Work out convexity and explain what it means
The idea
Duration alone predicts a linear price response to a yield change, but the true relationship curves. Convexity measures that curvature, and adding a convexity term to the duration estimate — one half times convexity times the squared yield change — corrects most of the remaining error. For an option-free bond convexity is positive, so the correction is always upward: the bond gains more from a yield fall than it loses from an equal rise. This is valuable, so higher convexity commands a lower yield among bonds of equal duration, and the difference only matters for large yield moves.
Formulas
Price change with the convexity adjustment
%Δprice ≈ −ModDur × Δyield + ½ × Convexity × (Δyield)²
- %Δprice
- the percentage change in the full price
- ModDur
- modified duration
- Convexity
- the curvature of the price-yield relationship
- Δyield
- the change in yield, as a decimal
The squared term is positive whichever way yields move, which is what it means to say convexity is worth having. Add it for large yield moves; for a few basis points it changes almost nothing.
A practice question
For an option-free bond, is convexity positive or negative?
- Positive — correct
- NegativeWhy not: Negative convexity requires an embedded option — a callable bond at low yields, or a mortgage-backed security facing prepayment.
- It depends on the couponWhy not: The coupon changes the *size* of convexity but never its sign. Every option-free bond is positively convex.
The price-yield curve bends away from the tangent in the holder's favour in both directions, which is positive convexity.
A harder one
A bond has modified duration 8 and convexity 100. Yields fall by 1%. What is the estimated price change?
- Approximately 8.0%Why not: The duration term alone. The convexity correction adds 0.5 x 100 x 0.01², which is another 0.5%.
- Approximately 8.5% — correct
- Approximately 9.0%Why not: A correction of a full point, which would need the convexity term computed without the one-half or without squaring the yield change.
Duration gives 8.0% and the convexity term adds 0.5 x 100 x 0.01 squared = 0.5%. The adjustment is small for a one-point move and grows with the square of the change.
The hardest tier
Two bonds have identical duration and different convexity. What should their yields look like?
- The more convex bond should yield less, since its convexity is valuable — correct
- The more convex bond should yield more, to compensate for the curvatureWhy not: Curvature is a benefit rather than something needing compensation. The holder gains more when yields fall and loses less when they rise.
- Their yields should be identical, since duration is the sameWhy not: Duration is the first-order term only. Anything an investor values is paid for in yield.
Anything an investor values is paid for in yield, and convexity is no exception.
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