Risk Transfer Testing of Reinsurance Contracts

1
Risk Transfer Testing of Reinsurance Contracts

• A Summary of the Report by the CAS Research
  Working Party on Risk Transfer Testing
• CAS Ratemaking Meeting
• March 2008
• David L. Ruhm, FCAS

2
Background

• AAA Committee on Property and Liability Financial
  Reporting (COPLFR) requested input on risk
  transfer testing, 2005
• CAS formed Working Party on Risk Transfer Testing
  to respond to AAA request (Michael Wacek, chair)
• Working Party Report issued, Summer 2005
• More developments since see AAA and NAIC
  websites

3
Background, continued

• Paper on Working Party Report published in
  Variance, Spring 2007 (Ruhm Brehm)
• Paper briefly describes 2 risk measurement
  methods in Working Party Report
• Expected reinsurer deficit (ERD)
• Right-tailed deviation (RTD)
• Paper also describes risk coverage ratio (RCR)
  method, which is related to ERD

4
Scopes of WP report, Variance paper

• Working Party took accounting rules as given
• Merits of accounting rules not debated
• Focus was on risk transfer testing methods
• Variance paper provides a brief summary of some
  key material from WP Report
• Also includes risk coverage ratio (RCR)
• Interested parties should read the full WP Report

5
Risk measurement Practical uses

• Better risk control, including ERM context
• You can manage only what you can measure
• Pricing and strategic planning
• Ensure expected profit is adequate compensation
  for amount of risk assumed
• Risk-based capital allocation
• Capital risk ? adequate price adequate ROC

6
Risk measurement Accounting

• If a contract transfers risk it can receive
  insurance accounting treatment
• If not, premiums are treated as deposits and
  net results are amortized into earnings over time
• Insurance accounting is often preferred
• Risk transfer requirements are similar for GAAP
  and Stat
• GAAP FAS 113
• Stat SSAP 62

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SSAP 62 highlights

• Reinsurer must assume significant insurance
  risk
• Requires non-remote probability of significant
  variation in amount timing of payments by
  reinsurer
• Reasonably possible that reinsurer may realize
  a significant loss
• Based on NPV of all cash flows between ceding
  assuming companies under reasonably possible
  outcomes (emphasis added).

8
WP proposed testing framework

• Three-step process
• 1. Determine if contract transfers substantially
  all the risk if so, stop.
• Assumed downside essentially same as cedants
  original
• 2. Determine whether or not risk transfer is
  reasonably self-evident if so, stop.
• E.g., cat x/s, x/s w/no loss sensitive features
• 3. Calculate recommended risk metrics and compare
  values to critical threshold values.

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Expected reinsurer deficit (ERD)

• Uses probability distribution of net economic
  outcomes (NPV of cash flows)
• Critical point 0 gain economic breakeven
• Formula
• ERD pT / P
• p probability of net loss
• T average conditional loss severity
• P expected premium

10
Expected reinsurer deficit (ERD)

• Concepts inherent in ERD
• Risk zone is area in distribution where
  economic loss exists in terms of negative NPV
• Risk loss frequency x average loss severity
• Base in denominator expected premium, measuring
  risk per 1 premium

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ERD example

• Simple example of ERD calculation
• Aggregate excess 250m excess of 500m
• Settlement 1 year after inception
• Investment yield 4.00 (1-yr risk-free rate
  available at inception)
• Premium 10m at inception

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ERD example

• Loss distribution (dollars in 000)
• Ceded loss Probability NPV(gain)
• 0 96 10,000
• 50,000 2 ( 38,077)
• 150,000 1 (134,231)
• 250,000 1 (230,385)
• 5,000 Expected value 5,192
• Condl loss severity (110,193)

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ERD example

• Simple example of ERD calculation, continued
• Probability of net loss p 4
• Average conditional loss severity
• (38,077 x 2 134,231 x 1 230,385 x 1) / 4
• T TVaR(96) 110,193
• ERD pT / P (4) (110,193) / 10,000 44.1
• By comparison, 10 chance of 10 loss 1.0 ERD

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ERD steps

• 1. Produce the probability distribution of net
  present value gain, including all flows (real
  examples have more flows).
• 2. Identify the risk zone part of the
  distribution containing net losses.
• 3. Measure probability of loss and average
  conditional severity when it occurs.
• 4. Apply the ERD formula.

