the effective duration and convexity of liabilities ...gepa.0000032567... · the effective duration...

The Geneva Papers on Risk and Insurance Theory, 29: 75–108, 2004c© 2004 The Geneva Association

The Effective Duration and Convexity of Liabilitiesfor Property-Liability Insurers Under StochasticInterest Rates

KEVIN C. AHLGRIM [email protected] of Finance, Insurance and Law, Illinois State University, 328 Williams Hall, Normal,IL 61790-5480, USA

STEPHEN P. D’ARCY [email protected] of Finance, University of Illinois, 1206 S. Sixth Street, Champaign, IL 61820, USA

RICHARD W. GORVETT [email protected] of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA

Abstract

Managing interest rate risk for property-liability insurers requires appropriate measurement of the sensitivity ofliabilities to movements in interest rates. Most prior studies have assumed that interest rates shift in a parallelfashion and that the cash flows from liabilities are unaffected by interest rate changes. This article recognizesthat unpaid property-liability (P-L) insurance losses are inflation-sensitive, that movements in interest rates willaffect future claim payouts due to the correlation between interest rates and inflation and that interest rates arestochastic. The effective duration and convexity of P-L insurance liabilities calculated based on this approachare substantially lower than those measured using traditional approaches, which has important implications forasset-liability management by P-L insurers.

Key words: duration, term structure model, property-liability insurance

JEL Classification No.: E43, G2

1. Introduction

The standard approach to measuring the interest rate sensitivity of insurance liabilitieshas been to calculate the Macaulay or modified duration. These methods are based on thefollowing three assumptions:

(1) the yield curve is flat,(2) any interest rate change is a parallel shift in the yield curve, and(3) the cash flows do not change as interest rates change.

Each of these assumptions conflicts with reality. First, even a casual perusal of interest ratesindicates that rates vary depending on the length of time the funds are committed. Typically,

76 AHLGRIM, D’ARCY AND GORVETT

interest rates on long-term commitments are higher than for shorter time periods (resultingin a “normal,” or positively sloped, yield curve), although inverted, or even humped, yieldcurves do occur occasionally (Ahlgrim, D’Arcy and Gorvett [1999]). Thus, the assumptionabout a flat yield curve is not valid. Second, interest rate movements are more complexthan a simple parallel shift. Litterman and Scheinkman [1991] identify three componentsof interest rate movements, which they term level, steepness, and curvature. Although thelevel factor, which would correspond with a parallel shift, accounts for approximately 90percent of the variance in interest rates over the period 1984–1988, the factors representingchange in both steepness and curvature were also found to have a significant impact. Brandtand Chapman [2002] measure the level factor by the yield on short term (six month) bonds,the steepness as the difference between the yields on ten year and six month bonds andthe curvature as the sum of yields on six month and 10 year bonds minus twice the levelof two year bonds. Third, P-L insurance liability cash flows are not fixed in value butare simply estimates of future payments. These payments will change as interest rateschange due to the correlation between interest rates and inflation (D’Arcy and Gorvett[2001]).

The purpose of this article is to determine the impact of incorporating contemporary termstructure models and interest sensitive cash flows on measures of the interest rate sensitivityfor insurance liabilities. These values should be lower when based on term structure modelsand interest sensitive cash flows for two reasons. Under accepted term structure models,mid term and long term interest rates are less volatile than short term rates. Since many ofthe payments from liabilities will be made after a number of years, the overall change of thenet present value for a given short term interest rate change will be reduced. Similarly, if thecash flows from liabilities increase as interest rates increase, then the impact of an interestrate change on the net present value of the cash flow will also be reduced. By incorporatingboth of these adjustments, the interest sensitivity of insurance liabilities is found to be 60–84percent lower than the standard measures would indicate. Approximately two-thirds of thisreduction is due to the use of term structure models; the remainder results from recognizingthe interest sensitivity of the loss reserves. The magnitude of this adjustment indicatesthat standard approaches are not accurate enough to use for financial risk management ofproperty-liability insurers.

Increases in financial volatility during the 1970s and 1980s led many institutions tobegin analyzing their overall exposure to financial risks. One general type of financialrisk management technique, asset-liability management (ALM), is a process under whichinstitutions analyze the combined impact of risks on both assets and liabilities to determinewhether the effects tend to offset or amplify each other. Van der Meer and Smink [1993]catalog a variety of ALM approaches, and classify them based on several factors, includingthe degree of ongoing decision-making involved and whether they are based on preservingvalue or rate of return.

Within the financial services industries, many ALM techniques were initially developedfor banks, and were later adopted by life insurers. Recently, property-liability insurers havealso begun to utilize ALM, and other financial risk management, methods. An extensiveanalysis of financial risk management as performed by insurers is presented in Santomeroand Babbel [1997]. One of the challenging issues in ALM for property-liability insurers is

THE EFFECTIVE DURATION AND CONVEXITY OF LIABILITIES 77

the measurement of the interest rate sensitivity of liabilities. P-L insurance liabilities, whichprimarily reflect unpaid losses, are contingent liabilities that depend upon the cost to repairor replace items, or to provide medical care and compensate for other bodily injury losses.These liabilities are associated with losses that either have already occurred but have not yetbeen settled (loss reserves), or have not yet even happened (unearned premium reserve). Inmany cases, the future payments emerging from a loss reserve will be for medical expensesor lost wages that have not yet been incurred, even though the incident that gives rise tothe claim has already occurred. Alternatively, the reserve might relate to a non-economicloss, such as pain and suffering, whose value will be determined at some point in the future.These types of future payments will be impacted by future inflation to a degree which isdependent upon several factors—e.g., the level of the coverage-specific inflation rate, andthe rate at which claim costs become “fixed” in value. Since inflation is widely recognizedas being correlated with interest rates, this makes the determination of the interest sensitivityof these cash flows a rather complex issue.

The interest rate sensitivity of the liabilities of property-liability insurers has been dealtwith in several different ways. Most analyses (Campbell [1995]; Panning [1995]; Hodesand Feldblum [1996]; Tzeng, Wang and Soo [2000]) assume that the cash flows emanatingfrom liabilities are not sensitive to interest rate changes. Several studies (Staking and Babbel[1995, 1997]; Choi[1992]) have examined the interest sensitivity of cash flows associatedwith liabilities, but have not found convincing evidence of a significant relationship. Thesestudies, though, used the Taylor separation method (Taylor [1986, 2000]) that divides paidloss development into two components, inflation and real loss development. In this method,the inflation component affects all loss payments made in a given year by the same amount,regardless of the original accident year. In essence, any loss that had not yet been paid,regardless of how long ago it had occurred or at what stage of the settlement process itwas in, is assumed to be fully sensitive to inflation. This assumption could be affecting theresults of the Staking and Babbel and Choi studies. An alternative approach to reflectingthe impact of inflation on loss development has been proposed by D’Arcy and Gorvett[2001]. In their model, loss reserves become gradually “fixed” in value over the settlementperiod. Inflation affects the portion of unpaid losses that have not yet been fixed in value(for example, medical treatment that has yet to be provided). This method provides a betterexplanation of the actual relationship between inflation and loss development, and leadsto a method for determining the cash flow sensitivity of liabilities for property-liabilityinsurers.

The correct approach to determining the impact of interest rate changes on the liabilitiesof P-L insurers needs to reflect both the interest rate sensitivity of the cash flows and a termstructure model of interest rates. Several researchers (Staking and Babbel [1995, 1997];Choi [1992]; Tzeng, Wang and Soo [2000]) have utilized stochastic interest rate modelsto value property-liability insurance liabilities, but did not reflect the interest sensitivityof the cash flows from liabilities. Noris and Epstein [1988] and Briys and de Varenne[1997] address both the interest sensitivity of cash flows and the stochastic interest rate forlife insurance applications. This paper performs a similar application for property-liabilityinsurance, providing a new approach to measure the magnitude of the interest rate sensitivityof liabilities, which is a critcial step in asset-liability management.


2. Literature review

Prior studies on interest rates and inflation

A number of studies have examined the relationship between interest rates and inflation.Irving Fisher [1930] proposed that nominal interest rates equal the anticipated inflation rateplus the desired real rate of return (the interest rate after adjustment for inflation). In thisformulation, termed the Fisher effect, the real interest rate is a constant and the nominalinterest rate is:

rn = (1 + i)(1 + rr ) − 1 (1)

where rn = the nominal interest rate, i = the inflation rate, and rr = the real interest rate.Fama and Bliss [1987] note a mean-reverting tendency of interest rates. Mishkin [1990]

finds that short term nominal interest rates provide information about the term structure ofreal interest rates and longer term interest rates provide information about future inflationrates. Fama [1990] finds that expected inflation and expected real returns are inverselyrelated. Fama [1984] finds that forward interest rates on U.S. Treasury bills have someexplanatory power for predicting future spot rates. Fama and Schwert [1977] test the Fishereffect and determine that nominal returns on government debt vary with expected inflation,but not unexpected inflation. The expected real returns are not related to the expectedinflation rate.

