Duration & Convexity
Duration measures how sensitive a bond's price is to yield changes. Convexity captures the curvature that duration misses.
Bond Parameters
Duration Approximation vs Actual Price
Duration gives a linear estimate (straight line). The actual curve bends — that's convexity. Adding convexity correction makes the estimate much more accurate.
Price Change: ±100 Basis Points
Convexity asymmetry: A 100bp yield drop gives a price gain of +$81.76, while a 100bp yield rise gives a loss of only $-74.39. The gain is larger than the loss — this is convexity at work.
Macaulay Duration — Weighted Average Time
Each bar represents how much a cash flow contributes to duration. The principal (final bar) typically dominates. Duration is the balance point (fulcrum).
Macaulay Duration
The weighted average time until you receive a bond's cash flows, where each cash flow is weighted by its present value. Think of it as the “balance point” on a seesaw of cash flows.
Modified Duration
Converts Macaulay duration into a direct measure of price sensitivity. Modified duration tells you: for a 1% yield change, the bond price moves approximately by this percentage.
Price change approximation using modified duration:
The negative sign means price moves opposite to yield. This is a linear approximation — it works well for small yield changes but gets less accurate for larger moves. That's where convexity comes in.
Convexity
Convexity measures the curvature of the price-yield relationship. It's the second-order correction that makes the approximation more accurate.
Full price change approximation with convexity:
Duration alone assumes a straight line. It overestimates losses when yields rise and underestimates gains when yields fall. The error grows with larger yield moves.
The convexity term () is always positive for vanilla bonds. It bends the estimate to match the actual curve, especially for large yield moves.
Key Intuitions
Higher coupon → Lower duration
More cash comes back sooner (via coupons), so the weighted average time is shorter. A zero-coupon bond has the highest duration — its entire return is at maturity.
Longer maturity → Higher duration
Cash flows are spread further into the future, increasing the weighted average time. A 30-year bond is much more rate-sensitive than a 2-year bond.
Convexity is always your friend
For vanilla bonds, convexity is always positive. This means you gain more from yield drops than you lose from yield rises. Investors value convexity and will pay a premium for it.
DV01 = Dollar Value of a Basis Point
The actual dollar change in price for a 1 basis point (0.01%) yield move. Traders use DV01 to hedge and manage interest rate risk across portfolios.