How to calculate the bond duration with the convexity rule
About this lesson
In this video, we are going to discuss what the convexity of a bond is and how it affects the interest rate risk of a bond.
Why duration isn’t enough
Duration estimates price changes as if the price–yield relationship were a straight line. In reality it’s a curve, bowed toward the origin. That curvature is convexity.
Because of convexity, duration alone:
- overstates the price fall when yields rise, and
- understates the price rise when yields fall.
The formula
ΔP ÷ P ≈ −D* × Δy + ½ × Convexity × (Δy)²
Convexity for annual cash flows:
Convexity = [1 ÷ (P × (1 + y)²)] × Σ CFt ÷ (1 + y)t × (t² + t)
Example
The 3-year bond with an 8% coupon at a 10% yield: price $950.26, modified duration 2.5249, convexity 8.94.
If the yield rises from 10% to 11% (Δy = 0.01):
| Method | Estimated change |
|---|---|
| Duration only: −2.5249 × 0.01 | −2.525% |
| Duration + convexity: −2.525% + ½ × 8.94 × 0.01² | −2.480% |
| Actual (new price $926.69) | −2.481% |
Adding convexity makes the estimate almost exact.
Why investors like convexity
The convexity term is always positive for ordinary bonds. Prices rise more when yields fall than they drop when yields rise by the same amount. Between two bonds with the same duration and yield, the one with more convexity is worth more.
- Convexity is higher for longer maturities and lower coupons.
- It matters most for large yield changes.
- Callable bonds and mortgage-backed securities can have negative convexity: their price gains are capped when yields fall, because they’re likely to be repaid early.
Next chapter: the optimal portfolio.