Measuring the Interest Rate Sensitivity of Loss

Reserves

- Stephen P. DArcy, FCAS, MAAA, Ph.D.
- Richard W. Gorvett, FCAS, MAAA, ARM, Ph.D.
- University of Illinois
- at Urbana-Champaign
- Casualty Actuarial Society
- Miami 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 didnt, 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 2.4
- 1982 13.7 10.5 - 3.2
- 1994 5.8 7.8 2.0
- 1999 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/(1r)

Macaulay and Modified Duration

Duration is the Slope of the Tangency Line for

the Price/Yield Curve

Price

Price-yield curve for financial instrument

r

Yield

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

(No Transcript)

Computing Convexity

- Take the second derivative of price with respect

to the interest rate

Assumptions Underlying Macaulay and Modified

Duration

- Cash flows do not change with interest rates
- But this does not hold for
- Collateralized Mortgage Obligations (CMOs)
- Callable bonds
- P-L loss reserves due to inflation-interest

rate correlation - Flat yield curve
- But generally, yield curves are upward-sloping
- Interest rates shift in parallel fashion
- But short term interest rates tend to be more

volatile - than 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 Formula

- Proportion of loss reserves fixed in value as of

time t - f(t) k (1 - k - m) (t / T) n
- k portion of losses fixed at time of loss
- m portion of losses fixed at time of settlement
- T time from date of loss to date of payment

1

m

Proportion of Ultimate Payments Fixed

nlt1

n1

ngt1

k

0

1

0

Proportion of Payment Period

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 1.0
- r 5 rr,i 0.40
- Exposure growth rate 10
- Automobile Workers
- Insurance Compensation
- Macaulay duration 1.52 4.49
- Modified duration 1.44 4.27
- Effective 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 insurers surplus to changes in

interest rates - DS S DA A - DL L
- DS (DA - DL)(A/S) DL
- where D duration
- S surplus
- A 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 (?)

Measuring the Interest Rate Sensitivity of Loss

Reserves

- Stephen P. DArcy, FCAS, MAAA, Ph.D.
- Richard W. Gorvett, FCAS, MAAA, ARM, Ph.D.
- University of Illinois
- at Urbana-Champaign
- Casualty Actuarial Society
- Miami 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 didnt, 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 2.4
- 1982 13.7 10.5 - 3.2
- 1994 5.8 7.8 2.0
- 1999 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/(1r)

Macaulay and Modified Duration

Duration is the Slope of the Tangency Line for

the Price/Yield Curve

Price

Price-yield curve for financial instrument

r

Yield

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

(No Transcript)

Computing Convexity

- Take the second derivative of price with respect

to the interest rate

Assumptions Underlying Macaulay and Modified

Duration

- Cash flows do not change with interest rates
- But this does not hold for
- Collateralized Mortgage Obligations (CMOs)
- Callable bonds
- P-L loss reserves due to inflation-interest

rate correlation - Flat yield curve
- But generally, yield curves are upward-sloping
- Interest rates shift in parallel fashion
- But short term interest rates tend to be more

volatile - than 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 Formula

- Proportion of loss reserves fixed in value as of

time t - f(t) k (1 - k - m) (t / T) n
- k portion of losses fixed at time of loss
- m portion of losses fixed at time of settlement
- T time from date of loss to date of payment

1

m

Proportion of Ultimate Payments Fixed

nlt1

n1

ngt1

k

0

1

0

Proportion of Payment Period

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 1.0
- r 5 rr,i 0.40
- Exposure growth rate 10
- Automobile Workers
- Insurance Compensation
- Macaulay duration 1.52 4.49
- Modified duration 1.44 4.27
- Effective 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 insurers surplus to changes in

interest rates - DS S DA A - DL L
- DS (DA - DL)(A/S) DL
- where D duration
- S surplus
- A 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 (?)