Macaulay Duration: The Concept
Duration was introduced by economist Frederick Macaulay in his 1938 study "Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States Since 1856." Macaulay defined duration as the weighted average time to receive all of a bond's cash flows, where each cash flow is weighted by its present value as a share of the total bond price.
Intuitively, duration answers the question: "On average, how long do I have to wait to get my money back?" A bond with a duration of 7 years is one where — after weighting each coupon and the final principal repayment by how much it contributes to total value — the average receipt of cash is seven years from now.
Two rules follow immediately:
- A zero-coupon bond's duration equals its maturity. Since all cash flow comes at maturity and there are no intermediate coupons, the weighted average wait is simply the number of years to maturity.
- A coupon bond's duration is always shorter than its maturity. The periodic coupon payments arrive before maturity, pulling the weighted average forward in time.
Modified Duration: Measuring Price Sensitivity
Modified duration converts Macaulay duration into a practical tool for measuring price sensitivity. It estimates the percentage change in a bond's price for a 1 percentage point (100 basis point) change in yield:
Approximate price change % ≈ −Modified Duration × Change in Yield
For example: a bond with a modified duration of 8 will lose approximately 8% of its price if yields rise by 1%. If you hold a £100,000 bond portfolio with a duration of 8 and yields rise 1%, you are looking at roughly an £8,000 mark-to-market loss.
This is why duration is the primary tool for managing interest rate risk in bond portfolios. Reducing duration reduces rate sensitivity; extending duration increases it.
Convexity: The Non-Linearity
Modified duration is a linear approximation. In reality, the price-yield relationship is curved (convex), not straight. Convexity is the measure of this curvature, and it works in bondholders' favour:
- When yields fall, bond prices rise by more than duration alone predicts.
- When yields rise, bond prices fall by less than duration alone predicts.
This asymmetry — prices rise faster than they fall — is called positive convexity and it is a property that benefits bondholders. Investors pay for convexity through lower yields on highly convex bonds.
Practical Implications: The 2022 Bond Market
The practical significance of duration was dramatically illustrated in 2022. Central banks globally raised interest rates at the fastest pace in decades to combat inflation. Long-duration government bonds — those with maturities of 20–30 years and modified durations of 15 or more — fell 30–40% in price. The US 30-year Treasury lost more than 40% of its value from its 2020 peak to late 2022.
Portfolios and funds (including some pension funds and bank treasury portfolios) that had concentrated in long-duration bonds to maximise income in a low-rate world absorbed severe losses when rates rose sharply. Silicon Valley Bank's 2023 collapse was partly attributable to unrealised losses in a long-duration bond portfolio funded by short-term deposits — a mismatch that duration analysis would have flagged clearly.
Short-duration bonds — money market funds, 1–2 year bills — experienced minimal price impact from the same rate moves, underscoring how profoundly maturity and coupon structure determine rate sensitivity.