107 September 2004

Practical Issues in Stochastic ModellingSeptember 2004Nigel Knowles (Standard Life)Michael Payne (Scottish Widows)

207 September 2004

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

307 September 2004

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

407 September 2004

The liability model

• Model points• Asset model calibration

507 September 2004

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

607 September 2004

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?

707 September 2004

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).

807 September 2004

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

907 September 2004

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.

1007 September 2004

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)?

1107 September 2004

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.

1207 September 2004

Figure 1. Short-term versus long-term implied volatility

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• In the absence of anything else assume market rates up to 5 years and a stable forward rate thereafter?

1307 September 2004

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.

1407 September 2004

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

1507 September 2004

Figure 2. Impact of different volatilities

0.980.990.991.001.001.011.011.021.021.031.031.04

Guarantee cost

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Vol + 4%

1607 September 2004

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

1707 September 2004

Figure 3. Impact of different fixed interest matching strategies

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1807 September 2004

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.

1907 September 2004

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?

2007 September 2004

Agenda

• Modelling assumptions– Dynamic behaviour– Dynamic decisions

2107 September 2004

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.

2207 September 2004

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

2307 September 2004

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 reasonable7.4.51 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.

2407 September 2004

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

2507 September 2004

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

2607 September 2004

Regular Bonus Decisions

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

2707 September 2004

Systems and controls

• Testing and checking• Model controls• Output analysis

2807 September 2004

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

2907 September 2004

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

3007 September 2004

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

3107 September 2004

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

3207 September 2004

Communication

• How should the results be presented ?– Single point estimate– Confidence Intervals– Percentile Graphs

3307 September 2004

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”

3407 September 2004

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