1. Fixed Income Term structure of interest rates and the yield curve

2. Learning outcomes Examine the relationship between the bond value or price Evaluate yield spreads and the yield curve Discuss the yield to maturity 12/23/20112

3. relationship between the bond value or price This relationship is not linear, but convex to the origin. 12/23/20113

4. Yield to maturity (YTM) (also known as the [Gross] Redemption Yield (GRY)) If the current price of a bond is given, together with details of coupons and redemption date, then this information can be used to compute the required rate of return or yield to maturity of the bond. YTM is the average annual rate of return the bond investors expect to receive from the bond till its redemption 12/23/20114

5. Example A bond paying a coupon of 7% is redeemable in five years at par (sh100) and is currently trading at sh106.62. Estimate its yield (required rate of return) assuming: Annual payments Semiannual payments 12/23/20115

6. Answer: Annual coupons: yield =5.452% Semiannual coupon: yield=5.469% Note! For the exam always give an annualized yield if the coupon is not annual 12/23/20116

7. YTMs for bonds are normally quoted in the financial press, based on the closing price of the bond. For example, a yield often quoted in the financial press is the bid yield. The bid yield is the YTM for the current bid price (the price at which bonds can be purchased) of a bond. 12/23/20117

8. Yield spreads Yield spreads are simply differences between the yields of any two debt securities or types of debt securities. The interest rate level is one of the most important determinants of portfolio value. For this reason it is important to know what determines interest rates and who is involved 12/23/20118

9. interest rate policy tools of a central bank 1.The discount rate is the rate at which banks can borrow reserves from the Central Bank. A lower rate tends to increase bank reserves, encourage lending, and decrease interest rates. A higher discount rate has the opposite effect, raising rates. 12/23/20119

10. 2.Open market operations is the trading of Treasury securities by CBK in the open market. When CBK buys securities, cash replaces securities in investor accounts, more funds are available for lending, and interest rates decrease. Sales of securities by CBK have the opposite effect, reducing cash balances and funds available for lending as well as increasing rates. 12/23/201110

11. 3.Bank reserve requirements are the % of deposits that banks must retain as reserves. By increasing this %, CBK effectively decreases the funds available for lending. This decrease in amounts available for lending will tend to increase interest rates. A decrease in the % reserve requirement will increase the funds available for loans and tends to decrease interest rates. 12/23/201111

12. 4.Persuading banks to tighten or loosen their credit policies. By asking banks to alter their lending policies, CBK attempts to affect their willingness to lend. Encouraging lending will tend to decrease rates and vice versa. The most commonly used policy tool is open market operations. 12/23/201112

13. Yield curves Yield curves are a plot of yields by years to maturity. There are four general shapes that we use to describe yield curves: 1. Normal or upward sloping. 2.Inverted or downward sloping. 3.Flat. 4.Humped. 12/23/201113

14. Yield Curve Shapes 12/23/201114

15. 12/23/201115

16. Yield curves can take on just about any shape, so don’t think these examples are the only ones observed. These four are representative of general types, and you need to be familiar with what is meant by an “upward sloping” or “normal” yield curve and by an "inverted” or “downward sloping” yield curve. 12/23/201116

17. Term structure of interest rates and the yield curve The yield to maturity is calculated implicitly based on the current market price, the term to maturity of the bond and amount (and frequency) of coupon payments. However, if a corporate bond is being issued for the first time, its price and/or coupon payments need to be determined based on the required yield. 12/23/201117

18. The required yield is based on the term structure of interest rates and this needs to be discussed before considering how the price of a bond may be determined. It is incorrect to assume that bonds of the same risk class, which are redeemed on different dates, would have the same required rate of return or yield. 12/23/201118

19. In fact, it is evident that the markets demand different annual returns or yields on bonds with differing lengths of time before their redemption (or maturity), even where the bonds are of the same risk class. This is known as the term structure of interest rates and is represented by the spot yield curve or simply the yield curve. 12/23/201119

20. For example, a company may find that if it wants to issue a one-year bond, it may need to pay interest at 3% for the year, if it wants to issue a two-year bond, the markets may demand an annual interest rate of 3.5%, and for a three-year bond the annual yield required may be 4.2%. 12/23/201120

21. Hence, the company would need to pay interest at 3% for one year; 3.5% each year, for two years, if it wants to borrow funds for two years; and 4.2% each year, for three years, if it wants to borrow funds for three years. In this case, the term structure of interest rates is represented by an upward sloping yield curve. 12/23/201121

22. The normal expectation would be of an upward sloping yield curve on the basis that bonds with a longer period of maturity would require a higher interest rate as compensation for risk. Note here that the bonds considered may be of the same risk class but the longer time period to maturity still adds to higher uncertainty. 12/23/201122

