Capital Allocation for Reinsurance Pricing

presentation by

Ira Robbin

Casualty Actuaries in Reinsurance

Seminar on Reinsurance, Boston

May 19-20, 2008

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Ground Rules and Disclaimers

If anything I say gets me in trouble in the future, you are all witnesses that I never said it. Nothing I say should be taken too seriously. Don’t come near to violating Anti-trust guidelines!No statements of corporate opinion will be made or should be inferred. Ask questions to clarify the material anytime.If you rely on ideas contained in this presentation and lose your shirt, remember I am not in the clothing trade.

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Overview of Discussion

Overall goal: raise awareness of issues that arise in allocating capital for reinsurance treaty pricing applications

Will provide opinions on answers to some key questions RORAC contextRisk metricsCapital for Property CAT treatiesCapital for Casualty treatiesTreaty featuresLoss ModelsGauntlet of testsConclusion: It is harder than you think to get this right!

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Basic Context

Capital allocation in pricing Hypothetical division: not an actual segregation of real funds Nothing is actually allocated: total capital stands behind all

contractsTheoretical required amount for each dealUse in corporate pricing process

Company may decline to write/ impose extra authority clearances on deals with pricing below indicated

Useful in price monitoring: follow indicated vs market price Creates incentives – are these the ones that are intended?

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RORAC Pricing Approach

Return on Risk-Adjusted Capital (RORAC) Theoretically required capital, not actual!

Indicated Price is price to achieve target ROE Target ROE is set by management

Should be the same for all deals and LOBs Should be sum of risk-free rate+ risk margin

Contrast with Risk-Adjusted Return on Capital (RAROC)

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Risk-Sensitive Pricing from RORAC

More risk More capitalMore capital Higher theoretical price needed

Higher price needed to cover margin on larger amount of capital

Risk-sensitive capital leads to risk-sensitive theoretical target pricing Actual market price driven by supply and demand

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Other Risk Pricing Approaches

Process versus parameter risk Theory says “No charge for process risk” Theory CAT gets a small capital allocation wrong

Non-Diversifiable risk pricing + CAPM CAPM produces a target return: r = r0 + ( rm – ro) ￚBeta is Cov of outcome with stock market

A RAROC approach, not a RORAC approach. Theory Return on CAT should equal stock market return,

on any amount of capital wrongRole of capital: amount of capital impacts insurance return, but is essentially irrelevant in CAPM stock pricing

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Capital Allocation Issues

Choice of risk metric Calibration Differing risks by LOB

Property – CAT risk Casualty – Mass tort/reserve risk/capital duration

Treaty provision adjustments Eg. reinstatements

Stand alone basis vs treaty impact on portfolio Allocation of diversification benefits

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Risk Metric Classification Properties

Coherent or not Coherence = Sensible scaling, shifting, and

diversification benefit properties Tail Focused vs Full Distribution

Capital consumption perspectives Explicable

Can it be sold to management/ financial gurus? Do the parameters have any intuitive meaning?

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Risk Metrics

Variance and Standard Deviation – not coherent Includes favorable and adverse deviations

VaR – Value at Risk – not coherent VaR(A+B) can exceed VaR(A) + VaR(B)

TVaR – Tail Value at Risk- is coherent TCE = Tail conditional expectation- is coherent Captures events in the extreme tail

Wang transform –is coherent What does the power parameter represent?

Capital Allocation by Percentile Layer-Bodoff method “Hold capital for the 250 year event” versus “Hold capital

even for the 250 year event”

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Calibration

Total capital over all treaties should reflect management’s overall risk/return perspective Regulatory and rating agency constraints

IRIS and RBC S&P and Best’s

Eliminate capital for investment risk – assume risk-free rate in pricing modelDuration of capital for long-tail lines – need for capital to cover reserve inadequacy.

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Property CAT Treaty Capital

Per Occ Loss vs Annual Agg Loss Capital needed to cover large loss event OEP or Capital needed to cover a bad year AEP Example: Katrina vs KRW

Treaty Loss vs Treaty Impact on portfolio Stand-alone treaty – no credit for diversification Should pricing be used to manage aggregation? OEP Impact is sometimes $0 or negative

Danger with OEP Impact Promotes writing of risky business in low PML zones

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Casualty Treaty Capital

Reserve risk Total historic reserve inadequacy is not random In concept, only need capital for inherent reserve

uncertaintyDuration of capital

How to reflect long-term commitment of capital? ROE = PVI/PVE is one solution See “IRR, ROE, and PVI/PVE” paper by Robbin

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Treaty Provision Adjustments

How to measure downside risk by treatyTreaty initial loss distribution Treaty provisions

Reinstatements, Swing Rating, Sliding Scale Commission, Loss Corridor, Profit Commission, etc.

Provisions can impact commissions and premiums as well as losses

Some may not change expected amounts Some reduce downside risk; others share gains

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Model UW Loss to Capture Net Treaty Risk

UW Loss = Loss + Expense - PremiumLoss alone does not fully describe treaty risk

Doesn’t capture impact of treaty features UW Loss provides a more complete picture

General way to handle different features Same risk for alternative deals with same UW Loss

distribution Note sign convention: negative UW Loss is a gain See Robbin and DeCouto paper, “Coherent Capital for

Treaty ROE Calculations”

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Reinsurance Loss Models

Attritional loss May have lower truncation ￚ e.g no loss below 25% LR

Usually described via lognormal, gamma, and other well-known programmable distributions.

Excess loss Low frequency/high severity potential Focus on per risk XOL impact

CAT Need to convert event lists to event frequency and severity

per event

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Simulation Modeling of Loss

A brute-force adaptable approach Model validation

Run single trials and extreme cases- check sample output Black box syndrome

Confuse number of trials with accuracy of parameters Neglect possibility structure is wrong

Practical concerns Convergence issue - keep running till the answers stabilize? Reproducibility – fix the random seed? Pricing alternatives – is differential larger than error bar?

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Modeling Losses via Points and Probabilities (PnP)

Insurance loss distributions suitable for PnP modeling Mass at zero No mass below a truncation point Conditional distribution described by a mix of tractable

parametric models ( gamma, lognormal, pareto and so forth) Technique

Choose 100 points of interest including zero Compute Limited Expected Values (LEVs) Derive Probs to match LEVs

Reproducible

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Gauntlet of Tests

LOB effectsChange in share

Does capital change in proportion to share? Change in reinsurance rate adequacy

Should rate improvement decrease required capital? Net rated deals

How much capital is needed for ceding commission? Reinstatements

Do they reduce or increase reinsurer risk?

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Conclusions

Allocating capital is difficult Presents major theoretical and practical challenges

Know before you go Run all current treaties through any proposed model Have line pricing actuaries look at pricing

differentials – what incentives will it create? Calibrate in advance

The proof of the capital method is in the pricing!