1

Using Stochastic Models in Risk and Capital

Management in Life Assurance

Tuesday 5th April 2005

Craig Turnbull

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Agenda

• Introduction: Developments in the use of (internal) stochastic models in life assurance– Why now? Who wants it?– How does it work?– What questions is it used to answer?

• Assessing Risk-Based Capital for With-Profits Business – Quantifying risks and their interaction

• Using Models as a Capital Management Tool– Identifying and appraising candidate solutions

• Questions and Answers

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Introduction:Developments in the use of (internal) stochastic models

in life assurance

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What Developments?

• Global life assurance industry developing large-scale internal stochastic asset-liability models– Sophisticated arbitrage-free multi-asset models– Complex liability models

• Dynamic management rules, ‘000s model points, etc

• Particularly in UK life industry and the top 20 multinational insurance groups

5

Why Now?

• Regulatory compulsion (UK only)• Greater appreciation of risks in

guarantees in life & pensions business

• Less capital / risk appetite than 5 years ago

• Appreciation that life / pensions ALM falling behind banking industry

• Technology– Cheaper, faster

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Who Wants It?• Regulators

– FSA• Market-consistent guarantee costs (RBS / Pillar 1)• Risk-based capital assessment (ICA / Pillar 2)

– Stochastic modelling approach required in US and Canada– Will other regulators follow FSA regime?

• Accountants– IAS, FRS 27 (FRS 17)– European Embedded Value

• Credit rating agencies– Risk-based capital adequacy – Calculation and communication

• Internal management– Economic capital allocation and performance measurement– Risk / capital management– Product design / pricing

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What can it deliver?

• Quantification of costs, risks and capital requirements– Relative size of drivers– Risk dynamics

• Diversification, interaction, non-linearity

• Identification and appraisal of candidate management solutions– Informing trade-offs

8

How Does it Work?

Office - Specific Liability Features,

Management Strategies

(Market – Consistent) Economic Scenario

Generator

Model Office Software

Market-Consistent Balance Sheet /

Capital Assessment /etc

Market Prices / Best-

Estimates

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Assessing Risk-Based Capital Requirements

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Approaches to measuring RBC

• What approaches can be taken to assessing risk-based capital requirements for insurance liabilities?– Run-Off

• Capital required to fund projected cashflow shortfalls with a specified level of confidence

– Value-At-Risk• Capital required to fund a future market-consistent

liability value with a specified level of confidence– Funding the cost of transferring market risk to market

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With-Profit Implementation challenges

• Run-Off– Estimating long-term asset return tails

• Scarcity of relevant data

– Projecting market-consistent balance sheet forward over multiple time horizons

• Important if m-c balance sheet is a driver of decision rules

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With-Profit Implementation challenges

• VaR– Estimating 1-year asset return extreme tails

• Conditional on recent market behaviour, option prices?

– Nested simulations required (in theory!!)– Practical (approximate) implementation

approaches

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Recent 1-yr FTSE 100 Option-Implied Volatilities

0%

5%

10%

15%

20%

25%

30%

35%

Mar-03 Jun-03 Sep-03 Dec-03 Mar-04 Jun-04 Sep-04 Dec-041-Y

r F

TS

E 1

00

Op

tio

n-I

mp

lie

d V

ol

100%

85%

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Implied Equity Falls

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

Mar-03 Jun-03 Sep-03 Dec-03 Mar-04 Jun-04 Sep-04 Dec-04

Imp

lie

d E

qu

ity

Fa

ll 95th Percentile

99.5th Percentile

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Individual Capital Assessment

• Predominantly VaR-style definitions used currently– Capital required to produce 99.5% confidence

that realistic liabilities are funded after one year– Given the above difficulties, how is VaR being

implemented for With-Profits?• Unconditional asset modelling

• Broadly two implementation approaches for VaR– Univariate

– Multivariate

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ICA for With-Profits – Univariate Approach

• Calculate 99.5th percentile events for each risk factor, and obtain capital requirements for each risk factor

• Calculate total capital requirement by applying a correlation matrix to the capital requirements for each risk factor

• This assumes:– Risks are linear

– Risks do not interact

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ICA for With-ProfitsMultivariate Approach

1. Estimate sensitivities of realistic balance sheet to each risk factor

2. Use these to project RBS to end-year (using stochastic asset model)

3. Read off 99.5th percentile discounted loss

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Illustrative Example

• Liability is a 10-yr equity total return put option with strike at-the-spot

– Interest rate of 5%

– Volatility of 20%

– Nominal of £1,644m

– Current market value of put option of £100m

• Assume assets backing guarantee cost are invested in equities

– And any assets required in excess of guarantee cost are invested in cash

19RBC under Univariate approach:

Risk Contributions• 99.5th percentile equity return is -36%

– Liability increases from 100 to 235– Assets fall from 100 to 64– Equity capital requirement is 163

• [(235-100) – (100-64)]/ 1.05

• 99.5th percentile rise in option-implied equity vol is 5%

– Liabilities increase from 100 to 160– Assets do not change in value– Vol capital requirement is 57

• 99.5th percentile interest rate fall is 1.5%– Liabilities increase from 100 to 157– Assets do not change in value– Interest rate capital requirement is 54

20RBC under Univariate approach:

Allowing for diversification• Sum of capital requirements is £274m• But this assumes perfect correlation• Assume correlations of:

