1. Financial Risk Management of Insurance Enterprises
Interest Rate Sensitivity of Assets and Liabilities of a Property-Liability Insurer

2. Assumptions Underlying Macaulay and Modified Duration
Cash flows do not change with interest rates
This does not hold for:
Collateralized Mortgage Obligations (CMOs)
Callable bonds
Loss reserves
Flat yield curve
Generally yield curves are upward sloping
Parallel shift in interest rates
Short term interest rates tend to be more volatile than longer term rates

3. Effective Duration Accommodates interest sensitive cash flows
Can be based on any term structure
Allows for non-parallel interest rate shifts

4. Prior Related Research (1)
Taylor Separation Method (1986)
Allows for a separate inflation component to loss payments
Inflation affects all payments made in given year
Babbel, Klock and Polachek (1988)
Macaulay duration reasonable approximation
Staking (1989), Babbel and Staking (1995, 1997)
Calculate effective duration of liabilities based on a modification of the Taylor Separation Method
Determine that most insurers operate in the least efficient range of interest rate risk

5. Prior Related Research (2)
Choi (1991)
“nobody knows how to model the cash flows as a function of interest rates or inflation rates (because interest rates are closely related to inflation rates) in the property-liability insurance industry.”
Hodes and Feldblum (1996)
“A mathematical determination of the loss reserve duration is complex.”
Assumes loss reserves are not interest rate sensitive

6. Interest Sensitive Cash Flows
Interest rates and inflation are correlated
Inflation can increase future loss payments
Loss reserve consists of future payments
Portion has already been “fixed” in value
Medical treatment already received
Property damage that has been repaired
Remainder subject to inflation
General damages to be set by jury
Future medical treatment

7. 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
n<1
n=1
n>1 k 1
Proportion of Payment Period

8. Non-Parallel Shifts
A change in the short term interest rate does not shift the long term rate as much

9. Term Structure Models - 1
Cox, Ingersoll, and Ross (CIR)
Mean-reverting, square-root diffusion process
κ = speed of reversion
r = current short term interest rate
θ = long run mean of short term interest rate
σ = volatility factor
dz = standard normal distribution 20 20

10. Term Structure Models - 2
Hull-White
No-arbitrage model
a = speed of reversion
r = current short term interest rate
θ = time dependent mean reversion level
σ = volatility factor
dz = standard normal distribution 20 20

11. Term Structure Models - 3
Two factor real interest rate model
κ1,2 = speed of reversion
r = current short term interest rate
l = long run mean of the short term rate
μ = ultimate long-term mean of the short term rate
σ1,2 = volatility factors
dz 1,2 = standard normal distributions
One factor mean reverting inflation model 20 20

12. Calculation of the Interest Rate Sensitivity of Loss Reserves (1)
Generate multiple interest rate paths based on the term structure model
For each path, calculate the loss payments that will develop
Determine the present value of each set of cash flows by discounting by the relevant interest rates
Calculate the average present value over all interest rate paths

13. Calculation of the Interest Rate Sensitivity of Loss Reserves (2)
Calculate the present value based on the initial interest rate, the initial interest rate plus 100 basis points and the initial interest rate minus 100 basis points
Calculate the effective duration based on:
Calculate the effective convexity based on:

14. Calculation of the Change in Economic Value of a Cash Flow
V = (-1)(Effective Duration)(r)
+ (1/2)(Convexity)(r)2

22. Impact of Increase in Interest Rates on Assets and Liabilities (from Table 8)

23. Example Using Two-Factor Model
Assets Million
Liabilities Million
Surplus Million
If interest rates increase 200 basis points
Assets decline by 600 x 3.87%, or 23.2 million
Liabilities decline by 450 x 1.47%, or 6.6 million
Surplus declines by 16.6 million, or 11.2%
Assets were supposed to be in a duration matched investment!

24. Conclusion
Traditional methods of ALM for property-liability insurers overestimate the sensitivity of liabilities to interest rate changes
Applying traditional measures leads to a mismatch of assets and liabilities
An increase in interest rates will reduce the value of insurers’ assets much more than it will reduce the value of its liabilities
Insurers are more exposed to interest rate risk than they realize