The Effective Duration of Property-Liability Insurance Liabilities with

Stochastic Interest Rates

Stephen P. D’Arcy, FCAS, Ph.D.

Richard W. Gorvett, FCAS, Ph.D.

University of Illinois

at Urbana-Champaign

Presented at the ARIA Annual Meeting

August, 2000

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 substantiallyExamples of Intermediate Term U.S. Bond Rates:

1/1 12/31 1979 8.8% 10.3% 1.5%1980 10.3 12.5 2.21982 14.0 9.9 -4.11994 5.2 7.8 2.61999 4.7 6.5 1.8

Are Property-Liability Insurers Exposed to Interest Rate Risk?

YES!

Long term liabilities

Medical malpractice

Workers’ compensation

General liability

Assets

Significant portion of assets invested in long term bonds

Measures of Interest Rate Risk

Macaulay duration

Recognizes that the sensitivity of the price of a fixed income asset is approximately related to the weighted average time to maturity

Modified duration

Negative of the first derivative of the price/yield curve

Macaulay duration/(1+interest rate)

Modified Duration is the Negative of the Slope of a Tangency Line to

the Price-Yield Curve

Price

Yieldr

Price-yield curve fora fixed income bond

Assumptions Underlying Macaulay and Modified Duration

1 Cash flows do not change with interest ratesThis does not hold for:

Collateralized Mortgage Obligations (CMOs)Callable bondsLoss reserves

2 Flat yield curveGenerally yield curves are upward sloping

3 Parallel shift in interest ratesShort term interest rates tend to be more volatile than longer term rates

Effective Duration

1 Accommodates interest sensitive cash flows

2 Can be based on any term structure

3 Allows for non-parallel interest rate shifts

Prior Related Research (1)Taylor Separation Method (1986)

Allows for a separate inflation component to loss paymentsInflation 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 MethodDetermine that most insurers operate in the least efficient range of interest rate risk

Prior Related Research (2)

Choi (1991)

“nobody knows how to model the cash flows as a function of interest rates or inflation rates (because interest rates are closely related to inflation rates) in the property-liability insurance industry.”

Feldblum and Hodes (1996)“A mathematical determination of the loss reserve duration is complex.”Assumes loss reserves are not interest rate sensitive

Interest Sensitive Cash Flows

Interest rates and inflation are correlatedInflation can increase future loss paymentsLoss reserve consists of future payments

Portion has already been “fixed” in valueMedical treatment already receivedProperty damage that has been repaired

Remainder subject to inflationGeneral damages to be set by juryFuture medical treatment

A Possible “Fixed” Cost FormulaProportion 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 lossm = portion of losses fixed at time of settlementT = time from date of loss to date of payment

Proportion of Payment Period

0 1

Proportionof UltimatePaymentsFixed

1

0k

m

n=1n<1

n>1

Term Structure

Cox, Ingersoll, and Ross (CIR)Mean-reverting, square-root diffusion process

α = speed of reversionr = current short term interest rateR = long run mean of short term interest rateσ = volatility factordz = standard normal distribution

dzrdtrRdr )(

Non-Parallel Shifts

A change in the short term interest rate does not shift the long term rate as much

Calculation of the Effective Duration of Loss Reserves (1)

1. Generate multiple interest rate paths based on the CIR model

2. For each path, calculate the loss payments that will develop

3. Determine the present value of each set of cash flows by discounting by the relevant interest rates

4. Calculate the average present value over all interest rate paths

Calculation of the Effective Duration of Loss Reserves (2)

5. 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

6. Calculate the effective duration based on:

Effective Duration = (PV--PV+)/(2PV0)(Δr)

Duration of Liabilities

Based on Steady State Operations and a 6% Interest Rate: Auto Liab. WC Other Liab.

Macaulay Duration 2.00 4.05 3.96Modified 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 1.65

Assumptions Underlying Effective Duration Calculation

Fixed Cost Parameters

k = .15

m = .10

n = 1.0

Impact of Inflation

Embedded Inflation Rate = 5%

Future Inflation = .05 + .46 X Short Term Interest Rate

CIR Interest Rate Parameters

α = .25

r = .06

R = .07

σ = .08

Additional Research

Other term structure models

Vasicek

Hull-White

Sensitivity of parameter estimates