Aıt-Sahalia [1999] examines whether interest rates follow a diffusion process (continu-ous time Markov process), given that only discrete-time interest rates are available. Thiswork finds that neither short-term interest rates nor long-term interest rates follow Markovprocesses, but the slope of the yield curve is a univariate Markov process and a diffusionprocess. Chapman and Pearson [2001] provide a comprehensive review of term structuremodels. They conclude that volatility increases with the level of the short term interest rateand, within normal interest rate ranges, mean reversion is weak. They also conclude thatmore research is needed to determine which interest rate model is best.

The general conclusion that can be drawn from this research is that no term structuremodel is clearly best and that, although interest rates and inflation are related, the relationshipis difficult to quantify based on available data. Thus, in this paper three term structuremodels will be tested, a one-factor equilibrium model, a one-factor no-arbitrage model anda two factor model, and the sensitivity of the results to different parameter values will beexamined.

Prior studies on the interest sensitivity of insurance liabilities

Tilley [1988] provides an excellent, non-mathematical discussion regarding the problem ofimmunization for insurance companies and pension plans. A study by the Financial AnalysisCommittee of the Casualty Actuarial Society [1989] examines the effect of a mismatch ofassets and liabilities for property-liability insurers but ignores the interest sensitivity of lossreserve development.


Staking [1989] and Babbel and Staking [1995, 1997] utilize a modification of the TaylorSeparation Method to project the total cash flows from claim payments. This method as-sumes that inflation in a given year affects all unpaid losses for a given line equally, regardlessof the accident year. The relationship between the market value of the firm and its lever-age and surplus duration is then measured. The results of these relationships are displayedgraphically by a saddle-shaped curve illustrating leverage, surplus duration, and the Tobin’sQ value (market to book value). While the mean value of leverage, 3.47, lies along the crestof the saddle, suggesting that on average insurers adopt a leverage ratio that maximizes themarket value of the firm, the mean value of surplus duration, 9.68, lies near the minimumvalues of the curve. This means if surplus duration were any lower or higher than the averagevalue, the market value of the firm would increase. This result indicates that insurers wereoperating at a surplus duration level that minimized the firm’s value, meaning either surplusduration is not measured accurately, or that insurers need to look at duration much moreclosely.

Choi [1992] addresses the issue of the interest rate sensitivity of loss payments also usingthe Taylor Separation model, and finds weak evidence of a link between inflation and losspayments, but no link between nominal interest rates and loss payments. Reitano [1992,1996] analyzes immunization for an insurance company based on non-parallel shifts in theyield curve. Although Reitano does not consider interest-sensitive liabilities, it does illustratethe complexity involved in attempting to immunize the surplus of an insurance companywhen interest rates change in realistic patterns. Babbel [1995] illustrates the problemsassociated with calculating duration incorrectly and explains that the proper approach formeasuring interest rate sensitivity is to use a stochastic interest rate model and reflect theinterest sensitivity of cash flows. Briys and de Varenne [1997] demonstrate the differencesbetween effective and Macaulay duration for a portfolio of universal life insurance policies.Li and Panjer [1994] calculate the duration of life and annuity cash flows using severalinterest rate models, including the Vasicek and Cox-Ingersoll-Ross models, showing thatthe duration based on stochastic models is smaller than duration based on constant interestrates, with the difference most pronounced for longer-term contracts. They also point outsome of the problems associated with using stochastic interest rate models, including thechoice of a particular model, the estimation of parameters, and the problems generated byaccounting conventions.

These studies clearly demonstrate the importance of using a stochastic interest rate modeland the need to reflect the interest rate sensitivity of the cash flows emanating from insuranceliabilities. The remainder of this paper describes several measures of interest rate sensitivity,the three stochastic interest rate models used in this study, a model that reflects the inflation-sensitivity of property-liability insurance loss reserves and the results of applying thesemodels.

3. Measuring interest rate sensitivity: Duration and convexity

Duration is a first order approximation of the sensitivity of the economic value of a cashflow to interest rate changes. Several different measures of duration have been proposed.Macaulay and modified duration are used for cash flows that are not sensitive to interest


rates. Effective duration is used when the cash flows can change as interest rates change. Inorder to determine the effective duration of a cash flow, the present value of the cash flowis calculated in three ways: with the original term structure of interest rates and then with aterm structure that is generated if the instantaneous interest rate is increased and decreasedby a specific amount (r ±�r ). The effective duration is then calculated as:

Effective duration = ED0 = PV− − PV+2PV0(�r )

where PV− = the present value of the expected cash flows if the instantaneous interest ratedeclines by �r , PV+ = the present value of the expected cash flows if the instantaneousinterest rate increases by �r , and PV0 = the initial present value of the expected cash flowsbased on the original term structure.

Fabozzi [1998] proposes a methodology for calculating the effective duration of aninterest-sensitive cash flow that assumes both a flat yield curve and parallel shifts. Wilmott[1998] uses simulation to value interest-sensitive cash flows. This approach is especiallyuseful when the cash flow depends on the particular interest rate path, not just the terminalvalue. Under this approach, one interest rate path is generated at random through simulation.The cash flow resulting from that interest rate path is then calculated and discounted basedon that interest rate path to determine one market value. The process is then repeated a largenumber of times to generate an entire distribution of market values. Since the cash flowsfrom insurance liabilities are path dependent (they vary based on the particular interest ratepath, not just the terminal value of the interest rate), then simulation is the most effectivemethod for determining the sensitivity of liabilities to interest rate changes.

Since the relationship between the economic value of a cash flow and interest rates isnormally curved, the approximation of the economic value to interest rate changes basedon effective duration is generally useful only for small changes in interest rates. Convexityis a second order approximation of this relationship, providing a more accurate estimate ofinterest rate sensitivity.1 For interest-sensitive cash flows, the effective convexity must becalculated. The effective convexity formula is:

Effective convexity = PV− + PV+ − 2PV0

PV0(�r )2

Duration and convexity are applied in ALM in an attempt to balance the effects of interestrate changes on the firm’s assets and liabilities so they tend to offset each other. In an extremecase, the surplus of a firm can be immunized from interest rate changes by setting the interestrate sensitivity of surplus to zero. The relationship would be determined as follows:

�sS = �AA −�LL

where � = the sensitivity of a the subscripted value to interest rates, S = Surplus, A =Assets, L = Liabilities.


For the purposes of this paper, �V (the change in economic value of a cash flow) willbe estimated by considering the effect of both duration and convexity as follows:

�V = (−1)(Effective Duration)(�r ) + (1/2)(Convexity)(�r )2

4. Interest rate models

Determining effective durations of assets and liabilities in a stochastic interest rate environ-ment requires employing a model for interest rates (also known as term structure models).There are a variety of such models in use among both academics and practitioners, and noone model—or even one type of model—has come to the forefront of interest rate theory orpractice. Instead, numerous models have been considered and empirically tested. The basicdistinctions among interest rate models are between equilibrium and no-arbitrage models,the number of factors involved in the model and whether the model depicts nominal or realinterest rates.

Equilibrium models develop a process for the short-term interest rate as an outgrowthof a broad economic framework. This framework then also provides information aboutthe entire term structure. These models, while not necessarily reflecting current interestrates, frequently have a simplified structure that allows for analytic valuation of the en-tire term structure based one or two factors. Examples of common equilibrium interestrate models include Vasicek [1977] and Cox, Ingersoll and Ross [1985] (hereinafter re-ferred to as “CIR”). Chan, Karolyi, Longstaff and Sanders [1992] compare eight differentequilibrium term structure models and provide estimates of the parameter values for eachmodel.

One potential problem with equilibrium models is that they will not necessarily accuratelyrepresent today’s term structure of interest rates. Thus, such models can erroneously indicatearbitrage opportunities, situations where an investor can be guaranteed to earn a risk-freeprofit with no initial investment. No-arbitrage models are designed to avoid this difficulty:they can be parameterized to precisely fit the current term structure. Examples of no-arbitrage interest rate models include Ho and Lee [1986], Hull and White [1990], andHeath, Jarrow and Morton [1992]. The Ho and Lee model is based on two parameters, theshort-rate standard deviation and the market price of risk in the short-term rate. Hull andWhite developed a no-arbitrage model that incorporated mean reversion. Heath, Jarrow andMorton [1992] developed an even more general term structure model that starts with thecurrent forward rate curve and allows stochastic changes in the term structure to occur. (SeeHull [2000] for an extensive description of term structure models.)

