Learning Module 1 The Term Structure and Interest Rate Dynamics

Learning Module 1 The Term Structure and Interest Rate Dynamics EXAMPLE 1 Spot and Forward Prices and Rates (1) Consider a two-year loan beginning in one year (A1, B3). The one-year spot rate is 27%0.07. The three-year spot rate is 2329%0.09. 1. Calculate the one-year discount factor: DFA DF Solution: Using Equation 1, DF (10.07) 0.9346 2. Calculate the three-year discount factor: DF = DF3 Solution: DF311 (1+0.09% 0.7722 3. Calculate the forward price of a two-year bond to be issued in one year: FAB-AF13 Solution: Using Equation 2, 0.77220.9346xF F-0.7722-0.9346-0.8262. 4. Interpret your answer to Problem 3. Solution: The forward contract price of DF1,3 = 0.8262 is the price agreed on today. to be paid one year from today for a bond with a two-year maturity and a risk-free unit-principal payment (e.g., $1. €1, or £1) at maturity in three years. As shown in the solution to 3, it is calculated as the three-year discount factor, DF = 0.7722, divided by the one-year discount factor, DF= 0.9346. EXAMPLE 2 Spot and Forward Prices and Rates (2) The spot rates for three hypothetical zero-coupon bonds (zeros) with maturities of one, two, and three years are given in the following table. Maturity (7) 1 2 3 Spot rates % 10% 1. Calculate the forward rate for a one-year zero issued one year from today, ha Solution: fis calculated as follows (using Equation 4): (1+2)(1)(1) (1+0.10)2 = (1+0.09) (1) 0.10 109 111.01% 2. Calculate the forward rate for a one-year zero issued two years from today, Solution: 2.1) is calculated as follows: (1+2)=(1+2)(1+2) (1+0.11)3(1+0.10)(1+2) 1-1-13.03% 1.10 3. Calculate the forward rate for a two-year zero issued one year from today, ha Solution: is calculated as follows: (12)-(1+2)(1+2) (1+0.11) = (1+0.09) (1+2)² 1-12.01% 4. Based on your answers to 1 and 2, describe the relationship between the spot rates and the implied one-year forward rates. Solution: The upward-sloping zero-coupon yield curve is associated with an upward-sloping forward curve (a series of increasing one-year forward rates because 13.03% is greater than 11.01%). This dynamic is explained further in the following discussion. EXAMPLE 3 Spot and Forward Prices and Rates (3) 1. Given the data and conclusions for 21.1.1 and 21 from Example 2: 21-9% =11.01% 2. Determine whether forward rates rise or fall as the initiation date, A, for the forward rate is later. Solution: The spot rates imply an upward-sloping yield curve, z3 > 22 > 21. Thus, the forward rates will rise with increasing A. This relationship was shown in Example 2, in which/1,1 = 11.01% and 2,1 = 13.03%. EXAMPLE 4 Spot and Forward Prices and Rates (4) Given the spot rates 219%, 22 10%, and 23 11%, as in Example 2 and Example 3: 1. Determine whether the forward rate f₁₂ is greater than or less than the long-term rate, zj Solution: The spot rates imply an upward-sloping yield curve, z3 >>2₁, or in general 282. Thus, the forward rate will be greater than the long-term rate, or JA.B-AB Note from Example 2 that /12 12.01% > 23=11%. 2. Determine whether forward rates rise or fall as the initiation date, A, for the forward rate is later. Solution: The spot rates imply an upward-sloping yield curve, 23 > 22 > 21. Thus, the forward rates will rise with increasing A. This relationship was shown in Example 2, in which 1,1 11.01% and 2,1 = 13.03%. WHAT IS BOOTSTRAPPING? Because the practical details of deriving the zero-coupon yield are beyond the scope of this reading, the concept of bootstrapping may be best shown using a numerical illustration. Suppose the following yields are observed for annual coupon sovereign debt: Par Rates: One-year par rate = 5%, two-year par rate = 5.97%, three-year par rate =6.91%, four-year par rate 7.81%. From these data, we can bootstrap zero-coupon rates. Zero-Coupon Rates: Given annual coupons, the one-year zero-coupon rate equals the one-year par rate because it has one cash flow, whereas two-year and longer maturity bonds have coupon payments prior to maturity. The derivation of zero-coupon rates begins with the two-year maturity. The two-year zero-coupon rate is determined by using z₁ = 5% and solving for z₂ in the following equation for of one monetary unit of current market value: 1 0.0597 10.0597 (1.05) (1+22)2 In the equation, 0.0597 and 1.0597 represent payments from interest and principal and interest, respectively, per unit of principal value. The equation implies that z₂ = 6%. We have bootstrapped the two-year spot rate. Continuing with forward substitution, the three-year zero-coupon rate can be bootstrapped by solving for z3 using the known values of the one-year and two-year spot rates of 5% and 6%: 1 0.0691 0.0691 10.0691 + (1.05) (1.06)2 (1+2) + Thus, z3 = 7%. Finally, we solve for the four-year zero-coupon rate, 24 1= 0.0781 0.0781 0.07811+0.0781 (1.05) (1.06)2 (1.07)3 (1+24) In summary, z₁ = 