“How many millionaires do you know who have become wealthy by investing in savings accounts? I rest my case.”
– Robert G. Allen
Charles Lamson
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Formula for Bond Yield
Because it is tedious to determine the bond yield to maturity through trial-and-error (see last post), an appropriate answer can also be found by using Formula 2.  The answer of 11.94 percent is a reasonably good approximation of the exact yield to maturity of 12 percent. We use the prime (') symbol after Y to indicate the answer based on Formula 2 is only an approximation. Note that the numerator of Formula 2 represents the average annual income over the life of the bond and the denominator represents the average investment. That is, in the numerator, we take the annual interest payment of $110 and add that to the average annual change in the bond value over 15 years, which is computed as $4.52. This percentage of the original price of $932.21 and the final value of $1,000 that we will receive at maturity to get the average investment over the 15-year holding period.  It should be pointed out that, in computing yield to maturity on a bond, financially oriented handheld calculators and software programs for personal computers can be extremely helpful. A future will present material related to calculators. As we have discussed throughout, the yield to maturity is the required rate of return bondholders demand. More importantly for our purposes here, it also indicates the current cost to the corporation to issue bonds. In the prior example, the corporation had issued bonds at 11 percent, but market conditions changed and the current price of the bond fell to $932.21. At this current price, the ongoing yield to maturity increase to 12 percent (11.94 percent, using the approximation method). If the corporate treasurer were to issue new bonds today, he or she would have to respond to the current market demanded rate of 12 percent rather than the initial yield of 11 percent. Only by understanding how investors value bonds in the marketplace can the corporate financial officer properly assess the cost of that source of financing to the corporation. Semiannual Interest and Bond Prices We have been assuming that interest was paid annually in our bond analysis. In actuality most bonds pay interest semi-annually. Thus a 10 percent interest rate bond may actually pay $50 twice a year instead of $100 annually. To make the conversion from an annual to semi-annual analysis, we follow these steps.
Assume a 10 percent, $1,000 par value bond has a maturity of 20 years. The annual yield to maturity is 12 percent. In following the three steps above we would show:
 In computing the price of the bond issued, on a semiannual analysis, we show: Present Value of Interest Payments---We take the present value of a $50 annuity for 40 periods. The semiannual discount rate is 6 percent. Using Table 1 from Part 31 of this analysis:  Present Value of Principal Payment at Maturity---We take the present value of $1,000 after 40 periods, using a 6 percent discount rate. Note that once we go to a semi-annual analysis with the interest payments, we consistently follow the same approach in discounting back the principal payment; otherwise we would be using semiannual and annual calculations on the same bond. Using Table 2 from Part 31 of this analysis:  We now and together the two values to get total present value. Total Present Value---  The answer of $849.30 is slightly below what we found previously for the same bond, assuming an annual interest rate $850.90. This value was previously shown in Part 32 of this analysis. In terms of accuracy, the semiannual analysis is a more acceptable method and is the method used in bond tables.  *MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 277-279* end |
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