Another classification of term structure models is the number of “factors” employed. One-factor models represent the interest rate process as having just one source of uncertainty—forexample, the short-term interest rate. This somewhat limits the shape and movements ofthe entire term structure of interest rates.2 In a multi-factor model, the interest rate processinvolves two or more sources of uncertainty. For example, the short-term rate might be onefactor, and the mean to which the short-term rate reverts over time might be another. Sucha framework is more flexible than a one-factor model, and can provide a wider variety ofpotential term structure movements.


An additional classification of term structure models is whether they apply to nominal orreal interest rates. Real interest rates represent the difference between nominal interest ratesand the expected inflation rate. Real interest rates cannot be accurately measured ex ante,since there is no published concensus inflation rate, and ex post measurements are subject toerror since there is no evidence that inflation can be accurately forecasted. Hibbert, Mowbryand Turnbull [2001] utilize a two factor Vasicek model for real interest rates, with the shortrate and the long term rate representing the two factors.

In this paper, we have chosen to employ three different interest rate models: two one-factor models, a CIR equilibrium model and a no-arbitrage Hull-White model, which bothapply to nominal interest rates, and a two factor equilibrium model of real interest rateswhich is combined with a separate model of inflation. The choice of one-factor models isbased on Ahlgrim [2001] that finds that these models adequately capture the interest ratevolatility for property-liability insurance applications. The two factor model allows realinterest rates and inflation to be determined separately.

Cox-Ingersoll-Ross model

The CIR model involves a mean-reverting process with volatility proportional to the square-root of the level of interest rates. Mathematically, the continuous-time version of the model is

dr = κ(θ − r ) dt + σ√

rdz

where r is the instantaneous (short-term) interest rate, κ is the speed of mean reversion, θis the long-run mean to which r tends to revert over time, σ is a volatility parameter, anddz is a simple Brownian motion process. The discrete-time version of CIR is

�r = κ(θ − r )�t + σ√

rε√�t

where �t is the discrete time interval and ε is a random sampling from a standard normaldistribution.3 It is this process that is simulated in this paper, as described below.

Hull-White model

Hull and White [1990] develop a no-arbitrage interest rate model of the following form:

dr = a

[θ (t)

a− r

]dt + σdz

where a is the speed of mean reversion, σ is the volatility of interest rate changes, and θ isthe time-dependent mean reversion level. The movement of the short-term interest rate isdetermined by the shape of the existing forward rate curve, as follows:

θ (t) = Ft (0, t) + aF(0, t) + σ 2

2a(1 − e−2at )

The inclusion of time dependent mean reversion in the Hull-White model allows the move-ment of the short rate to match the forward curve implied by the existing term structure.


Thus, when using the current term structure as an input, there are no implied arbitrageopportunities present in the assumed structure of the Hull-White model.

Two factor model of real interest rates

This two-factor term structure model allows the short-term rate (denoted by r ) to revert toa long-term rate (denoted by l) that is itself stochastic.

drt = κ1(lt − rt ) dt + σ1dz1

dlt = κ2(µ− lt ) dt + σ2dz2

The discrete time version of the model is:

�rt = κ1(lt − rt )�t + σ1ε1t

√�t

�lt = κ2(µ− lt )�t + σ2ε2t

√�t

The parameters used for the CIR, the Hull-White and the two factor models are displayedin Table 3. Only four parameters are needed for the CIR model, κ , θ , σ and r . The values forthe mean reversion speed, κ = .25, and volatility, σ = .08, are based on values determinedin CKLS [1992]. The initial short term interest rate, r = 3.62%, is the rate on the shortestterm (2 month) U.S. Treasury strip available on June 9, 2001. The long run mean interestrate, θ = 6.6%, was selected to minimize the mean squared error between actual yields asof June 8, 2001, and the yield curve generated by the CIR model.

The Hull-White model requires two parameters, a and σ , and the entire initial yield curve.The mean reversion rate, a = .1, and the volatility, σ = .015, are based on the factors usedin various applications of the Hull-White model, including Hull [2000]. The initial yieldcurve was based on the U.S. Treasury strips (coupon interest) as of the close of trading onJune 8, 2001, with some modifications. First, the longest term for which the U.S. Treasuryissues bonds is 30 years, and the longest maturity strip currently traded has a maturity ofFebruary, 2031. Since initial yields are needed for 32 years in order to generate the forwardrate curves long enough for the modeled time period, values for years 31 and 32 had to beselected. Second, the current policy of the U.S. Treasury is to retire long term debt, whichhas limited the supply of the longest term bonds, raising prices and lowering interest rates,which produces a humped yield curve. Due to the limited supply of the longest term bonds,the actual yield curve implies extremely low, and even negative, forward interest rates forthe last few years. Therefore, the selected values are the actual yields for the first 27 years,and then the smoothed values to reflect a consistent decline in yields for years 28 through 32.

The two factor model requires five parameters, which are calculated based on the follow-ing regressions on monthly data from 1982 to 2001:

rt+1 = α1 + α2lt + α3rt + ε′1t

lt+1 = β1 + β2lt + ε′2t


Traditional OLS regressions are not possible since the short rate process is dependentupon the long rate. Two-stage least squares estimation is necessary. In order to estimate theshort-rate equation, estimates for the long-rate must first be obtained.

Stage 1: lt+1 = β1 + β2lt + ε′2tStage 2: �rt+1 = α1(lt − rt ) + ε′1t

The resulting parameters for the real interest rate were selected from the regressionresults. Inflation (denoted by q) is modeled separately from the real interest rate. Inflationis assumed to follow an Ornstein-Uhlenbeck process of the form:

dqt = κ(µq − qt ) dt + σdBq

In discrete form this is:

�qt = qt+1 − qt = κq (µq − qt )�t + εqσq

√�t (1)

Fisher [1930] provides a thorough presentation of the interaction of real interest ratesand inflation and their effects on nominal interest rates. He argues that nominal interestrates compensate investors not only for the time value of money, but also for the erosionof purchasing power that results from inflation. In the two-factor model presented here, thecombined movements of real interest rates and inflation determine the underlying processfor nominal interest rates. Subsequently, when using simulation to determine economicvalues, the path of inflation is allowed to exert its effects separately on a security’s cashflows and the evolution of nominal interest rates, which includes compensation for inflation,determines the present value of projected cash flows.

5. A model of cash flow sensitivity to inflation and interest rates

The focus of this analysis is on the economic value of insurer liabilities. This value isdetermined by discounting the expected future cash flows emanating from these liabilitiesby the appropriate discount rate. In order to determine the economic value of liabilities,statutory values for loss reserves, loss adjustment reserves, and unearned premium reservesmust be used. In this paper, we concentrate on loss reserves, since they typically comprisethe greatest part of overall P-L insurance company liabilities.4 The cash flows for each ofthese components are sensitive to interest rate changes, although to varying degrees. Thestatutory loss reserves are the expected value of the nominal payments that will be made inthe future for losses that have already occurred without any discounting.

D’Arcy and Gorvett [2001] have proposed a model that reflects the relationship betweenunpaid losses and inflation. In this model, a distinction is made between unpaid losses thatare “fixed” in value, and therefore not subject to future inflation, and the portion of the lossthat is still subject to future inflation. For example, a first party claim for damage coveredunder a homeowners policy would be fixed in value once the repair estimates have been


obtained, although the claim would not be paid until the repairs are completed. An exampleof a loss reserve that is completely subject to future inflation would be a medical malpracticeclaim in which the claimant will have to undergo corrective surgery in the future. The costof this treatment will reflect future medical cost inflation between the current time and thetime of the surgery.

More commonly, unpaid claims are partially fixed and partially subject to future inflation.Aggregate loss reserves would consist of a portfolio of claims with differing proportions offixed and variable components.

A representative function that displays these attributes is:

f (t) = k + {(1 − k − m)(t/T )n} (2)

where f (t) represents the proportion of ultimate paid claims “fixed” at time t , k = theproportion of the claim that is fixed in value immediately, m = the proportion of the claimthat is not fixed in value until the claim is settled, n = 1 for the linear case, n < 1 for theconcave case, n > 1 for the convex case, and T = the time at which the claim is fully andcompletely settled.

