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Measuring the Interest Rate Sensitivity of Loss Reserves

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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 Illinois – PowerPoint PPT presentation

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Title: Measuring the Interest Rate Sensitivity of Loss Reserves


1
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

2
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

3
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

4
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

5
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)

6
Macaulay and Modified Duration

7
Duration is the Slope of the Tangency Line for
the Price/Yield Curve
Price
Price-yield curve for financial instrument
r
Yield
8
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

9
(No Transcript)
10
Computing Convexity
  • Take the second derivative of price with respect
    to the interest rate

11
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

12
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

13
The Liabilities of Property-Liability Insurers
  • Major categories of liabilities
  • Loss reserves
  • Loss adjustment expense reserves
  • Unearned premium reserves

14
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.

15
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)

16
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

17
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
18
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

19
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

20
Why is Duration Important?
  • Corporations attempt to manage interest rate risk
    by balancing the duration of assets and
    liabilities

21
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

22
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

23
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

24
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 (?)
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