– -0.3 between equities / interest rates– -0.4 between equities / option-implied vol– +0.1 between interest rates / implied vol

• Implies capital requirement of £185m– Diversification benefit of 32%

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RBC under Multivariate approach

• Use a number of sensitivity tests:– 20% equity fall increases liabilities from 100 to 159– 40% equity fall increases liabilities from 100 to 259– 0.85% interest rate fall increases liabilities from 100

to 130– 2% option-implied interest rate rise increases

liabilities from 100 to 124– Could use many more, e.g. 20% equity fall after 1%

interest rate fall, etc…

• Use ‘greeks’ to project liabilities in each 1-yr asset simulation

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Asset / Liability Projection as a function of equity returns

-300

-200

-100

0

100

200

300

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

UK Equity Return

Ass

ets

- L

iab

ilit

ies

(£m

's)

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Asset / Liability Projection as a function of vol changes

-300

-200

-100

0

100

200

300

-6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5% 6%

Option-implied Equity Vol Change

Ass

ets

- L

iab

ilit

ies(

£m)

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Capital Requirement as a function of equity return

0

50

100

150

200

250

300

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

UK Equity Return

Ad

dit

ion

al

Ca

pit

al

Re

qu

ire

d

(£m

's)

ICA

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RBC: Concluding Thoughts

• Current implementations of the multivariate approach produce similar capital requirements to univariate approach

– In example, capital requirements were £187m and £185m

• But mulitvariate approach is inherently more flexible and transparent

– Sophistication can be developed incrementally– More useful as a risk management tool (identifying

and appraising candidate management solutions)

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Correlations: An Aside

• Most life offices are exposed to falls in equities and falls in interest rates

– (Also true for Defined Benefit pension funds)

• Negative correlation assumption between equities and interest rates implies ‘natural hedge’

– i.e. Big diversification benefit

• What if we reduce equity / interest rate correlation?

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Impact of Correlations on RBC

0

50

100

150

200

250

300

Individual -0.3 correlation 0 correlation +0.3 correlation

Option-Implied VolatilityInterest RatesEquitiesTotal

89 7562

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RBC Benefit of Removing Interest Rate Risk

-10

-5

0

5

10

15

20

25

-0.3 correlation 0 correlation +0.3 correlation

Removing interest rate risk changes capital requirement to 193

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What’s the ‘right’ correlation?

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

%C

orr

ela

tio

n

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Using Stochastic Models as a Capital Management

Tool

31Appraising hedging solutions

Matching the risk exposures

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1% Equity Fall 0.25% IR Rise 0.5% Equity Vol Rise

Cha

nge

in B

alan

ce S

heet

(£m

's)

Increase in GuaranteeCosts

Equity FuturesPosition (Short £65m)

5-Yr Bond FuturesPosition (Short £25m)

Option strategy (Equityput option, equity andbond futures)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1% Equity Fall 0.25% IR Rise 0.5% Equity Vol Rise

Increase in GuaranteeCosts

Equity FuturesPosition (Short £65m)

5-Yr Bond FuturesPosition (Short £25m)

Option strategy (Equityput option, equity andbond futures)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1% Equity Fall 0.25% IR Rise 0.5% Equity Vol Rise

Increase in GuaranteeCosts

Equity FuturesPosition (Short £65m)

5-Yr Bond FuturesPosition (Short £25m)

Option strategy (Equityput option, equity andbond futures)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1% Equity Fall 0.25% IR Rise 0.5% Equity Vol Rise

Increase in GuaranteeCosts

Equity FuturesPosition (Short £65m)

5-Yr Bond FuturesPosition (Short £25m)

Option strategy (Equityput option, equity andbond futures)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1% Equity Fall 0.25% IR Rise 0.5% Equity Vol Rise

Increase in GuaranteeCosts

Equity FuturesPosition (Short £65m)

5-Yr Bond FuturesPosition (Short £25m)

Option strategy (Equityput option, equity andbond futures)

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0

5

10

15

20

25

30

35

40

45

50

Asset sharemix

Cash Equityfutures(RCM)

Equityfutures(ICA)

Optionstrategy

Assets in excess of asset share

Ca

pit

al

req

uir

em

en

t (£

m's

)

ICA

RCM

Appraising hedging solutions

Estimating economic capitalNeutralising equity exposure:

reductions in ICA and RCMOption strategy

improves gamma and vega matches:

significant reduction in ICA, no impact on

RCM

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Monitoring and managing a hedging strategy

• Liability risk exposures will change over time as financial markets move

• Any hedge is unlikely to be static for long periods. The extent to which this is the case will depend on choice of hedging solution – e.g. how well matched is equity gamma?

• Hedging performance can be regularly monitored (e.g. quarterly) and, when appropriate, re-balanced.

800

900

1000

1100

1200

1300

1400

1500

0% 10%Equity Fall

Re

alis

tic

Gu

ara

nte

e C

os

t

Resilience Scenario Yield Curve

Current Yield Curve

93

194 e.g. cash guarantee’s equity delta can double when the yield curve falls by 100bp.

Impact of Interest Rate and Equity Market Interaction on Realistic Guarantee Cost

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Concluding Thoughts

• Changes in regulatory / accounting / rating agency regimes mean significant step towards convergence in various capital / value / profit measures

• Reduces constraints to managing economic risks

• New valuation tools allow capital market solutions to be more effective at mitigating market risks in life assurance business

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Questions and answers