Chapter 6 APPENDIX B. The Yield Curve and the Law of One Price. Valuing a Coupon Bond with Zero-Coupon Prices

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1 196 Part Interest Rates and Valuing Cash Flows Chapter 6 APPENDIX B The Yield Curve and the Law of One Price Thus far, we have focused on the relationship between the price of an individual bond and its yield to maturity. In this section, we explore the relationship between the prices and yields of different bonds. Using the Law of One Price, we show that given the spot interest rates, which are the yields of default-free zero-coupon bonds, we can determine the price and yield of any other default-free bond. As a result, the yield curve provides sufficient information to evaluate all such bonds. Valuing a Coupon Bond with Zero-Coupon Prices We begin with the observation that it is possible to replicate the cash flows of a coupon bond using zero-coupon bonds. Therefore, we can use the Valuation Principle s Law of One Price to compute the price of a coupon bond from the prices of zero-coupon bonds. For example, we can replicate a three-year, 0 bond that pays 10% annual coupons using three zero-coupon bonds as follows: Coupon bond: $ year zero: -year zero: 3-year zero: $1100 Zero-coupon Bond portfolio: $1100 We match each coupon payment to a zero-coupon bond with a face value equal to the coupon payment and a term equal to the time remaining to the coupon date. Similarly, we match the final bond payment (final coupon plus return of face value) in three years to a three-year, zero-coupon bond with a corresponding face value of $1100. Because the coupon bond cash flows are identical to the cash flows of the portfolio of zero-coupon bonds, the Law of One Price states that the price of the portfolio of zerocoupon bonds must be the same as the price of the coupon bond. To illustrate, assume that current zero-coupon bond yields and prices are as shown in Table 6.7 (they are the same as in Example 6.1). We can calculate the cost of the zero-coupon bond portfolio that replicates the three-year coupon bond as follows: Zero-Coupon Bond Face Value Required Cost 1 Year Years Years * = Total Cost: $ By the Law of One Price, the three-year coupon bond must trade for a price of $1153. If the price of the coupon bond were higher, you could earn an arbitrage profit by selling

2 Chapter 6 Bonds 197 TABLE 6.7 Maturity 1 Year Years 3 Years 4 Years Yields and Prices (per Face Value) for Zero-Coupon Bonds YTM 3.50% 4.00% 4.50% 4.75% Price $96.6 $9.46 $87.63 $83.06 the coupon bond and buying the zero-coupon bond portfolio. If the price of the coupon bond were lower, you could earn an arbitrage profit by buying the coupon bond and selling the zero-coupon bonds. Valuing a Coupon Bond Using Zero-Coupon Yields To this point, we have used the zero-coupon bond prices to derive the price of the coupon bond. Alternatively, we can use the zero-coupon bond yields. Recall that the yield to maturity of a zero-coupon bond is the competitive market interest rate for a risk-free investment with a term equal to the term of the zero-coupon bond. Since the cash flows of the bond are its coupon payments and face value repayment, the price of a coupon bond must equal the present value of its coupon payments and face value discounted at the competitive market interest rates (see Eq. 5.7 in Chapter 5): Price of a Coupon Bond = (6.4) where CPN is the bond coupon payment, YTM n is the yield to maturity of a zero-coupon bond that matures at the same time as the nth coupon payment, and FV is the face value of the bond. For the three-year, 0 bond with 10% annual coupons considered earlier, we can use Eq. 6.4 to calculate its price using the zero-coupon yields in Table 6.7: This price is identical to the price we computed earlier by replicating the bond. Thus, we can determine the no-arbitrage price of a coupon bond by discounting its cash flows using the zero-coupon yields. In other words, the information in the zero-coupon yield curve is sufficient to price all other risk-free bonds. Coupon Bond Yields P = PV(Bond Cash Flows) CPN 1 YTM 1 P = = $1153 Given the yields for zero-coupon bonds, we can use Eq. 6.4 to price a coupon bond. In Section 6.1, we saw how to compute the yield to maturity of a coupon bond from its price. Combining these results, we can determine the relationship between the yields of zero-coupon bonds and coupon-paying bonds. Consider again the three-year, 0 bond with 10% annual coupons. Given the zerocoupon yields in Table 6.7, we calculate a price for this bond of $1153. From Eq. 6.3, the yield to maturity of this bond is the rate y that satisfies: P = 1153 = CPN (1 YTM ) Á CPN FV (1 YTM n ) n 100 (1 y) (1 y) (1 y) 3