Problems arise in developing parameters for this model. Although insurers maintainextensive detail about loss development, they do not calculate the rate at which the specialdamages (e.g. property damage and medical costs) become fixed in value. There is alsono information available about how non-economic losses (for pain and suffering and otherintangible losses) were valued, whether they were set by a multiple of medical expensesor whether they were based on the cost of living at the time of settlement. Thus, there isconsiderable uncertainty about the appropriate values for these parameters.

The model proposed above applies to loss reserves. Loss adjustment expense reservescould be modeled similarly, but the parameters are likely to be very different. In general,insurers pay loss adjustment expenses as they are incurred, rather than waiting until the lossis settled. Since this paper is concerned with the inflation sensitivity of liabilities, only theloss adjustment reserves are important. Thus, it can be expected that the loss adjustmentexpense reserve is more likely to reflect services that have not yet been provided, andtherefore more sensitive to future inflation, than loss reserves.

The inflation sensitivity of the unearned premium reserve is much easier to determine.The statutory unearned premium reserve is the pro-rata portion of the written premiumthat applies to the unexpired coverage period. The only uncertain element of the unearnedpremium reserve is the amount for losses and LAE. This depends of future losses, which inturn depend on future inflation. Thus, the economic value of the unearned premium reserveis fully sensitive to future inflation.

As described above, a significant component of the liabilities of property-liability insurersare inflation-sensitive. It is through this inflation sensitivity that the cash flows from theliabilities are, in turn, interest rate sensitive. Thus, estimates of both inflation and interestrates are required to measure the sensitivity of insurance liabilities to interest rate changes.

Historical patterns show a variety of different inflation trends. Prior to the 1940s, pricestended to cycle over time, generally in line with wars or famines, but demonstrated noconsistent long term drift. Thus, in the United States, consumer prices in 1942 were ap-proximately the same as in 1865, although society experienced bouts of both inflation and


deflation during the intervening period. Subsequent to World War II, inflation appears tohave become ingrained into the system, as prices have tended to drift higher each year. Someobservers credit (or blame) Keynesian economic policies that encourage deficit spendingby governments for this structural change. The pattern and levels of inflation vary widelyby nation, with some countries experiencing extremely high annual rates of inflation (100–400%) over extended periods of time, while other countries have kept inflation rates at muchlower levels.

Based on the Fisher effect, the inflation rate can be determined as a function of thenominal interest rate:

i = (1 + rn)/(1 + rr ) − 1

Different interest rates are all highly correlated. The relationships between individualinterest rates for different maturities of U.S. Treasury debt, as well as for average interestrates, with the inflation rate are shown on Table 1. The interest rate on three month Treasurybills has the highest correlation with inflation.

For the one factor CIR and Hull-White models it is necessary to relate the inflation rate tointerest rates. The output from each interest rate model is used to generate an inflation ratethat is then used to determine the cash flow from the liabilities. Prior researchers have shownthat the term structure is not an especially good predictor of future inflation. However, for thepurposes of this research we do not need to forecast future inflation, only measure inflationas it is reflected by a particular interest rate path. The interest rate models described aboveare used to generate a large number of interest rate paths. For each path, the correspondinginflation rate is determined as follows:

i = α + βrs (3)

where i = the inflation rate, and rs = the short term interest rate.This relationship is used for the one factor CIR and Hull-White models to obtain a general

inflation rate; in this case inflation is a determistic value based on the generated interest

Table 1. Correlation of inflation rate with U.S. treasury interest rates for various maturities April 1953–February2001.

Inflation 3 month 1 year 3 year 5 year 10 year Average

Inflation 1

3 month 0.576786 1

1 year 0.5614445 0.990516 1

3 year 0.507995 0.960875 0.984385 1

5 year 0.489875 0.940449 0.968042 0.996409 1

10 year 0.467207 0.912592 0.943023 0.984123 0.995059 1

Average 0.52802 0.973941 0.990385 0.998156 0.992773 0.979405 1


Table 2. Linear regression based on monthly data. April 1953–February 2001,Dependent variable: Inflation rate, Independent variable: 3 month interest rate.

Regression statistics

Multiple R 0.577

R Square 0.333

Adjusted R Square 0.332

Standard Error 3.277

Observations 575

Coefficients Standard error t stat

Intercept −0.445 0.301 −1.475

3 month rate 0.833 0.049 16.902∗

∗Significance level at .001.

rate. In the two factor model, however, inflation is generated as an independent stochasticprocess as described in the prior section. In both cases, the inflation rate that is determinedis the general inflation, which is then used for claim inflation. Historically, though, theinflation rate that has impacted insurance claims has varied from the general inflation ratedepending on the particular coverage being considered. Masterson [1968] documented thistrend by examining different components (medical, auto repair costs, etc.) of the consumerprice index, providing information that allows insurers to develop by-line inflation rates.In this paper we have modeled only general inflation since the approach is applied to theaggregate loss reserves of an insurer, not a specific line of business.

The general inflation rate is measured based on the Consumer Price Index (CPI) (sea-sonally adjusted, as determined for all urban consumers). The short-term interest rate is the3-month Treasury Bill rate (secondary market).5 The results of a least squares linear regres-sion, based on monthly data from April, 1953 through February, 2001, with the inflationrate as the dependent variable and the short term interest rate as the independent variablesare shown on Table 2. The coefficient for the short-term interest rate is highly significant.The R-squared value of this relationship is 33.3%, indicating that this variable explains asignificant portion of the variation in inflation.

To generate the general inflation rate for this model, first an interest rate path is simulatedas described above. The inflation rate is determined based on Eq. (3), with the parametersfrom the regression as shown in Table 2:

i = −.0044 + .83rs

6. Results

When the cash flows from an asset or liability are path-dependent, the appropriate methodfor measuring the effective duration or convexity of the instrument is through the useof simulation. This process follows four steps, which are described in detail along withan example of the process, in the Appendix. First, a large number of interest rate paths


Table 3. Interest rate model parameters.

CIR

k speed of mean reversion 0.25

θ long run mean interest rate 0.066

σ volatility 0.08

r initial interest rate 0.0362

The values for the speed of mean reversion and volatility are taken from CKLS [1992].

The initial interest rate is the yield on the shortest maturity U.S. Treasury strip (August 2001) as ofJune 8, 2001.

The long run mean interest rate is the value that produces the lowest mean square error when fitting the currentyield curve to the yield curve generated by the CIR model (for years 1–19, since after that point theyield curve slopes downward, which cannot be modeled by the one-factor CIR model).

Hull-White

a mean reversion rate 0.1

σ volatility 0.015

The values for the mean reversion and volatility are taken from Hull (2000).

Initial yield curve values are based on closing asked prices for U.S. Treasury strips (coupon interest) on June 8,2001, with minor adjusments.

The actual and selected yields are listed below.

Maturity Year Actual yield Implied forward rate Selected yield Implied forward rate

Aug 2001 0 0.0362 – 0.0362 –

May 2002 1 0.0376 0.0456 0.0376 0.0456

May 2003 2 0.0416 0.0536 0.0416 0.0536

May 2004 3 0.0456 0.0601 0.0456 0.0601

May 2005 4 0.0492 0.0572 0.0492 0.0572

May 2006 5 0.0508 0.0592 0.0508 0.0592

May 2007 6 0.0522 0.0627 0.0522 0.0627

May 2008 7 0.0537 0.0585 0.0537 0.0585

May 2009 8 0.0543 0.0642 0.0543 0.0642

May 2010 9 0.0554 0.0644 0.0554 0.0644

May 2011 10 0.0563 0.0674 0.0563 0.0674

May 2012 11 0.0573 0.0657 0.0573 0.0657

May 2013 12 0.0580 0.0684 0.0580 0.0684

May 2014 13 0.0588 0.0686 0.0588 0.0686

May 2015 14 0.0595 0.0685 0.0595 0.0685

May 2016 15 0.0601 0.0714 0.0601 0.0714

May 2017 16 0.0608 0.0642 0.0608 0.0642

May 2018 17 0.0610 0.0628 0.0610 0.0628

May 2019 18 0.0611 0.0649 0.0611 0.0649

(Continued on next page.)


Table 3. (Continued).