5%, 226%, 237%, and 248%. EXAMPLE 5 Spot Rate and Yield-to-Maturity Recall from earlier examples the spot rates z₁ = 9%, 2= 10%, and 11%. Let y be the YTM. 1. Calculate the price of a two-year annual coupon bond using the spot rates. Assume the coupon rate is 6% and the face value is $1,000. Next, state the forrnula for determining the price of the bond in terms of its YTM. Is z greater than or less than ya? Why? Solution: Using the spot rates, Price 560 $1,060 (1+0.09) (1+0.10 $931.08 Using the YTM. 560 $1,060 5931.08 (1) (1) Price Note that y, is used to discount both the first and second-year cash flows. Because the bond can have only one price, it follows that 2₁ << because Jy is a weighted average of 21 and 22 and the yield curve is upward sloping. Using a calculator, one can calculate the YTM as y 9.97%, which is less than 10% and greater than 219%, just as we would expect. Note that Jy is much closer to z₂ than to z, because the bond's largest cash flow occurs in Year 2, thereby giving a greater weight than 2, in the determination of 2. Calculate the price of a three-year annual coupon-paying bond using the spot rates. Assume the coupon rate is 5% and the face value is £100. Next, write a formula for determining the price of the bond using the YTM. Is 2 greater or less than yy? Why? Solution: Using the spot rates, Price 65 11009 1010 1011 £105 E85.49 Using the yield-to-maturity, 6105 EXS.49. 15 Note that y, is used to discount all three cash flows. Because the bond can have only one price, y must be a weighted average of 2 and 2 Given that the yield curve is upward sloping in this example, y32. Using a calculator to compute YTM, 2310.93%, which is less than 23 11% and greater than 9%-just as we would expect, because the weighted YTM must lie between the highest and lowest spot rates. Note that y, is much closer to 2 than it is to 2 or 2, because the bond's largest cash flow occurs in Year 3, thereby giving z, a greater weight than 21 and 22 in the determination of 33. EXAMPLE 6 Yield and Return Concepts 1. When the spot curve is upward sloping, the forward curve: A. lies above the spot curve. B. lies below the spot curve. C. is coincident with the spot curve. Solution: A is correct. Points on a spot curve can be viewed as an average of single-period rates over given maturities, whereas forward rates reflect the marginal changes between future periods. 2. Which of the following statements concerning the YTM of a default-risk-free bond is most accurate? The YTM of such a bond: A. equals the expected return on the bond if the bond is held to maturity. B. can be viewed as a weighted average of the spot rates applying to its cash flows. C. will be closer to the realized return if the spot curve is upward-sloping rather than flat through the life of the bond. Solution: B is correct. The YTM is the discount rate that, when applied to a bond's promised cash flows, equates those cash flows to the bond's market price and the fact that the market price should reflect discounting promised cash flows at appropriate spot rates. 3. When the spot curve is downward sloping, a later initiation date results in a forward curve that is: A. closer to the spot curve. B. a greater distance above the spot curve. C. a greater distance below the spot curve. Solution: C is correct. This answer follows from the forward rate model as expressed in Equation 6. If the spot curve is downward sloping (upward sloping), a later initiation date will result in a forward curve that is a greater distance below (above) the spot curve. See Exhibit 2 and Exhibit 3. EXAMPLE 7 Active Bond Portfolio Management 1. The "rolling down the yield curve strategy is executed by buying bonds whose maturities are A. equal to the investor's investment horizon. B. longer than the investor's investment horizon. C. shorter than the investor's investment horizon. Solution: B is correct. A bond with a longer maturity than the investor's investment horizon is purchased but then sold prior to maturity at the end of the invest ment horizon. If the yield curve is upward-sloping and yields do not change the bond will be valued at successively lower yields and higher prices over time. The bond's total return will exceed that of a bond whose maturity is equal to the investment horizon. 