Financial Risk Management of Insurance EnterprisesApplications of DurationWhy Worry About Interest Rate Risk?Assumptions Underlying Macaulay and Modified DurationAssuming Parallel ShiftsAn IllustrationPartial DurationInterpreting Partial DurationExampleKey RatesTypical Key RatesApplications of Key Rate DurationsCash Flows Change with Interest RatesEffective DurationEffective Convexity (Note – Fabozzi includes a 2 in denominator)Calculation of the Change in Economic Value of a Cash FlowExample – Fixed Cash FlowExample – Variable Cash FlowExample – Variable Cash Flow 2Estimated Impact of Change in Economic Value for 100 Basis Point Rise in Interest RateSurplus DurationSurplus Duration and Asset-Liability ManagementAre Property-Liability Insurers Exposed to Interest Rate Risk?The Liabilities of Property-Liability InsurersLoss ReservesWhat Portion of the Loss Reserve is Affected by Future Inflation (and Interest Rates)?How to Reflect “Fixed” Costs?A Possible “Fixed” Cost Formula“Fixed” Cost Formula ParametersLoss Reserve Duration ExampleExample of ALM for a Hypothetical WC InsurerConclusionNextFinancial Risk Management of Insurance EnterprisesDuration and Convexity – Part 2Applications 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 durationWhy 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 months t iMarch 1980 9.15% 13.00% 3.85%July 1981 9.84 14.49 4.65Oct 1982 15.33 10.88 - 4.45May 198410.30 13.34 3.04April 1986 11.34 7.16 - 4.18Dec 1995 7.80 5.63 - 2.17Assumptions Underlying Macaulay and Modified Duration•Cash flows do not change with interest ratesThis does not hold for:–Collateralized Mortgage Obligations (CMOs)–Callable bonds–P-L loss reserves – due to inflation-interest rate correlation•Flat yield curveGenerally, yield curves are upward-sloping•Interest rates shift in parallel fashionShort term interest rates tend to be more volatilethan longer term ratesAssuming 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 directionsAn 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 duration42.194.185105.1100205.1100194.18505.110005.1100322DPPartial 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 period93.094.185105.1100249.094.185105.110013221DDInterpreting 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 changes2211- valuebondin change PercentagerDrD 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%185.15 price Actual12.185price Predicted0.44%- 0.010.93-(-0.01)-0.49change price Predicted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 durationsTypical 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 yearsApplications 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 riskCash Flows Change with Interest Rates•Effective Duration•Effective ConvexityEffective Durationr)(2PVPV - PV = ED = duration Effective+-00Effective Convexity(Note – Fabozzi includes a 2 in denominator)200)(2 rPVPV PVPV =convexity Effective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.)Example – Fixed Cash FlowCash flow of $1000 in 10 yearsNo interest rate sensitivityCurrent interest rate = 10%Macaulay Duration = 10Modified Duration = 9.0909Convexity = 90.909Example – Variable Cash FlowCash flow of $1000 occurs at x years if r = x%Current r = 10%, cash flow at year 10PV = 1000/(1.10)^10 = 385.5433∆r = 50 basis pointsPV_= 422.2463PV+= 350.5065Effective Duration = 18.6075Effective Convexity = 172.8719Example – Variable Cash Flow 2Cash 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 $1025If interest rates fall to 9.5%, cash flow is $975. PV = 1000/(1.10)^10 = 385.5433∆r = 50 basis pointsPV_= 393.4263PV+= 377.6601Effective Duration = 4.0894Effective Convexity = -0.0169Estimated Impact of Change in Economic Value for 100 Basis Point Rise in Interest Rate V = (-1)(Effective Duration)(r)+ (1/2)(Convexity)(r)2Fixed cash flow -8.6%Variable cash flow 1 -17.7%Variable cash flow 2 -4.1%Surplus Duration•Sensitivity of an insurer’s surplus to changes in interest ratesDS S = DA A - DL LDS = (DA - DL)(A/S) + DLwhere D = duration S = surplusA = assetsL = liabilitiesSurplus Duration and Asset-Liability Management•To “immunize”