Maturity Year Actual yield Implied forward rate Selected yield Implied forward rate

May 2020 19 0.0613 0.0573 0.0613 0.0573

May 2021 20 0.0611 0.0611 0.0611 0.0611

May 2022 21 0.0611 0.0611 0.0611 0.0611

May 2023 22 0.0611 0.0611 0.0611 0.0611

May 2024 23 0.0611 0.0515 0.0611 0.0515

May 2025 24 0.0607 0.0532 0.0607 0.0532

May 2026 25 0.0604 0.0526 0.0604 0.0526

May 2027 26 0.0601 0.0467 0.0601 0.0467

May 2028 27 0.0596 0.0347 0.0596 0.0457

May 2029 28 0.0587 0.0301 0.0591 0.0447

May 2030 29 0.0577 -0.0063 0.0586 0.0437

Feb 2031 30 0.0555 0.0581 0.0427

– 31 – 0.0576 0.0417

– 32 – 0.0571

Two Factor

Inflation:

q initial inflation rate 0.01


µ long run mean inflation rate 0.048

σ volatility 0.04

Real interest rate (Hull-White 2-factor):

Short process


σ volatility 0.1

r initial short term interest rate 0.01

Long process


µ long run mean interest rate 0.028

σ volatility 0.1

l initial long term interest rate 0.015

are generated. Second, the associated cash flow for each interest rate path is determined.Third, the present value of each cash flow is calculated by discounting that cash flowat the corresponding interest rates. Finally, the overall present value of the instrument isdetermined by calculating the average of all the simulated values. This entire process isrepeated three times, once for the initial interest rate (to determine PV0), once with theinitial interest rate increased by a selected amount (to determine PV+) and once with theinitial interest rate decreased by the same amount (to determine PV−).


The impact of interest rate changes on property-liability loss reserves is illustrated bycombining the model of the interest rate sensitivity of loss reserves described in Eq. (2)with each of the three stochastic term structure models described above. Effective durationand convexity values for a representative property-liability insurance company situationare then calculated. These values are compared with the conventional measures of durationand convexity, which reflect neither the interest sensitivity of loss reserves nor stochasticterm structure models, but are based simply on a parallel shift of a non-flat term structure.(Although Macaulay and modified duration are sometimes calculated based on a flat yieldcurve, in this comparison all duration and convexity measures are determined based on sim-ilar initial yield curves to prevent the differences from being generated by that distinction.)Additionally, the effective duration convexity measures for liabilities are compared with thesame calculations for several asset portfolios.

The loss payout pattern for all lines combined was determined based on the industryaggregate all-lines-combined Schedule P exhibit in Best’s Aggregates and Averages [1999].Losses unpaid after 10 years were assumed to be paid over the next 20 years, generatinga 30 year payment pattern. This payment pattern was applied consistently for 30 accidentyears under a no-growth assumption to determine the current loss reserves and remainingloss payment patterns for each year. The model for determining the interest sensitivity ofloss reserves was then applied to each accident year individually to determine the fixed andinflation sensitive component of the loss reserve. The parameters of the base case modelwere k = 0.25, m = 0.25, and n = 1.00. The sensitivity of results to each of theseparameters was also tested.

A simulation for each interest rate model was run using @Risk for 1,000 iterationswith sampling based on the Latin Hypercube method to increase sampling efficiency.Figures 1, 2 and 3 show the mean values for the three interest rate paths used to determine

Figure 1. CIR interest rate curves: Mean values of 1000 simulations.


Figure 2. Hull-White interest rate curves: Mean values of 1000 simulations.

effective duration and convexity (initial, plus and minus 100 basis points) for each model.For the CIR model (figure 1), all three interest rate curves are smooth and upward slop-ing that gradually converge on the same value due to the mean reversion factor. For theHull-White model (figure 2), the interest rate curves are not as smooth and are humpshaped, reflecting the actual term structure reflected by yields on U.S. Treasury strips(essentially zero coupon bonds) as of June 9, 2001. These interest rate curves also grad-ually converge, but at a lower level than the CIR model paths. For the two factor model(figure 3), the real interest rate, the inflation rate and the nominal interest rate paths areall displayed. These curves have the same general shape as the CIR model. A compari-son of the range of simulated interest rate values for each year is shown in figures 4, 5and 6, which illustrate selected percentile values for the initial interest rate curve. Sev-eral differences among the models are notable. Although the range of interest rates after30 years is comparable for each model and the values diverge at about the same rate, thecurves under the CIR model level out, whereas the curves under the Hull-White model aredeclining for each percentile after approximately 15 years. The Hull-White model gener-ates a more flexible pattern of interest rates, since it is based on the entire current termstructure and not on a limited number of parameters that constrain the shape of the termstructure, as is the case for the CIR model. Finally, the two-factor model generates a signif-icant number of negative interest rates throughout the 30 year period. This generally occurswhen the inflation rate is a negative value, and in absolute terms exceeds the real interestrate.

The effective duration and convexity, as well as the Macaulay and modified duration, aredisplayed in Table 4. The effective duration and convexity measures reflect both the interestsensitivity of the loss reserve payments and stochastic interest rates. The Macaulay andmodified duration and the conventional convexity measure assume loss payments are not


Figure 3. Two-factor interest rate curves: Mean values of 1000 simulations.

Figure 4. Range of simulated interest rate paths CIR model. (based on 1000 simulations)

interest sensitive and yield curves shift in a parallel manner. To avoid differences resultingsimply from the initial yield curve, Macaulay and modified duration and the traditionalconvexity measure are calculated based on the same initial yield curves as used for all theinterest rate models. Both effective duration and effective convexity are much lower thanthe traditional measures of duration and convexity. This result should not be surprising;


Figure 5. Range of simulated interest rate paths: Hull-White model. (based on 1000 simulations)

Figure 6. Range of simulated interest rate paths: Two-factor model. (based on 1000 simulations)

intuitively, an increase in interest rates, holding expected cash flows constant, will decreasethe present value of the cash flow more than if the cash flows increase with interest rates.The impact of the differences is quite significant. For example, if interest rates were toincrease by 200 basis points, the economic value of loss reserves will reduce by 2.08 percent(based on the CIR model), 2.50 percent (based on the Hull-White model) or 1.47 percent(based on the two-factor model) as calculated based on the effective duration and convexity


Table 4. Duration and convexity measures of loss reserves base case parameters.

Interest sensitive paymentsbased on stochastic Fixed payments based

term structure model on parallel yield curve shift

Hull- Two CIR yield Hull-White Two factorCIR White factor curve yield curve yield curve

Effective duration 1.07 1.33 0.75 Macaulay duration 4.05 4.01 3.99

Effective convexity 2.84 7.62 1.69 Modified duration 3.89 3.84 3.87

Convexity 35.49 34.61 35.13

Impact of an increase in interest rates of:

100 basis points −1.05% −1.29% −0.74% −3.72% −3.67% −3.69%

200 basis points −2.08% −2.50% −1.47% −7.08% −6.99% −7.04%

300 basis points −3.08% −3.63% −2.17% −10.09% −9.96% −10.03%

measures. However, if the impact were calculated based on the traditional modified durationand convexity measures, these values would be expected to decrease by approximately7 percent. If an insurer used modified duration to establish an Asset-Liability Managementstrategy that immunized itself from interest rate risk, it would discover that it was stillsubject to interest rate risk.

The lower values for effective duration and effective convexity determined here resultfrom two separate effects. One is the use of stochastic term structure models that have meanreverting tendencies, so the effect of the shock to the short term rate dampens out overtime, and does not have as large effect on longer term interest rates. The other componentis allowing the cash flows to change. In order to determine the relative magnitude of theseeffects, the effective duration and effective convexity are calculated using the same interestrate paths, but holding the loss payments constant. This is done by setting k, the proportionof the loss reserve that is fixed in value when the loss occurs, equal to 1.00. For the CIRmodel, effective duration is 58 percent lower than modified duration solely as a result of theinterest rate model, compared with declines of 70–75 percent when the impact of inflationon loss reserves is included. For the Hull-White model, effective duration is 38 percentlower than modified duration solely as a result of the interest rate model, compared withdeclines of 60–70 percent when the impact of inflation on loss reserves is included. Forthe two-factor model, the relative proportion of the effect of the interest rate model andthe inflation sensitivity of the loss reserves depends heavily on the assumed relationshipbetween the interest rate shock and inflation. If the change in interest rates is fully reflectedin the inflation rate (when the short term interest rate is increased 100 basis points, theinflation rate is likewise increased 1 percentage point, in line with the Fisher effect), theninterest rate model alone accounts for a 59.1 percent decline in effective duration, comparedto a 73–84 percent decline overall.