15
Comparisons to other metrics

• Other popular metrics have a similar structure
• Based on distribution of a key financial item
• Specific threshold point of the distribution
• Measurement of frequency and/or severity
• VaR (value-at-risk)
• Key financial item net gain / (loss) of capital
• Threshold point Percentile, such as 5th
• Measurement is severity of percentile point
• What level of loss is possible at an outside
  chance?
• 10/10 rule VaR(90) gt 10 of premium
• Fixes frequency independently of particular
  contracts details
• Doesnt measure severity beyond percentile

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Comparison to other metrics

• TVaR (tail value-at-risk), CTE (conditional tail
  expectation)
• Key financial item net gain in capital, or net
  economic gain
• Threshold point Percentile, such as 5th
• Measurement is average severity beyond percentile
  point (tail)
• Whats the average loss of capital in the worst
  5 of cases?
• Fixes frequency independently of particular
  contracts details
• Doesnt capture the likelihood of a net loss
• ERD connection T TVaR(1-p), p probability of
  loss
• 10/10 rule A contract passing 10/10 will pass a
  1 ERD test, but not the other way around cat
  excess example

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Risk coverage ratio (RCR)

• Replace ERDs premium denominator with expected
  gain from NPV distribution (EG in formulas
  below)
• Formulas
• As risk per 1 of return
• RCR, form pT / EG
• As expected profit per unit of risk assumed
• RCR EG / pT
• All components come from the economic gain
  distribution
• Risk / return metric on economic value

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RCR example

• Same example as above
• Probability of net loss p 4
• Average conditional loss severity T 110,193
• EG Expected gain 5,192
• RCR pT / EG (4) (110,193) / 5,192
  84.9
• Risk concentration embedded in expected return
  84.9

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Advantages / applications

• Advantages of ERD and RCR
• Cutoff point is economic breakeven, rather than a
  statistical percentile
• Realized impact of risk on companies is in
  dollar, rather than percentile, terms
• Includes all loss events, rather than only the
  most extreme events
• Captures both frequency and severity in one
  metric
• RCR is not affected by traded dollars in
  premium
• RCR measures the risk/return tradeoff in terms of
  economic gain
• Applications of RCR
• Risk-based pricing
• Risk-based capital allocation (see paper for
  reference)

20
Right-tailed deviation (RTD)

• Some Working Party members prefer risk measures
  based on distributional transforms over ERD
• Transforms may have added benefits, some added
  complexity
• Right-tailed deviation (RTD) proposed by Shaun
  Wang
• Define F(x) 1 1 F(x) 0.5
• F is F with the tail stretched out a
  risk-loaded distribution
• F(x) F(x), which means E E
• RTD E E risk load

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RTD example

• Loss distribution (dollars in 000)
• Ceded loss F(x) F(x)
• 0 96 80
• 50,000 98 86
• 150,000 99 90
• 250,000 100 100
• Expected value 5,000 34,000
• RTD 34,000 - 5,000 29,000

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RTD example

• RTD risk transfer test
• Maximum qualified premium a(RTD)
• a parameter could be between 3 and 5 WP observed
  4 may be too low.
• In example, using a 5
• Maximum qualified premium 145m

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RTD advantages

• F(x) is a new loss distribution all the
  usual methods apply
• Easy to risk-price layers of coverage
• Other advantages see Wangs papers
• Maximum qualified premium concept opens door to
  qualifying part of premium in some cases, instead
  of all or nothing

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Conclusion

• The WP Report is a significant contribution to
  the literature on risk transfer
• Defined a structured process to narrow down
  contracts that have to be tested
• Described two risk metrics that appear superior
  to the 10-10 test ERD and RTD
• 1 ERD suggested as one possible threshold

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Conclusion

• Further research recommended
• Level 1 Consensus thresholds
• Level 2 Other methods, including quantitative
  definitions of terms and incorporating parameter
  uncertainty
• (Paper only) 3rd research area Develop the
  actuarial perspective on risk transfer,
  independent of current accounting rules.