Measuring Value in Reinsurance

1. Measuring Value in Reinsurance Gary Venter Guy Carpenter Instrat

2. Recovery Method • Compare premiums to recoveries • If you recovered more than you paid, reinsurance was effective

3. Recovery Method, con’t. • Try on your homeowners policy • Most insureds get no value by this measure • Ignores value of security provided • Ignores reduced cost of financing • Same for reinsurance • Few good deals disappear quickly

4. No Recovery is Best Result • Frequent recoveries indicate retention is too low • Trading dollars • Losing on expenses • Compare to buying at the 1 in 20 level • More efficient • With good luck, could be over 20 years between recoveries • Net results better with such luck - but no recoveries

5. Value of Reinsurance Comes from Stability Provided • Stability increases earnings • Replaces other sources of capital • Reduces cost of other financing • Thru improved claims-paying ratings • Stability increases earnings multiple • Improves analysts’ confidence in projected earnings

6. Stability ExampleStandard Deviation Reduces from 5% to 2% One in 200 combined ratio reduces from 120% to 111% Probability of exceeding 110% reduces from 22.5% to 5.5% But expected combined increases to 106.5 from 106.0

7. Reinsurance as Financing • Capital is needed to finance losses • Bonds • Sale of equity • Retained earnings • Reinsurance • Reinsurance reduces need for other financing • Total loss potential can be covered by a combination of reinsurance, surplus, and contingent financing • Provides a convenient source of capital

8. Reducing Cost of Other Financing • Lines of credit, conditional debt, debt all less expensive if repayment risk is less • Similar to your homeowners policy’s impact on mortgage costs • Risk reflected in debt ratings

9. Impact on Claims Ratings • S&P and Best’s Ratings depend on financing available, including reinsurance • Strength of financing directly relates to claims paying ability • Higher rating improves access to markets and in some cases allows higher rate levels

10. Median Results by S&P Rating

11. Cost vs. Value • Measure cost by recovery method, or better by premium less average expected recoveries • Compare costs to value • Measure value by stability provided • Cost/Benefit analysis requires quantitative modeling

12. ABCD Insurance • $33M of property and liability coverages • $14M in casualty coverages • Expected loss ratio 78% • $19M in property covers • Expected loss ratio 63% • Total expected losses are $22.9M • Expense ratio 23% • Total expected combined ratio 92%. • 4M x 1M casualty cover: $4.41M • 17M x 3M per-risk property cover: $2.36M • Cat program: 95%of 24M x 1M for $1.53M with one reinstatement at 100% • The cat program is designed to cover at least up to the 1 in 250 year cat event • Total $8.3M in ceded premiums prior to any reinstatementpremiums

13. Alternative Program • Stop loss 20M x 30M for $2M • Which is better? • How can you tell? • Simulate many years of possible results, test how bad it can be

14. Compare Average Results

15. What about the Bad Years? Current Program Stop Loss No Reinsurance Probability Net Underwriting Income*

16. Probability BARE Current Stop Loss 0.00% -49,263,333 -23,198,963 -32,243,333 0.25% -25,817,548 -12,416,243 -9,439,234 0.50% -21,827,529 -10,377,108 -6,311,695 0.75% -17,837,510 -8,337,973 -3,184,156 1.00% -13,847,491 -6,298,838 -56,618 1.25% -12,641,527 -5,703,459 237,924 1.50% -11,677,654 -5,290,176 286,117 1.75% -10,713,781 -4,876,893 334,311 2.00% -9,749,908 -4,463,610 382,505 4.00% -5,892,701 -2,551,287 575,365 6.00% -3,602,653 -1,315,561 689,867 8.00% -2,008,347 -409,204 769,583 10.00% -686,845 284,986 835,658 12.00% 416,042 951,819 890,802 14.00% 1,448,699 1,464,523 942,435 16.00% 2,415,661 1,919,933 990,783 18.00% 3,226,822 2,388,329 1,251,605 20.00% 3,905,868 2,802,539 1,925,868 22.00% 4,554,807 3,190,684 2,574,807 24.00% 5,209,039 3,549,185 3,229,039 25.00% 5,513,974 3,713,920 3,533,974 26.00% 5,832,081 3,880,394 3,852,081 28.00% 6,371,517 4,205,322 4,391,517 30.00% 6,891,421 4,514,526 4,911,421 32.00% 7,401,904 4,827,688 5,421,904 34.00% 7,856,716 5,146,708 5,876,716 36.00% 8,321,687 5,428,461 6,341,687 38.00% 8,761,854 5,694,960 6,781,854 40.00% 9,208,534 5,962,559 7,228,534 42.00% 9,639,097 6,244,632 7,659,097 44.00% 10,021,333 6,495,969 8,041,333 46.00% 10,439,457 6,780,995 8,459,457 48.00% 10,823,625 7,026,301 8,843,625 50.00% 11,191,515 7,269,232 9,211,515 In Table Format

17. Stability versus CostProblem with Standard Deviation 1 in 500 1 in 100 1 in 20 1 in 4 1 in 4 Net Underwriting Income 1 in 20 1 in 100 1 in 500

18. Combined Ratio Comparison No Reinsurance Stop Loss Current Program Probability Combined Ratio

19. Cost/benefit Example Bare Current Stop Loss Option B 1 in 250 probability level Benefit Efficient Frontier Cost

20. Value Summary • Value of reinsurance is in stability gain • Usually has some cost in long run • Stability increases value, so can be cost effective • Can compare efficiencies of alternatives by comparing cost vs. containment of adverse results