23. However, it is entirely normal for yield curves to be of many different shapes dependent on the perceptions of the markets on how interest rates may change in the future. Three main theories have been advanced to explain the term structure of interest rates or the yield curve: expectations hypothesis, liquidity preference hypothesis and market- segmentation hypothesis. 12/23/201123

24. Theories on the term structure of interest rates There are three theories on the term structure of interest rates namely: 1.Pure (unbiased) expectations theory 2.Liquidity preference theory 3.Market segmentation theory 12/23/201124

25. Pure (Unbiased) Expectations Theory This theory suggests that forward rates are solely a function of expected future spot rates. In other words, long-term interest rates equal the mean of future expected short-term rates. This implies, for example, that an investor could earn the same return by investing in a 5- year bond or by investing in a 3-year bond and then a 2-year bond after the 3-year bond expires. 12/23/201125

26. Therefore, the implications for the shape of the yield curve under the pure expectations theory are: 1.If the yield curve is upward sloping, short- term rates are expected to rise. 2.If the curve is downward-sloping, short-term rates are expected to fall. 3.A flat yield curve implies that the market expects short-term rates to remain constant. 12/23/201126

27. The pure expectations theory fails to consider the riskiness of bond investing. In particular it fails to recognize: Price risk—the uncertainty associated with the future price of a bond that may be sold prior to its maturity. Reinvestment risk—the uncertainty associated with the rate at which bond cash flows can be reinvested over an investment horizon. 12/23/201127

28. Hence, the pure (or unbiased) expectations theory does not recognize the risk difference between investing in a 1-year bond and sequentially investing in two 6-month bonds. 12/23/201128

29. Liquidity preference theory This theory addresses the limitations of the pure expectations theory by proposing that forward rates reflect investors’ expectations of future spot rates plus a liquidity premium to compensate them for exposure to interest rate risk. This suggests that this liquidity premium is positively related to maturity: a 25-year bond has a larger liquidity premium than a 5-year bond. 12/23/201129

30. Therefore, a positive-sloping yield curve may indicate that either: (1) the market expects future interest rates to rise; or (2) that rates are expected to remain constant (or even fall), but the addition of the liquidity premium results in a positive slope. A downward-sloping yield curve indicates falling short term rates according to the liquidity theory. 12/23/201130

31. Market segmentation theory Is based on the idea that investors and borrowers have preferences for different maturity ranges. Under this theory, the supply of bonds (desire to borrow) and the demand for bonds (desire to lend) determine equilibrium yields for the various maturity ranges. 12/23/201131

32. Institutional investors may have strong preferences for maturity ranges that closely match their liabilities. Life insurers and pension funds may prefer long maturities due to the long-term nature of the liabilities they must fund. A commercial bank that has liabilities of a relatively short maturity may prefer to invest in shorter-term debt securities. 12/23/201132

33. Another argument for the market segmentation theory is that there are legal or institutional policy restrictions that prevent investors from purchasing securities with maturities outside a particular maturity range. A somewhat weaker version of the market segmentation theory is the preferred habitat theory. 12/23/201133

34. Under this theory, yields also depend on supply and demand for various maturity ranges, but investors can be induced to move from their preferred maturity ranges when yields are sufficiently higher in other (non- preferred) maturity ranges. 12/23/201134

35. The preferred habitat theory provides an interpretation of the implied forward rate that is similar to that of the liquidity theory. However, there are two subtle differences: 1.The premium is a positive or negative risk premium related to supply and demand for funds at various maturities, not necessarily a liquidity premium. 2.This risk premium is not necessarily related to maturity. 12/23/201135

36. yield spread measures There are three different yield spread measures as follows: 1.Absolute yield spread (AYS) 2.Relative yield spread(RYS) 3.Yield ratio (YR) 12/23/201136

37. Absolute yield spread 1. The absolute yield spread is simply the difference between yields on two bonds. This simple measure is sometimes called the nominal spread. Absolute yield spreads are usually expressed in basis points AYS= higher-yield bond yield minus lower-yield bond yield 12/23/201137

38. Relative yield spread The relative yield spread is the absolute yield spread expressed as a percentage of the yield on the benchmark bond. relative yield spread = absolute yield spread/yield on the benchmark bond 12/23/201138

39. Yield ratio The yield ratio is the ratio of the yield on the subject bond to the yield on the benchmark bond. yield ratio = subject bond yield/benchmark bond yield Note that the yield ratio is simply one plus the relative yield spread. 12/23/201139

40. Example Consider two bonds, X and Y. Their respective yields are 6.50% and 6.75%. Using bond X as the benchmark bond, compute the absolute yield spread, the relative yield spread, and the yield ratio for these bonds. 12/23/201140

