΅Y#3SF9e%c&WVi)) Jj[MX\no kNK_z+G0ifm7`S3W :eĔTJ iC?S^h/>^UY[ZzHgbg?>}裎EGk@l8RFJNnh)1]b_]+(N$zgl !'dQX @^ɞc(r N]!w*#BOάs+Ϝ˸C4V)s z=狗|V4ks6Zee7Kx$D8c>&Avp7=Ď_=rq,|U$x[+kf9C:ĊC%>o?I˔?G3Da9k! &!_&͛:7G\"7-V\i1,)silvp6n\9'NqWn.phçò"\9)fysr2&XKbC΄']OC3a7q9ceM;(_--d٘+q!㭅ʙAp8qfAv aV;o|3BEIJתJ6-RBr#KED5bd\5i!|/'/2ZWcX~/g'>ӈFvWڼLD[0F[>~LtV{#|ĚQfNrDH;i0/LkȤvzpշlƨ8<Cs*9Xl+KȬ',w#LDL`\<cƾ'>MC,#_{ڄ0[_!Lބ멱{1\NakpmMBND>KSef BU47(B-s"Q0iz^9!Py>m<r8fn-ca(&{qauBCaJ)g5x.3P s}4'H5BpSƨCecϧ #騰U0;.:~ugUBɽؐsW ktTV$ΐ0V$U(WU7¼Wt,#a,T#t!Y a,Nh; a,>)1cbdBs>lD yT"Q 53w{ûoȨ9|yAB->xD{DFq*Tߜ=uIb~F1k/jrIFX1VjHםcyBmM9{̛ XP;Dï Ǩ=ҳ֝:G.rzю)ZBuD](KQDpngvB }u}?5 uBujuF4m =h{RuAJ@ۗPWDYRމQ7D[5tf[|fa)B#xdt}@Ηm&QcF|ȢFԠ_>6eM_;N/oLY~~Y߿A2fW |dF-}rWlPr-b4A B,FUá~G>S6MuY(\qu/$Ow. 4y1JC2=^#d1.?QiUBfgay+>ppD'-ttDO˷gO(W/BZ;ߨz8 uCUU;x#2g=Vi6浲 `Z9h~r\Ĕu6Z!ms6# lKk:-K %գ^~6ޠyC`ϱmmÏYMVdbZ"m"1VZ-&JwXi|+ޗˈj\IVȵXiZn&Jr;1V;[9ǘ/a郷rP8ES*l+q?lߡl*wߛkcO-+vMXYۗU-+R `[VYJj1VZM4b/lcޟ(>x+])vjczH-J!;+@j1VZ}bx+ꂘBV?XiU(fc\@VkXi[Xi$cS+"zb#cK J&r1VZcU 1VZu'*Q#JXiScU?&J$M+2+Xi)+cHFVcd b/cYKVeb*cUHV3ebtQvq?ϸWy +>j+XiU! HV/XiCcN,Jݲ41VZJMV{e1VZUSIVhb: c!ٜ+51VZ}!!J1bR$J<ٟ+XiNq9+㉱|+iXiuL+d|bXiuB.&Jb:%Wck\CMElr1VZ{<@"G 8O.j\ϭt w"C#?5${Bn2WV:5GGJg㎌#_Je89&]QtAYKfSKUL!qB[{='$)WWC GOKTZ-`$~8,]2)it@NmmYm=6z[jWрV@tղZ6w3l:*өf,Idn%㞒Ia=>ޤeZp8|UMin7A[m|إC{B3e (3gNYzt~:;x}N1Ԕmе cqGx9,V*P'bhxKo_i<IB#"IlAy}ːL1#eklm_A GY7d~H k;(50ѥQkgʟ]C۴zCr2EGUIKmMyMOwW]^;0kp~x {WY7[yr|tvA 1( X Chart M$, 4 On-screen Show+University of Illinois at Urbana-ChampaignR3 Times New RomanSymbolBlank PresentationMicrosoft Graph 2000 ChartMicrosoft Equation 3.0bThe Effective Duration of Property-Liability Insurance Liabilities with Stochastic Interest RatesWhy Worry About Duration?$Why Worry About Interest Rate Risk?@Are Property-Liability Insurers Exposed to Interest Rate Risk? Measures of Interest Rate Risk[Modified Duration is the Negative of the Slope of a Tangency Line to the Price-Yield Curve6Assumptions Underlying Macaulay and Modified DurationEffective DurationPrior Related Research (1)Prior Related Research (2)Interest Sensitive Cash Flows A Possible Fixed Cost FormulaTerm StructureNon-Parallel Shifts;Calculation of the Effective Duration of Loss Reserves (1);Calculation of the Effective Duration of Loss Reserves (2)Duration of Liabilities6Assumptions Underlying Effective Duration CalculationAdditional ResearchFonts UsedDesign TemplateEmbedded OLE Servers Slide Titles$_5Steve D'ArcySteve D'Arcy. .-2 Stochastic Interest Rates &.-- -- @Times New Roman- .42 Stephen P. DArcy, FCAS, Ph.D. . .62 GRichard W. Gorvett, FCAS, Ph.D. & . .(2 :University of Illinois . .2 ' at Urbana . . 2 -. .2 Champaign . .=2 E$Presented at the ARIA Annual Meeting $ . .2 {tAugust, 2000 .--"System-&TNPP & ՜.+,0 $, 4 On-screen Show+University of Illinois at Urbana-ChampaignR3 Times New RomanSymbolBlank PresentationMicrosoft Graph 2000 ChartMicrosoft Equation 3.0bThe Effective Duration of Property-Liability Insurance Liabilities with Stochastic Interest RatesWhy Worry About Duration?$Why Worry About Interest Rate Risk?@Are Property-Liability Insurers Exposed to Interest Rate Risk? Measures of Interest Rate Risk[Modified Duration is the Negative of the Slope of a Tangency Line to the Price-Yield Curve6Assumptions Underlying Macaulay and Modified DurationEffective DurationPrior Related Research (1)Prior Related Research (2)Interest Sensitive Cash Flows A Possible Fixed Cost FormulaTerm StructureNon-Parallel Shifts;Calculation of the Effective Duration of Loss Reserves (1);Calculation of the Effective Duration of Loss Reserves (2)Duration of Liabilities6Assumptions Underlying Effective Duration CalculationAdditional ResearchFonts UsedDesign TemplateEmbedded OLE Servers Slide Titlesbedded OLE Servers Slide Titles$_<5Steve D'ArcySteve D'Arcy !