Convexity

What is convexity?

Convexity describes the curvature of the relationship between a bond’s price and its yield. While duration estimates the linear sensitivity of price to small interest-rate changes, convexity captures how that sensitivity itself changes as yields move — a second-order, non‑linear effect.

Key points

• Duration gives a first-order (linear) estimate of price change for small yield moves.
• Convexity adds a second-order correction that improves accuracy for larger yield changes.
• Positive convexity benefits bondholders when yields fall and cushions price declines when yields rise.
• Negative convexity amplifies losses when yields rise and is common in some mortgage‑backed securities (MBS) because of prepayment behavior.

How convexity works

Bond prices and yields move inversely: when yields fall, prices rise; when yields rise, prices fall. Duration assumes this relationship is a straight line. In reality the price-yield curve is curved (convex), so the change in price for a given yield move depends on where you are on that curve.

A commonly used approximation for price change ΔP given a yield change Δy:
ΔP ≈ -Duration × Δy + 0.5 × Convexity × (Δy)^2
The second term (convexity) adjusts the duration estimate, improving accuracy for larger rate shifts.

Duration vs. term to maturity

• Duration measures price sensitivity to interest-rate changes (effective weighted time to cash flows). Higher duration → greater sensitivity and higher interest-rate risk.
• Term to maturity is simply how long until the bond’s principal is repaid. Two bonds with the same maturity can have different durations depending on coupon structure.

Rule of thumb: a 1% rise in rates tends to reduce a bond’s price roughly by its duration in percent (e.g., 5‑year duration → ~5% price drop), but this ignores convexity.

Positive and negative convexity

• Positive convexity: Duration falls as yields rise (or duration rises as yields fall). Price increases accelerate when yields decline, and price decreases are less severe when yields rise. Most plain‑vanilla fixed‑rate bonds exhibit positive convexity.
• Negative convexity: Duration increases as yields rise (and decreases as yields fall). Price falls can be larger when yields rise than the price gains when yields fall. MBS and callable bonds often show negative convexity because prepayments or calls shorten expected cash flows when rates fall.

Example

Two bonds, both $100,000 face and 5% coupon:
* Bond A: 5‑year maturity, duration ≈ 4 years.
* Bond B: 10‑year maturity, duration ≈ 5.5 years.

If yields rise 2%:
* Duration estimate: Bond A ≈ −8% price change (4 × 2%), Bond B ≈ −11% (5.5 × 2%).
* Because Bond B has higher convexity, its actual price decline will be smaller than the duration‑only estimate once the convexity correction is applied.

Why rates and prices move in opposite directions

New bonds must offer yields competitive with market rates. When market rates rise, existing bonds with lower coupons become less attractive, so their prices drop until their yields match prevailing rates. Conversely, falling market rates make existing higher‑coupon bonds more valuable, pushing their prices up.

Practical uses

• Portfolio managers use convexity to measure and manage exposure to interest‑rate risk.
• Investors match convexity characteristics to risk preferences: higher convexity generally implies greater potential upside when rates fall but may also imply more sensitivity to yield changes.
• Hedging strategies often target both duration and convexity to better control portfolio behavior across a range of interest‑rate scenarios.

Factors that affect convexity

• Coupon rate: higher coupons typically lower convexity.
• Maturity: longer maturities generally increase convexity.
• Embedded options (calls, prepayments): can create negative convexity.

Short FAQs

Q: What is the simplest difference between duration and convexity?
A: Duration is a linear estimate of price change for small yield moves; convexity adjusts that estimate for curvature and larger moves.

Q: Which securities commonly have negative convexity?
A: Mortgage‑backed securities and callable bonds, because borrower prepayments or issuer calls reduce expected cash flows when rates fall.

Q: What is “roll‑down”?
A: Roll‑down (roll‑down return) is selling a bond as it ages and moves down the yield curve, potentially capturing capital gains as its yield declines toward the spot yield for its shorter remaining term.

Bottom line

Convexity refines the duration-based view of interest‑rate risk by accounting for the curvature in the price‑yield relationship. It is essential for accurately estimating bond price changes for non‑infinitesimal yield moves and for managing fixed‑income portfolios across different rate environments. Understanding both duration and convexity helps investors match bond exposure to their risk and return objectives.