Bond and Note Valuation and Related Interest Rate Formulas

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Bond Valuation - Page 68% per year. This means, of course, that the bond pays $8 interest per year (per 100). Supposeadditionally, that 10 years have elapsed since the original issue of the bond, leaving only 20years to maturity (exactly). Finally, suppose that market interest yields on equivalent 20-yearbonds are now at 10%. What will be the market value of this bond?Q: Before you answer, let me make sure I have this straight. The bond is being resold with20 years remaining to maturity, so it must compete with other financial assets that alsohave ten years of life remaining, including new 20-year notes. Is that correct?A: Yes. Therefore the effective yield must be competitive with these other financial assets.Since the coupon rate, 8%, is fixed for the life of the bond, this bond must sell at adiscount (a price below par) to effectively raise its yield to 10% for the purchaser. Andagain, what is being valued is a cashflow stream consisting of 20 remaining interestpayments and the final redemption value of the bond (100).Here is the elementary bond formula that will allows us to calculate the value of this bond:9.MVCC1rCC 21r 1rC1rPar1r3 n 11r n nMV = present market value (asked price).C = the coupon payment (amount) equal to the coupon rate times par.r = prevailing interest rate on equivalent securities.n = number of years to maturity.Expressing the same formula in summation notation:10.MVnCPar ii1 1r 1r nFinally, to solve for our hypothetical example:11.MV208 ii11.10 1.10 100 82. 9720Because interest rates are higher than at the time this bond was first issues, the bond is trading ata discount for the price of 82.97. This means that if this bond was originally issued in thedenomination of $10,000, it is now worth $8,297.Readers with a good math background will recognize that formulas (9) and (10) are a geometric
Bond Valuation - Page 7series. Therefore the elementary bond formula can be reduced to a simpler formula that is easierto calculate on a calculator or in a computer program:12.MV 1 C 11rrn Par 1 n1r The hypothetical problem used in this section would be solved as 1 1201.10 1 13. MV 8 100 82. 9720 0.10 1.10 The derivation of this formula is shown in the Appendix A.Note that the values for equations (11) and (13) agree, as they should.VII. Elementary Bond Valuation - Periodic Interest PaymentsThe formula above is just a beginning approximation. It does not take into account the facts thatbonds usually pay interest either twice per year, quarterly or monthly. For example U.S. TreasuryBonds pay interest twice per year. The formula for periodic payments is a slight modification offormula (9) above:MV Cm1mC m1rmC m1rmC m1rmC m1rmPar1r14. 3 n 1r n n m2where variables have the same values as above exceptm = the number of times per year that interest is paid.n = number of remaining coupon payments rather than number of remaining years.Note that C still equals the annual value of the coupon payment (the coupon rate times par) andthat the final term is unchanged from formula (9), given that n is equal to m times the number ofyears.The summation notation version of this equation needs only a small modification of equation(10):
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