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSoUVnpqrstuvwxyz{|}~Root EntrydO)`,0TPicturesCurrent UserEDSummaryInformation(PowerPoint Document(3DocumentSummaryInformation8.SGraph.Chart.804Microsoft Graph 2000 Chart}0Equation Equation.30,Microsoft Equation 3.0b/0DTimes New Romanٸbbv0b( 0DSymbolew Romanٸbbv0b( 0 `. @n?" dd@ @@`` /X"$gFjFjSGÖ2$&J`Oi\c$@g45d5dv0bppp@<4BdBd` 0bڸRbuʚ;2Nʚ;<4!d!db{ 0bb<4ddddb{ 0bb___PPT9/0z ?%)aThe Effective Duration of Property-Liability Insurance Liabilities with Stochastic Interest Rates bUa(: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^j $ 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#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: 1/1 12/31 D 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 1.8p?Zw / >?Are Property-Liability Insurers Exposed to Interest Rate Risk? 6YES! Long term liabilities Medical malpractice Workers compensation General liability Assets Significant portion of assets invested in long term bondsMeasures of Interest Rate RiskMacaulay 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)$ ZModified Duration is the Negative of the Slope of a Tangency Line to the Price-Yield Curve [UZ(5Assumptions 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 -01Ug0U02U+02U!03UL0U- f * ! LmEffective DurationzAccommodates interest sensitive cash flows Can be based on any term structure Allows for non-parallel interest rate shiftsD+1#2-3{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 riskjZ!n#+1&b 2Prior 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 NwU6 Interest Sensitive Cash FlowsInterest 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 infTlation General damages to be set by jury Future medical treatmentHZUL"< >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 paymentZ9U#UU8 Term StructureCox, Ingersoll, and Ross (CIR) Mean-reverting, square-root diffusion process = speed of reversion r = current short term interest rate R = long run mean of short term interest rate = volatility factor dz = standard normal distributionZ, Non-Parallel ShiftsRA change in the short term interest rate does not shift the long term rate as muchSS: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" Z: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)(r)" *! Duration of Liabilities1Based on Steady State Operations and a 6% Interest Rate: Auto Liab. WC Other 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. WC Other Liab. Effective Duration 1.14 1.63 1.652Z? *bD , 5Assumptions Underlying Effective Duration CalculationFixed Cost Parameters k = .15 m = .10 n = 1.0 Impact of Inflation Embedded Inflation Rate = 5% Future Inflation = .05 + .46 X Short Term Interest RateIW CIR Interest Rate Parameters = .25 r = .06 R = .07 = .08 JFHAdditional ResearchTOther term structure models Vasicek Hull-White Sensitivity of parameter estimates UU0/r` < !"#$%&'()*+,-/0123456789:;<=>?@ABCDF Oh+'0Php4 LX x SThe EffectiveThe Effective Duration of Liabilities for Property-Liability InsurersStephen P. D'ArcyffCD:\Program Files\Microsoft Office\Templates\Blank Presentation.potb Steve D'Arcyles17vMicrosoft PowerPointoso@0V|U@<@0TXG g ?& &&#TNPP 2OMi & TNPP &&TNPP - "-- !-- "-&G& - &Gy& --8 -- @Times New Roman- .:2 NC"The Effective Duration of Property$# & !. . 2 Nc-. .?2 P%Liability Insurance Liabilities with # # &. .-2 Stochastic Interest Rates &.-- -- @Times New Roman- .42 Stephen P. DArcy, FCAS, Ph.D. . .62 GRichard W. Gorvett, FCAS, Ph.D. & . .(2 :University of Illinois . .2 ' at Urbana . . 2 -. .2 Champaign . .=2 E$Presented at the ARIA Annual Meeting $ . .2 {tAugust, 2000 .--"System-&TNPP & ՜.+,0