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Chapter 6 Bonds, Bond Prices, and the Determination of Interest Rates Conceptual and Analytical Problems Consider a U.S. Treasury Bill with 270 days to maturity. If the annual yield is 3.8 percent, what is the price? Answer: You are an officer of a commercial bank and wish to sell one of the bank’s assets—a car loan—to another bank. Using equation A5 in the Appendix to Chapter 4, compute the price you expect to receive for the loan if the annual interest rate is 6 percent, the car payment is $430 per month, and the loan term is five years. Answer: The present value of the payments can be found by using equation A5 in the appendix to Chapter 4: The monthly payment, C, is given as $430 per month and there are 60 months in the five year horizon. The annual interest rate is 6 percent, so the monthly rate in decimal form is im = (1.06)1/12 – 1 = .00487 Thus, the value of the car loan is Your financial adviser recommends buying a 10-year bond with a face value of $1,000 and an annual coupon of $80. The current interest rate is 7 percent. What might you expect to pay for the bond (aside from brokerage fees)? Answer: The value of the bond has two components: the present value of the coupon payments and the present value of the return of principal at maturity. This is: In this expression, C is the coupon payment, i is the interest rate, n is the number of periods the coupon payments are made, and F is the face value. Using the information in the question, we have Consider a coupon bond with a $1,000 face value and a coupon payment equal to 5 percent of the face value per year. If there is one year to maturity, find the yield to maturity if the price of the bond is $990. Explain why finding the yield to maturity is difficult if there are two years to maturity and you do not have a financial calculator. Answer: a. The yield to maturity can be found by equating the current price of the bond to the present value of the coupon payment plus the present value of the face value when both payments are due in one year. Specifically, so that i = .061 or 6.1 percent. b. If there are two years to maturity, then we would need to solve The presence of the quadratic term makes this equation much more time- consuming to solve without a financial calculator. Which of these $100 face value one-year bonds will have the highest yield to maturity and why? A 6 percent coupon bond selling for $85. A 7 percent coupon bond selling for $100. An 8 percent coupon bond selling for $115. Answer: a. b. c. Option (a) has the highest yield to maturity. The yield to maturity depends both on the coupon payment and any capital gain or loss arising from the difference between the selling price and the face value of the bond. While (a) has the lowest coupon rate, it is selling below face value, and so there is a capital gain. Option (b) is selling at face value, so there is no capital gain and option (c) is selling above face value and so there is a capital loss. As the calculations above show, the combination of the coupon payment and the capital gain on option (a) produces the highest yield to maturity. You are considering purchasing a consol that promises annual payments of $4. If the current interest rate is 5 percent, what is the price of the consol? You are concerned that the interest rate may rise to 6 percent. Compute the percentage change in the price of the consol and the percentage change in the interest rate. Compare them. Your investment horizon is one year. You purchase the consol when the interest rate is 5 percent and sell it a year later, following a rise in the interest rate to 6 percent. What is your holding period return? Answer: a. P falls by 16.7%; i rises by 20% c. Suppose you purchase a 3-year, 5-percent coupon bond at par and hold it for two years. During that time, the interest rate falls to 4 percent. Calculate your annual holding period return. Answer: The total holding period return over the two years consists of two coupon payments of $5 each plus the capital gain from the rise in the price of the bond due to the interest rate fall. The price at which you sell the bond after two years will be 5/1.04 + 100/1.04 = $100.96. Holding period return over two years = 10/100 + (100.96 - 100)/100 = .1096 or 10.96%. The total payoff on the bond for which you paid $100 is $110.96. To calculate the annual rate of return, we refer to the footnote on p. 140. It is assumed, for simplicity, that the first-year coupon is not reinvested for the second year. The annual rate of return is [(110.96/100)1/2 – 1] = .0534 or 5.34 percent. Because the interest rate fell during the holding period and you made a capital gain, the annual holding period return is higher than the coupon rate. In a recent issue of the Wall Street Journal (or on www.wsj.com or an equivalent financial website), locate the prices and yields on U.S. Treasury issues. For one bond selling above par and one selling