Managing Underwriting Risk & Capital John J. Kollar, FCAS, MAAA, CPCU July 30, 2003

Enterprise Risk Management Investment Risk Underwriting Risk Catastrophe Risk

Underwriting Risk Loss volatility (including catastrophic risk) Correlation between lines (dependencies) Risk of adverse development for long-tailed lines Underwriting cycles=varying margins Varying real prices Varying underwriting selectivity

Constructing an Underwriting Risk Model (URM) Fit claim severity distributions by line. Use individual claim severity Aggregate data of many insurers By line By settlement lag to estimate industry parameters for claim severity distributions

Constructing URM (Cont’d) Fit claim frequency distributions by line. Use net expected losses, exposures By insurer By line By settlement lag To estimate Industry parameters for the claim frequency distributions Correlations between lines

URM Input Insurer expected losses Policy Limits Reinsurance By line By settlement lag Policy Limits Reinsurance Use industry estimates for other parameters. Include insurer’s catastrophe loss distributions (catastrophe modeler). Can adjust losses by an economic scenario generator.

Calculating an Insurer’s Underwriting Risk Via URM Use the collective risk model. Separate claim frequency and severity analysis. For each line of insurance: Select a random claim count.  Use industry claim frequency parameters. Select random claim size for each claim.  Use industry claim severity parameters.  Adjust for policy limits and reinsurance.

Calculating an Insurer’s Underwriting Risk Via URM (Cont’d) The aggregate loss for all lines = sum of all the random claim amounts for all lines. Reflect the correlation between lines of insurance. Repeat the above thousands of times (simulation) or use Fourier transforms to calculate the insurer’s aggregate loss distribution. Total By profit center

Calculating an Insurer’s Underwriting Risk Via URM (Cont’d) What is the tolerance for risk? Regulatory requirement (RBC) “A” rating from a rating agency Tradition, etc. Select a statistical measure of risk that corresponds to the tolerance for risk. Value at risk Tail value at risk Standard deviation, etc. Determine total capital for underwriting from the aggregate loss distribution using the selected measure of risk.

Allocating (Cost of) Capital Calculate marginal capital for each profit center. Calculate the sum of the marginal capital for all capital centers. Diversification multiplier equals the total capital divided by the sum of the marginal capital. Allocated capital for each profit center equals the product of the diversification multiplier and the marginal capital for the profit center.

Minimizing the Cost of Financing Cost of capital = return x capital Discounted Post tax By profit center Cost of reinsurance = frictional cost For each program Cost of financing = cost of reinsurance + cost of capital For each program/profit center Minimize the total cost of financing

Setting Combined Ratio Targets by Line Expected losses Expected expenses Investment income Cost of financing

Planning Underwriting Strategy Add policies/portfolios that increase the return on capital. Drop policies that cause a drop in return on capital.