2. A bond will be overvalued if the expected spot rate in A. equal to the current forward rate. B. lower than the current forward rate. C. higher than the current forward rate. Solution: C is correct. If the expected discount rate is higher than the forward rate, then the bond will be overvalued. The expected price of the bond is lower than the price obtained from discounting using the forward rate. 1. Assume a flat yield curve of 6%. A three-year £100 bond is issued at par, paying an annual coupon of 6%. What is the portfolio manager's expected return if he predicts that the yield curve one year from today will be a flat 767 A. 4.19% B. 6.00% C. 5.83% Solution: A is correct. Expected return will be less than the current YTM of 6% if yields increase to 7%. The expected return of 4 19% is computed as follows: 100 -14.19% 4. A forward contract price will increase if A. future spot rates evolve as predicted by current forward rates. B. future spot rates are lower than what is predicted by current forward rates. future spot rates are higher than what is predicted by current forward rates. Solution: B is correct. The forward rate model can be used to show that a change in the forward contract price requires a deviation of the spot curve from that predicted by today's forward curve. If the future spot rate is lower than what is predicted by the prevailing forward rate, the forward contract price will increase because it is discounted at an interest rate that is lower than the originally anticipated rate. HISTORY OF THE US SWAP SPREAD SINCE 2008 The fact that governments generally pay less than private entities do in order to borrow suggests that swap spreads should always be positive. However, the 30-year Treasury swap spread turned negative following the collapse of Lehman Brothers Holdings Inc. in September 2008. Strong demand for duration combined with tighter liquidity and greater counterparty risk were widely cited as reasons for this phenomenon. For the period shown, the 30-year Treasury swap spread hit a record low (-62 bps intramonth) during November 2008. The 30-year Treasury swap spread was at or above zero for more than a year before becoming negative once again (see Exhibit 5). A recent study by the Federal Reserve Bank of New York (Boyarchenko, Gupta, Steele, and Yen, 2018) suggests that negative swap spreads have persisted because of increased regulatory capital requirements among swap dealers following the financial crisis. PREFERRED HABITAT AND QE The term "quantitative easing" (QE) refers to an unconventional monetary policy used by central banks to increase the supply of money in an economy when central bank and/ or interbank interest rates are already close to zero. The first of several QE efforts by the US Federal Reserve began in late 2008, following the establishment of a near-zero target range for the federal funds rate. Since then, the Federal Reserve has greatly expanded its holdings of long-term securities via a series of asset purchase programs, with the goal of putting downward pressure on long-term interest rates and thereby making financial conditions even more accommodative. Exhibit 8 presents information regarding the securities held by the Federal Reserve on 20 September 2007 (when all securities held by the Fed were US Treasury issuancel and on 29 October 2014 (when the Federal Reserve ended its third round of QE). Exhibit 8: Securities Held by the US Federal Reserve (US$ billions) 20 Sep 2007 29 Oct 2014 Securities held outright 780 4,219 US Treasury 780 2,462 0 Bills 267 2,347 Notes and bonds, nominal 472 36 115 16 Notes and bonds, inflation-indexed 5 0 0 Inflation compensation 40 Federal agency Mortgage-backed securities 1,718 As Exhibit 8 shows, the Federal Reserve's security holdings on 20 September 2007 consisted entirely of US Treasury securities, and about 34% of those holdings were short-term in the form of T-bills. On 29 October 2014, only about 58% of the Federal Reserve's security holdings were Treasury securities, and none were T-bills. Furthermore, the Federal Reserve held well over US$1.7 trillion of mortgage-backed securities (MBS), which accounted for 41% of all securities held. Prior to the QE efforts, the yield on MBS was typically in the 5%-6% range it declined to less than 2% by the end of 2012. Concepts related to preferred habitat theory could possibly help explain that drop in yield. The purchase of MBS by the Federal Reserve reduced the supply of these securities that was available for private purchase. Assuming that many MBS investors are either unwilling or unable to withdraw from the MBS market because of their comparative experience and expertise in managing interest rate and repayment risks of MBS versus option-free bonds, MBS investing institutions would have a "preferred habitat in the MBS market. If they were unable to meet investor demand without bidding more aggressively, these buyers would drive down yields on MBS. The Federal Reserve's purchase of MBS also resulted in a reduction in MBS yields. If a homeowner prepays on a mortgage, the payment is sent to MBS investors on a pro-rata basis. Although investors are uncertain about when such a prepayment will be received, prepayment is more likely in a declining interest rate environment. EXAMPLE 9 Traditional Term Structure Theories 1. Many fixed-income portfolio managers are limited in or prohibited from high-yield bond investments. When a bond is downgraded from an investment-grade to a high-yield (junk) rating, it is referred to as a fallen