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An interesting property of Macaulay duration is revealed by letting N, the number of periods to maturity, get large and approach infinity. In equation 6.15, the general expression in equation 6.13 is simplified to apply to a coupon date (i.e., t/T = 0).


As N approaches infinity, the denominator in the second term gets larger faster than the numerator because N is an exponent in the former and a coefficient in the latter. That whole second term goes to zero and the Macaulay duration becomes just (1 +y)/y. Such bonds, known as perpetuities, are rare but do exist. For instance, in the U.K. bonds called consols paying a fixed interest payment forever have been in existence since the 18 th century – and they're still out there.

Now consider bonds trading at a premium – their coupon rate is higher than the yield to maturity, c > y. The numerator in the second term in equation 6.15 is always positive, as is the denominator, so the Macaulay duration is less than (1 + y)/y. For longer maturities, other things being equal, the duration increases and approaches the perpetuity threshold monotonically from below. The same pattern holds for bonds that continue to trade at par value on coupon dates because the coupon rate equals the yield, c = y. If we subtracted the t/T term for dates between coupon payments, we would have a “saw-tooth” pattern. As days go by during the period, the duration would decline smoothly (assuming no change in the yield) but then jump up after the coupon is paid.

Things get interesting for bonds trading at a discount – the coupon rate is less than the yield, c < y. When N is large enough, the numerator becomes negative. When that happens the Macaulay duration is greater than (1 + y)/y. But eventually for an even larger N, the duration has to approach the perpetuity threshold monotonically from above. These relationships between Macaulay duration and maturity are illustrated in Figure 6.1. For a zero-coupon bond, it's a 45-degree line because they are equal. For a perpetuity, it's a horizontal line at (1 + y)/y. The relationship is positive – the longer the maturity, the higher the duration – for most coupon bonds having a stated maturity, but not always.

Let's revisit the problem of the fixed-income strategist considering two 6% annual payment bonds, one maturing in 20 years and the other in 30 years. Both are priced to yield 20%, and the strategist anticipates a 100 basis point drop. The easiest way to assess the choice is to do the basic bond pricing. We're presumably on a coupon date so equation 3.5 from Chapter 3 will work fine. So will a financial calculator or Excel.

Relationships between Macaulay Duration and Maturity

FIGURE 6.1 Relationships between Macaulay Duration and Maturity

It's true – given the same coupon rate and yield, the 20-year bond actually does have the higher percentage price increase for the same drop in yield, 5.85% compared to 5.46%. Try to explain this without appealing to duration. I have tried to do so but cannot. To me, it is just a bond math curiosity. It finally makes sense once we calculate the Macaulay durations using equation 6.15 (because t/T = 0).

The Macaulay duration on the 20-year bond is 6.66 but just 6.21 on the 30-year. The modified durations can be used to estimate the anticipated percentage price increase.

Notice that duration signals correctly the bond that has the greater price appreciation if the market responds as the strategist expects. However, the estimated changes are off by about 30 basis points compared to the actual results. By itself, duration is a conservatively biased estimate for the risk in a long position on a fixed-income bond – it overestimates the loss when the yield goes up and underestimates the gain when the yield goes down. The convexity adjustment improves the estimate in each case.

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