Bond Duration¶

OVERVIEW¶

Bond duration measures a bond's price sensitivity to interest rate changes. A bond with a duration of 7 loses approximately 7% of its value when rates rise 1%, and gains approximately 7% when rates fall 1%. Duration is expressed in years but functions as a risk multiplier, not a literal time-to-payback measure.

Note

Longer maturities, lower coupons, and lower yields all increase duration. Zero-coupon bonds have duration equal to their maturity since 100% of the payment arrives at a single future date.

FEATURES¶

Duration Types¶

Type	Formula	Primary Use
Macaulay Duration	`Σ [t × PV(CFₜ)] / Price`	Weighted-average time to receive cash flows
Modified Duration	`Macaulay Duration / (1 + y/n)`	Direct price-sensitivity approximation
Effective Duration	Models shifting expected cash flows	Callable bonds, mortgage-backed securities
DV01	Dollar value of a 1 basis point move	Futures and swaps hedging

Price Change Approximations¶

Approximation	Formula	When to Use
Linear (small moves)	`% ΔPrice ≈ −Modified Duration × Δy`	Small rate moves, quick estimates
Convexity-adjusted (large moves)	`% ΔPrice ≈ −Modified Duration × Δy + 0.5 × Convexity × Δy²`	Large rate swings, higher accuracy

Tip

For large yield moves, always apply the convexity adjustment. Ignoring convexity understates gains in a rally and overstates losses in a selloff.

HOW TO USE¶

Interpreting Duration on Tapeboard¶

Reading duration values

A modified duration of 8.7 on a 10-year Treasury note means a 1% yield rise produces approximately an −8.7% price change
A long-duration ETF (e.g. 20+ year Treasuries) with duration 16–17 implies a 16–17% price swing per 1% yield move
A 2-year note with duration near 1.9 is far less rate-sensitive than a long-duration fund

Applying duration across asset classes

Bond and rates traders — Use duration-matching or immunization to align a portfolio's aggregate rate exposure to a target
Equity traders — Treat unprofitable growth companies as long-duration assets; their distant cash flows reprice aggressively when yields spike, following the same discounting mechanics as bonds
Options and futures desks — Match DV01 exposure using Treasury futures or interest rate swaps to hedge rate risk

Calculating Duration¶

Macaulay Duration

D = Σ [t × PV(CFₜ)] / Price

t — time until each cash flow
PV(CFₜ) — present value of that cash flow
Price — current market price (sum of all PV(CFₜ))

Modified Duration

Modified Duration = Macaulay Duration / (1 + y/n)

y — yield to maturity
n — number of compounding periods per year

LIMITATIONS AND MISCONCEPTIONS¶

Limitation	Detail
Parallel shift assumption	Duration assumes every maturity moves by the same amount. Curve steepening or flattening produces outcomes duration alone cannot predict.
Fixed cash flow assumption	Duration does not apply directly to bonds with embedded options. Use effective duration for callable bonds and mortgage-backed securities.
Linear approximation	Duration is a straight-line estimate. For large yield swings, omitting the convexity term produces meaningful error.