Practical Issues in Stochastic Modelling

1. Practical Issues in Stochastic Modelling September 2004 Nigel Knowles (Standard Life) Michael Payne (Scottish Widows)

2. Agenda • Overview of realistic balance sheets • The liability model • Model points • Asset model calibration • Modelling assumptions • Dynamic behaviour • Dynamic decisions • Systems and controls • Communication • Current Topics • Summary

3. Overview of realistic balance sheets • With-profits portfolios contains many complex liabilities that look similar to put options • The aim of the Regulations is to place a realistic (market consistent) value on the contingent claims embedded in the with-profits portfolios of major UK life offices • Hedging costs (the office does the replication itself) • The cost of purchasing hedging instruments • Actuaries need to learn a little about financial economics

4. The liability model • Model points • Asset model calibration

5. The liability model • Closed form Black-Scholes is one possible method of assessing option costs • Monte Carlo simulation is an alternative • Already have projection engines for the model office (Financial condition reports) • Add ESG • Dynamic decisions • Future premiums

6. The complexity of these models is unparalleled for many insurers • Long projection periods • Monthly for 30+ years • Large numbers of model points • Lengthy run times • The requirement to warehouse large volumes of data • Better buy shares in Compaq?

7. Model points • We cannot project 5m policies or premium tranches monthly for 40 years in a stochastic environment. • Use model points instead • “Reasonably” heterogeneous clusters of policies • How should we create a set of model points that is representative of the portfolio? • First principles degrees of freedom “moneyness” (asset share, term outstanding, term in-force) and contract type • Model point budgets (dynamic decisions might have reference to all the portfolio at once so cannot look at just one half).

8. Model points • Sparse data (can we drop small data items or distant ones where the intrinsic value tiny?) • Smoothing needs a lot less clumping to be meaningful. • What if you cannot (or don’t want to) model everything? • Scaling (approximately modelled – small contracts?) • Scaling to statutory reserve versus WPBR

9. Asset model choice/calibration • What does market consistency mean? 7.4.168 R The market-consistent asset model in PRU 7.4.167R(1): (1) means a model that delivers prices for assets and liabilities that can be directly verified from the market; and (2) must be calibrated to deliver market-consistent prices for those assets that reflect the nature and term of the with-profits insurance liabilities of the with-profits fund.

10. Asset model choice/calibration • What does market consistency mean? • Are we replicating options (including the expenses of setting up trading strategies) Or • Are we buying options (estimate price charged by third parties)?

11. Asset model choice/calibration • Price is not defined by a single model so any calibrated model is unlikely to fit all observable data. • There are few instruments of the right term and duration that look like an insurer’s portfolio • Extrapolation from short-term over 40 years is a “Herculean” assumption. • Is it spurious to worry about “moneyness” structures over 40 years? • Constant volatility often assumed even though we know it isn’t • Data suggests stability over the long-term.

12. Figure 1. Short-term versus long-term implied volatility • In the absence of anything else assume market rates up to 5 years and a stable forward rate thereafter?

13. Asset model choice/calibration • Calibrate models to prices versus properties of instruments such as implied volatility – is this market consistent? • Unlikely to reproduce closely the prices of swaps if calibrate to Gilt-edged securities and vice versa.

14. Asset model choice/calibration • Crucial point is to sensitivity test the outcome; • If deduce that it is not quite as sensitive to equity volatility anyway if we have 50-60% equity backing ratios – getting it “wrong” by 20 basis points is not material

15. Figure 2. Impact of different volatilities

16. Asset model choice/calibration • Crucial point is to sensitivity test the outcome; • Separate model features from real world features; • Consols process in 2 factor model will tend to dominate long bond volatilities • Lognormal models are skew • Get the basics right first; • Not very sensitive to interest rate model choice of fixed interest if matched by duration

17. Figure 3. Impact of different fixed interest matching strategies

18. Asset model choice/calibration • Crucial point is to sensitivity test the outcome; • Separate model features from real world features; • Get the basics right first; • Remember that assumptions about payments on surrender or other management actions may have a far more significant impact.

19. Asset model choice/calibration • Cannot avoid the maths and delegate responsibility for completeness and appropriateness • Martingale tests for all asset classes • How many simulations should be used? • Central limit theorem/standard errors • How big an unexplained item do you want in your analysis of surplus? • How big are you prepared “unexpected” movements to be?