below par (assuming they both exist), compute the current yield and compare it to the coupon rate and the ask yield printed in the paper. (LO2) Answer: a. For the Treasury bond due August 15, 2025 with a coupon of 6.875%, on November 25, 2016, the bond price was 136.4141 and the asked yield 2.25%. That means the current yield was 6.875/136.4141 × 100 = 5.04%, so the coupon rate > current yield > asked yield, in line with Table 6.1 when the bond price is above the face value of 100. b. For the Treasury bond due August 15, 2025 with a coupon of 2.000%, on November 25, 2016, the bond price was 97.3516 and the asked yield 2.338%. That means the current yield was 2.000/97.3516 × 100 = 2.05%, so the coupon rate < current yield < asked yield, in line with Table 6.1 when the bond price is below the face value of 100. In a recent issue of the Wall Street Journal (or on www.wsj.com), locate the yields on government bonds for various countries. Find a country whose 10-year government bond yield was above that on the U.S. 10-year Treasury bond and one whose 10-year yield was below the Treasury yield. What might account for these differences in yields? Answer: As of Thursday, December 1, 2016, the 10-year U.S. Treasury yield was 2.441%, while the 10-year government bond yields in Germany and Australia were 0.373% and 2.790%, respectively. Because the default risk of all three governments is very low, the yield differentials most likely reflect differences in long-run inflation expectations, with inflation in Germany expected to be lower than in the United States, while inflation in Australia is expected to be a bit higher. A 10-year zero-coupon bond has a yield of 6 percent. Through a series of unfortunate circumstances, expected inflation rises from 2 percent to 3 percent. Assuming the nominal yield rises in an amount equal to the rise in expected inflation, compute the change in the price of the bond. Suppose that expected inflation is still 2 percent, but the probability that it will move to 3 percent has risen. Describe the consequences for the price of the bond. Answer: Price (with 2% expected inflation) = 100/(1.06)10 = $55.84 Price (with 3% expected inflation) = 100/(1.07)10 = $50.83 The price has fallen by $5.01 There is increased inflation risk. Investors will require compensation for taking on additional risk, so the price will fall and the yield will rise. As you read the business news, you come across an advertisement for a bond mutual fund – a fund that pools the investments from a large number of people and then purchases bonds, giving the individuals “shares” in the fund. The company claims their fund has had a return of 13½ percent over the last year. But you remember that interest rates have been pretty low – 5 percent at most. A quick check of the numbers in the business section you’re holding tells you that your recollection is correct. Explain the logic behind the mutual fund’s claim in the advertisement. Answer: There are two possible explanations for the high return. The first is that the mutual fund is investing in relatively risky bonds and is being compensated for taking on this risk with higher returns. The second is that the fund was holding bonds during a period when interest rates were falling, so the holding period return far exceeded the interest rate. Remember that when interest rates fall, the prices of bonds rise, giving the owner a capital gain. If interest rates are now low, then the likelihood is that they will rise, causing a capital loss to the owners. Chances are that if the high return is a consequence of the interest rate decline, not only will it not be repeated, it is likely to be followed by a low or even negative return when interest rates rise. You are sitting at the dinner table and your father is extolling the benefits of investing in bonds. He insists that as a conservative investor he will only make investments that are safe, and what could be safer than a bond, especially a U.S. Treasury bond? What accounts for his view of bonds? Explain why you think it is right or wrong. Answer: Like most people, your father believes that the government guarantee means that he will get his investment back. He’s right that the U.S. Treasury is extremely unlikely to default. But he’s wrong about interest-rate and inflation risk. The value of the bond will fluctuate when the interest rate changes (moving inversely) and the purchasing power of the coupon and principal repayment will fluctuate with inflation. So, the bond is not risk free. Consider a one-year, 10-percent coupon bond with a face value of $1,000 issued by a private corporation. The one-year risk-free rate is 10 percent. The corporation has hit on hard times, and the consensus is that there is a 20 percent probability that it will default on its bonds. If an investor were willing to pay at most $775 for the bond, is that investor risk-neutral or risk averse? Answer: If the bond were risk free, it would pay off $1,100 