Can you walk through the derivation of the second-order bond price approximation?

Question

AcadiFi · Accepted Answer

Bond price P is a function of yield y. We can expand P(y + Δy) around y using a Taylor series:

P(y + Δy) = P(y) + P'(y) × Δy + ½ × P''(y) × (Δy)² + higher-order terms

Rearranging for the percentage price change:

ΔP / P = [P'(y) / P] × Δy + ½ × [P''(y) / P] × (Δy)²

Now define:
- Modified Duration (MD) = −P'(y) / P → P'(y) = −MD × P
- Convexity (C) = P''(y) / P → P''(y) = C × P

Substituting:

ΔP / P = −MD × Δy + ½ × C × (Δy)²

That's the formula. The negative sign on duration reflects the inverse price-yield relationship (P'(y) is negative). Convexity is positive for standard bonds because the price-yield curve is convex from above (curves upward when viewed as P on y-axis against y).

**Derivation of convexity**: For a fixed-coupon bond with cash flows CFt at time t:

P(y) = Σ CFt / (1+y)^t

First derivative: P'(y) = Σ −t × CFt / (1+y)^(t+1)
Second derivative: P''(y) = Σ t(t+1) × CFt / (1+y)^(t+2)

Convexity = P''/P = Σ [t(t+1) × CFt / (1+y)^(t+2)] / P

The t(t+1) weighting explains why: (a) long-maturity bonds have high convexity (large t²), (b) zero-coupon bonds maximize convexity for given duration (all weight at t=T), (c) bonds with late cash flows (low coupons) have higher convexity than high-coupon bonds of same maturity.

**Higher-order terms**: The Taylor series has a third-order term involving P'''(y) × (Δy)³/6, which captures 'skew' in the price-yield relationship. Rarely used in practice because (Δy)³ is tiny for realistic moves, but quants computing options-on-bonds sometimes include it. For standard duration/convexity applications, the two-term expansion covers >99% of the price change for moves up to 200-300 bps.

**Intuition**: Duration is the slope; convexity is the curvature. If you drew the price-yield curve on a chart, duration tells you the tangent line's slope at current yield, and convexity tells you how much the curve bends away from that tangent as you move along yield axis.

Can you walk through the derivation of the second-order bond price approximation?

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