We can derive specific formulas for the various duration statistics by calculating carefully the first partial derivative of the bond pricing equation 6.1 with respect to a change in the yield per period. As much fun as it is to do the calculus and work though the ensuing algebra, the step-by-step process is relegated to the Technical Appendix. A general formula for the Macaulay duration statistic is shown in equation 6.13.
Here the coupon rate per period is denoted c, where c = PMT/FV.
Let's go back to the 4%, annual payment, 4-year corporate bond priced at 99.342 to yield 4.182% that we first saw in Chapter 3. Suppose that one month has transpired since then, and the bond remarkably is still yielding 4.182%. Its Macaulay duration is 3.691, found using y = 0.04182, N = 4, c = 0.04, and t/T = 30/360 assuming the 30/360 day-count convention.
That last assumption about the day-count is important – duration is the link between the change in market value and the change in yield, so how the yield is quoted matters (i.e., its periodicity and day-count convention). Change one of those assumptions and you get a slightly different duration statistic.
Now suppose that the yield on the bond increased to 4.650% during the month that has gone by instead of remaining the same. The Macaulay duration would be 3.689.
The impact from the higher yield is not much (3.691 versus 3.689) but does signal an inverse relationship between duration and yield. So, two factors are in play in reducing the duration from 3.774 one month ago (when t/T = 0 and y = 0.04182) to 3.689 now (when t/T = 30/360 and y = 0.04650). I discuss this duration drift due to the passage of time and the change in yield further in Chapter 10 when we consider bond portfolio strategies.
At this point you might be thinking (or recalling): “Isn't Macaulay duration the weighted average time to maturity?” In fact, it can be calculated as a weighted average of the times to receipt of cash flow, whereby the weights are the shares of market value corresponding to each payment date. This is the weighted-average formula for Macaulay duration, shown in equation 6.14 – its derivation is also relegated to the Technical Appendix.
Let's redo the last calculation to confirm that the Macaulay duration is 3.689. Enter PMT = 4, FV = 100, y = 0.04650, N = 4, and t/T = 30/360.
Note that 97.676 in the denominator is the price of the bond at the beginning of the period if its yield had been 4.650%.
The modified duration for this bond is 3.525 (= 3.689/1.04650). In general, the modified duration is the Macaulay duration divided by one plus the yield per period, but in this case 4.650% is quoted for annual compounding. The numerical difference between the Macaulay and modified duration statistics depends on the level of interest rates and the periodicity. As rates are lower and/ or as the periodicity increases, the difference diminishes. In fact, if the yield is quoted for continuous compounding, the Macaulay and modified durations are the same. In any case, once one is known, the other is easily obtained.
These duration calculations can be confirmed on Excel. The DURATION and MDURATION financial functions deliver the annualized Macaulay and modified duration statistics. Assume arbitrarily that the 4% annual payment bond matures on December 15, 2017, and the current settlement date is January 15, 2014, one month since the last coupon date (December 15, 2013) on a 30/360 basis.
The entry items are the settlement date, maturity date, annual coupon rate, annual yield to maturity, periodicity, and the code for the day-count convention (0 for 30/360, 1 for actual/actual).
The Macaulay duration of a zero-coupon bond is found by setting c = 0 in 6.12 or PMT = 0 in 6.14. In either case, it reduces to just N – (t/T), the time to maturity measured in periods. The high duration on a long-term, zero-coupon bond sheds further light on the Chapter 2 story about TIGRS, CATS, and TIONS. Investment banks in the 1980s profitably transformed coupon-bearing Treasury bonds into synthetic Treasury zeros because some investors were interested in “buying duration,” not yield.
Consider a 12%, semiannual payment, 28-year Treasury bond priced back then at 94 to yield 12.792% (s.a.). Its duration is 16.20 (in terms of semiannual periods), found by entering y = 0.12792/2 = 0.06396, c = 0.12/2 = 0.06, N = 28 * 2 = 56, and t/T = 0 into 6.13.
The annualized Macaulay duration on this long-term 28-year coupon bond is just 8.10, which is the 16.20 semiannual periods divided by two periods in a year.
Note that I intentionally avoided saying that the duration of the bond is 8.10 years. There are some circumstances when it is convenient to interpret duration in terms of time, but in general it's better to think of it as the interest rate sensitivity factor. We can say that this bond will be about twice as sensitive to a shift in yield as one having duration of 4.05.1 say “about” because duration alone is just the first-order approximation and neglects the convexity term. We can say that this 28-year, 12% coupon bond is the price-risk equivalent of an 8.10-year zero-coupon bond because their prices should respond about the same on a percentage basis given an equivalent shift in their yields to maturity.
Suppose an investment bank back in the day created a 28-year zero-coupon bond via coupon stripping and sold it to a hedge fund manager for a deeply discounted price of just 3.80 (percent of par value) to yield 12.027% (s.a.).
That yield is 76.5 basis points lower than the coupon bond, (0.12792 – 0.12027 = 0.00765). However, the zero-coupon bond has an annual Macaulay duration statistic of 28.00 – that's about 3.5 times higher than 8.10. We often describe duration as a measure of risk, but it also is an opportunity statistic. In this example, the hedge fund was positioned for much greater percentage price appreciation than on the 28-year coupon bond – in fact, about 3.5 times higher if yields fall on each by the same amount. A speculative investor having a short-term horizon doesn't care about the lower yield to maturity – all the action is in the duration.