in one year’s time - $100 coupon payment and $1,000 face value of the bond. If there is a 20% risk of default, then the expected value of these payment flows associated with the bond are ($1,100 × 0.8) + ($0 × 0.2) = $880 The present value of $880 in one year’s time is $880/1.1 = $800. This would be the price a risk-neutral investor would be willing to pay. If the investor is willing to pay at most $775 for the bond, he or she requires compensation for bearing the risk associated with the bond and so is risk averse. If, after one year, the yield to maturity on a multi-year coupon bond that was issued at par is higher than the coupon rate, what happened to the price of the bond during that first year? Answer: The price of the bond fell below par. When a bond is at par, the yield to maturity equals the coupon rate. If the yield to maturity rises, the price of the bond falls. If you buy the bond below par, the capital gain you receive by holding it to maturity is included along with the coupon payments, so the yield to maturity is higher than the coupon rate alone. Use your knowledge of bond pricing to explain under what circumstances you would be willing to pay the same price for a consol that pays $5 a year forever and a 5-percent, 10-year coupon bond with a face value of $100 that only makes annual coupon payments for 10 years. Answer: The price you are willing to pay for a bond reflects the present value of the payment flows from the bond. In this case, if i = 5%, the present value of the payment flows for both these bonds would be $100. Intuitively, while the consol makes coupon payments forever, the 10-year coupon bond pays back the principal at maturity, which then can be reinvested. Assuming you have no reason to believe that rates will rise or fall over the 10-year period, you would be indifferent between these bonds. If you are certain that rates will be higher in ten years, you would prefer the 10-year coupon bond, whose proceeds can then be reinvested at the higher rate while the value of the consol would have fallen. Similarly, if you are certain that rates will be lower in ten years, you would prefer the consol. You are about to purchase your first home and receive an advertisement regarding adjustable-rate mortgages (ARMs). The interest rate on the ARM is lower than that on a fixed rate mortgage. The advertisement mentions that there would be a payment cap on your monthly payments and you would have the option to convert to a fixed-rate mortgage. You are tempted. Interest rates are currently low by historical standards and you are anxious to buy a house and stay in it for the long term. Why might an ARM not be the right mortgage for you? Answer: There are several factors to consider. First, with a fixed rate mortgage, your payments are fixed over the life of the loan. The interest rate on this mortgage is higher because the lender is assuming the interest rate risk. The ARM has a lower interest rate in part because you will assume risk associated with interest rate movements over the life of the loan (your payments will rise if rates rise). Given that interest rates are currently relatively low, it is more likely that they will rise, pushing up your payments. This problem is more likely to be an issue the longer you plan to stay in the house. Second, converting later to a fixed-rate mortgage from an adjustable rate loan often involves restrictions and fees. Third, payment caps may limit how much your monthly payments can rise, but may be associated with negative amortization if your payments don’t cover the interest costs of your loan. Shortages are added to the principal of your loan, pushing up your costs. Use the model of supply and demand for bonds to illustrate and explain the impact of each of the following on the equilibrium quantity of bonds outstanding and on equilibrium bond prices and yields: A new website is launched facilitating the trading of corporate bonds with much more ease than before. Inflationary expectations in the economy fall evoking a much stronger response from issuers of bonds than investors in bonds. The government removes tax incentives for investment and spends additional funds on a new education program. Overall, the changes have no effect on the government’s financing requirements. All leading indicators point to stronger economic growth in the near future. The response of bond issuers dominates that of bond purchasers. Answer: a. The new website would increase the relative liquidity of bonds, shifting the bond demand curve to the right, increasing the equilibrium price of bonds and reducing yields. The equilibrium quantity of bonds outstanding rises. b. For a given nominal interest rate, a fall in inflationary expectations increases the real interest rate, shifting the bond supply curve to the left and the bond demand curve to the right. If the response of the bond issuers is relatively stronger, the supply curve shift will dominate and the quantity