Chapter 05 App

# Chapter 05 App - Chapter 5 Appendix 5A How to Value Bonds...

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Chapter 5 How to Value Bonds and Stocks 5A-1 The Term Structure of Interest Rates, Spot Rates, and Yield to Maturity In the main body of this chapter, we have assumed that the interest rate is constant over all future periods. In reality, interest rates vary through time. This occurs primarily because inF ation rates are expected to differ through time. To illustrate, we consider two zero coupon bonds. Bond A is a one-year bond and bond B is a two-year bond. Both have face values of \$1,000. The one-year interest rate, r 1 , is 8 per- cent. The two-year interest rate, r 2 , is 10 percent. These two rates of interest are examples of spot rates. Perhaps this inequality in interest rates occurs because inF ation is expected to be higher over the second year than over the ± rst year. The two bonds are depicted in the following time chart: 2 8% Bond A \$1,000 10% Bond B \$1,000 1 0 Year 1 Year 2 We can easily calculate the present value for bond A and bond B as follows: P V A 5 \$925.93 5 \$1,000 ______ 1.08 PV B 5 \$826.45 5 \$1,000 ______ (1.10) 2 Of course, if PV A and PV B were observable and the spot rates were not, we could determine the spot rates using the PV formula, because: PV A 5 \$925.93 5 \$1,000 _______ (1 1 r 1 ) r 1 5 8% and: PV B 5 \$826.45 5 \$1,000 ________ (1 1 r 2 ) 2 r 2 5 10% Now we can see how the prices of more complicated bonds are determined. Try to do the next example. It illustrates the difference between spot rates and yields to maturity. Appendix 5A www.mhhe.com/rwj EXAMPLE 5A.1 On the Spot Given the spot rates r 1 equals 8 percent and r 2 equals 10 percent, what should a 5 percent coupon, two-year bond cost? The cash f ows C 1 and C 2 are illustrated in the Following time chart: 2 8% \$50 10% \$1,050 1 0 Year 1 Year 2 The bond can be viewed as a portFolio oF zero coupon bonds with one- and two-year maturities. ThereFore: PV 5 \$50 ________ 1 1 0.08 1 \$1,050 __________ (1 1 0.10) 2 5 \$914.06 (A.1) ( continued )

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5A-2 Part II Valuation and Capital Budgeting Graphing the Term Structure The term structure describes the relationship of spot rates with different maturities. Figure 5A.1 graphs a particular term structure. In Figure 5A.1 the spot rates are increasing with longer maturities—that is, r 3 . r 2 . r 1 . Graphing the term structure is easy if we can observe spot rates. Unfortunately this can be done only if there are enough zero coupon government bonds. A given term structure, such as that in Figure 5A.1, exists for only a moment in time— say 10:00 a.m., July 30, 2006. Interest rates are likely to change in the next minute, so that a different (though quite similar) term structure would exist at 10:01 a.m. We now want to calculate a single rate for the bond. We do this by solving for y in the following equation: \$914.06 5 \$50 _____ 1 1 y 1 \$1,050 _______ (1 1 y) 2 (A.2) In Equation A.2, y equals 9.95 percent. As mentioned in the chapter, we call y the yield to maturity on the bond. Solving for y for a multiyear bond is generally done by means of trial and error.
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Chapter 05 App - Chapter 5 Appendix 5A How to Value Bonds...

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