Enterprise Risk Modeling

1. Enterprise Risk Modeling Getting the Risk Right – Problems and Pitfalls Gary Venter, July 2002

2. Overview • Common problems and options for improving enterprise models • Capturing the risk • Key details needed to get risk right • Capital need and capital allocation • Critical to business managers • Alternative methods may improve rationality of approach

3. Issues • Assets • Reserves • Parameter risk and event risk • Correlation • Capital needed and allocation

4. 1. Asset Issues • Arbitrage-free models • No reward without some risk • Probabilistic reality • Modeled scenarios consistent with historical patterns • Balancing asset and underwriting risk

5. Arbitrage-Free Yield Curves • Long-term rates built from market expectations of short-rate changes plus a risk charge • Financial theory specifies required features of the risk charge • Called market price of risk • Adds a usually upward drift to the short rate to get longer term rates

6. Why No Arbitrage Is Important • Key element of modern financial analysis • Part of getting right distribution of scenarios • Having arbitrage possibilities in scenario set distorts any optimization towards the arbitrage strategies

7. Balancing Asset and Underwriting Risk • Look at efficient investment frontier and how that changes with different reinsurance programs • Can review offsetting insurance risk with investment risk for optimal balance by adjusting reinsurance program to fit best to investment portfolio

8. Constrained Asset Efficient Frontier with Current Reinsurance Program Frontier of Constrained After-Tax Operating Income / Assets 1-year horizon: 2003 0.01 0.005 0 -0.005 -0.01 Mean -0.015 -0.02 Frontier -0.025 Company -0.03 0.02 0.03 0.04 0.05 0.06 0.07 Std Deviation

9. Probability of Returns on Frontier Vary reinsurance and investments Ef. fr Q01 Q05 Variability of portfolios on after tax OI/Assets frontier Q25 Q75 0.18 Q95 Q99 0.15 0.12 0.09 0.06 0.03 0.00 -0.03 -0.06 -0.09 -0.12 -0.15 0.02 0.03 0.04 0.05 0.06 0.07 STD

10. 2. Reserve Issues • Loss reserving models • UEPR and current underwriting risk • Time capital must be held

11. Loss Reserving Models • Actuaries start with development factors and Bornheutter method • Many more models are out there • Key issue is measuring correlation between inflation and development • E.g., see 1998 PCAS Testing the Assumptions of Age-to-Age Factors

12. Six Questions Give 64-Way Classification of Reserve Models • Do the losses that emerge in a period depend on the losses already emerged? • Is all loss emergence proportional? • Is emergence independent of calendar year events? • Are the parameters stable? • Are the disturbance terms generated from a normal distribution? • Do all the disturbance terms have the same variance?

13. Testing and Simulating ModelsLive Data Example SSE Model Params Simulation Formula 157,902 CL 9 qw,d = fdcw,d + e 81,167 BF 18 qw,d = fdhw + e 75,409 CC 9 qw,d = fdh + e 52,360 BF-CC 9 qw,d = fdhw + e 44,701 BF-CC+7 qw,d = fdhwgw+d + e Some models fit better with fewer parameters Simulation and so development risk depends on model Best fitting model has future paid responsive to future inflation

14. UEPR and Current Underwriting Risk • Different from loss reserve risk • Backward projection of reserve risk does not model the risk situation • Can be quantified through risk elements • Frequency risk • Severity risk • Correlation among lines • Risk usually considered in terms of uncertainty about ultimate results, not just one year of stated values • Metarisk model designed to measure this risk gross and net of reinsurance

15. Time Period Reserve Capital is Needed • Capital needed to support an accident year until it runs off • Declining capital needed as losses settle • Looking at capital needed for just one year of runoff is generally felt to understate reserve capital need • Modelers sometimes understate this capital and thus allocate too little to long-tailed lines

16. Development Factors Accident Year 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 1984 2.672 1.578 1.214 1.058 1.044 1.011 1.018 1.011 1.002 1.001 1.000 1.000 1985 2.581 1.505 1.224 1.070 1.024 1.015 1.004 1.022 1.005 1.000 1.013 1986 2.853 1.397 1.355 1.126 1.052 1.016 1.007 1.000 1.001 1.010 1987 2.514 1.595 1.215 1.126 1.063 1.030 1.006 1.017 1.000 1988 2.794 1.501 1.264 1.135 1.044 1.009 1.003 1.000 1989 2.613 1.464 1.230 1.072 1.019 1.017 1.022 1990 2.471 1.601 1.193 1.120 1.021 1.010 1991 2.690 1.469 1.304 1.115 1.023 1992 2.742 1.715 1.165 1.159 1993 2.679 1.440 1.195 1994 2.605 1.498 1995 2.580 Zone Rated Development

