by J. Huston McCulloch

The "maturity" of a bond is the remaining time to its last payment. However, a coupon-bearing bond in fact represents payments with several maturities, most of them shorter than the final maturity, so that final maturity greatly overstates the effective average maturity of the bond. In 1938, Frederick Macaulay pointed out that an unweighted average maturity of the bond's payments was not a very useful alternative to final maturity either, since the later coupons have smaller present discounted value than the earlier coupons. For very long-term bonds, even the principal might represent only a small portion of the value of the bond. Accordingly, Macaulay proposed using present-value-weighted-average-maturity as the single number that best represented its effective maturity, and coined the term "Duration" as a shorthand for this concept.

Macaulay in fact gave two definitions of duration: Stricly speaking, he defined duration to be the present value weighted average maturity, using the full term structure of zero-coupon interest rates on bonds of comparable quality and terms. This is what might appropriately be called "exact duration."

However, since the full term structure cannot be inferred from a quotation on a single bond, and since in any event computers didn't exist in 1938 that could easily extract the full term structure from the prices on a series of bonds with comparable quality and terms, Macaulay also gave a better-known approximation to exact duration that simply uses a bond's own internal rate of return to discount all its payments, as if the yield curve were flat. This approximate duration might called "internal duration," and is usually what is meant by "Macaulay duration." Although zero-coupon yield curves such as those on this website could in principle be used to compute exact duration for each bond, the difference between exact duration and internal duration is ordinarily quite small, and internal duration is adequate for most practical purposes.

The yield that is relevant for computing a bond's internal duration is its own yield to maturity. Similarly, the yield curve that is relevant for computing a bond's exact duration is the zero-coupon yield curve on bonds of similar terms and quality. Thus, nominal US Treasury bond payments should be discounted with nominal US Treasury yields. Default-free yen- or euro- or baht-denominated bond payments should be discounted with default-free yen- or euro- or baht-denominated interest rates, respectively. Payments on US dollar "junk bonds" should be discounted with their own junky nominal dollar interest rates, since these incorporate the market's discount on the prospect that their promised payments will actually be made.

Likewise, the duration of TIPS should be computed by discounting their US CPI-denominated payments with their own US Treasury real interest rates. If the inflation premium is positive, as it has been, this implies that TIPS will generally have longer durations than nominals of comparable final maturity. Long duration is a big attraction if one is investing for retirement or other expenditures several decades in the future.

There is therefore no way a portfolio of TIPS could have a duration of 1.6 or even 4.0 years, if its average maturity in any sense was 14.2 or 12.7 years, as reported by Vanguard and 59 Wall St. above. I suspect that what they are calling "average maturity" is really Macaulay duration, and that what they are calling "duration" is something else entirely.

It can readily be shown that a bond's internal duration equals the semielasticity of the bond's price with respect to its own continuously compounded yield to maturity, just as a zero-coupon bond's unambiguous maturity is equal to the semielasticity of its price with respect to its own continuously compounded yield. Similarly, its exact duration is equal to the semielasticity of its price with respect to a parallel shift in the appropriate zero-coupon yield curve.

It is now well known that the yield
curve cannot be counted on to always move in a parallel fashion,
since if this were the case, riskless
arbitrage opportunities would be generated.
Still, this does not
lessen the fact that if it *were* to move in a parallel
fashion (something that cannot be ruled out without unrealistic
side assumptions even if it does not
*have* to move in this fashion), the derivative of the log
of price with respect to the shift would be the exact duration.
Likewise, the yield curve cannot always remain flat (as assumed by
internal duration) and still shift without generating
arbitrage opportunities. Nevertheless, this does not rule
out the possibility that it *could*
start off flat and shift to a new flat level. And even if the yield curve is not flat,
it is still a mathematical identity that a bond's internal duration equals
the semielasticity of its price with respect to its own internal
rate of return.

The semielasticity with respect to the bond's semiannually compounded quoted yield equals internal duration divided by 1 plus half the quoted yield. This semielasticity is commonly called "modified duration." Although it is no longer a true average maturity, it differs from duration per se by only a few percent, and so cannot account for anything like the discrepancies between "average maturity" and "duration" on the Vanguard and 59 Wall St. websites.

Apparently these practitioners are erroneously calling "duration" their
estimate of the semielasticity of the portfolio's price with respect to
some *nominal* interest rate, perhaps a comparable-maturity Treasury rate,
based on some assumption about or estimate of the
correlation between real and nominal rates.
While this might be of interest to some investors as a volatility index, it is not duration
in any sense. Nominal dollar interest rates, even nominal Treasury rates,
are as irrelevant for computing TIPS durations as yen- or euro- or baht-denominated
interest rates are for computing durations on nominal Treasuries.

The exact duration of a bond portfolio is the value-weighted average of the exact durations of its component bonds. However, the internal duration of a portfolio is not, strictly speaking, the value-weighted average of the internal durations. Rather, it is the value-weighed average of the maturities of all its payments, using the internal rate of return on the portfolio as a whole. Nevertheless, the difference between these two values would be quite small, so that it would not be misleading to refer to the value-weighted average of the individual internal durations as the "portfolio duration" or "average duration". To avoid misunderstandings, this could be called the "portfolio Macaulay duration" or "average Macaulay duration."

Duration per se relates to securities whose future payments are of a known size (in some numeraire such as dollars, euros or CPI units) and known maturity. However, callable bonds and certain Collateralized Mortgage Obligations (CMOs) incorporate options whose exercise prospects are sensitive to interest rates. It is harmless to speak of the "duration" of these securities as the interest semielasticity of their price, taking into account the likely reaction of the option component to their own yield. To avoid confusion, however, this inferred semielasticity could perhaps be called "effective duration." Thus, some CMOs have effective durations that are longer than their own final maturity, while others actually have negative effective durations.