Investments
Bond duration and interest-rate risk
Duration converts a change in yield into an approximate change in a bond's price, and convexity corrects the estimate.
Definition
Interest-rate risk is the risk that a change in market yields changes the value of a bond or a portfolio of bonds. Duration measures the first-order sensitivity of price to yield: the present-value-weighted average time to the cash flows, converted into a percentage price change per unit of yield.
Modified duration is the practical form. Convexity is the second-order term that corrects the estimate, making gains from falling yields a little larger and losses from rising yields a little smaller than duration alone predicts.
Intuition
A bond is a set of promised cash flows. The further away those promises are, the more their present value moves when the discount rate changes, so duration is really a measure of how far away the money is.
When to use it
- Measuring the mark-to-market risk of a bond holding.
- Matching the duration of assets to the duration of liabilities, known as immunisation.
- Sizing a hedge with interest-rate futures or swaps.
When not to use it
- For very large yield moves, where duration alone is no longer accurate.
- When the cash flows are not fixed: callable, prepayable, or credit-risky bonds break the assumption.
Formula
Macaulay duration
D = Σ [t × PV(CFt)] ÷ Price
The present-value-weighted average time to the cash flows, in years.
Formula
Modified duration
D* = D ÷ (1 + y ÷ m)
The percentage price change for a one-point change in yield.
Formula
Price change with convexity
ΔP ≈ P × [−D* × Δy + ½ × C × Δy²]
A second-order estimate of the price move for a change in yield.
Worked example
A ten-year bond
৳1,000 face value, 6% annual coupon, ten years to maturity, priced at a 6% yield.
- 1
Price
Coupon equals yield, so the price is exactly ৳1,000.
- 2
Duration
Macaulay duration is about 7.80 years; modified duration is 7.80 ÷ 1.06 = 7.36 years.
- 3
One-point shock
A rise to 7% predicts a fall of about 7.36%, to roughly ৳926 before convexity.
- 4
Convexity
Convexity of about 69.74 adds back roughly ৳3.9, giving ৳929.89 against an exact ৳929.76.
Duration turns a yield view into a price move; convexity makes the estimate accurate enough to use for a one-point shock.
Common mistakes
- Reading duration as a time to maturity; it is always shorter than maturity for a coupon bond.
- Using modified duration without converting the yield for coupon frequency.
- Hedging once and forgetting that duration drifts as time passes and yields move.
How it connects
requires
Learn the source first; the target assumes it.
- Bond pricing and yield: You need a price before you can measure its sensitivity.
- Time value of money: Duration is a present-value-weighted average time.
is used in
The source appears as an input to the target.
- Financial risk management: Duration feeds market-risk measurement.
relates to
Useful to study together, in either order.