17. Murphy Method for Triangle Risk • Residuals from fit give estimated sigma2 • Also estimate variance of each lag’s factors • Accident year n variance of ultimate losses = process variance + parameter variance • process var(n) = process var(n–1) * factor(n)2 + est. sig(n)2 * cum dvlp(n–1). Start with process var(1) = last actual * est sig(1)2 • param var(n) = var of factor(n) * cum dvlp(n)2 + mean sq factor(n) * param var(n–1). Start this with param var(1) = last actual2 * var of factor(1)

18. Resulting Runoff Risk CV’s of ultimate losses by accident year: • 0.073 0.071 0.049 0.035 0.022 0.019 0.020 0.016 0.013 0.013 0.011 The 99th percentile loss is above the mean by: • 18.3% 17.6% 11.9% 8.3% 5.2% 4.5% 4.7% 3.8% 3.0% 3.1% 2.6%

19. Select Risk Measure • Cost of capital for risk A = c*cov(A, market) + d*cov(A, company) , or • Cost of capital for risk A = a*corr(A, market)* (std dev A) + b*corr(A, company)*(std dev A) • Assumed correlation structure: • Correlations for Unit with:Market Company • Loss 20% 35% • Investment Income 80% 50%

20. Assumed weighting coefficients Investment Income CV Total capital costs exceed total profits

21. Is This Line Profitable Enough? • Usual test is to compare costs on a discounted basis • Capital cost could be considered an outgoing cash flow each year • At any interest rate over 2%, present value of annual capital cost is less than present value of underwriting cash flow • A lot of work needed on risk measures and weights – probably fixed correlations wrong

22. 3. Parameter Risk • All loss risk not coming from known frequency and severity fluctuations • Includes estimation risk, projection risk, and event risk • Systematic risk – does not reduce by adding volume • For large companies this could be the largest risk element, comparable to cat risk before reinsurance and greater than cat risk after reinsurance

23. Projection Risk • Change in risk conditions from recent past • In part due to uncertain trend • Can include change in exposures • More driving as gas prices change and other transportation looks risky • New types of fraud become more prevalent

24. Measuring Risk from Uncertain Trend

25. Impact of Projection Risk J on Aggregate CV(CV is ratio of standard deviation to mean)

26. Translating CV Effect to Loss RatioProbabilities E(LR)=65, 3 E(N)’s

27. Estimation Risk • Data is never enough to know true probabilities for frequency and severity • Statistical methods quantify how far off estimated parameters can be from true • More data and better fits both reduce this risk – but never gone

28. Estimation Risk – Pareto Example

29. Other Parameter Risk – “Events” • One or several states decide to “get tough” on insurers • Consumer groups decide company has been unfair and wins in court • Court rules that repairs must use replace-ment parts from original car makers only • Mold is suddenly a loss cause • Biggest writer in market decides it needs to increase market share and reduce surplus so it lowers rates and others follow • Rating downgrade These are big bucks risks and can dwarf others. Hard to predict in future, but must be considered an ongoing risk source and build into random effects.

30. 4. Correlation Issues • Correlation is stronger for large events • Multi-line losses in large events • Modeled by copula methods • Quantifying correlation • Degree of correlation • Part of spectrum correlated • Measure, model, or guess

31. Modeling via Copulas • Correlate on probabilities • Inverse map probabilities to correlate losses • Can specify where correlation takes place in the probability range

32. Gumbel Copula Correlates Large Losses

33. Heavy Right Tail Copula Even More So

34. Normal Copula Doesn’t

35. Quantifying Dependency • Directly measure degree of and location of dependency • Fit to copulas by matching measurement functions • Model dependency through generating process • For example losses and asset returns could be fed by inflation

36. Concentration Measurement Functions for Right and Left Tail – Conditional Probability of Both in Tail if One Is

37. Using Measurement Functions in Fitting

38. 5. Capital Needed and Allocation • RAROC or RORAC? • Economic Capital Target • Coherent Measures of Risk • Matching Capital and Return • Allocation Methodologies

