Duration

Modified duration is the first-order sensitivity of a bond's price to yield changes, converting a yield move directly into an approximate return.

1. The idea

Macaulay duration is the weighted-average time to a bond's cash flows, weighted by the present value of each payment. It represents the bond's effective maturity, balancing coupons received early against principal received at the end.

Modified duration $D_{m o d} = D_{M a c} / (1 + y / m)$ (for $m$ compounding periods per year) converts this into a direct price sensitivity: it measures the approximate percentage price change per unit change in yield.

The key first-order approximation is $Δ P / P \approx - D_{m o d} Δ y$ : a bond with duration 7 loses about 7% of its value for a 100bp rise in yields. This linear approximation degrades for large yield moves, which is where convexity provides the next correction.

2. Why it matters

Duration is the primary risk metric for fixed-income desks, used to size hedges and to compare interest-rate sensitivity across bonds of different maturities and coupons.

3. Core mathematics

Macaulay duration

D_{M a c} = \frac{\sum _{t} t \cdot P V ( C F _{t} )}{\sum _{t} P V ( C F _{t} )}

Modified duration

D_{m o d} = \frac{D _{M a c}}{1 + y / m}

Price sensitivity

\frac{Δ P}{P} \approx - D_{m o d} Δ y

4. How to recognize it

Think of this when…

'Approximate the percentage price change for a given yield move'
Comparing interest-rate risk of bonds with different maturities or coupons
Any bond hedge sizing question using duration

5. Worked example

Worked example

A bond has modified duration 8 and price $1000. Yields rise by 50 basis points. Approximate the new price.

1
Apply the duration approximation.
$\frac{Δ P}{P} \approx - D_{m o d} Δ y = - 8 (0.005) = - 0.04$
2
Convert to a dollar change.
$Δ P \approx - 0.04 (1000) = - 40$
3
New price.
$P_{n e w} \approx 960$
Answer: Approximate new price ≈ $960 (a 4% decline)

6. Common mistakes

Using Macaulay duration directly in the price-sensitivity formula instead of modified duration
Forgetting that duration approximations break down for large yield changes (need convexity)
Sign errors — rising yields lower bond prices, so the relationship is negative

7. Quick check

Higher duration means the bond is:

The duration approximation $Δ P / P \approx - D_{m o d} Δ y$ is:

8. Practice

6 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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