of bonds outstanding will fall. Regardless of the relative size of the shifts, the equilibrium price of bonds will rise and yields will fall. c. The removal of tax incentives on investment would make investment more costly, reducing the supply of bonds by corporations, shifting the supply curve to the left. As there is no change in the financing requirements of the government, the supply of government bonds doesn’t change. Equilibrium quantity falls. Equilibrium bond prices rise and yields fall. d. A business cycle upturn increases business investment opportunities, shifting the bond supply curve to the right. Wealth also increases, shifting the bond demand curve to the right. If the supply shift dominates, equilibrium bond prices fall and yields rise. The equilibrium quantity of bonds outstanding increases. Suppose that a sustainable peace is reached around the world, reducing military spending by the U.S. Government. How would you expect this development to affect the U.S. bond market? Answer: As the government’s need to issue bonds to finance military spending is reduced, the supply of government bonds will fall, shifting the supply curve to the left. Bond prices will increase and yields will fall. Use the model of supply and demand for bonds to determine the impact on bond prices and yields of expectations that the real estate market is going to weaken. Answer: If we think of real estate as an alternative investment to bonds, expected weakness in the real estate market implies an increase in the relative return on bonds. Bond demand shifts to the right, increasing the equilibrium bond prices and lowering yields. Suppose there is an increase in investors’ willingness to hold bonds at a given price. Use the model of the demand for and supply of bonds to show that the impact on the equilibrium bond price depends on how sensitive the quantity supplied of bonds is to the bond price. Answer: The sensitivity of bond supply to changes in the price of bonds is reflected in the slope of the supply curve. The more sensitive quantity supplied is to a movement in the price, the flatter the supply curve and the smaller the impact on the equilibrium price for any given shift in the demand curve. Under what circumstances would purchase of a Treasury Inflation Protected Security (TIPS) from the U.S. government be virtually risk free? Answer: Purchasing a Treasury Inflation Protected Security (TIPS) would be virtually risk free if you purchased a bond whose maturity exactly matched your investment horizon. The default risk of holding a U.S. government-issued bond is very low while inflation risk is eliminated by the inflation-indexed nature of the TIPS. If you know your investment horizon with certainty and purchase a bond whose maturity matches that horizon, you eliminate interest rate risk, as you are confident that the bond will be redeemed at par when it matures. Interest rate movements that cause the price of the bond to change before it matures will not affect you. In the wake of the financial crisis of 2007-2009, negative connotations often surrounded the term mortgage-backed security. What arguments could you make to convince someone that they may have benefitted from the growth in securitization over the past 30 years? Answer: If the person you are trying to convince is a borrower, they may have received a lower mortgage interest rate due to the increased liquidity provided by securitization. If they are from a small town, they may have found it easier to get a mortgage as securitization broadened the potential sources of funds for their loan. If they are an investor, you might point to the opportunities for diversification provided by securitization. During the euro-area sovereign debt crisis, the spread between the yields on bonds issued by the governments of geographically peripheral European countries (such as Greece, Ireland, Italy, Portugal, and Spain) and those on bonds issued by Germany widened considerably. Use the model of supply and demand for bonds to illustrate how this could be explained by a change in investors’ perceptions of the relative riskiness of peripheral sovereign versus German bonds. Answer: Investor worries about the possibility of default on bonds issued by relatively indebted peripheral euro-area governments increased during the crisis. This can be illustrated with a leftward shift of the bond demand curve, lowering the price and raising the yield. For a given German bond yield, this would increase the spread of peripheral government bond yields above those on German bonds, reflecting the need for a larger risk premium to compensate investors. Not long after the United Kingdom’s vote to leave the European Union, the yields on some British Government bonds (called gilts) turned negative. Assuming that these bonds were issued with a positive coupon rate, would you expect their market prices to be above, below or equal to their face value? Explain your choice. Answer: If the