FI 11 Yield-Based Bond Duration Measures and Properties

Duration is the standard numerical gauge of interest rate risk: the exposure of a bond’s price to movements in interest rates. Several duration measures exist, and they fall into two families. Yield duration measures track how a bond’s price responds to a change in the bond’s own yield-to-maturity, and they treat the promised cash flows as certain. Curve duration measures instead track sensitivity to shifts in a benchmark yield curve, and they allow for less certain cash flows, such as those on bonds that can default. This lesson stays entirely within the first family.

Four yield duration statistics do the work here: Macaulay duration, modified duration, money duration, and the price value of a basis point. Earlier lessons showed that a bond faces two opposing exposures, price risk on one side and reinvestment risk on the other, and that holding a bond for a span equal to its Macaulay duration keeps them in balance. The measures below build on that idea and focus on price risk.

Price moves inversely with yield

Bond prices fall when yields rise and rise when yields fall, but the size of that reaction differs sharply from bond to bond. Consider three instruments introduced earlier, all quoted in the same currency: a one-year zero-coupon Australian government bond, a five-year Bright Wheels Automotive Corporation (BRWA) bond paying a 3.2% semiannual coupon, and a thirty-year Romanian government bond paying a 4.625% annual coupon. Plotting each bond’s price against a range of yields produces three very different lines. The one-year zero traces an almost flat line, while the thirty-year bond traces a steep, downward-sloping line.

The contrast is clearest in the numbers. Suppose every bond’s yield climbs from 2% to 3%.

The same one-point yield move costs the short zero about 1 percent of its value and the long Romanian bond about 17 percent. A picture helps investors see this, but a single number that ranks bonds by price sensitivity is far more useful. That single figure is the first derivative, or slope, of the line that relates price to yield. A nearly flat line, like the Australian bond’s, has a small slope and low interest rate risk; a steep line, like the Romanian bond’s, has a large slope and high interest rate risk. The measures that follow put this slope on a common footing.

For an option-free bond, price equals the discounted sum of every promised cash flow, using the yield per period r as the discount rate.

Candidates are not required to work the calculus, but the logic is worth tracing. Differentiating the price with respect to r gives the rate of change of price for a change in yield. Factoring a common term out of that derivative leaves a bracketed expression in which each cash flow’s present value, taken as a share of the bond price and weighted by its time to receipt, appears. Dividing the whole derivative by price converts it to a percentage change, and the bracketed piece divided by price is exactly the Macaulay duration, MacDur, defined in an earlier lesson. What remains is a compact result: the percentage price change per unit change in yield equals the negative of Macaulay duration divided by one plus the yield per period. Dropping the negative sign gives modified duration, ModDur.

Because modified duration captures the link between price and yield, it estimates the percentage change in a bond’s full price for a change in its annualized yield-to-maturity.

Two points on notation deserve emphasis. The change refers to the full price, also called the dirty price, which includes accrued interest. And the approximation sign is deliberate: this is a straight-line estimate of a relationship that is actually curved, so it is only accurate for small yield moves. The negative sign restates that price and yield travel in opposite directions. Spreadsheet users can obtain modified duration directly with the MDURATION function, whose inputs match those of the DURATION function used for Macaulay duration.

Approximate modified duration

When Macaulay duration is already known, modified duration follows immediately. For bonds whose Macaulay duration is hard to pin down, because of embedded options or default risk, an alternative estimates the slope of the price-yield curve directly. Shift the yield up and down by the same small amount, the ΔYield, reprice the bond at each, and call the results PV+ (yield raised) and PV− (yield lowered). The slope, scaled to a percentage of the starting price PV0, is the approximate annualized modified duration.

Modified duration speaks in percentages. Money duration converts that sensitivity into currency units. It multiplies the annualized modified duration by the bond’s full price, and it can be quoted per 100 of par or for the entire position. Practitioners in the United States usually call it dollar duration.

Price value of a basis point

The price value of a basis point (PVBP) estimates the change in a bond’s full price for a one-basis-point change in yield. It uses the same repricing idea as the approximate modified duration: reprice the bond after cutting the yield by 1 bp and again after raising it by 1 bp, then halve the difference.

Zero-coupon and perpetual bonds

A zero-coupon bond pays a single amount, its face value at maturity, so that one cash flow carries a present-value weight of 1.0. Its Macaulay duration therefore equals its time-to-maturity, and its modified duration is time-to-maturity divided by one plus the yield. A perpetuity, or perpetual bond, never matures and pays a fixed coupon indefinitely unless it is called; there is no maturity value. Non-callable perpetuities are rare, yet their Macaulay duration takes an elegant closed form.

Floating-rate notes and loans

Interest on a floating-rate instrument tracks a market reference rate (MRR) plus a fixed quoted margin, and it resets on scheduled dates. Because each reset realigns the coupon with current rates, interest rate risk lives only in the interval between resets. The Macaulay duration of a floater is thus just the fraction of the current period still remaining until the next reset.

For a 180-day period with 57 days elapsed, the Macaulay duration is (180 − 57) / 180 = 0.683333. Since coupon periods rarely run beyond six months, floaters carry very low duration, which is why investors reach for them to trim the duration of a fixed-income portfolio.

For a traditional fixed-rate bond, the Macaulay and modified yield duration statistics are driven by a small set of features: how long until it matures, how large its coupon is, where its yield sits, and how much of the present coupon period has already gone by. A closed-form expression for Macaulay duration, carried over from an earlier lesson, makes these dependencies visible. The same behavior carries through to modified duration, money duration, and the price value of a basis point.

The coupon and yield effects share one cause. A lower coupon or a lower yield lifts the weight of the final maturity payment and lightens the weight of the nearer cash flows, which lengthens the weighted-average time to receipt and raises duration. So a lower-coupon bond has higher duration than an otherwise identical higher-coupon bond, and a lower yield raises duration as well.

Maturity and the shape of duration

Longer maturity usually means higher duration, and for bonds priced at par or at a premium this always holds. In the closed-form expression the second bracketed term stays positive whenever the coupon is at least the yield, so (c − r) is not negative; the Macaulay duration then sits below the (1 + r)/r ceiling and creeps up toward it as maturity lengthens. Discount bonds behave more strangely. When the number of periods is large and the coupon lies below the yield, so (c − r) turns negative, the duration can climb past (1 + r)/r, reach a peak, and then fall. The counterintuitive implication is that a very long-dated discount bond can carry less interest rate risk than a shorter one.

Duration between coupon dates

Hold the yield fixed and let time pass within a coupon period. The braced expression in the closed-form formula does not change, so Macaulay duration slides down smoothly as t/T grows, then springs back up by a small step the moment a coupon is paid and t/T resets. Repeating period after period traces a saw-tooth path that trends downward overall.