1. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-1 Chapter 15 The Term Structure of Interest Rates

2. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-2 Term Structure of Interest Rates Often called the yield curve. All else equal, the term structure is the relationship at a specific time between yields on securities and their maturities. Treasury securities yield curve is the best way to control for extraneous factors.

3. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-3 Yield Curve Shapes Downward sloping or inverted Short-term yields are greater than long-term yields. Upward sloping or normal Short-term yields are smaller than long-term yields. Flat Short-term yields and long-term yields are basically the same.

4. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-4 Yield Curves and the Business Cycle Downward sloping yield curve often appears at the peak of the business cycle. Investors seem to anticipate recession with the lower demand for funds and a decrease in interest rates. During sluggish economic performance but towards the end of a recession, the yield curve has often been upward sloping. Investors anticipate the end of a recession and an increase in future rates.

5. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-5 The Unbiased (Pure) Expectations Theory: Assumptions All else equal, investors are indifferent between owning a single long-term security or a series of short-term securities over the same time period. All investors hold common expectations about the course of short-term rates. On average, investors are able to predict rates accurately.

6. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-6 There are no taxes, information or transactions costs in the financial markets. Investors are free to exchange securities of varying maturities quickly and without penalty.

7. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-7 Pure Expectation Theory Mathematics where: 1 r n = observed rate in period 1 for a security with n years to maturity 1 r 1 = observed rate in period 1 for a security with 1 year to maturity 2 r 1 = the expected rate of a 1 year security at the beginning of period 2 n r 1 = the expected rate for a 1 year security at the beginning of period n

8. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-8 OBSERVED AND EXPECTED 1-YEAR YIELDS AND PREMIUMS AS OF JANUARY 2000 The hypothetical data on observed and expected 1-year rates and the liquidity premiums can be used to estimate the shape of the yield curve. BILL PURCHASED BILL MATURES OBSERVED OR EXPECTED ANNUAL YIELD (%) LIQUIDITY PREMIUM (%) January 2000 January 2001 January 2002 January 2003 January 2001 January 2002 January 2003 January 2004 8.50% observed ( 1 r 1 ) 9.50% expected ( 2 r 1 ) 11.00% expected ( 3 r 1 ) 11.75% expected ( 4 r 1 ) 0.00% (on 1-year) 0.35% (on 2-year) 0.45% (on 3-year) 0.50% (on 4-year)

9. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-9 What is the yield to maturity on a four-year Treasury security bought in January 2000, i.e., what is 1 r 4 ?. 10180 or 10.180%

10. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-10 What are the spot yields on securities with two- and three-year maturities as of January 2000?.08999 or 8.999%.09662 or 9.662%

11. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-11 Strategy: Buy a two-year security in January 2000 (annual yield of 8.999%) and two successive one-year T-bills in 2002 and 2003 with 11% and 11.75% yields, respectively. What is the holding period return? 1 + i H = [(1.08999)(1.08999)(1.1100)(1.1175)] (1/4) i H = 1.10180 -1 =.1018 or 10.18%

12. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-12 Strategy: Buy a three-year security in January 2000 (annual yield of 9.662%) and a one-year T-bill in 2003 with a 11.75% yield. What is the holding period return? 1 + i H = [(1.09662)(1.09662)(1.09962)(1.1175)] (1/4) i H = 1.10180 -1 =.1018 or 10.18%

13. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-13 As shown, if the pure expectation theory is correct, the average annual yield an investor could obtain over the period 2000-2004 is the same, regardless of the investment strategy chosen. This will be true as long as: all proceeds are reinvested; and expectations about future rates remain constant during the period.

14. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-14 Hypothetical Observed Yield Curve, January 2000 Because long-term yields are the average of expected short-term yields, if short-term rates are expected to increase, the pure expectations theory holds that the term structure will be upward-sloping.

15. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-15 Liquidity Preference Theory Development of theory is based on the belief that most investors find long-term securities riskier than short-term securities. Investors demand a term premium to induce them to hold long-term securities instead of a series of short-term securities.

16. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-16 Liquidity Premium Theory Mathematics where: L t = liquidity premium for holding a t-period security instead of a one-year security t r = the expected return on a t-period security

17. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-17 1 + 1 r 2 = [(1.0850)(1.0950 + 0.0035)] (1/2) 1 r 2 = 1.0917 - 1 =.0917 or 9.17% 1 + 1 r 3 = [(1.0850)(1.0950 + 0.0035)(1.1100.0045)] (1/3) 1 r 3 = 1.0993 - 1 =.0993 or 9.93% 1 + 1 r 4 = [(1.0850)(1.0950 + 0.0035)(1.1100 +.0045)(1.1175 +.0050] (1/4) 1 r 4 = 1.1050 - 1 =.1050 or 10.50% Hypothetical Example: Liquidity Preference

18. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-18 Hypothetical Observed Yield Curve with Liquidity Premium If short-term rates are expected to increase and if investors demand a premium for holding long-term securities, the long-term yields will be higher than if expectations alone were considered.

19. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-19 If investors demand a premium for holding short-term securities and if…. short-term rates are expected to increase, long- term yields will be higher than if expectations alone were considered. short-term rates are expected to decrease slightly, the yield curve could be slightly upward-sloping. short-term rates are expected to decrease sharply, the slope of the yield curve will be less steep than if expectations alone were considered.

20. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-20 The Modified Expectations Theory The expectations of borrowers and lenders and their conflicting maturity preferences put pressure on long-term rates, producing an upward sloping curve. Differs from pure expectations only in the motivation for determining spot rates. Otherwise : expectations of falling rates produce a downward- sloping curve; and expectations of rising rates produce a upward- sloping curve.

21. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-21 The Segmented Market Theory Argues that there is really no term structure. Short-term yields result from the interaction of individuals and institutions in the short- term segment. The same is true for long-term yields. Laws, regulations, or institutional objectives may prevent many market participants from borrowing or lending in every segment. Thus, maturity is of no concern.

22. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-22 Preferred Habitat Theory Investors’ time preference for spending versus saving influences their choice among securities. Investors will lend in markets other than their preferred one only if a premium exists to induce them to switch.

23. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-23 Short-term and long-term yield differentials are only partially explained by the expectations hypothesis. Supply and demand imbalances in various markets may result in positive or negative premiums added to the pure expectations rate to induce shifts from one segment to another.

24. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-24 Interest Rate Forecasting With the Pure Expectations Theory The forward rate is the implied expected rate calculated from an existing yield curve. where: t r 1 = one-year forward rate as of the beginning of any future period (t)

25. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-25 HYPOTHETICAL SPOT RATES ON TREASURY SECURITIES AS OF JUNE 2000 The hypothetical data on observed yields can be used to infer 1-year forward rates. MATURITY DATESPOT YIELD (%)NOTATION June 200112.50% 1 r 1 June 200211.85% 1 r 2 June 200311.00% 1 r 3 June 200410.90% 1 r 4 June 200510. 50% 1 r 5

26. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-26 11.20% One-year Forward Rate Expected to Prevail at the Beginning of Year 2

27. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-27 Interest Rate Forecasting With the Pure Liquidity Premium Theory where: t r 1 = one-year forward rate as of the beginning of any future period (t) adjusting for liquidity premium

28. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-28 10.70% One-year Forward Rate Expected to Prevail at the Beginning of Year 2 Adjusting for the Liquidity Premium Assume investors expected a premium of 0.5% for holding a two-year security in 2000.

29. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-29 Setting Institutional Interest Rates: Application to CD and Mortgage Rates Scenario Using one-year CDs to fund 5 year mortgages. Upward-sloping yield curve suggests the cost of one-year CDs will be rising in the future. According to the liquidity preference theory, Treasury rates may include a liquidity premium.

30. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-30 A default premium should be required for holding mortgages instead of Treasury securities. The segmented markets and preferred habitat theories suggest competitive pressures should be considered in setting both rates.

31. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-31 Steps in Setting the CD and Mortgage Rates Calculate the series of one-year forward rates implied by the existing yield curve. Remove the liquidity premiums embedded in the current term structure to avoid overestimating expected one-year CD rates. Use the adjusted series of forward rates to anticipate one-year CD rates.

32. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-32 Estimate the administrative cost of servicing deposits. Add administrative cost to CD base rate. Make any adjustments to CD rates to account for competitive pressures. Estimate the premiums required to compensate institution for holding mortgages.

33. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-33 Add the desired profit margin to obtain the estimated annual return required to cover all costs. Calculate the five-year geometric average to obtain the mortgage rate to be charged. Make adjustments to the mortgage rate to account for competitive pressures.

34. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-34 USING THE TERM STRUCTURE TO SET INSTITUTIONAL INTEREST RATES The hypothetical data on observed yields, liquidity premiums, administrative costs, risk premiums, and profit markup used to estimate the appropriate rate to charge on a mortgage loan. MATURITY (YEARS) OBSERVED YIELD ON TREASURY SECURITIES UNADJUSTED 1-YEAR FORWARD RATE (FROM EQUATION 15.4 ESTIMATED LIQUIDITY PREMIUM ESTIMATED 1-YEAR RATE WITHOUT THE LIQUIDITY PREMIUM 10.08000.08000.00000.0800 20.08250.08500.00500.0800 30.09500.12040.01000.1104 40.10250.12530.02500.1003 50.11000.14050.03500.1055

35. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-35 ANNUAL CD COST (INCLUDES ADMINISTRATIVE MARKUP OF 0.25%) ESTIMATED PREMIUM REQUIRED TO HOLD MORTGAGES ESTIMATED REQUIRED ANNUAL RETURN (INLUDING A PROFIT MARKUP OF 0.75% PER YEAR) 10.08250.01000.1000 20.08250.02500.1150 30.11290.03000.1504 40.10280.04000.1003 50.10800.04500.1605 YEARS ESTIMATION OF REQUIRED ANNUAL YIELD ON MORTGAGES i m = 0.1350 or 13.50%

36. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-36 Using Forward Rates to Set Loan Credit Risk Premiums Requires the calculation of the: forward rate for next year’s one-year Treasury bond; and forward rate for next year’s corporate bond with the desired credit risk classification. The probability of no default for the corporate bond is calculated by taking the ratio of (1 + forward rate for treasuries) to (1+ forward rate for the corporate bond).

37. Copyright © 2000 by Harcourt, Inc. All rights reserved. 15-37 Hypothetical Example : Wall Street Journal quotes in 2002 Expected 1-Year T-bill = 4.961% Expected 1-Year Corporate rate = 5.360% Size of Risk Premium = (5.36% - 4.961%) = 0.4% Probability of not defaulting = (1.4961)/(1.5360) = 0.9962 Probability of defaulting = [1-(.9962)] = 0.0038 or 0.38% Any one-year loan with 0.38% probability of default should have a risk premium 40 basis points.