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Transcript
Measuring the Duration of Liabilities Stephen P. D’Arcy, FCAS, MAAA, Ph.D. University of Illinois at Urbana-Champaign Casualty Actuarial Society Asset-Liability Management Session San Diego, CA May 20, 2002 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 – P-L liabilities – due to inflation-interest rate correlation • Flat yield curve Generally, yield curves are upward-sloping • Interest rates shift in parallel fashion 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 liabilities • Need to reflect the inflationary impact on future loss and LAE payments of interest rate changes A Further Refinement: 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 The Liabilities of PropertyLiability Insurers • Major categories of liabilities: – Loss reserves – Loss adjustment expense reserves – Unearned premium reserves A Model for the Interest Sensitivity of Loss Reserves • D’Arcy and Gorvett PCAS 2000 • Divides loss reserves into “fixed” and inflation sensitive portions 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 Proportion of Ultimate Payments Fixed k 1 m n<1 n=1 n>1 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 Adjustment Expense Reserves • LAE on losses that have already occurred • Primarily future expenses • Sensitive to interest rate changes Unearned Premium Reserves • Statutory reserve for the unexpired portion of premiums • Economic value is the future losses on current policies • Since these losses have not occurred yet, they would be completely interest rate sensitive Approach for Measuring Interest Rate Sensitivity of Insurance Liabilities 1. Select a term structure (interest rate) model 2. Generate multiple interest rate paths based on the selected model 3. For each path, calculate the loss and LAE payments that will develop 4. Determine the present value of each set of cash flows by discounting by the relevant interest rates 5. Calculate the average present value over all interest rate paths 6. This average is PV0 Measuring Interest Rate Sensitivity of Insurance Liabilities (2) 7. Shock the initial instantaneous interest rate by increasing, and decreasing, it by 100 basis points 8. Repeat steps 2-5 to determine PV+ and PV9. Calculate the effective duration based on: Effective Duration = (PV--PV+)/(2PV0)(Δr) Results of Effective Duration Calculations Effective duration is less than modified duration – Cash flows change • Higher interest rate → higher cash flows • Lower interest rate → lower cash flows – Longer term interest rates don’t move as much as short term rates with most term structure models Illustrative Example: Duration of Loss Reserve Liabilities Aggregate Industry – All Lines Combined • Based on steady-state operations and a 4% initial short-term interest rate: Macaulay Duration: Modified Duration: 4.24 4.08 • Based on CIR and interest-sensitive cash flows: Effective Duration: 1.70 Stochastic Interest Rates • Cox-Ingersoll-Ross (CIR) term structure model – “Equilibrium” model – Mean-reverting, square-root diffusion process dr a (b r ) dt a = r = b= σ= dz = r dz speed of reversion to long-run mean current short-term interest rate long-run mean of short-term interest rate volatility factor standard Wiener process Assumptions Underlying Illustrative Effective Duration Calculation Fixed Cost Parameters k = 0.15 m = 0.10 n = 1.00 CIR Interest Rate Parameters a = .25 r = .04 b = .05 σ = .08 Impact of Inflation: Embedded inflation rate = 2.5% Future claim inflation = 4% + .40 x short-term interest rate Why is Duration Important? • Corporations attempt to manage interest rate risk by balancing the duration of assets and liabilities Surplus Duration • Sensitivity of an insurer’s surplus to changes in interest rates D S S = DA A - D L 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 Implications • Use the same approach to measure the interest sensitivity of both assets and liabilities • A company may choose a duration mismatch • Need to determine if the compensation for accepting interest rate risk is adequate