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
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
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
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
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
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
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).
WP proposed testing framework
• Three-step process– 1. Determine if contract transfers “substantially all
the risk” – if so, stop.• Assumed downside essentially same as cedant’s 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.
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
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
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
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
Cond’l loss severity ($110,193)
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
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.
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%) > 10% of premium– Fixes frequency independently of particular contract’s details– Doesn’t measure severity beyond percentile
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”)
– “What’s the average loss of capital in the worst 5% of cases?”
– Fixes frequency independently of particular contract’s details
– Doesn’t 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
Risk coverage ratio (RCR)
• Replace ERD’s premium denominator with expected gain from NPV distribution (“E[G]” in formulas below)
• Formulas:As risk per $1 of return:
RCR, % form = pT / E[G]
As expected profit per unit of risk assumed:RCR = E[G] / pT
• All components come from the economic gain distribution• Risk / return metric on economic value
RCR example
• Same example as above– Probability of net loss = p = 4%
– Average conditional loss severity = T = $110,193
– E[G] = Expected gain = $5,192
– RCR % = pT / E[G] = (4%) (110,193) / 5,192 = 84.9%
– Risk concentration embedded in expected return = 84.9%
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)
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
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
$34,000RTD = $34,000 - $5,000 = $29,000
RTD example
• RTD risk transfer test:Maximum qualified premium = α(RTD)
• α parameter could be between 3 and 5; WP observed 4 may be too low.
• In example, using α = 5:Maximum qualified premium = $145m
RTD advantages
• F*(x) is a new “loss” distribution – all the usual methods apply– Easy to risk-price layers of coverage– Other advantages – see Wang’s papers
• “Maximum qualified premium” concept opens door to qualifying part of premium in some cases, instead of “all or nothing”
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
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.