Duration: Definition and Use in Fixed‑Income Investing

Duration measures a bond’s sensitivity to changes in interest rates and estimates how long, in years, it takes for an investor to be repaid a bond’s price through its cash flows. It is a central tool for assessing interest‑rate risk: higher duration means greater price sensitivity to interest‑rate movements.

Key takeaways
* Duration is different from time to maturity: maturity is the linear time until principal is repaid, while duration is a weighted average of cash‑flow timing and changes with interest rates.
* Higher duration → greater price sensitivity to interest‑rate changes.
* Portfolio duration = weighted average of the durations of individual holdings.
* Common variants: Macaulay duration, modified duration, dollar duration, effective duration, and key‑rate duration.

How duration works
* Duration quantifies the expected percentage change in a bond’s price for a unit change in yield (with modified duration) or gives the weighted average time to receive cash flows (with Macaulay duration).
* Factors that increase duration:
* Longer time to maturity.
* Lower coupon rate (payments come later, on average).
* Example rule of thumb: A bond with a five‑year duration will lose about 5% of its value if yields rise by 1 percentage point (all else equal).

Macaulay duration
* Definition: the weighted average time (in years) until a bond’s cash flows are paid, weighting each payment by its present value.
* Formula (conceptual):
Macaulay Duration = [sum over f of (present value of cash flow f × time to f)] / (present value of all cash flows)
* Use: helpful for comparing timing of cash flows across bonds; expressed in years.
* Note: for a zero‑coupon bond, Macaulay duration equals its time to maturity because all cash flows occur at maturity.

Modified duration
* Definition: converts Macaulay duration into an estimate of price sensitivity to changes in yield.
* Formula:
Modified Duration ≈ Macaulay Duration / (1 + YTM / m)
where YTM = yield to maturity (as a decimal) and m = number of coupon periods per year.
* Interpretation: approximate percentage change in price for a 1 percentage‑point (100 bps) change in yield. For small yield moves, %ΔPrice ≈ −Modified Duration × ΔYield.
* Convexity: because the price–yield relationship is curved, modified duration is a linear approximation; convexity measures the curvature and improves accuracy for larger yield moves.

Worked example (summary)
* Three‑year bond, face value $100, 10% coupon paid semiannually ($5 every six months), YTM = 6%:
* Macaulay duration (calculated from present values of all coupon and principal payments) ≈ 2.684 years.
* Modified duration ≈ 2.61.
* If bond price ≈ $100, a 1% rise in yields would reduce price by roughly 2.61% (≈ $2.61).

Other duration measures
* Dollar duration: the dollar change in a bond’s value for a 1% change in yield = Modified Duration × Price × 0.01. Useful for position‑sizing and hedging.
* Effective duration: used for bonds with embedded options (callable/putable), accounting for changes in expected cash flows as yields move.
* Key‑rate duration: measures sensitivity to yield changes at specific maturities along the curve, useful for analyzing yield‑curve shifts.

Strategies using duration
* Long‑duration strategy: concentrate in bonds with higher duration to benefit when rates fall (capital gains), but face greater downside if rates rise.
* Short‑duration strategy: favor bonds with lower duration to reduce interest‑rate exposure if rates are expected to rise.
* Portfolio management: match portfolio duration to investment horizon or liabilities (immunization), or adjust duration to express a view on future interest‑rate moves.

Practical uses
* Match bond holdings to your risk tolerance and expected holding period.
* Use duration to size hedges (e.g., how much interest‑rate futures or swaps are needed to neutralize exposure).
* Combine duration with convexity to better estimate price changes for larger yield movements.

Explain like I’m five
Duration tells you how much a bond’s price will wobble when interest rates change. The longer the average wait for your bond payments, the bigger the wobble.

Frequently asked questions
Q: Is duration the same as maturity?
A: No. Maturity is when the bond repays principal. Duration is the weighted average timing of all payments and reflects interest‑rate sensitivity.

Q: Which bonds have the highest duration?
A: Long‑maturity, low‑coupon (or zero‑coupon) bonds have the highest durations.

Q: How do I reduce interest‑rate risk?
A: Shorten portfolio duration—buy shorter‑term bonds, higher‑coupon bonds, or use derivatives to hedge duration.

Summary
Duration is a fundamental metric for fixed‑income investors. It quantifies interest‑rate risk, guides bond selection and hedging decisions, and—when used with measures like convexity and key‑rate duration—helps manage and understand how a bond or portfolio will respond to changes in the yield environment.