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

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

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

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

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

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

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

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

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

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

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

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

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:

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

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

Example Using Two-Factor Model Assets 600 Million Liabilities 450 Million Surplus 150 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!

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