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

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

Issues 1. Assets 2. Reserves 3. Parameter risk and event risk 4. Correlation 5. Capital needed and allocation

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

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

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

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

Constrained Asset Efficient Frontier with Current Reinsurance Program Frontier of Constrained After-Tax Operating Income / Assets 1-year horizon: Std Deviation Mean Frontier Company

Probability of Returns on Frontier STD Variability of portfolios on after tax OI/Assets frontier Ef. fr Q01 Q05 Q25 Q75 Q95 Q99 Vary reinsurance and investments

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

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

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?

Testing and Simulating Models Live Data Example SSE Model Params Simulation Formula 157,902 CL 9 q w,d = f d c w,d + e 81,167 BF 18 q w,d = f d h w + e 75,409 CC 9 q w,d = f d h + e 52,360 BF-CC 9 q w,d = f d h w + e 44,701 BF-CC + 7 q w,d = f d h w g w+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

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

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

Zone Rated Development

Murphy Method for Triangle Risk  Residuals from fit give estimated sigma 2  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 actual 2 * var of factor(1)

Resulting Runoff Risk CV’s of ultimate losses by accident year:  The 99 th 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%

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  Loss20%35%  Investment Income80%50%

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

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

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

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

Measuring Risk from Uncertain Trend

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

Translating CV Effect to Loss Ratio Probabilities E(LR)=65, 3 E(N)’s

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

Estimation Risk – Pareto Example

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.

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

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

Gumbel Copula Correlates Large Losses

Heavy Right Tail Copula Even More So

Normal Copula Doesn’t

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

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

Using Measurement Functions in Fitting

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

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

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

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

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

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

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

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 X i each with mean m i  Then the co-measure for X i is:  CoR(X i ) = E[(X i – am i )g(x)|condition]  Note that CoR(X 1 )+CoR(X 2 ) = CoR(X 1 +X 2 ) and so the sum of the CoR’s of the n X i ’s is R(X)

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(X i ) = E[(X i – m i )g(X)| X>b] = E[S(b)(X – b)(X i – m i )/(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

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

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

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

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

b. Allocate by Price of Bearing Risk  Financial theory gives market price guidelines for risk bearing  Can be calibrated to insurance market situation  Business units can be evaluated by profit vs. risk-pricing standards  Can allocate capital in proportion to target profitability

c. Charge Capital Cost against Profits  Instead of return rate, subtract cost of capital from unit profitability  Use true marginal capital costs of business being evaluated, instead of an allocation of entire firm capital –If evaluating growing business 10%, charge the cost of the capital needed for that much growth –If evaluating stopping writing in a line, use the capital that the company would save by eliminating that line  This maintains financial principle of comparing profits to marginal costs

Fixed and Marginal Capital Costs  Company X buys a widget maker and pays a big fee each year for mortgage costs  Running it and producing widgets is cheap  How does it decide whether or not to make more widgets?  Compare revenue with marginal cost of production

Fixed and Marginal Costs  Similarly for insurance company –Expanding or contracting a business unit evaluated based on revenue vs. marginal costs, especially marginal cost of new capital needed or capital released –This includes capital needed for reserves and investment income on funds generated –Separate analysis needed for strategies for fixed costs

Calculating Marginal Capital Costs  Could use change in overall risk measure of firm that results from the marginal business  Or set capital cost of a business segment as the value of the financial guarantee the firm provides to the clients of the business segment

Value of Financial Guarantee  Cost of capital for subsidiary is a difference between two put options: –1. The cost of the guarantee provided by the corporation to cover any losses of the subsidiary –2. The cost to the clients of the subsidiary in the event of the bankruptcy of the corporation  Economic value added of the subsidiary is risk-adjusted profit less cost of capital –Profit risk adjusted to account for long-term average costs of highly unstable risks, like cats  If EVA is positive, it is worth growing the subsidiary

Allocation and Evaluation Summary  Allocating by risk measure straightforward but arbitrary  Using risk pricing appropriate for comparing profitability  Actual marginal surplus most useful for determining economic contributions of business units. This is not the same as allocation in proportion to marginal risk.  Leveraged mutual fund comparison is appropriate for evaluating return on total capital and the marginal contribution of each business unit to that

Conclusions  Asset models should be arbitrage-free and distributionally representative of history  Reserve risk requires alternative models and is easy to understate, both on time capital held and UEPR reserves  Parameter risk is a key issue for large companies and is difficult to quantify  Correlation should incorporate tail links to get true large loss risk  VAR is not the best overall capital standard, nor is allocation of total capital the best way to evaluate profitability  Getting the modeling right takes care and expertise, and is subject to many pitfalls