set of quantitative and qualitative criteria that would need to be developed.
Quantitative
Qualitative
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
Robustness of the Scenario Generator
Use of multiple interest rate models
Integration with Equity Market Risk
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. Shape distribution & inversion frequency (long – short spread)
Some examples of calibration criteria envisioned by the AAA Economic Scenario Subgroup, but still a work in progress…
Quantitative
Qualitative
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
Adherence to stylized facts, and/or commentary on desirable attributes

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

13. Yield Curve Interpolation
Example 1 continued
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)

14. 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. Commentary on the “Multiplicative shock” model
Example 2 continued
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)

16. 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. Intuitively, bond returns can be a simple function of interest rates:
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:
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
Is the interest rate of maturity m at time t

18. Seed value to ‘prime’ algorithm
Random Number Generator is always critical, but often overlooked! Desirable Characteristics
Seed value to ‘prime’ algorithm
Robustness of generator should be independent of seed
Reproducible
Ability to produce large number of values before repeating
Should be >> required number of samples
Long Periodicity
Free from bias, trends and autocorrelation
Pass statistical tests for ‘randomness’ (e.g., DIEHARD)
Independence
Statistical Routines
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. Despite a good quality RNG, sampling error inherent in Monte Carlo simulations can be significant, especially for tail measures
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):
The CTE Estimator
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. Several approaches are used to reduce sampling error
Several approaches are used to reduce sampling error. Example 1: use a ‘representative sample’ of stochastic scenarios
Significance Method / stratified sampling:
Define a significance measure, e.g. 
Calculate S for each of N stochastic scenarios
Sort the paths in ascending order based on S
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. Estimate Actual Values Design Control Variate Run Valuation Scenarios
Example 2: a ‘control variate’ approach identifies any valuation bias introduced by the sampling of the n scenarios
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
High precision of true distribution for control variate may be obtainable in closed form

22. 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. Sampling Error of AAR at CTE90 Random Sampling (1) vs
Sampling Error of AAR at CTE90 Random Sampling (1) vs. Representative Scenarios (2)

24. Reducing Sampling Error The Control Variate Approach: Results

25. CIA prescribed scenarios can produce more extreme valuation results than guideline CTE60-80 range from a stochastic valuation
Prescribed scenarios:
Grade from current to 5% flat after 20 years
Grade from current to 12% flat after 20 years
(i) Grade long rate from current to 12%, then to 5%, etc (ii) Short rate = 60% of long
(i) Grade long rate from current to 5%, then to 12%, etc (ii) Short rate = 60% of long
Same as #3 except short rate moves to 120% of long rate, then to 40%, etc
Same as #4 except short rate moves to 40% of long rate, then to 120%, etc
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 modifies and adds to these prescribed scenarios.