Why do callable bonds and MBS exhibit negative convexity, and what's the cost to investors?

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AcadiFi · Accepted Answer

For an option-free bond, the price-yield relationship is everywhere convex (curving up when plotted with price on y-axis) — meaning the second derivative P'' > 0. When yields fall, price rises at an increasing rate; when yields rise, price falls at a decreasing rate. That's the 'positive convexity' that helps holders asymmetrically. A callable bond gives the issuer the right to redeem at a fixed call price (usually par or slight premium). As yields fall below the call threshold, the call becomes increasingly likely; the bond's price cannot rise indefinitely because investors know it'll be called at the call price. The price-yield curve flattens and eventually bends downward at low yields — this is the negative convexity region. ```mermaid flowchart TD A[Yields Fall] --> B{Call Option
ITM?} B -->|Yes| C[Price Capped
Near Call Price] B -->|No| D[Normal Convex
Price Gain] C --> E[Negative
Convexity Region] D --> F[Positive
Convexity] ``` MBS show similar behavior: falling rates trigger homeowner refinancing, returning principal at par when investors would have preferred continuing coupons. The result: price appreciation is capped, duration shortens exactly when the investor wanted it to extend. **Convexity cost = option cost**: Mathematically, callable bond = option-free bond − call option (issuer bought call). The price of that embedded call is the 'option cost', and it shows up as a lower price (higher yield) for the callable vs option-free benchmark. That yield spread — often called 'option-adjusted spread' (OAS) when isolated from other spread components — compensates investors for bearing the negative convexity. **Quantifying the cost**: Z-spread − OAS = option cost. If a callable agency bond trades at Z-spread 55bps and OAS 30bps, the 25bps difference is the investor's compensation for the call optionality — the cost of negative convexity in yield terms. **Practical implication**: In a falling-rate regime, callable/MBS portfolios underperform duration-matched option-free portfolios. In a rising-rate regime, MBS underperform again (extension risk — duration lengthens as prepayments slow). This 'lose-lose' asymmetry is why MBS typically offer higher yields than Treasuries of same average life.

Why do callable bonds and MBS exhibit negative convexity, and what's the cost to investors?

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