Managing Underwriting Risk & Capital

1. ManagingUnderwriting Risk & Capital John J. Kollar, FCAS, MAAA, CPCU July 30, 2003

2. Enterprise Risk Management Enterprise Risk Investment Risk Underwriting Risk Catastrophe Risk

3. 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

4. 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 • By line • By settlement lag

5. 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 • By line • By settlement lag • Correlations between lines

6. URM Input • Insurer expected losses • 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.

7. 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.

8. 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

9. 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.

10. 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.

11. 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

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

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