3 198 Part Interest Rates and Valuing Cash Flows We can solve for the yield by using a financial calculator: N I/Y PV PMT FV Given: Solve for: 4.44 Excel Formula: RATE(NPER,PMT,PV,FV) RATE(3,100, 1153,1000) Therefore, the yield to maturity of the bond is 4.44%. We can check this result directly as follows: P = = $1153 Because the coupon bond provides cash flows at different points in time, the yield to maturity of a coupon bond is a weighted average of the yields of the zero-coupon bonds of equal and shorter maturities. The weights depend (in a complex way) on the magnitude of the cash flows each period. In this example, the zero-coupon bonds yields were 3.5%, 4.0%, and 4.5%. For this coupon bond, most of the value in the present value calculation comes from the present value of the third cash flow because it includes the principal, so the yield is closest to the three-year, zero-coupon yield of 4.5%. EXAMPLE 6.11 Yields on Bonds with the Same Maturity Problem Given the following zero-coupon yields, compare the yield to maturity for a three-year, zerocoupon bond; a three-year, coupon bond with 4% annual coupons; and a three-year coupon bond with 10% annual coupons. All of these bonds are default free. Solution Plan Maturity 1 Year Years 3 Years 4 Years Zero-coupon YTM 3.50% 4.00% 4.50% 4.75% From the information provided, the yield to maturity of the three-year, zero-coupon bond is 4.50%. Also, because the yields match those in Table 6.7, we already calculated the yield to maturity for the 10% coupon bond as 4.44%. To compute the yield for the 4% coupon bond, we first need to calculate its price, which we can do using Eq Since the coupons are 4%, paid annually, they are $40 per year for 3 years. The 0 face value will be repaid at that time. Once we have the price, we can use Eq. 6.3 to compute the yield to maturity. Execute Using Eq. 6.4, we have: P = = $ The price of the bond with a 4% coupon is $ From Eq. 6.4: $ = 40 (1 y) (1 y) (1 y) 3

4 Chapter 6 Bonds 199 We can calculate the yield to maturity using a financial calculator or spreadsheet: N I/Y PV PMT FV Given: Solve for: 4.47 Excel Formula: RATE(NPER,PMT,PV,FV) RATE(3,40, ,1000) To summarize, for the three-year bonds considered: Coupon Rate 0% 4% 10% YTM 4.50% 4.47% 4.44% Evaluate Note that even though the bonds all have the same maturity, they have different yields. In fact, holding constant the maturity, the yield decreases as the coupon rate increases. We discuss why below. Example 6.11 shows that coupon bonds with the same maturity can have different yields depending on their coupon rates. The yield to maturity of a coupon bond is a weighted average of the yields on the zero-coupon bonds. As the coupon increases, earlier cash flows become relatively more important than later cash flows in the calculation of the present value. The shape of the yield curve keys us in on trends with the yield to maturity: 1. If the yield curve is upward sloping (as it is for the yields in Example 6.11), the resulting yield to maturity decreases with the coupon rate of the bond.. When the zero-coupon yield curve is downward sloping, the yield to maturity will increase with the coupon rate. 3. With a flat yield curve, all zero-coupon and coupon-paying bonds will have the same yield, independent of their maturities and coupon rates. Treasury Yield Curves As we have shown in this section, we can use the zero-coupon yield curve to determine the price and yield to maturity of other risk-free bonds. The plot of the yields of coupon bonds of different maturities is called the coupon-paying yield curve. When U.S. bond traders refer to the yield curve, they are often referring to the coupon-paying Treasury yield curve. As we showed in Example 6.11, two coupon-paying bonds with the same maturity may have different yields. By convention, practitioners always plot the yield of the most recently issued bonds, termed the on-the-run bonds. Using similar methods to those employed in this section, we can apply the Law of One Price to determine the zerocoupon bond yields using the coupon-paying yield curve. Thus, either type of yield curve provides enough information to value all other risk-free bonds.

