The Effective Duration of Property-Liability Insurance Liabilities with Stochastic Interest Rates

1. The Effective Duration of Property-Liability Insurance Liabilities with Stochastic Interest Rates 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 August, 2000

2. Why Worry About Duration? Duration is an estimate of the sensitivity of a cash flow 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-and-Loan Industry didn’t, and look what happened to them Interest rates can and do fluctuate substantially Examples of Intermediate Term U.S. Bond Rates: 1/112/31 1979 8.8% 10.3% 1.5% 1980 10.3 12.5 2.2 1982 14.0 9.9 -4.1 1994 5.2 7.8 2.6 1999 4.7 6.5 1.8

4. Are Property-Liability Insurers Exposed to Interest Rate Risk? YES! 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 weighted average time to maturity Modified duration Negative of the first derivative of the price/yield curve Macaulay duration/(1+interest rate)

6. Modified Duration is the Negative of the Slope of a Tangency Line to the Price-Yield Curve Price-yield curve for a fixed income bond Price r Yield

7. Assumptions Underlying Macaulay and Modified Duration • Cash flows do not change with interest rates This does not hold for: Collateralized Mortgage Obligations (CMOs) Callable bonds Loss reserves • Flat yield curve Generally yield curves are upward sloping • Parallel shift in interest rates Short term interest rates tend to be more volatile than longer term rates

8. Effective Duration • Accommodates interest sensitive cash flows • Can be based on any term structure • Allows for non-parallel interest rate shifts

9. Prior Related Research (1) Taylor Separation Method (1986) Allows for a separate inflation component to loss payments Inflation affects all payments made in given year Babbel, Klock and Polachek (1988) Macaulay duration reasonable approximation Staking (1989), Babbel and Staking (1995, 1997) Calculate effective duration of liabilities based on a modification of the Taylor Separation Method Determine that most insurers operate in the least efficient range of interest rate risk

10. Prior Related Research (2) Choi (1991) “nobody knows how to model the cash flows as a function of interest rates or inflation rates (because interest rates are closely related to inflation rates) in the property-liability insurance industry.” Feldblum and Hodes (1996) “A mathematical determination of the loss reserve duration is complex.” Assumes loss reserves are not interest rate sensitive

11. Interest Sensitive Cash Flows Interest rates and inflation are correlated Inflation can increase future loss payments Loss reserve consists of future payments Portion has already been “fixed” in value Medical treatment already received Property damage that has been repaired Remainder subject to inflation General damages to be set by jury Future medical treatment

12. 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 1 m Proportion of Ultimate Payments Fixed n<1 n=1 n>1 k 0 1 0 Proportion of Payment Period

13. Term Structure Cox, Ingersoll, and Ross (CIR) Mean-reverting, square-root diffusion process α = speed of reversion r = current short term interest rate R = long run mean of short term interest rate σ = volatility factor dz = standard normal distribution

14. Non-Parallel Shifts A change in the short term interest rate does not shift the long term rate as much

15. Calculation of the Effective Duration of Loss Reserves (1) • Generate multiple interest rate paths based on the CIR model • For each path, calculate the loss payments that will develop • Determine the present value of each set of cash flows by discounting by the relevant interest rates • Calculate the average present value over all interest rate paths

16. Calculation of the Effective Duration of Loss Reserves (2) • Calculate the present value based on the initial interest rate, the initial interest rate plus 100 basis points and the initial interest rate minus 100 basis points • Calculate the effective duration based on: Effective Duration = (PV--PV+)/(2PV0)(Δr)

17. Duration of Liabilities Based on Steady State Operations and a 6% Interest Rate: Auto Liab.WCOther Liab. Macaulay Duration 2.00 4.05 3.96 Modified Duration 1.89 3.82 3.73 Based on CIR and interest sensitive cash flows: Auto Liab.WCOther Liab. Effective Duration 1.14 1.63 1.65

18. Fixed Cost Parameters k = .15 m = .10 n = 1.0 Impact of Inflation Embedded Inflation Rate = 5% Future Inflation = .05 + .46 X Short Term Interest Rate CIR Interest Rate Parameters α = .25 r = .06 R = .07 σ = .08 Assumptions Underlying Effective Duration Calculation

19. Additional Research Other term structure models Vasicek Hull-White Sensitivity of parameter estimates