1. CHAPTER 13 Measurement of Interest-Rate Risk for ALM What is in this Chapter? Introduction Gap Reports Contractual-Maturity Gap Reports Estimating Economic Capital based on Gap Reports Sophisticated interest-rate models

2. INTRODUCTION The purposes of measuring ALM interest- rate risk establish the amount of economic capital to be held against such risks How to reduce the risks by buying or selling interest-rate-sensitive instruments Although ALM risk is a form of market risk, it cannot be effectively measured using the trading- VaR framework

3. This VaR framework is inadequate for two reasons. First, the ALM cash flows are complex functions of customer behavior. Second, interest-rate movements over long time horizons are not well modeled by the simple assumptions used for VaR.

4. INTRODUCTION Banks use three alternative approaches to measure ALM interest-rate risk, as listed below: Gap reports （缺口報告） Rate-shift scenarios Simulation methods similar to Monte Carlo VaR

5. GAP REPORTS The "gap“ is the difference between the cash flows from assets and liabilities Gap reports are useful because they are relatively easy to create This measure is only approximate because gap reports do not include information on the way customers exercise their implicit options in different interest environments There are three types of gap reports: contractual maturity repricing frequency effective maturity

6. Contractual-Maturity Gap Reports A contractual-maturity gap report indicates when cash flows are contracted to be paid for liabilities, it is the time when payments would be due from the bank, assuming that customers did not roll over their accounts. For example, the contractual maturity for checking accounts is zero because customers have the right to withdraw their funds immediately.

7. Contractual-Maturity Gap Reports The contractual maturity for a portfolio of three-month certificates of deposit would (on average) be a ladder of equal payments from zero to three months. The contractual maturity for assets may or may not include assumptions about prepayments. In the most simple reports, all payments are assumed to occur on the last day of the contract

8. Repricing Gap Reports Repricing refers to when and how the interest payments will be reset

9. Effective-Maturity Gap Reports Although the repricing report includes the effect of interest-rate changes, it does not include the effects of customer behavior. This additional interest-rate risk is captured by showing the effective maturity. For example, the effective maturity for a mortgage includes the expected prepayments, and may include an adjustment to approximate the risk arising from the response of prepayments to changes in interest rates.

10. Effective-Maturity Gap Reports  Gap reports give an intuitive view of the balance sheet, but they represent the instruments as fixed cash flows, and therefore do not allow any analysis of the nonlinearity of the value of the customers' options.  To capture this nonlinear risk requires approaches that allow cash flows to change as a function of rates.

11. Estimating Economic Capital based on Gap Reports

13. For this analysis, we made several significant assumptions: We assumed that value changes linearly with rate changes We also assumed that the duration would be constant over the whole year

14. Estimating Economic Capital based on Gap Reports Finally, we assumed that annual rate changes were Normally distributed. These assumptions could easily create a 20% to 50% error in the estimation of capital The methods that do not require so many assumptions (please refer to Page 194 to 195)

15. SIMULATION METHODS Models to Create Interest-Rate Scenarios Randomly An important component in the simulation approach is the stochastic (i.e., random model used to generate interest-rate paths) This basic interest-rate model assumes that the interest rate in the next period (r t+1 ) will equal the current rate (r t ), plus a random number with a standard deviation of σ :

16. The basic interest-rate model This is inadequate for ALM purposes because over long periods, such as a year, the simulated interest rate can become negative This model also lacks two features observed in historical interest rates: rates are mean reverting heteroskedastic (their volatility varies over time)

17. Sophisticated interest-rate models Two classes of more sophisticated models have been developed for interest rates: general-equilibrium (GE) models and arbitrage-free (AF) models A general model for the GE approach has a mean-reverting term and a factor that reduces the volatility as rates drop

18. Sophisticated interest-rate models the speed of reversion >If κ is close to 1, the rates revert quickly > if it is close to 0, the model becomes like a random walk the level to which interest rates tend to revert over time The relative volatility of the distributions of interest rate > determines how significantly the volatility will be reduced as rates drop >If it was 0, the volatility would not change if rates changed >0.5 for Cox-Ingersoll-Ross model >1 for the Vasicek model

19. Sophisticated interest-rate models Values for the parameters θ, κ, σ, and γ can be determined from historical rate information using maximum likelihood estimation