1. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-1 Chapter 16 Interest Rate Risk Measurements and Immunization Using Duration

2. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-2 Interest Rate Risk Defined The potential variation in returns due to UNEXPECTED changes in interest rates.

3. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-3 Fundamental Principles of Financial Asset Pricing A single investor is unable to influence the price of a financial asset. The supply and demand for a financial asset also influences price. All else equal, the price of a riskier asset will be lower than that of a less risky one because financial market participants are risk averse. Risk aversion causes investors to demand higher expected rates of return for riskier investments.

4. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-4 Financial Asset Pricing where: y  = discount rate that makes the sum of the present value of assets’ cash flows (C t ) equal to its price (P 0 )

5. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-5 All else equal, prices and yields change simultaneously in the opposite direction. If y  rises, the price of the security falls. If y  falls, the price of the security rises.

6. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-6 Calculating Ex Ante Effective Annual Yields ( y  ) for Bonds The ex ante effective annual yield (y  ) that an investor expects to receive over the life of the bond is the discount rate that makes the present value of the bond’s future cash flows equal to its current price.

7. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-7 where: P 0 = price (market value) of the bond PVIFA= present value of the annuity factor [1-(1/(1+ y  ) n )/ y  )] PVIF= present value factor [1/(1+ y  ) n ]

8. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-8 In order to earn the ex ante effective annual yield, an investor would have to: hold the bond until maturity; and invest each coupon payment at the y  rate over the life of the bond.

9. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-9 Calculate y  for a bond with 15 years to maturity, a coupon rate of 7.5% and a price of 116% of $1,000 par value. P 0 = 1.160 × $1,000 = $1,160 Coupon Payment =.075 × $1,000 = $75 $1,160 = $75(PVIFA y , 15) + $1,000(PVIF y ,15) y  = 5.87% An investor will only receive an annual effective yield of 5.87% if the bond is held to maturity and coupons can be reinvested at a 5.87% rate each year.

10. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-10 Calculating Ex Post (Actual) Effective Annual Yield where: FV = future value of the cash flows received at the end of the life of the bond

11. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-11 where: FVIFA= future value of the annuity factor [ ((1+ y  ) n - 1)/ y  )]

12. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-12 What is the EAY for a bond with 15 years to maturity, a coupon rate of 7.5%, bought at a price of $1,160 and for which coupons where reinvested at a 5.87% rate? $75(FVIFA 5.87%,15 ) + $1000 $75(23.047) + $1000 = $2,728.50 ($2,728.50/$1,160) (1/15) - 1 (2.35185) 0.0067 - 1 = 0.0587 or 5.87%

13. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-13 If all coupon payments are reinvested at the y  rate and the bond is held until maturity, the EAY will be equal to the ex ante effective annual rate, y 

14. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-14 The Two Sides of Interest Rate Risk Reinvestment Rate Risk The risk of interest rates falling and having to reinvest coupon payment at a lower rate than y , resulting in a lower ex-post yield. Price or Market Value Risk The risk of rates rising and the market price of the bond falling if the bond must be sold prior to maturity, also resulting in a lower ex-post yield.

15. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-15 Example of Reinvestment Rate Risk Use the previous example to calculate EAY, assuming rates have dropped to 5% over the life of the bond after the bond was purchased. FV of cash flows $75(FVIFA 5%,15 ) + $1000 $75(21.579) + $1000 = $2,618.39 EAY = ($2,618.39/$1,160) (1/15) - 1 = (2.2573) 0.0067 - 1 = 0.0558 or 5.58% Because of reinvestment rate risk, the investor has a lower ex post yield than the expected y  of 5.87%.

16. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-16 Example of Market Price Risk Using the previous example, calculate the EAY for the bond if rates after 5 years rise to 6.5% and remain at that level for the remainder of the life of the bond. Assume that the investor will sell the bond after 5 years. Price of the bond in 5 years $75(PVIFA 6.5,10 ) + $1,000(PVIF 6.5,10 ) $75(7.188) +1000(0.532) = $1,071.89 FV of cash flows $75(FVIFA 5.87,5 ) + $1,071.89 $75(5.6225) + $1,071.89 = $1,493.58

17. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-17 EAY ($1,493.58/$1,160) (1/5) - 1 (1.2876).2 - 1 =.0519 or 5.19% Because of market price risk, the annual effective yield received ex post is 5.19% over 5 years, 68 basis points lower than the expected effective yield of 5.87%.

18. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-18 Bond Theorems Theorem I. Bond prices must move inversely to bond yields. Theorem II. Holding the coupon rate constant, percentage changes in bond prices are greater the longer term to maturity of a bond. Theorem III. The percentage price change for a bond will increase at a decreasing rate as N increases.

19. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-19 Theorem IV. (Convexity Theorem) Starting with a given market yield y , holding other factors constant, the rise in price with a fall in y  will be greater than the fall in price with the same absolute value rise in y . Theorem V. Holding N constant and starting from the same y , the higher the coupon rate, the smaller the percentage change in price for a given change in yield.

20. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-20 Summary A Bond’s Interest Rate Risk Depends On The current market yield (y  ) The coupon rate of the bond The maturity of the bond

21. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-21 Duration Defined It is the weighted average time over which the cash flows from an investment are expected. Duration is a measure of the effective maturity. For most securities, the cash benefits are received before the maturity date. Duration is also a measure of interest rate risk. Securities with higher durations will have a greater change in market value with a change in market rates.

22. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-22 where: C t = cash flow in time period t t = the time each cash flow comes in y  = current market yield

23. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-23 Find the duration for the following three bonds: Bond 1 has an 8% coupon rate, $1000 par, n=6 Bond 2 is a zero-coupon bond with a $1,000 par, n=6 Bond 3 has a 6% coupon rate, $1,000 par, n=6 BOND 1: 8% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY Year Cash FlowPVIF 8%PV Cash Flowst × PV Cash Flow 1 $80.9259$74.07 $74.07 2 80.8573 68.58 137.16 3 80.7938 63.50 190.50 4 80.7350 58.80 235.20 5 80.6806 54.45 272.25 6 1080.6302 680.62 4,083.72 Sum $1000.00 4,992.90 Duration = $4,992.90/$1,000 = 4.993 years

24. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-24 BOND 2: ZERO COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY Year Cash FlowPVIF 8%PV Cash Flowst × PV Cash Flow 1 $0.9259 $0 $0 2 0.8573 0 0 3 0.7938 0 0 4 0.7350 0 0 5 0.6806 0 0 6 1000.6302 630.20 3,781.20 Sum $630.20 $3,781.20 Duration = $3,781.20/$630.20 = 6 years

25. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-25 BOND 3: 6% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY Year Cash FlowPVIF 8%PV Cash Flowst × PV Cash Flow 1 $60.9259$55.55 $55.55 2 60.8573 51.44 102.88 3 60.7938 47.63 142.89 4 60.7350 44.10 176.40 5 60.6806 40.84 272.25 6 1060.6302 668.01 4,008.07 Sum $907.57 4,689.97 Duration = $4,689.97/$907.57 = 5.17 years Note that duration is a negative function of a bond’s coupon payment. The duration for zero coupon will always be equal to its maturity.

26. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-26 Alternative Formula for Duration to Use with Long-term Securities where:

27. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-27 For Bond 1, with N = 6 years, Coupon Payment = $80, y  = 8%, P 0 = $1,000 and PVIFA factor for 8% and 6 years = 4.6229 Dur = 6 - [($80/($1000×.08)]×[6-(1.08)(4.6229)] = 6 - (1) × (1.0073) = 4.993 years The same as was calculated using the regular formula.

28. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-28 Why is duration important for assessing an institution’s interest rate risk? Because for a given change in market yields, percentage changes in asset prices are proportional to the assets’ duration. where: %  P 0 = percentage change in price %  y  = percentage change in market yield y  = the original market rate prior to the change

29. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-29 What is the percentage change in price for Bond 1 with a duration of 4.993, a P 0 = $1,000, and y  = 8% if rates increase to 8.5%? - 4.99[0.50%/(1.08)] = - 2.31% Multiplying (0.0231 × $1,000), the bond’s price would fall by approximately $23.10.

30. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-30 Duration assumes a linear relationship between price changes and changes in market yields. Bond price changes follow a convex pattern with changes in market yields. Duration will overestimate the percent fall in price with a rise in rates. Duration wil1 underestimate the percent rise in price with a fall in rates.

31. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-31 Estimating Interest Rate Elasticity Interest rate elasticity is the percentage price change expected for a 1% change in market yields.

32. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-32 Calculate the interest rate elasticity for Bonds 1, 2, and 3. Bond 1: E = -4.993(0.08/1.08) = -0.3699 Bond 2: E = -6.000(0.08/1.08) = -0.4444 Bond 3: E = -5.170(0.08/1.08) = -0.3830

33. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-33 For Bond 1, for every 1% change in the bond’s yield of 8%, the bondholder could have expected a price change of 0.3699%. Bonds 2 and 3 have higher elasticity than Bond 1, which implies greater interest rate risk. In this case, elasticities describe the same risk ranking as their relative duration, because each bond has the same market yield of 8%. If market yields are different, calculating elasticities provides a better comparison of different securities’ relative interest rate risk.

34. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-34 Modified Duration Multiplying MD by any  y  gives the approximate change in value of that fixed-rate asset or liability with a change in y .

35. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-35 Limitations of Duration-Based Measures of Interest Rate Risk Duration assumes that there will be a parallel shift in the yield curve so that for a given level of default risk, yields across the entire term structure change equally. Duration assumes a flat yield curve because the same y  discount rate is used for early and late cash flows. As market rate (y  ) changes, so will a security’s duration.

36. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-36 Portfolio Immunization It is a duration based strategy that makes a portfolio immune to interest rate risk over a given holding period. Used to achieve a realized annual rate of return at the end of the holding period that is no less than the expected annual yield at the beginning of the period.

37. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-37 Suppose an insurance company issues a $1 million guaranteed investment contract that guarantees an annual effective yield of 8%. The company is considering two alternatives: 1. Bond 1: six-year bonds with an 8% coupon rate at $1,000 par value which sell for $1,000 per bond, with y  = 8%. 2. Bond 2: five-year bonds with an 8% coupon rate and $1,000 par value, which sell for $1,000 per bond, with y  = 8%.

38. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-38 Duration of Bond 1 = 4.993 (from earlier calculations) Duration of Bond 2 5 - [(80/80)] - [5 - (1.08)(3.9927)] 5 - 0.68788 = 4.312

39. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-39 ILLUSTRATION OF DURATION AND PORTFOLIO IMMUNIZATION. STRATEGY 1: DURATION MATCH; BUY BOND 1 WITH APPROXIMATELY A 5- YEAR DUATION Coupon payments reinvested at y  and bond sold at end of 5 years. Scenario (1) Market interest rates stay at y  = 8% for 5 years Ex post effective annual yield for Bond 1: FV of cash flows per bond at the end of year 5 80(FVIFA 5,8%) + (1080/1.08) = $1,469.33 Ex post effective annual rate is (1469.33/1000) 0.20 - 1 =.08 or 8% Scenario (2) Market interest rates stay at y  = 9% for 5 years Ex post effective annual yield for Bond 1: FV of cash flows per bond at the end of year 5 80(FVIFA 5,9%) + (1080/1.09) = $1,469.61 Ex post effective annual rate is (1469.61/1000) 0.20 - 1 =.08 or 8% Scenario (3) Market interest rates stay at y  = 7% for 5 years Ex post effective annual yield for Bond 1: FV of cash flows per bond at the end of year 5 80(FVIFA 5,7%) + (1080/1.07) = $1,469.41 Ex post effective annual rate is (1469.41/1000) 0.20 - 1 =.08 or 8%

40. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-40 STRATEGY 2: MATURITY MATCH; BUY BOND 2 WITH A MATURITY OF 5 YEARS Coupon payments reinvested at y  and bond sold at end of 5 years. Scenario (1) Market interest rates stay at y  = 8% for 5 years Ex post effective annual yield for Bond 2: FV of cash flows per bond at the end of year 5 80(FVIFA 5,8%) + 1000 = $1,469.33 Ex post effective annual rate is (1469.33/1000) 0.20 - 1 =.08 or 8% Scenario (2) Market interest rates stay at y  = 9% for 5 years Ex post effective annual yield for Bond 2: FV of cash flows per bond at the end of year 5 80(FVIFA 5,9%) + 1000 = $1,478.78 Ex post effective annual rate is (1478.78/1000) 0.20 - 1 =.0814 or 8.14% Scenario (3) Market interest rates stay at y  = 7% for 5 years Ex post effective annual yield for Bond 2: FV of cash flows per bond at the end of year 5 80(FVIFA 5,7%) + 1000 = $1,460.06 Ex post effective annual rate is (1460.06/1000) 0.20 - 1 =.0.786 or 7.86%

41. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-41 In this example, holding bonds with a duration of 5 years will lock in an annual return of 8% over the holding period. Reasoning: With immunization, the expected return on a portfolio is protected from both reinvestment and market value risk. Reinvestment risk exactly offsets market value risk.

42. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-42 Immunization Assumptions and Limitations Assumes that there is only one parallel shift in the yield curve immediately after the beginning of the period. Market shifts may occur in the middle of the investment period and an investor may be less than perfectly immunized. It is difficult to find an investment with a duration exceeding 10 years.

43. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-43 Differences in Funding Gap and Duration A financial institution’s funding gap serves as a measure of reinvestment risk for a given time period. Funding gap ignores: time value of money; and the effects of changes in the market value of assets and liabilities on a financial institution’s balance sheet. Duration gap provides a better overall measure of interest rate risk including market price as well as reinvestment risk.

44. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-44 Duration Gap where: DurA = duration of assets which is the weighted average of the duration of assets on the balance sheet DurL = duration of liabilities which is the weighted average of the duration of liabilities on the balance sheet

45. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-45 DGAP > 0IncreaseDecrease> Decrease Fall DecreaseIncrease >IncreaseRise DGAP < 0IncreaseDecrease <DecreaseRise DecreaseIncrease<IncreaseFall DGAP = 0IncreaseDecrease=Decrease Zero DecreaseDecrease=DecreaseZero Position  y  y  in Market Value AssetsLiabilitiesEquity DURATION GAP AND CHANGES IN THE MARKET VALUE OF EQUITY

46. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-46 Change in the Value of a Depository Institution’s Equity-to-Asset Ratio where: y  = average rate on assets

47. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-47 Calculating Excelsior Bank’s DGAP Assets CashDur = 0$10,000 6-month T-bills y  = 6% Dur =.5 years$40,000 10 year Fixed-Rate LoanDur = ?$50,000 Total Assets$100,000 Liabilities and Equity Transactions Deposit Dur = 0$40,000 (rate =2%) 1-year CD, rate = 7%Dur = 1$50,000 Equity$10,000 Total Liabilities and Equity$100,000

48. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-48 Duration of a Loan Coupon Payment (Loan Payment) Loan Amount/(PVIFA 10%, 10) = $50,000/6.1446 = $8,137.27 per year Dur = 10 - [($8,137.27/($50,000×.10)] × [10 - (1.10)(6.1446)] = 10 - (1.6275)(3.241) = 4.73 years

49. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-49 Duration of Assets ($10,000/$100,000)0 + ($40,000/$100,000)0.5 + ($50,000/$100,000)4.73 = 2.565 years Duration of Liabilities ($40,000/$90,000)0 + ($50,000/$90,000)1 = 0.555 years 2.565 yrs - ($90,000/$100,000)0.5556 yrs = 2.065 years

50. Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-50 Change in the Value of Equity from a 1% Rise in Rates - 2.065[0.01/(1.074)] = - 0.0192 or -1.92% y  = ($10,000/$100,000)0 + ($40,000/$100,000)0.06 + ($50,000/$100,000)0.10 = 0.074 or 7.4%