5 PART Integrative Case This case draws on material from Chapters 3 6. Adam Rust looked at his mechanic and sighed. The mechanic had just pronounced a death sentence on his road-weary car. The car had served him well at a cost of $500 it had lasted through four years of college with minimal repairs. Now, he desperately needs wheels. He has just graduated, and has a good job at a decent starting salary. He hopes to purchase his first new car. The car dealer seems very optimistic about his ability to afford the car payments, another first for him. The car Adam is considering is $35,000. The dealer has given him three payment options: 1. Zero percent financing. Make a $4000 down payment from his savings and finance the remainder with a 0% APR loan for 48 months. Adam has more than enough cash for the down payment, thanks to generous graduation gifts.. Rebate with no money down. Receive a $4000 rebate, which he would use for the down payment (and leave his savings intact), and finance the rest with a standard 48-month loan, with an 8% APR. He likes this option, as he could think of many other uses for the $ Pay cash. Get the $4000 rebate and pay the rest with cash. While Adam doesn t have $35,000, he wants to evaluate this option. His parents always paid cash when they bought a family car; Adam wonders if this really was a good idea. Adam s fellow graduate, Jenna Hawthorne, was lucky. Her parents gave her a car for graduation. Okay, it was a little Hyundai, and definitely not her dream car, but it was serviceable, and Jenna didn t have to worry about buying a new car. In fact, Jenna has been trying to decide how much of her new salary she could save. Adam knows that with a hefty car payment, saving for retirement would be very low on his priority list. Jenna believes she could easily set aside $3000 of her $45,000 salary. She is considering putting her savings in a stock fund. She just turned and has a long way to go until retirement at age 65, and she considers this risk level reasonable. The fund she is looking at has earned an average of 9% over the past 15 years and could be expected to continue earnings this amount, on average. While she has no current retirement savings, five years ago Jenna s grandparents gave her a new 30-year U.S. Treasury bond with a $10,000 face value. Jenna wants to know her retirement income if she both (1) sells her Treasury bond at its current market value and invests the proceeds in the stock fund, and () saves an additional $3000 at the end of each year in the stock fund from now until she turns 65. Once she retires, Jenna wants those savings to last for 5 years until she is 90. Both Adam and Jenna need to determine their best options. 00

6 Case Questions PART Integrative Case What are the cash flows associated with each of Adam s three car financing options?. Suppose that, similar to his parents, Adam had plenty of cash in the bank so that he could easily afford to pay cash for the car without running into debt now or in the foreseeable future. If his cash earns interest at a 5.4% APR (based on monthly compounding) at the bank, what would be his best purchase option for the car? 3. In fact, Adam doesn t have sufficient cash to cover all his debts including his (substantial) student loans. The loans have a 10% APR, and any money spent on the car could not be used to pay down the loans. What is the best option for Adam now? (Hint: Note that having an extra $1 today saves Adam roughly $1.10 next year because he can pay down the student loans. So, 10% is Adam s time value of money in this case.) 4. Suppose instead Adam has a lot of credit card debt, with an 18% APR, and he doubts he will pay off this debt completely before he pays off the car. What is Adam s best option now? 5. Suppose Jenna s Treasury bond has a coupon interest rate of 6.5%, paid semiannually, while current Treasury bonds with the same maturity date have a yield to maturity of % (expressed as an APR with semiannual compounding). If she has just received the bond s tenth coupon, for how much can Jenna sell her treasury bond? 6. Suppose Jenna sells the bond, reinvests the proceeds, and then saves as she planned. If, indeed, Jenna earns a 9% annual return on her savings, how much could she withdraw each year in retirement? (Assume she begins withdrawing the money from the account in equal amounts at the end of each year once her retirement begins.) 7. Jenna expects her salary to grow regularly. While there are no guarantees, she believes an increase of 4% a year is reasonable. She plans to save $3000 the first year, and then increase the amount she saves by 4% each year as her salary grows. Unfortunately, prices will also grow due to inflation. Suppose Jenna assumes there will be 3% inflation every year. In retirement, she will need to increase her withdrawals each year to keep up with inflation. In this case, how much can she withdraw at the end of the first month of her retirement? What amount does this correspond to in today s dollars? (Hint: Build a spreadsheet in which you track the amount in her retirement account each month.) 8. Should Jenna sell her Treasury bond and invest the proceeds in the stock fund? Give at least one reason for and against this plan.

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