Measuring the Interest Rate Sensitivity of Loss Reserves

Stephen P. D’Arcy, FCAS, MAAA, Ph.D.Richard W. Gorvett, FCAS, MAAA, ARM, Ph.D.

University of Illinoisat Urbana-Champaign

Casualty Actuarial SocietyMiami Beach, FL

May 7, 2001

Why Bother with “Duration”?

• Duration measures how sensitive the value of a financial instrument is 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 & Loan industry didn’t, and look what happened to them– Asset-liability “mismatch”

• Interest rates can and do fluctuate substantially• Examples of intermediate-term U.S. bond rates:

t 12/t-1 12/ t 1979 9.0% 10.4% 1.4%1980 10.4 12.8 2.41982 13.7 10.5 - 3.21994 5.8 7.8 2.01999 4.7 6.3 1.6

Are Property-Liability Insurers Exposed to Interest Rate Risk?

• Absolutely!!• 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 (present value) weighted average time to maturity

• Modified duration is the negative of the first derivative of price with respect to interest rates, divided by the price

• Modified duration = Macaulay duration/(1+r)

Macaulay and Modified Duration

durationMacaulay )1(

1

1

)1(

1

duration Modified

)1( where

)1(DurationMacaulay

1

r

Pr

CFt

Pr

P

r

CFP

Pr

CFt

ttt

tt

t

ttt

Duration is the Slope of the Tangency Line for the Price/Yield

CurvePrice

Yieldr

Price-yield curve forfinancial instrument

A Refinement: Also Consider Convexity

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

Duration: first-order approximation

Accurate only for small changes in interest rates

Convexity: second-order approximation

Reflects the curvature of the price-yield curve

Present Value of a $1 Million Ten Year Zero Coupon Bondwith 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

Bond Value Modified Duration Estimate Convexity Estimate

Computing Convexity

• Take the second derivative of price with respect to the interest rate

tt

tt

tt

i

CFtt

iP

Pi

CFtt

Pi

CFt

P

PConvexity

)1(

)1(

)1(

11

1

)1(

)1(

1

)1(i

1

i

2

2

12

2

Assumptions Underlying Macaulay and Modified Duration

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

• Flat yield curveBut: generally, yield curves are upward-sloping

• Interest rates shift in parallel fashionBut: short 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 loss reserves

• Need to reflect the inflationary impact on future loss payments of interest rate movements

The Liabilities of Property-Liability Insurers

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

Loss Reserves

• Major categories:– In the process of being paid– Value of loss is determined, negotiating over share of

loss to be paid– Damage is yet to be discovered– Continuing to develop: some of loss has been fixed,

remainder is yet to be determined

• Inflation, which is correlated with interest rates, will affect each category of loss reserves differently.

What Portion of the Loss Reserve is Affected by Future Inflation

(and Interest Rates)?

• If the damage has not yet occurred, then the future loss payments will fully reflect future inflation

• If the loss is continuing to develop, then a portion of the future loss payments will be affected by future inflation (and another portion will be “fixed” relative to inflation)

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 Period

0 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 Reserve Duration Example

For the values:k = .15 m = .10 n = 1.0r = 5% r,i = 0.40Exposure growth rate = 10%

Automobile Workers’ Insurance Compensation

Macaulay duration: 1.52 4.49Modified duration: 1.44 4.27Effective duration:1.09 3.16

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

Example of Asset-Liability Mgt. for a Hypothetical WC Insurer

Dollar Modified Effective

Value Duration Duration

Loss & LAE Reserve 590 4.271 3.158

UPR 30 3.621 1.325

Other liabilities 90 0.952 0.952

Total liabilities 710 3.823 2.801

Total assets 1,000

Asset duration to immunize surplus: 2.714 1.989

Conclusion

• Asset-liability management depends upon appropriate measures of effective duration (and convexity)

• Potentially significant differences between effective and modified duration values

• Critical factors and parameters– Line of business– Payment pattern– Correlation between interest rates and inflation– Interest rate model (?)