How much does a bond move when rates move 1%?
This is the question the whole course exists to answer, and you are one small step from it. Macaulay duration is a time. Divide it by one plus the periodic yield and you get a sensitivity.
The conversion
Modified duration = Macaulay duration ÷ (1 + y ÷ k)
where y is the annual yield and k the number of coupon periods per year. For our house bond — Macaulay 4.5459 years, annual coupons, 5% yield:
Modified duration = 4.5459 ÷ 1.05 = 4.33
For a US Treasury paying twice a year at a 5% yield you would divide by (1 + 0.05 ÷ 2) = 1.025 instead. Same idea, different denominator.
The one formula to carry
%ΔP ≈ − Modified duration × Δy
The minus sign is the price-yield seesaw, now with a number attached. Read it as: "for every 1% (100 basis points) that yields rise, this bond's price falls about 4.33%."
Watch it work
Our €1,000 bond, modified duration 4.33:
- Rates rise 100bp (5% → 6%): estimate −4.33% → €956.71. Reprice the bond properly at 6% and the true price is €957.88. Duration overstated the loss by €1.17.
- Rates fall 100bp (5% → 4%): estimate +4.33% → €1,043.29. True price: €1,044.52. Duration understated the gain by €1.22.
- Rates rise 25bp: estimate −1.08% → €989.18. True price €989.25. Off by seven cents.
Two things to notice. The estimate is excellent for small moves and drifts for large ones. And both errors land in the bondholder's favour — that is not luck, and Unit 3 names it.
What "duration" means in conversation
When a portfolio manager says "the fund runs a duration of 6.2", they mean modified duration almost every time. The number is inherited from a measure of years but is being used as a percentage-per-percentage sensitivity. Don't let the unit confuse you.
Duration in that sentence is sensitivity to the general level of yields. A bond also has a spread duration, its sensitivity to its own spread over the benchmark, which the Credit course uses. For a fixed-coupon bond the two are the same number, because a yield rise from either cause hits the same fixed cash flows. They separate on a floating-rate note. Its coupon resets to the reference rate every one or three months, so its price returns to par at each reset; its rate duration is the time to the next reset, weeks rather than years, while its spread duration runs to maturity. A five-year floater carries almost no rate risk and five years of spread risk.
Portfolios add up, by value
Duration is value-weighted additive. A portfolio of €600,000 in a bond with duration 3 and €400,000 in a bond with duration 9:
0.60 × 3 + 0.40 × 9 = 1.8 + 3.6 = 5.4
One number now describes the rate sensitivity of the whole book. That is why duration became the industry's common language: it aggregates.
The two assumptions you just made
- The move is small. The relationship is a curve; duration is a straight line drawn along it. Unit 3.
- The whole curve moved in parallel. Real curves twist — the 2-year and the 30-year rarely move by the same amount. Unit 4.
Neither assumption makes duration useless. Both make it a first-order answer that a professional knows to qualify.
In the data
A bond fund states its duration, because that is the number its holders need. The table is the iShares Core US Aggregate Bond ETF (AGG), a portfolio of thousands of US government, mortgage and corporate bonds.
Read the modified duration as this lesson's formula reads it: the fund's approximate percentage price move for a one-point change in yields. The effective duration beside it is the version that lets mortgage repayments speed up or slow down as rates move, which is why the two differ slightly. The market quotes prices and yields; duration is always computed from them.
Try it now
- Take the fund's modified duration from the table in the section above and compute what a 50bp rate move implies for its price. For a fund at duration 6.2 it would be 6.2 × 0.50% ≈ 3.1%.
- The chart below is one month of the US 10-year government yield, quoted in percent. Read the level at each end, convert the difference into basis points, and turn that into an estimated price move for a duration-6 portfolio.
- Build a two-bond portfolio on paper with a target duration of exactly 5, using bonds with durations 2 and 8. (Solve 2w + 8(1−w) = 5.)