yield on a bond with a positive coupon rate is negative, the price must be above the face value. Thinking about maturity, if the yield is negative, the investor must suffer a capital loss to offset the positive coupon payments. Therefore, the market price of the bond must be above the face value. Data Exploration Graph investors’ long-term expected inflation rate since 2003 by subtracting from the 10-year U.S. Treasury bond yield (FRED code: GS10) the yield on 10-year Treasury Inflation Protected Securities (FRED code: FII10). Do these market-based inflation expectations appear stable? Did the financial crisis of 2007-2009 affect these expectations? Answer: The indicated data plot is: With the exception of the downward spike in late 2008, inflation expectations by this measure appeared stable through 2014, fluctuating mostly in the range of 2.0 percent to 2.5 percent. The temporary plunge in expectations at the end of 2008 followed the Lehman bankruptcy, which ushered in the most intense episode of the 2007-2009 financial crisis. More recently, this measure of expected inflation has drifted lower, perhaps due to the persistent undershoot of the Fed’s 2 percent inflation target and to the modest growth of real GDP. Compare long-run market expectations of inflation with a consumer survey measure of one-year-ahead inflation expectations. Starting with the graph from Data Exploration Problem 1, add as a second line the University of Michigan survey measure of inflation expectations (FRED code: MICH) Why might these measures differ systematically? Answer: The indicated data plot is: We should not expect that one-year ahead consumer inflation expectations match 10-year-ahead investor inflation expectations, but they are broadly correlated. Interestingly, consumer short-term inflation expectations exceed investor long-term expectations virtually throughout this period. One reason may be that consumers focus more on the prices of goods that they buy frequently, such as food and gasoline where price changes are visible and frequent, and less on infrequent purchases (like electronics) for which inflation may be lower. In contrast, TIPS compensate investors for changes in the CPI that includes all these prices. How does the variability of annual inflation – an indicator of inflation risk – change over time? Graph the percent change from a year ago of the consumer price index (FRED code: CPIAUCSL) since1990 and visually compare the decades of the 1990s, the 2000s, and the period that began in 2010. Answer: The data plot is: After inflation declines to around 3 percent in the early 1990s, it appears less variable than in the 2000s, which includes the relatively large downward spike to negative inflation occurs during 2009. By visual inspection, if variability of inflation represents inflation risk, then this risk appears to change through time, and picked up temporarily during the financial crisis of 2007-2009. Even ignoring the Great Recession, inflation variability since 2000 appears higher than in the 1990s. Download the data from the graph you produced in Data Exploration Problem 3. Calculate the standard deviation of the annual inflation rate for the three time periods and compare these results against your visual assessment from Data Exploration Problem 3. Answer: The standard deviation from 1990 through 1999 is 1.13. From 2000 through 2009, it is 1.42, but this masks a lower standard deviation of 0.84 from 2000 through 2007:12 (the onset of the Great Recession). From 2010 through mid-2016, the standard deviation is 0.98. (Computations after this date will include additional observations and incorporate any data revisions.) These results appear consistent with the visual assessment in Data Exploration Problem 3. Economists sometimes exclude food and energy prices from the “headline” consumer price index and use the resulting “core” price measure to assess inflation prospects. For the period since 1990, plot on one graph the percent change from a year ago of the consumer price index (FRED code: CPIAUCSL) and the percent change from a year ago of the consumer price index excluding food and energy (FRED code: CPILFESL). Visually compare the variability of these two measures of inflation. Why might inflation, excluding food and energy, be a better predictor of future inflation than headline inflation? Answer: The data plot is: Because food and energy prices are relatively volatile, the “core” measure of inflation is less variable than the “headline” measure. Volatility tends to mask the trend of an economic indicator by adding statistical “noise” to the trend. Consequently, examining core inflation can reveal to economists the underlying trend of inflation. If that trend is stable, it can be useful in anticipating inflation prospects indicates more difficult problems. Solution Manual for Money, Banking and Financial Markets Stephen G. Cecchetti, Kermit L. Schoenholtz 9781259746741, 9780078021749, 9780077473075