angel, Because of restrictions, many pension funds sell fallen angels when they are downgraded from investment-grade to high-yield (junk). This coordinated selling action often results in depressed prices and attractive yields for the fallen angels. Which of the following reasons best explains why fallen angel yields often exceed otherwise identical bonds? A. The preferred habitat theory B. The segmented markets theory C. The local expectations theory Solution: B is correct. Market segmentation in this example results from the requirement that some fixed-income fund managers are prohibited or limited in their capacity to hold high-yield bonds. The segmentation results in selling pressure on fallen angels that depresses their prices. 2. The term structure theory in which investors can be induced by relatively attractive yields to hold debt securities whose maturities do not match their investment horizon is best described as the: A. preferred habitat theory. B. segmented markets theory. C. unbiased expectations theory. Solution: A is correct. Preferred habitat theory asserts that investors are willing to deviate from their preferred maturities if yield differentials encourage the switch. Segmented markets theory is more rigid than preferred habitat in that asset/liability management constraints force investors to buy securities whose horizons match those of their liabilities. The unbiased expectations theory makes no assumptions about maturity preferences. Rather, it con tends that forward rates are unbiased predictors of future spot rates. 3. The unbiased expectations theory assumes investors are: A. risk averse. B. risk neutral. C. risk seeking. Solution: B is correct. The unbiased expectations theory asserts that different ma- turity strategies, such as rollover, maturity matching, and riding the yield curve, have the same expected return. By definition, a risk-neutral party is indifferent about choices with equal expected payoffs, even if one choice is riskier. Thus, the predictions of the theory are consistent with the existence of risk-neutral investors. 4. Market evidence shows that forward rates are A. unbiased predictors of future spot rates. B. upwardly biased predictors of future spot rates. C downwardly biased predictors of future spot rates. Solution: B is correct. The existence of a liquidity premium ensures that the forward rate is an upwardly biased estimate of the future spot rate. Market evidence clearly shows that liquidity premiums exist, and this evidence effectively refutes the predictions of the unbiased expectations theory. 5. Market evidence shows that short holding-period returns on short-maturity bonds most often are: A. less than those on long-maturity bonds. B. about equal to those on long-maturity bonds. C greater than those on long-maturity bonds. Solution: A is correct. Although the local expectations theory predicts that the short-run return for all bonds will equal the risk-free rate, most of the evidence refutes that claim. Returns from long-dated bonds are generally higher than those from short-dated bonds, even over relatively short investment horizons. This market evidence is consistent with the risk-expected return trade-off that is central to finance and the uncertainty surrounding future spot rates. EXAMPLE 11 Building a Rate View Based On Economic Forecasts and Monetary Policy Morgan Salaz is a fixed income analyst responsible for advising fixed income clients about bond trading opportunities. In the current recessionary environment, the level of government bond yields is low and the term structure is nearly flat. Salaz's firm forecasts that after a brief recession, economic growth will return quickly during the coming 12 months. 1. Which of the following changes to the yield curve is consistent with Salaz's expectation of increasing economic growth over the coming year? A. Decrease in the level B. Decrease in the term spread of long-term rates over short-term rates C. Increase in the term spread of long-term rates over short-term rates Answer: C is correct. Economic growth forecasts impact long-term rates. The view that economic growth will return to robust levels is consistent with a shift to a positively sloped term structure. 2. Salaz also expects the Federal Reserve to decrease asset purchases of long-term bonds as the economic recovery continues. Which of the following scenarios is consistent with this view? The reduced asset purchases will likely: A. amplify the effect of increased economic activity on the term spread. B. dampen the effect of increased economic activity on the term spread. C. have no effect on the term spread. Answer. A. Reduced asset purchases constitute a negative shift in demand for longer-term bonds, which raises their yields. The reduced asset purchases of long-maturity bonds would add to the effect of greater economic activity, both of which will increase the term spread.