1. Measuring the Interest Rate Sensitivity of Loss Reserves Stephen P. D’Arcy, FCAS, MAAA, Ph.D. Richard W. Gorvett, FCAS, MAAA, ARM, Ph.D. University of Illinois at Urbana-Champaign Casualty Actuarial Society Miami Beach, FL May 7, 2001

2. Why Bother with “Duration”? Duration measures how sensitive the value of a financial instrument is to interest rate changes Duration is used in asset-liability management Properly applied, asset-liability management can hedge interest rate risk

3. Why Worry About Interest Rate Risk? The Savings & Loan industry didn’t, and look what happened to them –Asset-liability “mismatch” Interest rates can and do fluctuate substantially Examples of intermediate-term U.S. bond rates: t 12/t-1 12/ t  1979 9.0% 10.4% 1.4% 198010.4 12.8 2.4 198213.7 10.5 - 3.2 1994 5.8 7.8 2.0 1999 4.7 6.3 1.6

4. Are Property-Liability Insurers Exposed to Interest Rate Risk? Absolutely!! Long-term liabilities –Medical malpractice –Workers’ compensation –General liability Assets –Significant portion of assets invested in long term bonds

5. Measures of Interest Rate Risk Macaulay duration recognizes that the sensitivity of the price of a fixed income asset is approximately related to the (present value) weighted average time to maturity Modified duration is the negative of the first derivative of price with respect to interest rates, divided by the price Modified duration = Macaulay duration/(1+r)

6. Macaulay and Modified Duration

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

8. 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

10. Computing Convexity Take the second derivative of price with respect to the interest rate

11. Assumptions Underlying Macaulay and Modified Duration Cash flows do not change with interest rates But: t his does not hold for: –Collateralized Mortgage Obligations (CMOs) –Callable bonds –P-L loss reserves – due to inflation-interest rate correlation Flat yield curve But: generally, yield curves are upward-sloping Interest rates shift in parallel fashion But: short term interest rates tend to be more volatile than longer term rates

12. An Improvement: Effective Duration Effective duration: –Accommodates interest sensitive cash flows –Can be based on any term structure –Allows for non-parallel interest rate shifts Effective duration is used to value such assets as: –Collateralized Mortgage Obligations –Callable bonds –And now… property-liability insurance loss reserves Need to reflect the inflationary impact on future loss payments of interest rate movements

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

14. Loss Reserves Major categories: –In the process of being paid –Value of loss is determined, negotiating over share of loss to be paid –Damage is yet to be discovered –Continuing to develop: some of loss has been fixed, remainder is yet to be determined Inflation, which is correlated with interest rates, will affect each category of loss reserves differently.

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

16. 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

17. 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 ] k= portion of losses fixed at time of loss m= portion of losses fixed at time of settlement T= time from date of loss to date of payment Proportion of Payment Period 0 1 Proportion of Ultimate Payments Fixed 1 0 k m n=1 n<1 n>1

18. “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

19. Loss Reserve Duration Example For the values: k =.15m =.10n = 1.0 r = 5%  r,i = 0.40 Exposure growth rate = 10% Automobile Workers’ InsuranceCompensation Macaulay duration:1.524.49 Modified duration:1.444.27 Effective duration:1.093.16

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

21. 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

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

23. Example of Asset-Liability Mgt. for a Hypothetical WC Insurer Dollar Modified Effective Value Duration Duration Loss & LAE Reserve 590 4.271 3.158 UPR 30 3.621 1.325 Other liabilities 90 0.952 0.952 Total liabilities 710 3.823 2.801 Total assets 1,000 Asset duration to immunize surplus: 2.714 1.989

24. 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 –Interest rate model (?)