1. The Effective Duration of Liabilities for Property- Liability Insurers Stephen P. D’Arcy, FCAS, Ph.D. Richard W. Gorvett, FCAS, Ph.D. University of Illinois at Urbana-Champaign Presented at the ARIA Annual Meeting and the Actuarial Research Conference August 9, 1999

2. What is Duration? Duration measures how sensitive the value of a financial instrument is to interest rate changes Modified duration is the negative of the first derivative of price with respect to interest rates, divided by the price Modified duration assumes: Flat yield curve Cash flows are not affected by interest rates No embedded options

3. Duration is the Slope of the Tangency Line for the Price/Yield Curve Price Yield r Price-yield curve for financial instrument

4. A Refinement: Also Consider Convexity The larger the change in interest rates, the larger the misestimate of the price change using duration Duration: first-order approximation Accurate only for small changes in interest rates Convexity: second-order approximation Reflects the curvature of the price-yield curve

5. What is Effective Duration? Effective duration allows cash flows to be affected by interest rate changes Effective duration is used to value such assets as: Collateralized Mortgage Obligations (CMOs) Callable bonds

6. The Liabilities of Property- Liability Insurers Major categories of liabilities: Loss reserves Loss adjustment expense reserves Unearned premium reserves

7. Loss Reserves Major categories: In the process of being paid Value of loss is determined, negotiating over share of loss to be paid Damage has not yet occurred, but loss will be covered under an occurrence policy Loss has occurred and is continuing to develop Inflation, which is correlated with interest rates, will affect each category of loss reserves differently.

8. What Portion of the Loss Reserve is Affected by Future Inflation (and Interest Rates)? If the damage has not yet occurred, it will fully reflect future inflation If the loss is continuing to develop, then a portion of the loss will be affected by future inflation

9. How to Reflect “Fixed” Costs? “Fixed” here means that portion of damages which, although not yet paid, will not be impacted by future inflation Tangible versus intangible damages Determining when a cost is “fixed” could require –Understanding the mindset of jurors –Lots and lots of data

10. A Possible “Fixed” Cost Formula Proportion of loss reserves fixed in value as of time t: f(t) = k + [(1 - k - m) (t / T) n ] Proportion of Payment Period 0 1 Proportion of Ultimate Payments Fixed 1 0 k m n=1 n<1 n>1

11. “Fixed” Cost Formula Parameters Examples of loss costs that might go into k –Medical treatment immediately after the loss occurs –Wage loss component of an injury claim –Property damage Examples of loss costs that might go into m –Medical evaluations performed immediately prior to determining the settlement offer –General damages to the extent they are based on the cost of living at the time of settlement –Loss adjustment expenses connected with settling the claim

12. Loss Reserve Duration Example For the values: k =.15m =.10 n = 1.0  r,i = 0.40 r = 5%Exposure Growth Rate = 10% Automobile Workers’ InsuranceCompensation Modified duration:1.444.27 Effective duration:1.093.16

13. Why is Duration Important? Corporations attempt to manage interest rate risk by balancing the duration of assets and liabilities

14. Surplus Duration Sensitivity of an insurer’s surplus to changes in interest rates D S S = D A A - D L L D S = (D A - D L )(A/S) + D L where D = duration S = surplus A = assets L = liabilities

15. Surplus Duration and Asset-Liability Management To “immunize” surplus from interest rate risk, set: D S = 0 Thus, asset duration should be: D A = D L L / A An accurate estimate of the duration of liabilities is critical for ALM

16. Conclusion Asset-liability management depends upon appropriate measures of effective duration and convexity Potentially significant differences between effective and modified duration values Critical factors and parameters –Line of business –Payment pattern –Correlation between interest rates and inflation