Negative Convexity

Key takeaways

• Negative convexity occurs when a bond’s price–yield relationship is concave: price rises less when yields fall and falls more when yields rise.
• Common in callable bonds and mortgage-backed securities because embedded options (calls or prepayments) limit upside price gains as rates decline.
• Convexity is the second derivative of price with respect to yield; it refines duration-based price estimates and is useful for risk management.

What negative convexity means

Convexity measures how a bond’s duration changes as yields change. If convexity is negative, the price–yield curve bends downward (concave). For such bonds, a drop in interest rates does not raise price as much as expected (because of increased likelihood of calling or prepayment), and price sensitivity to rising rates can be larger. In practice, negative convexity increases interest‑rate risk and complicates hedging.

Examples of instruments that often show negative convexity:
* Callable corporate bonds (issuer can redeem at predetermined price).
* Mortgage‑backed securities (borrowers can prepay mortgages when rates fall).

How convexity is measured (simple finite‑difference approximation)

A practical approximation for convexity uses three observed prices around a small yield change dy:

Convexity ≈ [P(+) + P(−) − 2·P(0)] / [2·P(0)·(dy)^2]

Where:
* P(0) = current bond price
* P(+) = price if yield falls by dy
* P(−) = price if yield rises by dy
* dy = change in yield in decimal form (e.g., 1% = 0.01)

This yields an estimate of C = (1/P)·d^2P/dy^2.

Example
* P(0) = $1,000
P when yield falls 1% (dy = 0.01): P(+) = $1,035
P when yield rises 1%: P(−) = $970

Convexity ≈ (1,035 + 970 − 2·1,000) / (2·1,000·0.01^2)
Convexity ≈ 5 / 0.2 = 25

Using duration and convexity to estimate price change

A better approximation for percentage price change for a small yield change Δy is:

%ΔP ≈ −Duration × Δy + 0.5 × Convexity × (Δy)^2

Notes:
* Duration is typically the modified duration (sensitivity of price to yield).
* Convexity is the measure computed above (C ≈ (1/P) d^2P/dy^2).
* The convexity term corrects the linear (duration) estimate and is especially important for larger Δy.

Using the example convexity (C = 25) and Δy = 1% (0.01), the convexity contribution is:
0.5 × 25 × (0.01)^2 = 0.00125 = 0.125% of price ≈ $1.25 on a $1,000 bond.

Practical implications for investors

• Risk: Negative convexity increases downside sensitivity to rising rates and limits upside when rates fall — effectively making the bond behave worse than a non‑callable bond with similar duration.
• Hedging: Managing a portfolio with negatively convex instruments typically requires more active hedging and can be more expensive because convexity cannot be fully offset by linear duration hedges.
• Valuation: Always include convexity (not just duration) when estimating price changes for non‑linear instruments, especially for larger interest‑rate moves.

Bottom line

Negative convexity describes concave price behavior caused by embedded options (calls, prepayments). Measure it with a finite‑difference convexity approximation and use the convexity term together with duration to obtain more accurate price-change estimates and to manage interest‑rate risk effectively.