Evaluating Risk Transfer: The Bootstrap Model And Other Techniques · 2013. 4. 11. · – May...
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Evaluating Risk Transfer: The Bootstrap Model And Other
Techniques
Susan J. Forray, FCAS, MAAA [email protected]
262.796.3328 Casualty Loss Reserve Seminar
September 15, 2009
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Outline § Loss Distribution
– Parameterized • Loss Ratio
• Frequency/severity
– Bootstrap Model
§ Bootstrap Model – Reserve Variability
– Prospective (i.e., Risk Transfer) Application
§ Examples
§ Considerations in Selection of Loss Distribution § Other Applications
§ Questions
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Loss Distribution
§ Parameterized – Loss Ratio – E.g., lognormal
– Most common method
§ Parameterized – Frequency/Severity – Frequency: Poisson, binomial, negative binomial, normal, etc.
– Severity: Lognormal, gamma, inverse Gaussian, etc.
– Typically developed on a ground-up basis
§ Bootstrap model – Used frequently for
• Reserve variability • Capital modeling, etc.
– Can also provide a prospective loss distribution
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Bootstrap Model
§ Main application is reserve variability – Usually a retrospective model – Can be prospective
§ Basis of model – Uses entire triangle – Calculates “scaled Pearson residual” for each accident year/
development period – Assumption is that these are independent and identically distributed – Residuals then used to simulate triangle “as it could have been”
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Bootstrap Model (cont.) § Versions of model
– Original based on paid chain ladder only (given in England/Verrall paper)
– Multiple papers since then (England/Verrall, Pinheiro, etc.)
– More sophisticated models now exist • Incurred chain ladder
• Bornhuetter-Ferguson o Paid and incurred
o Uses lognormal or other distribution for a priori • Cape Cod
– Can weight methods together
– Multiple lines of business correlated, etc. § Typical uses
– Reserve distribution
– Capital requirements
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Example 1
§ Proposed Reinsurance Treaty – Per occurrence excess of loss
– No swing rating, corridors, etc.
§ Parameter Method Assumptions – Discounted ceded loss ratio lognormally distributed
• Mean of 75%
• CV of 0.3
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Ceded Discounted Loss Ratio
Example 1 – Lognormal Distribution
Mean
90th Percentile
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Ceded Discounted Loss Ratio
Lognormal
Reinsurer Deficit
Example 1 – Lognormal Distribution
ERD = 2.2%
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Example 1 – Lognormal Distribution EXPECTED REINSURER DEFICIT
Cumulative Discounted Discounted ReinsurerProbability Loss Ratio Profit / (Loss) Deficit
90% 104.6% -4.6% 4.6%91% 106.5% -6.5% 6.5%92% 108.5% -8.5% 8.5%93% 110.8% -10.8% 10.8%94% 113.4% -13.4% 13.4%95% 116.4% -16.4% 16.4%96% 120.1% -20.1% 20.1%97% 124.8% -24.8% 24.8%98% 131.3% -31.3% 31.3%99% 142.2% -42.2% 42.2%
Avg of Above xxx xxx 1.8%Expected Value xxx xxx 2.2%
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Example 1 – Lognormal Distribution
§ Average of scenario reinsurer deficits – Will understate expected value
– If sufficient points are included, will approximate the expectation
§ Expected Reinsurer Deficit – Under parameterized distribution, can be calculated directly from parameters
– E.g., lognormal: • ERD = exp(µ + σ2/2) x Φ[(µ + σ2) / σ] - Φ(µ/ σ2)
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Example 1 – Parameter Assumptions
§ Mean – Developed from cedant data
§ Coefficient of Variation – Developed loss ratios will typically understate this
• Inherently “expected value” estimates
• Small sample
• Includes volatility due in part to market forces (could overstate CV)
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Bootstrap Model Requirements
§ Minimum – Paid triangle
§ Also helpful – Incurred triangle
– Premium/exposure
– A Priori loss distribution (for Bornhuetter-Ferguson)
– Loss trends / on-level factors (for Cape Cod)
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Bootstrap Model Variance
§ Parameter Variance – “Recreates” paid/incurred triangles as they could have been
– Develops unpaids from these using standard development methods
– Chain ladder
– Bornhuetter-Ferguson
– Cape Cod
§ Process Variance – Simulates incremental unpaids
• Typically uses Gamma (proxy for overdispersed Poisson)
• Lognormal, etc. also an option
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Bootstrap Model Variance (cont.)