The accuracy of the effective duration and convexity measures depend on selecting anappropriate interest rate model and choosing the correct parameters both for the interest


rate model and the loss reserve model. Since, as discussed in Chapman and Pearson [2001],there is no consensus for a single interest rate model, we have used three popular modelsto illustrate the calculations, and the results are fairly similar. For the base case parameters,the effective duration is 1.07 based on the CIR model, 1.33 based on the Hull-White modeland 0.75 based on the two factor model. All are significantly lower than the Macaulay andmodified duration values calculated on the same initial yield curves. Similarly the effectiveconvexity is 2.84 based on the CIR model, 7.62 based on Hull-White and 1.69 basedon the two-factor model, compared with 35.49 and 34.33 and 35.13, respectively. Thus,although there may be disagreement over which interest rate model is most appropriate,in each case the resulting measures of the interest sensitivity of liabilities are significantlylower than traditional duration and convexity measures. The results indicate that ignoringthe interest sensitivity of the liabilities and assuming a parallel shift in yield curves willindicate that loss reserves are much more sensitive to interest rate changes than they actuallyare.

Several of the parameters used in this study are estimated either based on historicaldata or the results of limited testing. There is, for example, for the interest rate models noconsensus over the selection of a volatility parameter, the mean reversion parameter, andfor the CIR model, the long run mean rate. (See Lamoureux and Witte [2002], Pearson andSun [1994] and Stanton [1997] for a discussion of the different approaches used to estimatethese parameters.) Other parameters that affect the results are the values for k, m and r inthe loss reserve model and the coefficients for the relationship between interest rates andinflation. To test the sensitivity of the results to the volatility parameter, the simulations werererun using alternative parameter selections to illustrate the impact of changes in specificparameters. The results of these sensitivity tests are shown on Tables 5 (CIR model), Table 6(Hull-White model) and Table 7 (Two-factor model). Based on these results, the most criticalparameters are the mean reversion parameters (κ for the CIR model and a for the Hull-Whitemodel), the inflation coefficient β, and, in the loss reserve model, the portion of the claimpayment that is not fixed until the loss is settled, m, and the factor determining the shape ofthe function, n. Care should be taken in selecting these parameters and additional researchshould focus on determining appropriate values for these parameters especially. None ofthe parameters exhibit significant sensitivity under the two-factor model. However, for theentire range of likely values for the parameters, effective duration is less than one-halfof the modified duration and effective convexity is less than 38 percent of the traditionalconvexity measure. Thus, it is more important to utilize the approach outlined here thanto continue use the traditional measures of interest rate sensitivity due to concern over thecorrect parameter values to utilize.

7. Interest sensitivity of assets

Proper asset liability management requires a consistent measurement of the interest ratesensitivity of both assets and liabilities. Therefore, effective duration and convexity shouldbe determined for assets the same way that it is calculated for liabilities, with the same termstructure model. Although the information on insurers’ investment portfolios necessary toperform a complete analysis of the interest sensitivity of assets is not readily available,


Table 5. Summary of results for Cox-Ingersoll-Ross model.

Effective Macaulay Modified Ratio of eff. to Effective Traditional Ratio of eff. toduration duration duration mod. duration convexity convexity trad. convexity

“Fixed” function parameters

k = 0.10 0.96 4.05 3.89 0.247 2.45 35.48 0.069

k = 0.25∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

k = 0.40 1.18 4.05 3.90 0.302 3.26 35.50 0.092

m = 0.10 1.22 4.05 3.89 0.312 3.26 35.49 0.092

m = 0.25∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

m = 0.40 0.92 4.05 3.90 0.237 2.16 35.50 0.061

k,m = 0.00 1.13 4.05 3.89 0.291 2.88 35.47 0.081

k,m = 0.10 1.11 4.04 3.89 0.284 2.88 35.47 0.081

k,m = 0.25∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

k,m = 0.40 1.03 4.05 3.90 0.264 2.82 35.51 0.079

n = 0.50 1.21 4.05 3.90 0.310 3.32 35.51 0.093

n = 1.00∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

n = 2.00 0.91 4.05 3.89 0.234 2.36 35.49 0.066

CIR parameters

κ = 0.10 1.57 4.22 4.07 0.385 5.31 38.58 0.138

κ = 0.25∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

κ = 0.40 0.79 3.98 3.83 0.208 1.69 34.39 0.049

θ = 0.05 1.09 4.27 4.11 0.266 3.04 39.56 0.077

θ = 0.066∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

θ = 0.08 1.05 3.88 3.73 0.281 2.75 32.42 0.085

σ = 0.06 1.07 4.03 3.88 0.277 2.89 35.23 0.082

σ = 0.08∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

σ = 0.10 1.05 4.06 3.91 0.269 2.80 35.74 0.078

Inflation coefficient

β = 0.63 1.28 4.05 3.89 0.330 4.08 35.49 0.115

β = 0.83∗ 1.07 4.05 3.89 0.274 2.84 35.49 0.080

β = 1.03 0.84 4.05 3.90 0.216 2.38 35.50 0.067

Effects of stochastic interest rates

k = 1.00 1.63 4.04 3.89 0.420 4.98 35.38 0.141

∗Parameter values for the “base case.”All results based upon 1,000 simulation trials.

several calculations have been performed on sample fixed income portfolios to illustratethis approach. These results are displayed in Table 8.

The first portfolio consists of a non-callable five year 6 percent coupon bond, the nearestfull year bond that approximates the modified duration of the liabilities. This is the issue that


Table 6. Summary of results for Hull-White model.



k = 0.10 1.21 3.99 3.82 0.316 6.70 34.33 0.195

k = 0.25∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

k = 0.40 1.46 3.99 3.82 0.381 8.61 34.33 0.251

m = 0.10 1.53 3.99 3.82 0.401 8.98 34.33 0.262

m = 0.25∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

m = 0.40 1.14 3.99 3.82 0.298 6.37 34.33 0.186

k,m = 0.00 1.44 4.00 3.82 0.377 2.59 34.37 0.075

k,m = 0.10 1.41 3.99 3.82 0.368 7.97 34.33 0.232

k,m = 0.25∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

k,m = 0.40 1.26 3.99 3.82 0.330 7.30 34.33 0.213

n = 0.50 1.50 3.99 3.82 0.391 8.88 34.33 0.259

n = 1.00∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

n = 2.00 1.14 3.99 3.82 0.299 6.26 34.33 0.182

Hull-White parameters

a = 0.05 1.73 4.01 3.84 0.450 13.16 34.61 0.380

a = 0.10∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

a = 0.15 1.04 3.98 3.81 0.272 4.75 34.13 0.139

s = 0.005 1.33 4.02 3.85 0.346 5.08 34.73 0.146

s = 0.015∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

s = 0.025 1.33 3.94 3.78 0.352 12.91 33.52 0.385

Inflation coefficient

b = 0.63 1.59 3.99 3.82 0.416 8.94 34.33 0.260

b = 0.83∗ 1.33 3.99 3.82 0.348 7.62 34.33 0.222

b = 1.03 1.05 3.99 3.82 0.275 6.65 34.33 0.194


k = 1.00 2.37 4.00 3.82 0.620 5.64 34.37 0.164

∗ Parameter values for the “base case.”All results based upon 1,000 simulation trials.

insurers purchasing newly issued securities would obtain. The second portfolio consists ofa non-callable seven year 6 percent coupon bond, one that represents a slight, and typical,asset-liability mismatch. The third portfolio exactly matches the expected cash flow of theliabilities. For each portfolio, the three duration and two convexity measures are calculated.Since the cash flows from the assets are not interest rate sensitive, then the effective durationand effective convexity measures are higher than the corresponding values for loss reserves.In fact, the values for the cash flow matching portfolios equal the effects of stochastic interest


Table 7. Summary of results for two-factor model: Inflation and real interest rates are modeled separately.



k = 0.10 0.65 3.99 3.87 0.167 1.41 35.13 0.040

k = 0.25∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

k = 0.40 0.85 3.99 3.87 0.221 1.99 35.13 0.057

m = 0.10 0.88 3.99 3.87 0.228 1.99 35.13 0.057

m = 0.25∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

m = 0.40 0.62 3.99 3.87 0.160 1.42 35.13 0.041

k,m = 0.00 0.80 3.99 3.87 0.206 1.70 35.13 0.048

k,m = 0.10 0.78 3.99 3.87 0.201 1.70 35.13 0.048

k,m = 0.25∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

k,m = 0.40 0.72 3.99 3.87 0.186 1.70 35.13 0.048

n = 0.50 0.88 3.99 3.87 0.228 2.04 35.13 0.058

n = 1.00∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

n = 2.00 0.60 3.99 3.87 0.156 1.35 35.13 0.038

Inflation parameters

Mean reversion speed

a = 0.2 0.89 4.06 3.93 0.227 2.38 36.35 0.065

a = 0.4∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

a = 0.6 0.65 3.96 3.82 0.169 1.31 34.44 0.038

Mean reversion level

b = 0.036 0.77 4.18 4.05 0.189 1.76 38.57 0.046

b = 0.048∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

b = 0.060 0.73 3.83 3.70 0.198 1.65 32.15 0.051

Volatility

s = 0.025 0.74 3.96 3.83 0.194 1.69 34.46 0.049

s = 0.04∗ 0.75 3.99 3.87 0.194 1.69 35.13 0.048

s = 0.055 0.76 4.04 3.92 0.193 1.72 36.16 0.048


k = 1.00 1.55 3.99 3.87 0.401 3.77 35.15 0.107

rates shown on the prior exhibits, the ones that held the liability cash flows constant. Asshown on Table 8, even with a consistent calculation of effective duration and convexityboth for assets and liabilities, assets still remain twice as sensitive to interest rate changes asliabilities. Thus, in order to accurately manage assets and liabilities, insurers need to measurethe interest rate sensitivity of liabilities through the use of a model such as proposed in thispaper.


