## Answers to Chapter Review and Self Test Problems

7.1 Because the bond has a 10 percent coupon yield and investors require a 12 percent return, we know that the bond must sell at a discount. Notice that, because the bond pays interest semiannually, the coupons amount to \$100/2 = \$50 every six months. The required yield is 12%/2 = 6% every six months. Finally, the bond matures in 20 years, so there are a total of 40 six-month periods.

The bond's value is thus equal to the present value of \$50 every six months for the next 40 six-month periods plus the present value of the \$1,000 face amount:

Bond value = \$50 X [(1 - 1/1.0640)/.06] + 1,000/1.06"° = \$50 X 15.04630 + 1,000/10.2857 = \$849.54

Notice that we discounted the \$1,000 back 40 periods at 6 percent per period, rather than 20 years at 12 percent. The reason is that the effective annual yield on the bond is 1.062 - 1 = 12.36%, not 12 percent. We thus could have used 12.36 percent per year for 20 years when we calculated the present value of the \$1,000 face amount, and the answer would have been the same.

7.2 The present value of the bond's cash flows is its current price, \$911.37. The coupon is \$40 every six months for 12 periods. The face value is \$1,000. So the bond's yield is the unknown discount rate in the following:

\$911.37 = \$40 X [1 - 1/(1 + r)12]/r + 1,000/(1 + r)12

The bond sells at a discount. Because the coupon rate is 8 percent, the yield must be something in excess of that.

If we were to solve this by trial and error, we might try 12 percent (or 6 percent per six months):

Bond value

This is less than the actual value, so our discount rate is too high. We now know that the yield is somewhere between 8 and 12 percent. With further trial and error (or a little machine assistance), the yield works out to be 10 percent, or 5 percent every six months.

By convention, the bond's yield to maturity would be quoted as 2 X 5% = 10%. The effective yield is thus 1.052 - 1 = 10.25%.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

III. Valuation of Future Cash Flows

7. Interest Rates and Bond Valuation