Process Variance
Process &Parameter Variance
(Simulated)Parameter
Process &Parameter Variance
(Simulated) Parameter &Process Variance
Variance
(Simulated)
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Example 1 – Bootstrap Model
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Ceded Discounted Loss RatioBootstrap Lognormal
Mean
90th Percentiles
Bootstrap ERD = 3.3% Lognormal ERD = 2.2%
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Ceded Discounted Loss RatioBootstrap Lognormal
Example 1 – Cumulative Distributions
Mean
90th Percentiles
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Example 2
§ Proposed Reinsurance Treaty – Per occurrence excess of loss (as Example 1)
– Aggregate deductible of $20,000,000
– Aggregate limit of $40,000,000
– Ceded premium of $25 million (half of Example 1)
– Other assumptions the same
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Example 2 – Effect on Distribution
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Ceded Discounted Loss RatioBootstrap Lognormal
Aggregate Deductible
Aggregate Limit
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Ceded Discounted Loss RatioBootstrap Lognormal
Example 2 – Cumulative Distribution
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Ceded Discounted Loss RatioBootstrap Lognormal Reinsurer Deficit
Example 2 – Expected Reinsurer Deficit
Bootstrap ERD = 6.0% Lognormal ERD = 6.7%
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Example 2 – Bootstrap Model
§ Triangles of paid/incurred losses – Historical years restated under proposed treaty terms
– Losses stated prior to aggregates
– Include parameter & process variance in all triangle cells
§ Resulting distribution – Gross of aggregate deductible/limit
– Can adjust each simulated loss scenario for these
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Example 3 – Aggregate Excess
Small Ceded Expected Value, Large Risk Gross Distribution Net Distribution Ceded Distribution
Agg XS Layer
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Example 4 – Quota Share Reinsurance Constant Ceding Commission: Gross / Ceded / Net All Equal Variable Ceding Commission:
Ceding Commission
Gross Distribution
Net Distribution
Ceded Distribution
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Example 5 – Stop Loss Reinsurance
LOB “A”
LOB “B”
LOB “C”
Aggregate Distribution
Aggregate Excess Does Not Benefit Stop Loss
Stop Loss Layer
Aggregate Excess Does Benefit Stop Loss
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Considerations in Selection of Loss Distribution § Parameterized loss ratio distribution
– May be the simplest formulaically
– Difficult to estimate variance
§ Compound frequency/severity distribution – May be easier to estimate variance for these components
– Computationally more time-consuming
– May require simulation
§ Bootstrap model – Requires historical data
– Does not require variance or correlation assumptions
– May require working with numerous simulated scenarios
– May allow parameter estimation for parameterized distributions
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What If There’s No Risk Transfer?
§ To Account For As Reinsurance – Aggregate Cover
• Increase aggregate limit
• Decrease aggregate deductible
• Decrease ceded premium
– Quota Share
• Increase loss ratio cap
• Decrease ceded premium at higher percentiles
Ø Greater provisional ceding commission
Ø Lesser swing range
§ Use Deposit Accounting – Only option if treaty already in effect
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A Note on Other Applications § Reserving for aggregate deductibles / limits
– Example: • Per occurrence excess of loss treaty
• Developed accident year losses of $45 million
• $50 million aggregate limit – IBNR indications
• Judgmental provision, e.g.: o 30% likelihood of losses exceeding aggregate o $10 million expected value loss in excess of aggregate (if exceeded)
o Implies $3 million increase in net reserve
• Parameterized distribution o Should incorporate data to date
• Bootstrap model o Losses would most likely be stated gross of aggregate limit