Table 8. Interest rate sensitivity measures for assets.

Ratio of Ratio ofEffective Macaulay Modified eff. to mod. Effective Traditional eff. to trad.duration duration duration duration convexity convexity convexity

CIR

Single bond—Duration 2.51 4.47 4.31 0.584 8.17 23.93 0.341matching


Cash Flow matching 1.63 4.04 3.89 0.420 4.98 35.38 0.141

Hull-White




Two-factor model




Impact of an increase in interest rates of 200 basis points

Two-factorCIR Hull-White model

Assets

Single Bond—Duration −4.87% −5.85% −3.87%matching

Single Bond—Duration −5.40% −7.11% −4.10%matching

Cash Flow matching −3.17% −4.65% −3.03%

Loss Reserves −2.08% −2.50% −1.47%(from Table 4)

All results based upon 1,000 simulation trials.

8. Conclusion

Effective asset-liability management requires appropriate measures of the sensitivity of bothassets and liabilities to potential interest rate changes. Traditional duration measures such asMacaulay and modified duration and the commonly used convexity measure have importantlimitations: they assume a flat yield curve, parallel interest rate shifts, and independencebetween future cash flows and changes in interest rates. This research has attempted toovercome these limitations with regard to the interest rate sensitivity of property-liability


insurance liabilities by using effective duration and convexity measures in three stochasticinterest rate environments, in line with prior research applied to life insurance. The cal-culations are based on a model which specifies the degree to which P-L loss reserves areimpacted by inflation and interest rate changes over time.

The results indicate that, when stochastic interest rate models are used and the impact ofinterest rate movements on future cash flows is taken into account, the effective duration andconvexity of representative property-liability insurance liabilities are significantly less thantraditional measures would indicate. This difference can vary substantially for different pa-rameter values, but the general relationship holds for all values, and for all stochastic interestrate models tested. These findings are important for asset-liability management considera-tions in the property-liability insurance industry since they demonstrate that the economicvalue of loss reserves is much less sensitive to interest rate changes than previously indicated.Given the current maturity distribution of assets of property-liability insurers, companies aremore exposed to interest rate risk than conventional measures would indicate. Any strategyto manage interest rate risk by matching the duration of assets and liabilities would requirethat assets have a much shorter duration than traditional measures would indicate.

Appendix: Details on the effective duration calculations

This appendix provides more details on the four-step approach used to calculate the effectiveduration of P-L liabilities. The presentation will follow the outline previously described inSection 6.

(1) A large number of interest rate paths are generated. Each of the three term structuremodels is used to generate 1,000 interest rate paths. For the equilibrium models (CIR andtwo-factor), the parameters are estimated from the time series of historical interest ratemovements (and inflation data for the two-factor model), giving “realistic” interest ratescenarios. However, the Hull-White model is an arbitrage-free model. The parameters arechosen through calibration so that the resulting model is most consistent with the marketprices of traded securities (for the details of this calibration, see Ahlgrim [2001]). Given thedifficulties associated with estimating the market risk premium from arbitrage-free models,interest rates under the Hull-White term structure model are projected under the assumedrisk-neutral measure from the calibration.

A similar procedure is used to calculate the value and interest rate sensitivity of liabilitiesunder each of the term structure models presented in Section 4. To illustrate the calculations,this appendix focuses on using the two-factor term structure model. This model has a sepa-rate stochastic process for inflation and real interest rates. For completeness, the parametersof these processes are repeated here.

Inflation process

Initial inflation rate 1.0%

Reversion speed 0.4

Reversion level 4.8%

Volatility 4.0%


Two-factor real interest rate process

Short process Long process

Initial short rate 1.0% Initial long rate 1.5%

Reversion speed 6.1 Reversion speed 5.1

Volatility 10.0% Reversion level 2.8%

Volatility 10.0%

Using these parameters, a single path (scenario) of inflation and interest rates is generatedin quarterly increments for the next 30 years. Given that insurers report payment patternsin annual time steps, we compound the quarterly rates into annual rates of inflation andinterest. Payment patterns are used to project liability cash flows as described in step (2)below.

Using the initial value for the annual inflation rate and the parameters of the process, thenext quarter’s level of inflation is determined by iterating the assumed process (see Eq. (1)in Section 4):

qt+1 = qt + κq (µq − qt )�t + εqσq

√�t

= 0.010 + 0.4 × (0.048 − 0.010) × 0.25 + 0.040 ×√

0.25 × εq

where εq is a draw from a standardized normal random variable distribution. Suppose wesample this distribution at –0.68278. Using this draw, the rate of annual inflation during thesecond quarter is 0.0144%. The quarterly factor is calculated as (1+Annualized Rate)1/4,and the quarterly factors are accumulated over the entire year. This process is repeated eachquarter for 30 years. The following table illustrates this process for the first two years in theprojection.

Random Annualized Quarterly 1-year 1-yearQuarter draw 3-month rate (%) 3-month factor accums. Year rate (%)

1 1.0000 1.00249 1.00249

2 −0.68278 0.0144 1.00004 1.00253

3 1.32223 3.1374 1.00775 1.01030

4 0.14176 3.5872 1.00885 1.01924 1 1.9

5 0.59923 4.9070 1.01205 1.01205

6 0.15355 5.2034 1.01276 1.02496

7 0.10299 5.3690 1.01316 1.03845

8 1.72401 8.7601 1.02123 1.06048 2 6.0

A sample path of annual inflation over the entire 30-year projection period is shown inExhibit 1A, column (2).


Exhibit 1A . Determining the value of liabilities: One sample inflation/interest rate path.

Inflation Short real Nominal Projected Discount PresentYear rate % int. rate % int. rate % inflated payments factor value(1) (2) (3) (4) (5) (6) (7)

1 1.9 2.3 4.2 550.92 0.95925 528.46

2 6.0 3.0 9.2 339.63 0.87842 298.33

3 7.3 4.3 11.7 237.97 0.78656 187.18

4 6.5 −0.6 5.9 168.89 0.74295 125.48

5 6.2 1.6 7.9 126.09 0.68869 86.83

6 9.2 −2.5 6.7 100.13 0.64517 64.60

7 4.7 −0.4 4.2 81.65 0.61914 50.55

8 2.1 1.0 3.1 67.57 0.60061 40.58

9 5.1 6.8 12.2 59.51 0.53548 31.87

10 7.4 −2.8 4.8 53.97 0.51084 27.57

11 6.6 6.7 13.3 50.45 0.45085 22.74

12 6.3 6.2 12.6 46.74 0.40042 18.71

13 3.6 1.6 5.3 42.46 0.38032 16.15

14 1.2 2.4 3.7 37.72 0.36671 13.83

15 6.1 5.4 11.4 33.53 0.32920 11.04

16 10.5 −7.7 3.0 31.73 0.31973 10.15

17 8.8 5.9 15.0 29.66 0.27804 8.25

18 1.2 −0.9 0.4 26.63 0.27703 7.38

19 3.4 5.2 8.6 23.75 0.25500 6.06

20 5.0 3.8 8.8 20.93 0.23444 4.91

21 4.6 −4.2 0.5 17.99 0.23332 4.20

22 4.6 4.8 9.4 14.94 0.21322 3.19

23 7.0 5.1 12.1 11.91 0.19019 2.27

24 6.3 −0.1 6.1 8.70 0.17920 1.56

25 3.1 5.9 8.9 7.32 0.16463 1.21

26 5.2 −1.8 3.4 5.97 0.15914 0.95

27 3.0 1.5 4.6 4.52 0.15220 0.69

28 10.1 2.8 12.9 3.13 0.13475 0.42

29 8.3 −5.0 3.5 1.62 0.13014 0.21

30 7.7 1.0 8.8 0.91 0.11960 0.11

PV 1,575.46

A similar process determines the path of annual real interest rates. In the two-factormodel, two stochastic processes are followed at each quarter: one process tracks the long-term mean reversion level of interest while a second process simulates the short-end of theterm structure. The short rate is reverting to a level that is itself changing over time. Usingthe initial interest rate data and the parameters of the two-factor term structure model, and


assuming the random draws are −.59760 and .31601:

rt+1 = rt + κ1(lt − rt )�t + σ1ε1t

√�t

= 0.01 + 6.1 × (0.015 − 0.01) × 0.25 + 0.01 ×√

0.25 ×−0.59760 = −1.2%

lt+1 = lt + κ2(µ− lt )�t + σ2ε2t

√�t

= 0.015 + 5.1 × (0.028 − 0.015) × 0.25 + 0.01 ×√

0.25 × 0.31601 = 4.7%

After compounding the quarterly real interest rates to annual rates, the sample path ofreal interest rates is shown in Exhibit 1A, column (3).