21. Modeling Issues • Information risk • Correlation • Assets • Liabilities

22. Information Risk Part 1: Estimation Uncertainty • Probability distributions are defined by a few mathematical parameters • These are estimated from historical data • Possible errors in this estimation process can be quantified • Can simulate using the probability of the parameters being the right ones • Simulate parameters then losses

23. Quantifying Estimation Uncertainty for Severity High mean Heavy tail

24. Information Risk Part 2: Projection Uncertainty • Trending losses to time of exposure is also an uncertain process • Historical losses used in fitting have some inherent uncertainty • Fit is not perfect, so trend has more uncertainty • This uncertainty increases over time • Changes in economic environment can add still more randomness

25. Quantifying Uncertainty in Future Costs as Spread around Trend Line Spread increases in projection History Projection

26. Correlation:The Normal Copula • N(x) = N(x;0,1) • B(x,y;a) = bivariate normal distribution function,  = a • Let p(u) be the percentile function for the standard normal: • N(p(u)) = u, dN(p(u))/du = N’(p(u))p’(u) = 1 • C(u,v) = B(p(u),p(v);a) • C1(u,v) = N(p(v);ap(u),1-a2) • c(u,v) = 1/{(1-a2)0.5exp([a2p(u)2-2ap(u)p(v)+a2p(v)2]/[2(1-a2)])} • t(a) = 2arcsin(a)/p • a: 0.15643 0.38268 0.70711 0.92388 0.98769 • t: 0.10000 0.25000 0.50000 0.75000 0.90000

28. Heavy Right Tail Copula • C(u,v) = u + v – 1 + [(1 – u)-1/a + (1 – v)-1/a – 1]-a a>0 • C1(u,v) = 1 – [(1 – u)-1/a + (1 – v)-1/a – 1] -a-1(1 – u)-1-1/a • c(u,v) = (1+1/a)[(1–u)-1/a +(1– v)-1/a –1] -a-2[(1–u)(1–v)]-1-1/a • t(a) = 1/(2a + 1) • Can solve conditional distribution for v

30. Partial Perfect Correlation Copula Generator • Assume logical values 0 and 1 are arithmetic also • h : unit square  unit interval • H(x) = 0xh(t)dt • C(u,v) = uv – H(u)H(v) + H(1)H(min(u,v)) • C1(u,v) = v – h(u)H(v) + H(1)h(u)(v>u) • c(u,v) = 1 – h(u)h(v) + H(1)h(u)(u=v)

31. h(u) = (u>a) • H(u) = (u – a)(u>a) • t(a) = (1 – a)4

32. h(u) = ua • H(u) = ua+1/(a+1) • t(a) = 1/[3(a+1)4] + 8/[(a+1)(a+2)2(a+3)]

33. Tail Concentration Functions • L(z) = Pr(U<z,V<z)/z2 • R(z) = Pr(U>z,V>z)/(1 – z)2 • L(z) = C(z,z)/z2 • 1 - Pr(U>z,V>z) = Pr(U<z) + Pr(V<z) - Pr(U<z,V<z) • = z + z – C(z,z). • Then R(z) = [1 – 2z +C(z,z)]/(1 – z)2 • Generalizes to multi-variate case

35. Cumulative Tau • t = –1+40101 C(u,v)c(u,v)dvdu • J(z) = –1+40z0z C(u,v)c(u,v)dvdu/C(z,z)2 • Generalizes to multi-variate case

37. Need to produce arbitrage-free scenarios with yield curves generated according to probability of occurring Even three-factor model of short-term dynamics does not capture right distribution of all yield spreads Can get closer to right dynamics by adding two stochastic market risk factors Illustrated below by looking at historical distribution of yield spreads as a function of the short-term rate Assets: Start with Yield Curve

38. Historical Yield Spread 3-Month to 1-Year Flat with notable variability

39. Historical 1-Year to 3-Year Yield Spread Higher short rates produce lower or even negative yield spread, again with notable variability

40. One-factor CIR Model Compared to Historical • 3-month to 1-year spread • Too much slope • Too little spread

41. Three-Factor Model Compared to Historical • 3-month to 1-year spread • Slope and spread match historical

42. Three-Factor Compared to Historical Long Spread • 3-year to 10-year spread • Too much slope • Too little spread

43. Corrected by Adding Variable Market Price of Risk

44. Simulating reserve development requires knowledge of process that is generating development Many models just assume one Earlier paper developed a 64-way classification system for emergence models Can test data to see which model is generating the emergence for each segment of business Liabilities:Measuring Loss Development Risk

45. Is development a function of losses already emerged? Is development purely multiplicative? Are there calendar year (diagonal) effects? Are parameters stable - e.g. stable factors? Are disturbances normally distributed? Is there a constant variance of disturbances? Or is variance proportional to some measure? Test these by goodness of fit Six binary issues produce 26 = 64 classes

46. Fit Model Simulation Formula 157,902 CL qw,d = fdcw,d + e 81,167 BF qw,d = fdhw + e 75,409 CC qw,d = fdh + e 52,360 BF/CC qw,d = fdhw + e 44,701 BF-CC+ qw,d = fdhwgw+d + e Sample Emergence Models and Fits to a Data Set

47. Need to model movements of short-term rate as well as yield curve Most models in use do not capture all known dynamics Instrat has state-of-the-art interest rate model Measuring Interest Rate Risk

48. To get historical movements you need: Mean reversion Rate-sensitive volatility Stochastic changes in volatility Stochastic changes in mean reverting to These are incorporated in Instrat’s three-factor short-term rate model Simpler models do not capture full dynamics Short-Term Rate Dynamics

49. finis