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Calculating Yield to Maturity for the Treasury Bond

We found the value of the 6 percent coupon Treasury bond by discounting at a 5.6 percent interest rate. We could have phrased the question the other way around: If the price of the bond is $1,010.77, what return do investors expect? We need to find the yield to maturity, in other words, the discount rate r, that solves the following equation:

To find the yield to maturity, most people use a financial calculator. For our Treasury bond you would enter a PV of $1,010.77.4 The bond provides a regular payment of $60, entered as PMT = 60. The bond has a future value of $1,000, so FV = 1,000. The bond life is 3 years, so n = 3. Now compute the interest rate, and you will find that the yield to maturity is 5.6 percent. The nearby box reviews the use of the financial calculator in bond valuation problems.

Example 3 illustrates that the yield to maturity depends on the coupon payments that you receive each year ($60), the price of the bond ($1,010.77), and the final repayment of face value ($1,000). Thus it is a measure of the total return on this bond, accounting for both coupon income and price change, for someone who buys the bond today and holds it until maturity. Bond investors often refer loosely to a bond s yield. It s a safe bet that they are talking about its yield to maturity rather than its current yield. The only general procedure for calculating yield to maturity is trial and error. You guess at an interest rate and calculate the present value of the bond s payments. If the present value is greater than the actual price, your discount rate must have been too low, so you try a higher interest rate (since a higher rate results in a lower PV). Conversely, if PV is less than price, you must reduce the interest rate. In fact, when you use a financial calculator to compute yield to maturity, you will notice that it takes the calculator a few moments to compute the interest rate. This is because it must perform a series of trial-and-error calculations.

Figure 3.4 is a graphical view of yield to maturity. It shows the present value of the 6 percent Treasury bond for different interest rates. The actual bond price, $1,010.77, is marked on the vertical axis. A line is drawn from this price over to the present value curve and then down to the interest rate, 5.6 percent. If we picked a higher or lower figure for the interest rate, then we would not obtain a bond price of $1,010.77. Thus we know that the yield to maturity on the bond must be 5.6 percent. Figure 3.4 also illustrates a fundamental relationship between interest rates and bond prices:

When the interest rate rises, the present value of the payments to be received by the bondholder falls, and bond prices fall. Conversely, declines in the interest rate increase the present value of those payments and result in higher prices.

A gentle warning! People sometimes confuse the interest rate that is, the return that investors currently require with the interest, or coupon, payment on the bond. Although interest rates change from day to day, the $60 coupon payments on our Treasury bond are fixed when the bond is issued. Changes in interest rates affect the present value of the coupon payments but not the payments themselves.