Modified duration

What does modified duration measure?

Modified duration turns Macaulay duration from a length of time into a price sensitivity. It is the approximate percentage change in a bond’s full price for a 1 percentage point (100 bp) change in the bond’s own yield-to-maturity. A modified duration of 5 means the price falls by about 5% when the yield rises by 1 percentage point, and rises by about 5% when it falls by 1 point.

Two points to fix in mind. First, it is the full price that moves, meaning the flat price plus accrued interest. Second, it measures the response to the bond’s own yield, with cash flows assumed certain. Together with Macaulay duration, money duration and the price value of a basis point (PVBP, the next lesson) it makes up the family of yield duration measures.

Duration (interest rate risk)two families

• Yield duration Memorize cash flows certain; moves with the bond’s own yield
  • Macaulay duration Memorize weighted average time, in years
  • Modified duration Memorize % price change per 1 point of yield
  • Money duration Memorize currency change; next lesson
  • PVBP Memorize currency change per 1 bp; next lesson
• Curve duration Know the idea benchmark curve moves; cash flows may change; later modules

Graphically, modified duration is the slope of the price–yield curve at today’s yield, scaled by the price. The curve bends, so the straight tangent line is exact only at the point where it touches.

How do you get from Macaulay to modified duration?

Divide by one plus the yield per period. The yield per period is the annual yield divided by the number of coupons a year, so a semiannual bond divides by $1 + y/2$. If the Macaulay duration is annualized, the result is the annualized modified duration .

$$\text{ModDur} = \frac{\text{MacDur}}{1 + r}$$

Here $r$ is the yield per period. Once you have it, the price estimate is a single multiplication.

$$\%\Delta PV^{Full} \approx -\text{AnnModDur} \times \Delta y$$

The yield change $\Delta y$ goes in as a decimal: 50 bp is 0.0050. The minus sign records that price and yield move in opposite directions.

Exam tip The minus sign and the decimal are where marks are lost. A 35 bp fall in yield is +0.0035 × duration, a gain.

How do you approximate modified duration from prices?

When you have prices but no cash-flow table, estimate the slope directly. Move the yield down and up by the same small amount, reprice the bond, and measure how far the price swings. The result is the approximate modified duration .

$$\text{ApproxModDur} = \frac{PV_- - PV_+}{2 \times \Delta y \times PV_0}$$

$PV_-$ is the full price after the yield falls by $\Delta y$, $PV_+$ the full price after it rises by $\Delta y$, and $PV_0$ today’s full price. The numerator is the price swing across the two bumps, the $2\Delta y$ is the total yield distance, and dividing by $PV_0$ turns the swing into a percentage.

$$\text{ApproxMacDur} = \text{ApproxModDur} \times (1 + r)$$

Learn the pattern “price down minus price up, over 2 times the bump times the price”. The same pattern, run on a benchmark curve, defines effective duration in a later module.

Go deeper Why dividing by (1 + r) works

A bond’s price is the sum of its cash flows discounted at the yield. The slope of that sum with respect to the yield is minus the sum of each time to receipt times its present value, all divided by one plus the yield. Divide the slope by the price and the time-weighted sum becomes the Macaulay duration. So the percentage slope, modified duration, is Macaulay duration over $1 + r$. You do not need calculus on the exam, only the result.

Go deeper Negative yields, and the spreadsheet functions

If a yield is negative, $1 + r$ is below 1, so modified duration is larger than Macaulay duration. At a yield of −0.5% a Macaulay duration of 5.00 gives 5.00 ÷ 0.995 = 5.025.

In a spreadsheet, DURATION(settlement, maturity, coupon, yield, frequency) returns the annualized Macaulay duration and MDURATION(...) with the same arguments returns the annualized modified duration.