Financial Risk Management of Insurance Enterprises

1. Financial Risk Management of Insurance Enterprises Duration and Convexity – Part 2

2. Applications of Duration • Remember, ALM evaluates the interaction of asset and liability movements • Insurers attempt to equate interest sensitivity of assets and liabilities so that surplus is unaffected • Surplus is “immunized” against interest rate risk • Immunization is the technique of matching asset duration and liability duration

3. Why Worry About Interest Rate Risk? • The 1970s Savings & Loan industry didn’t • Asset-liability “mismatch” • Interest rates can and do fluctuate substantially • Examples of 7 Year U.S. T-bond interest rates: r at r at t t-12 monthsti March 1980 9.15% 13.00% 3.85% July 1981 9.84 14.49 4.65 Oct 1982 15.33 10.88 - 4.45 May 1984 10.30 13.34 3.04 April 1986 11.34 7.16 - 4.18 Dec 1995 7.80 5.63 - 2.17

4. 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 • P-L loss reserves – due to inflation-interest rate correlation • Flat yield curve Generally, yield curves are upward-sloping • Interest rates shift in parallel fashion Short term interest rates tend to be more volatile than longer term rates

5. Assuming Parallel Shifts • The assumption of parallel shifts in the yield curve is not plausible • In reality, short-term rates move more than long-term rates • Also, it is possible that the yield curve “twists” • Short-term and long-term rates move in opposite directions

6. An Illustration • There are two cash flows, 100 at the end of year 1 and 100 at the end of the second year • The interest rate is a flat 5% • Calculating modified duration

7. Partial Duration • Each term in the calculation tells us something about interest rate sensitivity • It is the sensitivity of the cash flow to that interest rate • In this example, define two “partial” durations • One for each cash flow period

8. Interpreting Partial Duration • Note that the sum of the partial durations is equal to the original duration calculation • Using partial duration, we can determine the interest rate sensitivity to any non-parallel shift in the yield curve • We can use partial duration to predict price changes

9. Example • From our two period cash flow, what is the change in value if the one year rate goes to 4% and the two year rate goes to 6%

10. Key Rates • Interest rates of “similar” maturities move in the same fashion • The 10 year rate and the 10½ year rate move similarly • Therefore, partial durations can be based on a few points on the yield curve • These are called key rates • Partial durations are sometimes referred to as key rate durations

11. Typical Key Rates • Popular key rates are: • 3 month and 6 month rate • 1 year • 2 years • 3 years • 5 years • 7 years • 10 years • 30 years

12. Applications of Key Rate Durations • Key rate durations are very useful for hedging purposes • Because multiple partial durations provide more information than a single duration number, insurers can determine their sensitivity to interest rates based on various parts of the yield curve • If the insurer is not immunized, it can use interest rate derivatives to hedge the risk

13. Cash Flows Change with Interest Rates • Effective Duration • Effective Convexity

14. Effective Duration

15. Effective Convexity(Note – Fabozzi includes a 2 in denominator)

16. Calculation of the Change in Economic Value of a Cash Flow V = (-1)(Effective Duration)(r) + (1/2)(Convexity)(r)2 (Note: if using Fabozzi convexity calculation, omit the (1/2) in the second term.)

17. Example – Fixed Cash Flow Cash flow of $1000 in 10 years No interest rate sensitivity Current interest rate = 10% Macaulay Duration = 10 Modified Duration = 9.0909 Convexity = 90.909

18. Example – Variable Cash Flow Cash flow of $1000 occurs at x years if r = x% Current r = 10%, cash flow at year 10 PV = 1000/(1.10)^10 = 385.5433 ∆r = 50 basis points PV_= 422.2463 PV+= 350.5065 Effective Duration = 18.6075 Effective Convexity = 172.8719

19. Example – Variable Cash Flow 2 Cash flow of $1000 occurs at 10 years if r = 10% Cash flow changes at ½ the percentage change that interest rates change (from 10%) If interest rates rise to 10.5%, cash flow is $1025 If interest rates fall to 9.5%, cash flow is $975. PV = 1000/(1.10)^10 = 385.5433 ∆r = 50 basis points PV_= 393.4263 PV+= 377.6601 Effective Duration = 4.0894 Effective Convexity = -0.0169

20. Estimated Impact of Change in Economic Value for 100 Basis Point Rise in Interest Rate V = (-1)(Effective Duration)(r) + (1/2)(Convexity)(r)2 Fixed cash flow -8.6% Variable cash flow 1 -17.7% Variable cash flow 2 -4.1%

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

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

23. 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

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

25. 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.

26. 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)

27. How to Reflect “Fixed” Costs? • “Fixed” here means that portion of damages which, although not yet paid, willnot 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

28. 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

29. “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

30. Loss Reserve Duration Example For the values: k = .15 m = .10 n = 1.0 r = 5% rr,i = 0.40 Exposure growth rate = 10% Automobile Workers’ InsuranceCompensation Macaulay duration: 1.52 4.49 Modified duration: 1.44 4.27 Effective duration: 1.09 3.16 Convexity 5.75 50.77 Effective convexity 1.99 16.04

31. Example of ALM for a Hypothetical WC Insurer Dollar Modified Effective ValueDurationDuration Loss & LAE Reserve 590 4.271 3.158 UPR 30 3.621 1.325 Other liabilities 900.9520.952 Total liabilities 710 3.823 2.801 Total assets 1,000 Asset duration to immunize surplus: 2.714 1.989

32. 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 (?)

33. Next • Review for first exam • First exam – February 27, 2008 • An introduction to stochastic processes • The use of stochastic movements in modeling interest rates • Using interest rate models to calculate duration and convexity