Measuring the Duration of Liabilities

Stephen P. D’Arcy, FCAS, MAAA, Ph.D.

University of Illinoisat Urbana-Champaign

Casualty Actuarial SocietyAsset-Liability Management Session

San Diego, CA

May 20, 2002

Assumptions Underlying Macaulay and Modified Duration

• Cash flows do not change with interest ratesThis does not hold for:– Collateralized Mortgage Obligations (CMOs)– Callable bonds– P-L liabilities – due to inflation-interest rate correlation

• Flat yield curveGenerally, yield curves are upward-sloping

• Interest rates shift in parallel fashionShort term interest rates tend to be more volatilethan longer term rates

An Improvement: Effective Duration

• Effective duration:– Accommodates interest sensitive cash flows– Can be based on any term structure– Allows for non-parallel interest rate shifts

• Effective duration is used to value such assets as:– Collateralized Mortgage Obligations– Callable bonds– And now… property-liability insurance liabilities

• Need to reflect the inflationary impact on future loss and LAE payments of interest rate changes

A Further Refinement: Convexity

The larger the change in interest rates, the larger the misestimate of the price change using duration

Duration: first-order approximationAccurate only for small changes in interest rates

Convexity: second-order approximationReflects the curvature of the price-yield curve

Present Value of a $1 Million Payment in Ten Yearswith Modified Duration and Convexity Estimates

$(400,000.00)

$(200,000.00)

$-

$200,000.00

$400,000.00

$600,000.00

$800,000.00

$1,000,000.00

$1,200,000.00

0 0.05 0.1 0.15 0.2 0.25 0.3

Interest Rates

Actual Value Modified Duration Estimate Convexity Estimate

The Liabilities of Property-Liability Insurers

• Major categories of liabilities:– Loss reserves– Loss adjustment expense reserves– Unearned premium reserves

A Model for the Interest Sensitivity of Loss Reserves

• D’Arcy and Gorvett PCAS 2000• Divides loss reserves into “fixed” and

inflation sensitive portions

How to Reflect “Fixed” Costs?

• “Fixed” here means that portion of damages which, although not yet paid, will not be impacted by future inflation

• Tangible versus intangible damages• Determining when a cost is “fixed” could require

– Understanding the mindset of jurors– Lots and lots of data

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 Period0 1

Proportionof UltimatePaymentsFixed

1

0k

m

n=1n<1

n>1

“Fixed” Cost Formula Parameters

• Examples of loss costs that might go into k– Medical treatment immediately after the loss occurs– Wage loss component of an injury claim– Property damage

• Examples of loss costs that might go into m– Medical evaluations performed immediately prior to

determining the settlement offer– General damages to the extent they are based on the cost of

living at the time of settlement– Loss adjustment expenses connected with settling the claim

Loss Adjustment Expense Reserves

• LAE on losses that have already occurred• Primarily future expenses• Sensitive to interest rate changes

Unearned Premium Reserves

• Statutory reserve for the unexpired portion of premiums

• Economic value is the future losses on current policies

• Since these losses have not occurred yet, they would be completely interest rate sensitive

Approach for Measuring Interest Rate Sensitivity of Insurance Liabilities

1. Select a term structure (interest rate) model2. Generate multiple interest rate paths based on the

selected model3. For each path, calculate the loss and LAE

payments that will develop4. Determine the present value of each set of cash

flows by discounting by the relevant interest rates5. Calculate the average present value over all

interest rate paths6. This average is PV0

Measuring Interest Rate Sensitivity of Insurance Liabilities (2)

7. Shock the initial instantaneous interest rate by increasing, and decreasing, it by 100 basis points

8. Repeat steps 2-5 to determine PV+ and PV-

9. Calculate the effective duration based on:Effective Duration = (PV--PV+)/(2PV0)(Δr)

Results of Effective Duration Calculations

Effective duration is less than modified duration– Cash flows change

• Higher interest rate → higher cash flows• Lower interest rate → lower cash flows

– Longer term interest rates don’t move as much as short term rates with most term structure models

Illustrative Example: Duration of Loss Reserve Liabilities

Aggregate Industry – All Lines Combined

• Based on steady-state operations and a 4% initial short-term interest rate:

Macaulay Duration: 4.24Modified Duration: 4.08

• Based on CIR and interest-sensitive cash flows:

Effective Duration: 1.70

Stochastic Interest Rates• Cox-Ingersoll-Ross (CIR) term structure model

– “Equilibrium” model– Mean-reverting, square-root diffusion process

a = speed of reversion to long-run meanr = current short-term interest rateb = long-run mean of short-term interest rateσ = volatility factordz = standard Wiener process

dzrdtrbadr )(

Assumptions Underlying Illustrative Effective Duration Calculation

Fixed Cost Parametersk = 0.15

m = 0.10n = 1.00

Impact of Inflation: Embedded inflation rate = 2.5%Future claim inflation = 4% + .40 x short-term interest rate

CIR Interest Rate Parametersa = .25r = .04b = .05σ = .08

Why is Duration Important?

• Corporations attempt to manage interest rate risk by balancing the duration of assets and liabilities

Surplus Duration

• Sensitivity of an insurer’s surplus to changes in interest rates

DS S = DA A - DL L

DS = (DA - DL)(A/S) + DL

where D = duration S = surplusA = assets

L = liabilities

Surplus Duration and Asset-Liability Management

• To “immunize” surplus from interest rate risk, set DS = 0

• Then, asset duration should be:DA = DL L / A

• Thus, an accurate estimate of the duration of liabilities is critical for ALM

Implications

• Use the same approach to measure the interest sensitivity of both assets and liabilities

• A company may choose a duration mismatch• Need to determine if the compensation for

accepting interest rate risk is adequate