1. MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY Chapter 5 Bank Management 5th edition. Timothy W. Koch and S. Scott MacDonald Bank Management, 5th edition. Timothy W. Koch and S. Scott MacDonald Copyright © 2003 by South-Western, a division of Thomson Learning

2. Duration GAP …f ocus on managing NII or the market value of equity, recognizing the timing of cash flows. Interest rate risk is measured by comparing the weighted average duration of assets with the weighted average duration of liabilities. If Asset duration > Liability duration interest rates  Market value of equity will fall.

3. Duration versus maturity 1.)1000 loan, principal + interest paid in 20 years. 2.)1000 loan, 900 principal in 1 year, 100 principal in 20 years. 1000 + int |-------------------|-----------------| 0 10 20 900+int 100 + int |----|--------------|-----------------| 0 1 10 20 What is the maturity of each? 20 years What is the "effective" maturity? 2.) = [(900/100) x 1]+[(100/1000) x 20] = 2.9 yrs 1 2 Duration, however, uses a weighted average of the present values.

4. Solve for  Price:  P  -Duration x [  y / (1 + y)] x P Price (value) changes Longer maturity/duration larger changes in price for a given change in i-rates. Larger coupon smaller change in price for a given change in i-rates. Duration …approximate measure of the price elasticity of demand

5. 债权的久期（ DURATION ）的含义 从本意来看，久期确实表示一种期限，是 以未来现金流现值占债权价值的比例作为 权重，对未来现金流所经历的时间进行的 加权平均时间； 但从其实际应用来看，债权的久期可以计 量利率变动对债权市场价格所产生的影响， 表达了一种弹性概念。

6. In general notation, Macaulay’s duration (D): Examples 1000 face, 10% coupon, 3 year, 12% YTM Measuring duration

7. If YTM = 5% 1000 face, 10% coupon, 3 year, 5% YTM

8. If YTM = 20% 1000 face, 10% coupon, 3 year, 20% YTM 只要债权未来的现金流不止一次，可以确定的是该种债权的久期肯定 比该种债权的到期期限短； 零息票折现债权的久期与其到期时间相等。

9. If YTM = 12% and Coupon = 0 1000 face, 0% coupon, 3 year, 12% YTM 1000 |-------|-------|-------| 0 1 2 3

10. Duration and modified duration Duration allows market participants to estimate the relative price volatility of different securities: Using modified duration: modified duration= Macaulay’s duration / (1+y) We have an estimate of price volatility: %change in price = modified duration x  y

11. Effective Duration …used to estimate a securities price sensitivity when the security contains embedded options. Effective duration is Where P i- = price if rates fall, P i+ = price if rates rise; P 0 = initial (current) price; i + initial market rate plus the increase in rate; i - = initial market rate minus the decrease in rate. Effective duration compares a security’s estimated price in a falling and rising rate environment.

12. Effective duration example: Consider a 3-year, 9.4 percent coupon bond selling for $10,000 par to yield 9.4 percent to maturity. The Macaulay’s duration for the option-free version of this bond with semiannual coupons and compounding was calculated to be 5.36 semiannual periods, or 2.68 years at the market rate of 4.7 percent semiannually. The modified duration was 5.12 semiannual periods or 2.56 years. A 30 basis point increase in rate to 5 percent semiannually will lower the price to $9,847.72. The callable bond’s effective duration for a 30 basis point (0.30 percent) semiannual movement in rates either up or down is 2.54:

13. Two types of interest rate risk Reinvestment rate risk... the risk that a bank can not reinvest cash flows from assets or refinance rolled over or new liabilities at a certain rate in the future Cost of funds versus the return on assets  Funding GAP, impact on NII Price Risk … changes in interest rates will also cause a change in the value (price) of assets and liabilities Longer maturity (duration)  larger change in value for a given change in interest rates  Duration GAP, impact on market value of equity

14. Reinvestment rate risk and price risk Reinvestment rate risk: If i-rate , then yield from reinvestment of the cash flows  and holding period return (HPR) increases. Price risk: If i-rate  and you sell the security prior to maturity, the price or value falls, hence HPR falls. Increases in i-rates will improve HPR from a higher reinvestment rate but reduce HPR from capital losses if the security is sold prior to maturity. An immunized security is one in which the gain from the higher reinvestment rate is just offset by the capital loss. This point is where your holding period equals the duration of the security.

15. There are four steps in duration gap analysis. 1. Management develops an interest rate forecast. 2. Management estimates the market value of bank assets, liabilities and stockholders’ equity. 3. Management estimates the weighted duration of assets and weighted duration of liabilities. The effects of both on- and off-balance sheet items are incorporated. These estimates are used to calculate duration gap. 4. Management forecasts changes in the market value of stockholders’ equity across different interest rate environments.

16. Duration GAP at the bank The bank can protect either the market value of equity (MVE) or the book value of NII, but not both. To protect the MVE the bank would set DGAP to zero: DGAP = DA - u x DL where DA = weighted average duration of assets DL = weighted average duration of liabs

17. Focus on the market value of equity (MVE) We know that:  MVE =  MVA –  MVL With  A i = – DA i [  y / (1+y) ] A i and  Lj = – DL j [  y / (1+y) ] L j Hence:  MVE = –[DA – (MVL / MVA) DL] [  y / (1+y)] MVA If we define a bank’s duration gap: (DGAP) = DA – (MVL / MVA) DL, then  MVE = – DGAP [  y / (1+y) ] MVA

18. Hypothetical Bank Balance Sheet

19. Calculating DGAP DA = (700/1000)*2.69 + (200/1000)*4.99 = 2.88 DL = (620/920)*1.00 + (300/920)*2.81 = 1.59 DGAP = 2.88 - (920/1000)*1.59 = 1.42 years What does 1.42 mean? The average duration of assets > liabilities, hence asset values change by more than liability values.

