YIELDS TO MATURITY ON ZERO-COUPON RONDS
After dealing with money market interest rate calculations in Chapter 1, zero- coupon bond yields are a welcome relief and a return to classic time-value-of- money theory. A pricing formula for zeros is shown in equation 2.1,
where PV = present value, or price, of the bond, FV = future value, which usually is 100 (percent of par value) at maturity, Years = number of years to maturity, PER = periodicity – the number of evenly-spaced periods in the year; and APRPER = yield to maturity, stated as an annual percentage rate corresponding to PER.
We can now use equation 2.1 to illustrate the yield calculations for the two TIGRS. Thirty-year TIGRS priced at $50 per $1,000 entail solving for APR2 = 10.239%, the annual yield on a semiannual bond basis for PV = 50, PV = 1,000, Years = 30, PER = 2.
Fourteen-year TIGRS are priced at $250 to yield 10.151% (s.a.).
Such problems are easily solved using the time-value-of-money keys on a financial calculator or a spreadsheet program.
In this chapter, I work with a 10-year zero-coupon corporate bond that is priced at 60 (percent of par value). Its yield to maturity is 5.174% (s.a.).
The assumption of two periods in the year, while totally arbitrary, is common in financial markets because the yield on the zero then can be compared directly to yields to maturity on traditional semiannual payment fixed- income bonds. However, there is no inherent reason why the annual yield on a zero-coupon bond cannot be calculated for quarterly, monthly, daily, or even hourly compounding. Those yields turn out to be 5.141%, 5.119%, 5.109%, and 5.108% using PER = 4,12, 365, and 365 * 24, respectively.
Alternatively, you could convert from any one periodicity to any other using equation 1.9 from Chapter 1.
There are times in bond math when it is convenient to assume continuous compounding. That is, there are assumed to be an infinite number of compounding periods in the year. Continuous-time finance is particularly useful in interest rate term structure and option valuation models. The formula for the APR given PER = 00 and the two cash flows PV and FV involves the natural logarithm (LN):
The 10-year zero-coupon bond priced at 60 has a yield annualized for continuous compounding equal to 5.108%, which rounded to the nearest one- tenth of a basis point is the same as hourly compounding.
A general formula for converting from an annual rate for periodic compounding to continuous compounding is shown in equation 2.3.
So, instead of working with the two cash flows, you could convert 5.174% (s.a.) directly to continuous compounding.
A conversion formula to go in the other direction uses the exponential (EXP) function.
The quarterly compounded annual yield of 5.141% can be obtained using PER = 4.