How duration and convexity describe interest-rate risk
A bond’s price is just the present value of its future coupons and principal, so when yields move, every one of those discounted cash flows moves with them. Duration packages that sensitivity into one number: it is the present-value-weighted average time to the bond’s payments, and it tells you roughly how much the price changes for a small shift in yield. But the relationship between price and yield is not a straight line — it bends. Convexity measures that bend, and adding it as a second-order term sharpens the estimate, especially for larger yield moves. If you want the underlying price and yield-to-maturity that feed these figures, start with thebond pricing & YTM calculator.
Duration and convexity
Macaulay = Σ [ t · PV(CFt) ] / price
Modified = Macaulay / (1 + y/m)
ΔP/P ≈ −Modified·Δy + ½·Convexity·(Δy)²
where t is the time in years until each cash flow,y is the yield, and m is the number of compounding periods per year. PV(CFt) is the present value of the cash flow at time t. The first term is the straight-line duration estimate; convexity is the second-order term that corrects for the curvature duration alone cannot capture.
What makes this calculator different
- It computes all three measures. Macaulay duration, modified duration, and convexity are calculated together from the bond’s cash-flow schedule, not estimated by a rule of thumb.
- It compares the estimates to reality. For a yield change you choose, it shows the duration-only estimate, the duration-plus-convexity estimate, and the bond’s actual reprice side by side — so you can see exactly how much the convexity term buys you.
- It draws the price-yield curve. The convex price-yield relationship is charted with the duration tangent line overlaid, making it visible why duration over- and understates moves as you travel along the curve.
- It builds intuition for the inputs. Change the coupon, maturity, or yield and watch duration respond — longer maturities and lower coupons push it up, higher yields pull it down.
Frequently asked questions
What is bond duration?+
Duration measures how sensitive a bond’s price is to changes in interest rates. Despite the name, it is not simply the bond’s maturity — it is a weighted average of the times at which the bond pays cash, with each time weighted by the present value of that payment. A bond with a duration of 7 will, to a first approximation, fall about 7% in price for every 1% rise in yield. It compresses everything about a bond’s timing — coupon, maturity, and yield — into a single risk number you can compare across bonds.
What’s the difference between Macaulay and modified duration?+
Macaulay duration is the present-value-weighted average time, in years, until you receive the bond’s cash flows. Modified duration takes that figure and rescales it so it reads directly as a price sensitivity: it equals Macaulay duration divided by (1 + y/m), where y is the yield and m is the number of compounding periods per year. Macaulay answers “on average, how long until I’m paid?”; modified answers “what percentage does my price move per 1% change in yield?” They are closely related, and for most bonds the two numbers are quite similar.
Why does a bond’s price fall when rates rise?+
A bond pays fixed coupons, so when market interest rates rise, newly issued bonds offer higher coupons and your older, lower-coupon bond becomes less attractive. To sell it, you must drop the price until its yield matches the new market level. Mechanically, the bond’s price is the present value of its future cash flows, and a higher discount rate shrinks every one of those present values. The longer until you receive a given payment, the more that discounting hurts — which is exactly why duration captures the size of the effect.
What is convexity and why does it matter?+
Convexity describes the curvature of the price-yield relationship. Duration assumes price changes in a straight line as yields move, but the true price-yield curve bends — it is convex. Because of that curve, duration alone overstates the price drop when yields rise and understates the price gain when yields fall, and the error grows as the yield move gets larger. Convexity is the second-order correction that captures this bending. Positive convexity is favorable to a bondholder, since it cushions losses and amplifies gains relative to the straight-line duration estimate.
Does a longer maturity or lower coupon mean higher duration?+
Yes — both push duration up, because both shift weight toward cash flows that arrive later. A longer maturity means the large principal repayment is further out, so more of the bond’s value sits in distant, rate-sensitive payments. A lower coupon means less cash comes back early, so proportionally more of the bond’s value is tied up in that final payment. A higher yield works the other way, discounting distant cash flows more heavily and pulling duration down. A zero-coupon bond is the limiting case: with a single payment at maturity, its Macaulay duration equals its maturity exactly.
Disclaimer: This calculator is foreducation and illustration only. Duration and convexity are model-based approximations of interest-rate risk and rely on simplifying assumptions about yield changes and cash-flow timing; their output is not a tradeable quote and real bond prices will differ. Nothing here is investment, tax, or trading advice.