20. 1 percent increase in all rates.

21. Calculating DGAP DA= (683 / 974) * 2.68 + (191 / 974) * 4.97 = 2.86 yrs DA= (614 / 906) * 1.00 + (292 / 906) * 2.80 = 1.58 yrs DGAP = 2.86 - (906 / 974) * 1.58 = 1.36 years What does 1.36 mean? The average duration of assets > liabilities, hence asset values change by more than liability values.

22. Using the relationship: We can define the change in the MVE as: In our case:  MVE = (-1.42) x [+0.01 / (1.10)] x 1,000 = -$12.90 Change in the Market Value of Equity

23. Positive and negative DGAPs Positive DGAP …indicates that assets are more price sensitive than liabilities, on average. Thus, when interest rates rise (fall), assets will fall proportionately more (less) in value than liabilities and the MVE will fall (rise) accordingly. Negative DGAP …indicates that weighted liabilities are more price sensitive than assets. Thus, when interest rates rise (fall), assets will fall proportionately less (more) in value that liabilities and the MVE will rise (fall).

24. An immunized portfolio … What is the minimum risk position? To eliminate the risk of changes in the MVE, how much must DA or DL change by? Change DA = -1.42 Change DL = +1.42/u = 1.54

25. Immunized portfolio DGAP = 2.88 – 0.92 (3.11) ≈ 0

26. Immunized portfolio: 1% increase in all rates

27. Stabilizing the book value of net interest income This can be done for a 1-year time horizon, with the appropriate duration gap measure: DGAP* MVRSA(1- DRSA) - MVRSL(1- DRSL) where MVRSA = cumulative market value of RSAs, MVRSL = cumulative market value of RSLs, DRSA = composite duration of RSAs for the given time horizon; equal to the sum of the products of each asset’s duration with the relative share of its total asset market value, and DRSL = composite duration of RSLs for the given time horizon; equal to the sum of the products of each liability’s duration with the relative share of its total liability market value. If DGAP* is positive, the bank’s net interest income will decrease when interest rates decrease, and increase when rates increase. If DGAP* is negative, the relationship is reversed. Only when DGAP* equals zero is interest rate risk eliminated. Banks can use duration analysis to stabilize a number of different variables reflecting bank performance.

28. Market value of equity sensitivity analysis MVE sensitivity analysis effectively involves the same steps as earnings sensitivity analysis. In MVA analysis, however, the bank focuses on: the relative durations of assets and liabilities, how much the durations change in different interest rate environments, and what happens to the market value of equity across different rate environments.

29. Embedded options sharply influence the estimated volatility in MVE Prepayments that exceed (fall short of) that expected will shorten (lengthen) duration. A bond being called will shorten duration. A deposit that is withdrawn early will shorten duration. A deposit that is not withdrawn as expected will lengthen duration.

30. Sensitivity of economic value of equity (MVE) versus most likely (zero shock) interest rate scenario Sensitivity of Economic Value of Equity measures the change in the economic value of the corporation’s equity under various changes in interest rates. Rate changes are instantaneous changes from current rates. The change in economic value of equity is derived from the difference between changes in the market value of assets and changes in the market value of liabilities.

31. Effective “duration” of equity By definition, duration measures the percentage change in market value for a given change in interest rates Hence, a bank’s duration of equity measure the percentage change in MVE that will occur with a 1 percent change in rates: Effective duration of equity 9.9 yrs. = 8,200 / 82,563

32. Asset / liability sensitivity and DGAP Funding GAP and Duration GAP are not directly comparable. Funding GAP examines various “time buckets” while DGAP represents the entire balance sheet. Generally, if a bank is liability (asset) sensitive in the sense that net interest income falls (rises) when rates rise and vice versa, it will likely have a positive (negative) DGAP suggesting that assets are more price sensitive than liabilities, on average.

33. Advantages of DGAP over Funding GAP DGAP analysis has the advantage of focusing on all cash flows from the underlying assets and liabilities and not just cash flows that are expected to arise over short time intervals. Interest rate risk can be summarized in one measure for the entire portfolio. 值得注意的是： DGAP 分析用于衡量利率变动对 股权市场价值的影响；静态 GAP 分析用于衡量利 率变动对净利差收入的影响。

34. Speculating on DGAP It is difficult to consistently alter either GAP or DGAP on the balance sheet and increase earnings or the market value of stockholders' equity. Whenever management chooses to change asset and liability maturities and/or durations in anticipation of rate changes, it is placing a bet against forward rates from the yield curve.

35. In expansionary stages rates rise until they reach a peak as the Federal Reserve tightens credit availability. In contractionary stages rates fall until they reach a trough when the U.S. economy falls into recession. Interest Rates and the Business Cycle The general level of interest rates and the shape of the yield curve appear to follow the U.S. business cycle. Time I n t e r e s t R a t e s ( P e r c e n t ) Expansion Contraction Expansion Long-TermRates Short-TermRates Peak Trough

36. 在经济周期的不同阶段，确实具有不同的 利率，而且利率可能与经济增长的速度不 同步。 银行经理在预测利率变动时必须关注经济 周期

37. 中国工商银行 2008 年就利率风险对股权市场价 值的影响进行如下处理 利用公允价值套期来规避由于市场利率变动导致金 融资产和金融负债公允价值变化所带来的影响。 对可供出售债券投资的利率风险以利率掉期合约 作为套期工具。