On a quarterly basis, the nominal interest rate is calculated as the sum of the real interestrate and inflation. The compounded annual nominal interest rate is shown in column (4) inExhibit 1A.

The projected inflation is then used to determine the cash flows of the insurer (step (2)below) while the nominal interest rates are used to discount the projected cash flows backto date of valuation (step (3) below).

(2) The associated cash flow for each interest rate path is determined. The cash flows ofthe P-L insurer are based on three components: (i) the projected payment pattern of claims,(ii) the portion of unpaid losses that are subject to inflation, and (iii) the rate of inflation.

The aggregate industry loss payment pattern is used to project the timing of unpaid losses.From this pattern of payments, we also consider how these losses become fixed over time,that is, how these claims are no longer subject to inflation. From the discussion in Section 5,Eq. (2), D’Arcy and Gorvett [2001] introduce a model indicating how unpaid losses becomefixed:

f (t) = k + {(1 − k − m)(t/T )n} (A.1)

We determine the portion of time that has elapsed since the accident date, the (t/T ) termin Eq. (A.1). We assume that, on average, accidents occur in the middle of the year whilethe last payment is settled at the end of the payment year.

To illustrate this process, consider claims from accident year 2003 which will be com-pletely paid in 2006. To determine the percentage of unpaid claims that have been fixed bythe end of 2004, consider the following timeline:


Therefore, 1 12 out of 3 1

2 years or 42.9% of the time has expired since the accident date.The following table shows a sample of these calculations.

Proportion of claim cohort life expired

Evaluation date (12/31 for each year)

Payment year 2004 2005 2006 2007

2005 0.600

2006 0.429 0.714

2007 0.333 0.556 0.778

2008 0.273 0.455 0.636 0.818

Then, by using Eq. (A.1), we determine the portion of unpaid claims that are still sensitiveto inflation. Consider the base case, where 25% of the claim is fixed on the accident date,another 25% of the claim remains sensitive to inflation until the settlement date, and theremaining 50% of the claim amount becomes fixed proportionally to the time elapsed sincethe accident date (n = 1.00). For claims incurred during accident year 2003 that will beultimately paid by the end of 2006, 42.9% of the time has elapsed by the end of 2004.Therefore, the amount of the unpaid loss that is fixed at the end of 2004 is:

f (t) = k + {(1 − k − m)(t/T )n} = 0.25 + {(1 − 0.25 − 0.25)(0.429)1}≈ 0.464 or 46.4%

The following table shows a sample of other calculations from accident year 2003 underthe base case.

Fixed proportion [ f (t)] for claim cohort

Evaluation date (12/31 for each year)

Payment year 2004 2005 2006 2007

2005 0.550 1.000 1.000 1.000

2006 0.464 0.607 1.000 1.000

2007 0.417 0.528 0.639 1.000

2008 0.386 0.477 0.568 0.659

The resulting rate of inflation is applied to that portion of the claims that are not fixed.Inflation is generated differently depending on the term structure used to generate interestrates. For one-factor term structure models, the realizations of the annual, nominal interestrate imply a path of inflation based on the assumed relationship shown in Eq. (4) fromSection 5. For the two-factor model, inflation is generated by its own stochastic process,which is projected simultaneously with the path of real interest rates.


The payment pattern used in this analysis extends for 30 years, requiring projectionsof claim cohorts from 30 different accident years. Projected claims are summed acrossaccident years to determine the total liability payment along each simulation path. Cashflow projections are shown in column (5) of Exhibit 1A.

(3) The present value of each cash flow is calculated by discounting that cash flow atthe corresponding interest rates. The inflation-sensitive cash flows are discounted alongeach simulated interest rate path to determine the value of the liabilities on the evaluationdate. The discount factors used to find the present value of the projected claim paymentsare based on the projected short-term, nominal interest rates from step (1) described above.These discount factors are shown in column (6) of Exhibit 1A and the present values of theclaims are shown in column (7).

(4) The overall present value of the instrument is determined by calculating the averageof all the simulated values. When interest rates and inflation are projected along a singlepath as in Exhibit 1A, the present value of the claim liabilities are 1,575.46 (shown atthe bottom of column (7)). This is the present of the liability along the unique interestrate path illustrated. This process is repeated 1,000 times to generate 1,000 distinct valuesfor the liability payments. The average value of these 1,000 projections determines thevalue of the liabilities under the existing interest rate environment. This is denoted PV0 inSection 3.

Simulation PV of liability

1 1,575.46

2 1,637.12

3 1,711.60

. .

. .

. .

1000 1,618.08

Average (PV0) 1, 629.42

To determine the interest sensitivity of the liabilities (or effective duration and con-vexity), we calculate the change in value of the liabilities as interest rates change. Theentire four-step process shown here is repeated twice more under different interest rateenvironments: (1) by shifting the initial interest rate(s) up 100 basis points (this deter-mines PV+) and (2) shifting the initial interest rates down by 100 basis points (todetermine PV−).

The manner in which initial interest rates are shifted is dependent on the specific termstructure model used in the analysis. For the CIR term structure model, shifting initialinterest rates means shocking the short-term rate. For Hull-White, the entire existing term


structure is shocked. Note that when using a one-factor model, shifting interest rates impliesa change in inflation by a predetermined amount based on the estimated regression Eq. (3)in Section 5.

When shifting interest rates for the two-factor model, however, one must determine therelative shifts of the nominal interest rate, the real interest rate, and the inflation rate. Inall likelihood, when the nominal interest rate changes by 100 basis points, the level ofinflation and the real interest rate may be affected simultaneously. However, empiricallymeasuring the separate effects on each of these factors is a difficult task. For the purposesof our calculations, we look at two extremes: (1) the shift in the nominal interest rate iscompletely driven by changes in initial inflation and (2) the shift in the nominal interest rateis completely driven by changes in the initial real interest rate (both the short and the longfactors are adjusted).

After shocking the initial interest rates, the calculation proceeds in the same manner:

(1) simulate 1,000 paths for inflation and interest rates over 30 years,(2) project the insurer cash flows along each path,(3) discount each cash flow to determine a present value of the liability on the simulated

path, and(4) average the results to determine the value of the claim payments.

The measures of interest rate sensitivity of the liabilities are then determined numericallyby:

Effective duration = ED0 = PV− − PV+2PV0(�r )

Effective convexity = PV− + PV+ − 2PV0

PV0(�r )2

Notes

1. Based on the Taylor series, any function can be approximated by a series of derivatives of the function valuedat another point. The more derivatives used in the approximation, the more accurate the estimate. Durationis the first derivative of the function; convexity is the second derivative. Estimating the effect of a change ininterest rates on a financial instrument with duration is, thus, a first order approximation. Including the effectof convexity generates a second order approximation.

2. Although, as Hull [2000] mentions, a one-factor model “does not, as is sometimes supposed, imply that theterm structure always has the same shape. A fairly rich pattern of term structures can occur under a one-factormodel.”

3. Actually, CIR [1985] find that, in continuous time, the randomness in the volatility term follows a non-centralchi-square distribution. However, as CKLS [1992] mention, this may not be the case in discrete time. Forsimplicity, we have assumed that randomness which follows a standard normal distribution is adequate for ourpurposes.

4. As of the end of 1999, loss reserves accounted for 52.9% of total liabilities, loss adjustment expense reservesaccounted for 11.1% and the unearned premium reserve accounted for 19.9% (for the aggregate P-L insuranceindustry (A.M. Best [2000])).

5. These data are all available from the Federal Reserve Bank of St. Louis web site http://www.stls.frb.org/fred/


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the effective duration and convexity of liabilities ...gepa.0000032567... · the effective duration...

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