Equation (1) defines the value of a bond that pays coupons on an annual basis and a principal at maturity.
The value of a bond paying a fixed coupon interest each year (annual coupon payment) and the principal at maturity, in turn, would be:
Where M = Number of years to maturity
With the coupon payment fixed each period, the C term in Equation 1 can be factored out and the bond value can be expressed as:
Bonds, of course, differ in the frequency in which they pay coupons each year, and many bonds have maturities less than one year. Also, when investors buy bonds they often do so at non-coupon dates. Equation (1), therefore, needs to be adjusted to take these practical factors into account.
Semiannual Coupon Payments
Many bonds pay coupon interest semiannually. When bonds make semiannual payments, three adjustments to Equation (1) are necessary: (1) the number of periods is doubled; (2) the annual coupon rate is halved; (3) the annual discount rate is halved. Thus, if our illustrative 10-year, 9% coupon bond trading at a quoted annual rate of 10% paid interest semiannually instead of annually, it would be worth $937.69:
Note that the rule for valuing semiannual bonds is easily extended to valuing bonds paying interest even more frequently. For example, to determine the value of a bond paying interest four times a year, we would quadruple the periods and quarter the annual coupon payment and discount rate. In general, if we let n be equal to the number of payments per year (i.e., the compoundings per year), M be equal to the maturity in years, RA be the discount rate quoted on an annual basis (simple annual rate), and R be equal to the periodic rate, then we can express the general formula for valuing a bond as follows:
CA = Annual coupon = (CR)(F)
The 10% annual rate in the first example and the 8% rate in the second is a simple annual rate: It is the rate with one annualized compounding. With one annualized compounding and a 10% annual rate, we earn 10% every year and a $100 investment would grow to equal $110 after one year: $100(1.10) $110. If the simple annual rate were expressed with semiannual compounding, then we would earn 5% every six months with the interest being reinvested; in this case, $100 would grow to equal $110.25 after one year: $100(1.05)2 $110.25. If the rate were expressed with monthly compounding, then we would earn 0.8333% (10%/12) every month with the interest being reinvested; in this case, $100 would grow to equal $110.47 after one year: $100[1 + (.10/12)]12 $110.47. If we extend the compounding frequency to daily, then we would earn 0.0274% (10%/365) daily, and with the reinvestment of interest, a $100 investment would grow to equal $110.52 after one year: $100[1 + (.10/365)]365 $110.52. Note that the rate of 10% is the simple annual rate, whereas the actual rate earned for the year is [1+(RA/n)]n-1. This rate that includes the reinvestment of interest (or compounding) is known as the effective rate.
When the compounding becomes large, such as daily compounding, then we are approaching continuous compounding with the n term in Equation (2.3) becoming very large. For cases in which there is continuous compounding, the future value (FV) for an investment of A dollars M years from now is equal to
where e is the natural exponent (equal to the irrational number 2.71828). Thus, if the 10% simple rate were expressed with continuous compounding, then $100 (A) would grow to equal $110.52 after one year: $100e(.10)(1) $110.52. (After allowing for some slight rounding differences, this is the value obtained with daily compoundings.) After two years, the $100 investment would be worth $122.14: $100e(.10)(2) $122.14.
Note that from the FV expression, the present value (A) of a future receipt (FV)is
If R .10, a security paying $100 two years from now would be worth $81.87, given continuous compounding: PV $100e-(.10)(2) $81.87. Similarly, a security paying $100 each year for two years would be currently worth $172.36:
Thus, if we assume continuous compounding and a discount rate of 10%, then the value of our 10-year, 9% bond would be $908.82:
It should be noted that most practitioners use interest rates with annual or semiannual compounding. Most of our examples in this book, in turn, will follow that convention. However, continuous compounding is often used in mathematical derivations, and we will make some use of it when it is helpful.
In general, the bond market is volatile, and fixed income securities carry interest rate risk. (As interest rates rise, bond prices usually fall, and vice versa. This effect is usually more pronounced for longer-term securities). Fixed income securities also carry inflation risk, liquidity risk, call risk and credit and default risks for both issuers and counterparties. Lower-quality fixed income securities involve greater risk of default or price changes due to potential changes in the credit quality of the issuer. Foreign investments involve greater risks than U.S. investments, and can decline significantly in response to adverse issuer, political, regulatory, market, and economic risks. Any fixed-income security sold or redeemed prior to maturity may be subject to loss.