Tell apart Macaulay duration, modified duration and money duration
The idea
Macaulay duration is the weighted average time to receipt of a bond's cash flows, with each weight the present value of that cash flow as a share of the price; it is measured in years. Modified duration divides it by one plus the yield per period and is the percentage price change for a one percentage point change in yield. Money duration, or dollar duration, multiplies modified duration by the full price and gives the change in money terms. The price value of a basis point is money duration scaled to a single basis point, and it is what a trader hedges with.
Formulas
Modified duration from Macaulay duration
ModDur = MacDur ÷ (1 + r)
- MacDur
- Macaulay duration, the weighted average time to the bond's cash flows
- r
- the yield per period
Macaulay duration is a time, in years. Modified duration is a percentage price change per 1% of yield. Dividing by one plus the periodic yield is the whole of the difference between them.
Money duration
MoneyDur = ModDur × PV
- ModDur
- modified duration
- PV
- the full price of the position, in money
Duration in currency rather than percent, so the estimate comes out in money: the price change is −MoneyDur × Δyield. Taken per basis point on 100 of par it is the price value of a basis point.
A practice question
What does Macaulay duration measure?
- The weighted average time to receipt of a bond's cash flows — correct
- The percentage price change for a change in yieldWhy not: That is modified duration, which is Macaulay duration divided by one plus the periodic yield.
- The bond's remaining term to maturityWhy not: Only for a zero-coupon bond. Any coupon pulls the weighted average time to receipt forward of the final date.
It is a time measure in years. Dividing it by one plus the periodic yield converts it to the price sensitivity measure.
A harder one
A bond has a Macaulay duration of 7.0 years and a yield of 4% with annual compounding. What is its modified duration?
- 7.28Why not: 7.0 x 1.04 — multiplied rather than divided. Modified duration is always slightly *below* Macaulay duration.
- 6.73 — correct
- 7.00Why not: The Macaulay figure unconverted. The division by 1.04 is what turns a time measure into a price sensitivity.
7.0 / 1.04 = 6.73. Modified duration is always slightly below Macaulay duration, and the gap widens as yields rise.
The hardest tier
Why does a trader hedging a position work in money duration rather than modified duration?
- Because modified duration cannot be computed for a portfolioWhy not: It can, as a value-weighted average. The objection is that it does not carry position size.
- Because money duration includes convexityWhy not: It does not — it is duration expressed in currency. Convexity is a separate adjustment applied on top.
- Because equal percentages can mean unequal risk — correct
Two positions of the same percentage sensitivity but different sizes carry very different amounts of risk. A hedge has to offset an amount rather than a rate, and percentages say nothing about size.
In the app: 9 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
Take the free diagnostic →