41. Answer absolute yield spread = 6.75% – 6.50% = 0.25% or 25 basis points relative yield spread = 0.25% / 6.50% = 0.038 = 3.8% yield ratio = 6.75% / 6.50% = 1.038 12/23/201141

42. The most commonly used yield spread is the absolute yield spread but it is the most simplistic in its assumptions What do you think are its weaknesses? 12/23/201142

43. A shortcoming of the absolute yield spread is that it may remain constant, even though overall rates rise or fall. In this case, the effect of rising or falling rates on spreads is captured by the relative yield spread or the yield ratio. Consider two yields that rise from 6.5% and 7.0% to 7.0% and 7.5%, respectively. The absolute yield spread remains constant at 50 basis points, while the relative spread falls from 7.69% to 7.14% and the yield ratio decreases from 1.077 to 1.071. 12/23/201143

44. Credit spreads A credit (or quality) spread is the difference in yields between two issues that are similar in all respects except for credit rating. An example of a credit spread is the difference in yields between long AA rated general obligation (GO) municipal bonds and long A rated GO municipal (an intra-market spread as well) 12/23/201144

45. These spreads show the effect of credit quality on yields and reveal the risk-return tradeoff the investor can expect 12/23/201145

46. Effect of embedded options on yield spreads A call option on a bond is an option the bond issuer holds and will only be exercised if it is advantageous to the issuer to do so. From the bondholder’s perspective, a non- callable bond is preferred to a bond that is otherwise identical but callable. Investors will require a higher yield on a callable bond, compared to the same bond without the call feature. 12/23/201146

47. The inclusion of a put provision or a conversion option with a bond will have the opposite effect; the choice of whether to exercise either of these options is the bondholder’s. Compared to an identical option-free bond, a putable bond will have a lower yield spread to Treasuries due to the value of the put feature “included” with the bond. 12/23/201147

48. Bonds that have less liquidity have higher spreads to Treasuries. Investors prefer more liquidity to less and will pay a premium for greater liquidity. Liquidity is affected by the size of an issue. Larger issues normally have greater liquidity because they are more actively traded in the secondary market. 12/23/201148

49. Empirical evidence suggests that issues with greater size have lower yield spreads. 12/23/201149

50. Concept checks 1.Under the pure expectations theory, an inverted yield curve is interpreted as evidence that: A. demand for long-term bonds is falling. B. short-term rates are expected to fall in the future. C. investors have very little demand for liquidity. 12/23/201150

51. 2. According to the liquidity preference theory, which of the following statements is least accurate? A. All else equal, investors prefer short-term securities over long-term securities. B. Investors perceive little risk differential between short-term and long-term securities. C. Borrowers will pay a premium for long-term funds to avoid having to roll over short-term debt. 12/23/201151

52. 3. With respect to the term structure of interest rates, the market segmentation theory holds that: A. an increase in demand for long-term borrowings could lead to an inverted yield curve. B. expectations about the future of short-term interest rates are the major determinants of the shape of the yield curve. C. the yield curve reflects the maturity demands of financial institutions and investors. 12/23/201152

53. 4. The most commonly used tool of CBK to control interest rates is: A. the discount rate. B. the bank reserve requirement. C. open market operations. 12/23/201153

54. 5.For two bonds that are alike in all respects except maturity, the relative yield spread is 7.14%. The yield ratio is closest to: A. 0.714. B. 1.0714. C. 107.14. 12/23/201154

55. 6.Assume the following yields for different bonds issued by a corporation: 1-year bond: 5.50%. 2-year bond: 6.00%. 3-year bond: 7.00%. If a 3-year Treasury bond is yielding 5%, then what is the absolute yield spread on the 3-year corporate issue? A. 0.40. B. 100 bp. C. 200 bp. 12/23/201155

56. 7.Assume the following corporate yield curve: 1-year bond: 5.00%. 2-year bond: 6.00%. 3-year bond: 7.00%. If a 3-year Treasury bond yielding 6% is the benchmark bond, the relative yield spread on the 3-year corporate is: A. 16.67%. B. 1.167. C. 14.28%. 12/23/201156

57. 8.If a regional investor is forecasting that the yield spread between regional Treasury bonds and regional corporate bonds is going to widen, which of the following beliefs would he be also most likely to hold? A. The economy is going to expand. B. The economy is going to contract. C. There will be no change in the economy. 12/23/201157

58. 9.For a Treasury bond and a corporate bond that are alike in all respects except credit risk, the yield ratio is 1.0833. If the yield on the corporate bond is 6.5%, the Treasury (benchmark) bond yield is closest to: A. 5.50%. B. 6.00%. C. 8.33%. 12/23/201158