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.