Measuring the Interest Rate Sensitivity of Loss Reserves

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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

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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

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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

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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

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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)

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Macaulay and Modified Duration

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Duration is the Slope of the Tangency Line for
the Price/Yield Curve
Price
Price-yield curve for financial instrument
r
Yield
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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

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Computing Convexity

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

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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

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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

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The Liabilities of Property-Liability Insurers

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

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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.

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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)

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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

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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
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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

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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

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Why is Duration Important?

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

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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

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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

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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

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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 (?)