39. RAROC or RORAC? • Capital, not return, usually risk-adjusted • Sometimes return adjusted to replace cat losses by expected • Return targets often do not reflect value of favorable insurance pricing and availability provided to mutual company policyholders

40. Economic Capital Target • Comparison to bond ratings • E.g., 99.97% chance of not defaulting • Measuring 1-year default probability accurately for large company almost impossible • Strongly affected by risk guesses made • Projecting out to tails of distributions with no data to tell if the tail is right • Single year default of A-rated insurer takes unusual circumstances not even in models, like Enron-type accounting, management fraud, ratings downgrade below A-, not meeting debt service, substantial hidden reserve deficiencies, etc. • More realistic to set probability target for partial surplus loss, such as: • 99% chance of not losing more than 20% of surplus

41. Coherent Measures of Risk • Mathematical consistency requirement for risk measures • VAR does not meet requirement • For instance, combination of independent risks can increase VAR beyond the sum of the individual VARs • TVAR does meet requirement • Average loss above VAR threshold • More relevant to policyholders • Other coherent measures being researched

42. Matching Capital and Return • Each business unit generates investment returns on cash flow and on capital supporting the business • That income is part of return of unit • That income and the capital needed to support those investments both need to be charged to the business unit to properly evaluate the unit’s economic contribution

43. Alternatives for Capital Allocation and Performance Measurement • Allocate by risk measure • Coherently • Incoherently • Allocate by price of bearing risk • Charge capital costs against profits • Marginal capital costs of the business • Value of risk guarantee of parent • Compare value of float generated by the business to a leveraged investment fund with the same risk

44. a. Allocate by Risk Measure • Pick a risk measure • Coherent, such as TVAR • Not coherent, such as VAR • Pick an allocation method • Maybe spread in proportion to marginal contribution to company risk • Or use the Kreps method of creating additive co-measures, like co-TVAR, that give 100% additive allocation and consistent splits to subunits

45. Definition of Co-Measures • Suppose a risk measure for risk X with mean m can be defined as: • R(X) = E[(X– am)g(x)|condition] for some value a and function g, and X is the sum of n portfolios Xi each with mean mi • Then the co-measure for Xi is: • CoR(Xi) = E[(Xi– ami)g(x)|condition] • Note that CoR(X1)+CoR(X2) = CoR(X1+X2) and so the sum of the CoR’s of the n Xi’s is R(X)

46. Example: EPD • If X is losses and b total assets, the expected policyholder deficit is EPD = E[(X – b)S(b)|X>b] where S(b)=1 – F(b) • Let a = 1 and g(x) = S(b)(X – b)/(X – m) • Then with condition = X>b, R(X) = EPD • CoEPD(Xi) = E[(Xi– mi)g(X)| X>b] = E[S(b)(X – b)(Xi– mi)/(X – m)|X>b] • Each portfolio gets a fraction of the overall deficit given by the ratio of its adverse losses to the total annual adverse losses in each scenario

47. Allocating Capital by CoEPD • Each portfolio charged in proportion to its contribution to overall default • Does not equalize portfolio expected default costs across portfolios • Additive across sub-portfolios and up to total losses • For instance, you could allocate capital for each line to state, then add up all lines to get total state capital

48. Example: TVaR • TVARq = E[X|X>xq] where F(xq) = q. Note that if xq = assets, then: • EPD = default probability * (TVARq –assets) • Thus TVaR at default and EPD rank all risks identically • For a=0, CoTVaRq(Xi) = E[Xi |X>xq] • Charges each portfolio for its part of total losses in those cases where total losses exceed threshold value

49. Coherence of TVaR • TVaR is a coherent measure, which means, among other things, that for a fixed q the sum of the TVaR’s of any collection of loss portfolios will be the same or greater than the TVaR of the combined portfolio • Not true for EPD or for VaR with fixed q • TVaR criticized for ignoring losses below threshold and for not differentiating among risks that have the same mean above thresh-old – other coherent measures better there

50. Problems with Allocation by Risk Measure • Arbitrary choices of measure and method • Business units will favor choices that favor them, and there will be no underlying theory to fall back on • Pricing to equalize returns may not tie in to risk pricing standards