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Chapter 6 APPENDIX B - Pearsoncmg.com

Chapter 6 APPENDIX B - Pearsoncmg.com

Chapter

Chapter 6 Bonds 179 TABLE 6.7 Yields and Prices (per $100 Face Value) for Zero-Coupon Bonds Maturity 1 Year 2 Years 3 Years 4 Years YTM 3.50% 4.00% 4.50% 4.75% Price $96.62 $92.45 $87.63 $83.06 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 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 P = PV1Bond Cash Flows2 = CPN 1 + YTM 1 + (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, $1000 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: P = 100 1.035 + 100 100 + 1000 + 2 1.04 1.045 3 = +1153 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 Given the yields for zero-coupon bonds, we can use Eq. 6.4 to price a coupon bond. In Section 6.3, 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 zerocoupon bonds and coupon-paying bonds. Consider again the three-year, $1000 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 11 + YTM 2 2 2 + g + CPN + FV 11 + YTM n 2 n 100 11 + y2 + 100 100 + 1000 + 2 11 + y2 11 + y2 3

180 Part 2 Interest Rates and Valuing Cash Flows We can solve for the yield by using a financial calculator or spreadsheet: N I/Y PV PMT FV Given: 3 1153 100 1000 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 = 100 1.0444 + 100 100 + 1000 + 2 1.0444 1.0444 3 = +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 zero-coupon 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. Maturity 1 Year 2 Years 3 Years 4 Years Zero-Coupon YTM 3.50% 4.00% 4.50% 4.75% Solution ◗ Plan 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. 6.4. Since the coupons are 4%, paid annually, they are $40 per year for three years. The $1000 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 = 40 1.035 + 40 40 + 1000 + 2 1.04 1.045 3 = +986.98 The price of the bond with a 4% coupon is $986.98. From Eq. 6.4: +986.98 = 40 11 + y2 + 40 40 + 1000 + 2 11 + y2 11 + y2 3 We can calculate the yield to maturity using a financial calculator or spreadsheet: N I/Y PV PMT FV Given: 3 986.98 40 1000 Solve for: 4.47 Excel Formula: RATE(NPER,PMT,PV,FV)RATE(3,40,986.98,1000)

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