How do duration and convexity measure interest rate risk? I need an intuitive explanation.

Duration and convexity together give you a complete picture of how a bond's price responds to interest rate changes. Think of duration as the first-order effect and convexity as the second-order correction.

Duration — The First-Order Effect:

Macaulay Duration is the weighted-average time to receive the bond's cash flows, where weights are the present value of each cash flow divided by the bond price. It's measured in years.

Modified Duration converts this into a price sensitivity measure:

Modified Duration = Macaulay Duration / (1 + YTM/n)

%ΔPrice ≈ -Modified Duration x ΔYield

Example:

A bond has a modified duration of 6.5. If yields rise by 50 basis points (0.50%):

%ΔPrice ≈ -6.5 x 0.0050 = -3.25%

So on a $1,000 bond, the price drops approximately $32.50.

Convexity — The Second-Order Correction:

Duration assumes a linear relationship between price and yield, but the actual relationship is curved (convex). For large yield changes, duration alone underestimates price increases and overestimates price decreases.

%ΔPrice ≈ (-ModDur x ΔYield) + (0.5 x Convexity x ΔYield^2)

Same example with convexity of 45:

%ΔPrice ≈ (-6.5 x 0.005) + (0.5 x 45 x 0.005^2)

= -3.25% + 0.056% = -3.19%

The convexity adjustment reduces the estimated loss from 3.25% to 3.19%. The effect grows dramatically for larger yield changes.

Intuitive summary:

Key principles:

• Higher coupon = lower duration (you get cash back sooner)
• Longer maturity = higher duration
• Positive convexity benefits bondholders (price rises more than duration predicts when rates fall, falls less when rates rise)
• Callable bonds have negative convexity at low yields — price is capped near the call price

Exam tip: If the question says "approximate" or involves a small yield change (<50bp), duration alone may suffice. If it specifies a large change or provides convexity data, use the full formula.

Master duration and convexity with our CFA Level I Fixed Income practice problems.