## Finding the Yield to Maturity More Trial and Error

Frequently, we will know a bond's price, coupon rate, and maturity date, but not its yield to maturity. For example, suppose we are interested in a six-year, 8 percent coupon bond. A broker quotes a price of \$955.14. What is the yield on this bond?

We've seen that the price of a bond can be written as the sum of its annuity and lump-sum components. Knowing that there is an \$80 coupon for six years and a \$1,000 face value, we can say that the price is:

\$955.14 = \$80 X [1 - 1/(1 + r)6]/r + 1,000/(1 + r)6

where r is the unknown discount rate, or yield to maturity. We have one equation here and one unknown, but we cannot solve it for r explicitly. The only way to find the answer is to use trial and error.

This problem is essentially identical to the one we examined in the last chapter when we tried to find the unknown interest rate on an annuity. However, finding the rate (or yield) on a bond is even more complicated because of the \$1,000 face amount.

We can speed up the trial-and-error process by using what we know about bond prices and yields. In this case, the bond has an \$80 coupon and is selling at a discount. We thus know that the yield is greater than 8 percent. If we compute the price at 10 percent:

Bond value = \$80 X (1 - 1/1.106)/.10 + 1,000/1.106 = \$80 X 4.3553 + 1,000/1.7716 = \$912.89

At 10 percent, the value we calculate is lower than the actual price, so 10 percent is too high. The true yield must be somewhere between 8 and 10 percent. At this point, it's "plug and chug" to find the answer. You would probably want to try 9 percent next. If you did, you would see that this is in fact the bond's yield to maturity. Our discussion of bond valuation is summarized in Table 7.1.

I. Finding the value of a bond

Bond value = C x [1 - 1/(1 + rf ]/r + F/(1 + rf where

C = Coupon paid each period r = Rate per period t = Number of periods F = Bond's face value

### II. Finding the yield on a bond

Given a bond value, coupon, time to maturity, and face value, it is possible to find the implicit discount rate, or yield to maturity, by trial and error only. To do this, try different discount rates until the calculated bond value equals the given value (or let a financial calculator do it for you). Remember that increasing the rate decreases the bond value.

Ross et al.: Fundamentals III. Valuation of Future 7. Interest Rates and Bond of Corporate Finance, Sixth Cash Flows Valuation

Edition, Alternate Edition

CHAPTER 7 Interest Rates and Bond Valuation 209 ## Sell Your Annuity

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