20. Agenda • Modelling assumptions • Dynamic behaviour • Dynamic decisions

21. Modelling assumptions • Models should allow for policyholder behaviour • 7.4.57 R Policyholder actions refer to the foreseeable actions that would be taken by the firm’s policyholders, taking into account: (1) the experience of the firm in the past; and (2) the changes that may occur in the future if options and guarantees become more valuable to policyholders than in the past.

22. Policyholder behaviour • Realism sought • Eliminate artificial releases of surplus arising from lapses • Allow for likely option take-up rates • WP Bonds spot guarantee dates at key policy anniversaries • Tax-free cash in GAOs • Pensions contracts and endowment assurances may have material guaranteed surrender values/early retirement options • Irrational policyholders vs. American options • American options cost more • Limited data in extreme conditions • Contradictory data • WP PVIF value offsets in some instances

23. Modelling assumptions • Dynamic management actions are available to mitigate risk • 7.4.50 R In calculating the risk capital margin for a with-profits fund, a firm may reflect, in its projections of the value of assets and liabilities under the scenarios in PRU 7.4.42R, the firm’s prospective management actions (see PRU 7.4.51R). • But assumptions need to be reasonable 7.4.51 R Prospective management actions refer to the foreseeable actions that would be taken by the firm’s management, taking into account: (1) an appropriately realistic period of time for the management actions to take effect; and (2) the firm’s PPFM and its regulatory duty to treat its customers fairly.

24. Dynamic management actions • Why do we want to the model to make dynamic decisions? • Complex office behaviour needs replicated by model if it’s to be realistic • More useful if it’s realistic • Capital requirements based on the output of these models • Unnecessary prudence undesirable

25. Investment Decisions • Manage exposure to risky assets • Explicit stress test of the balance sheet in line with RCM (-20% equity and/or –100bps gilt yields) • Run times • Complexity • Closed fund projections • Implicitly through proxy based on FTSE • Simple

26. Regular Bonus Decisions • Projected affordability/sustainability • Smoothing of changes from one year to the next • Portfolio-wide costs can be estimated

27. Systems and controls • Testing and checking • Model controls • Output analysis

28. Testing and checking • Start deterministic • Then scenarios • Then percentiles from stochastic • Worth the effort on • Accurate specifications • Detailed test plans • Extensive testing • Enlist the help of expert in their owe areas • Extreme returns break code • “Div almost zero” errors • Burst out of top of arrays • Taylor’s theorem relies on small interest rate deltas • Third parties can add value in reviewing • Simulation walkthrough

29. Model controls • It’s important as drives balance sheet (through WPICC) • Unit admin, quotation/projection and valuation systems all rolled into one! • Complexity means that it is infeasible to audit? • Sign off of the initial model • Ongoing audit trails and controls • Sign-off of master model for all purposes based on full stochastic run • Sign off of the model following a model update • Test grouped model points remain appropriate • Recalibrate and check again the economic scenario files

30. Output analysis • Analyses of change • BSM instructive • Crude deltas and other Greeks • Explanation of results from one period to the next • Changes in model • Changes to management actions • Changes in intrinsic component • Changes in time value component

31. Communication • Explaining the results is tricky! • Who is the audience ? • Actuarial / Technical • Executive • Regulator • What message are we trying to get across ? • What is the purpose of the results • What are the key messages from the results • What action should I take (if any) given this information

32. Communication • How should the results be presented ? • Single point estimate • Confidence Intervals • Percentile Graphs

33. Current Topic – FRED 34 • Results currently required to go to FSA within 3 months (of year end) • Many companies are completing within 6 – 8 weeks • Shareholder reporting means that results likely to be required within around 2 weeks • Results rolled forward • Risk of actual being significantly different than expected • Risk of R&A differing from FSA submission • Models and methodology still being developed • Many companies currently going through audits of RBS • Will auditors be happy to sign off on working day 10 “estimates”

34. Summary • Useful tool to understand dynamics of complex liabilities • Still in its infancy • Keeping things understandable is important • Thousands of model inputs and outputs • Complex models requires extensive testing • Strong auditable controls around master model developments • Tailor the output to the user • The key challenge is to develop a well controlled process that produces explainable results in time to meet tight shareholder (or other) reporting deadlines