Duration

Duration is the most important measure of interest rate risk in bonds. It tells you roughly how much a bond's price will change when yields change. A bond with a modified duration of 7 will lose about 7% of its value if yields rise by 1 percentage point, and gain about 7% if they fall by 1 point. Duration lets investors compare bonds with different coupons and maturities, build portfolios with a target level of rate risk and hedge that risk.

Macaulay duration#

Macaulay duration, introduced by Frederick Macaulay in 1938, is the weighted average time until a bond's cash flows are received, with each cash flow weighted by its present value.

Macaulay duration = Σ [t × PV(cash flow at t)] / bond price

It is measured in years. A zero coupon bond's Macaulay duration equals its maturity, because all cash comes at the end. Coupon bonds have shorter durations than their maturities, because some cash arrives earlier.

Modified duration#

Modified duration converts Macaulay duration into a price sensitivity:

modified duration = Macaulay duration / (1 + y / f)
% price change ≈ - modified duration × change in yield

where y is the yield and f is the number of coupon payments per year.

What affects duration#

Typical durations#

Figures depend on coupons and yields. In 2022, when 30 year yields rose by more than 2 percentage points, long Treasuries lost over 30%, consistent with their high duration. See Treasury Bills, Notes and Bonds.

Effective duration#

For bonds whose cash flows change with rates, such as callable bonds and mortgage backed securities, traders use effective duration, calculated by repricing the bond for small up and down shifts in yields:

effective duration = (P_down - P_up) / (2 × P_0 × Δy)

Mortgage securities can have "negative convexity": as rates fall, homeowners refinance, shortening duration just when investors want it longer.

Portfolio duration#

A portfolio's duration is the market value weighted average of its bonds' durations. Fund managers often target a duration close to their benchmark index and adjust it to express views on rates.

Using duration#

• Measure risk: a portfolio with duration 6 loses about 6% for a 1 point parallel rise in yields.
• Match liabilities: pension funds and insurers match asset duration to liability duration (immunisation).
• Hedge: offset duration with Treasury futures or swaps. See DV01 and Interest Rate Swaps.
• Compare bonds with different structures.

Limits#

• Small changes only: duration is a linear approximation; for large moves, add convexity.
• Parallel shifts: duration assumes all yields move by the same amount, while real yield curves twist and steepen. See Yield Curves and Yield Curve Trades: Steepeners, Flatteners and Butterflies.
• Credit spreads: corporate bonds also have spread duration, sensitivity to changes in credit spreads. See Credit Spreads.

Frequently asked questions#

What is bond duration?#

A measure of a bond's sensitivity to interest rate changes, expressed in years, indicating roughly how much its price changes for a 1 percentage point move in yields.

What is the difference between Macaulay and modified duration?#

Macaulay duration is the weighted average time to receive cash flows; modified duration adjusts it to measure the percentage price change for a change in yield.

Why do long term bonds have higher duration?#

Because more of their cash flows arrive far in the future, and distant cash flows are more sensitive to changes in discount rates.

Next, learn the curvature that duration misses in Convexity.