What Is Convexity?
What Is Convexity?
Convexity is the secret that duration keeps: a bond's price does not fall in a straight line as yields rise, and it does not rise in a straight line as yields fall. The curve is not linear—it bends.
Key takeaways
- Duration measures the linear (straight-line) relationship between yield changes and price changes.
- Convexity measures the curvature—how much the actual price-yield curve deviates from that straight line.
- Positive convexity (the normal case) means the price-yield curve is curved upward; gains from falling rates exceed losses from rising rates.
- Negative convexity (callable bonds, mortgages) means the curve bends downward; gains are capped and losses are amplified.
- For a given duration, positive convexity is a valuable feature; negative convexity is a cost you pay (usually justified by higher yield).
The limits of duration
Suppose you hold a bond with 5-year duration, yielding 4%. Duration says: if yields rise 1%, the bond falls roughly 5%. If yields fall 1%, the bond rises roughly 5%.
This works well for small moves. But test it with a large move. Suppose yields fall 3%. Duration predicts the bond rises 15% (5 × 3%). Let's check actual prices.
A standard 5-year bond yielding 4% starts at par value (£100). If yields fall to 1%, the bond's new price is not £115—it is £114.90 or so (the exact value depends on coupon and maturity). Duration slightly underestimated the gain.
Now suppose yields rise 3% to 7%. Duration predicts the bond falls to £85 (100 − 15). But the actual price is £85.10 or so. Duration slightly underestimated the loss.
Wait: the overestimation of losses and underestimation of gains is asymmetrical. The bond gained 14.90 when rates fell 3%, but only lost 14.90 when rates rose 3%. This asymmetry is convexity. The price-yield curve is not a straight line—it curves upward.
Visualizing the price-yield relationship
Imagine a graph with yield on the x-axis (0% to 10%) and bond price on the y-axis (50 to 150). The actual price-yield curve is a smooth, upward-bending arc. Duration is the slope of the curve at a specific yield level (usually your starting yield). If you draw a straight line tangent to the curve at your starting point, that line represents the duration approximation.
For small moves (0.25%–0.50% yield change), the straight-line approximation is accurate—the curve and the line stay close. For larger moves, the curve bends away from the line. A positive convexity curve bends upward and to the left (to the favorable side for the bondholder). A negative convexity curve bends downward and to the right (to the unfavorable side).
A positive convexity bond is like a call option embedded in your favor: you get the upside of falling rates (price appreciation) and protection on the downside (losses are smaller than duration predicts because the curve is above the duration line).
Positive convexity: the normal case
Most plain-vanilla bonds have positive convexity. The reason is mathematically elegant: the price-yield relationship is based on the present-value formula, which is inherently convex (bends upward).
For a bondholder, positive convexity is valuable. If you hold a positively convex bond:
- When rates fall and your bond appreciates, you gain more than duration predicts. In late 2023 and early 2024, when yields fell sharply, bonds with positive convexity delivered outsized gains.
- When rates rise and your bond depreciates, you lose less than duration predicts. You are cushioned by the upward curve.
This asymmetry means that over the long run, in a volatile environment, positive convexity works in your favor. This is why investors are willing to own longer-duration bonds even though rates might rise—the positive convexity acts as a free option.
Negative convexity: the costly case
Some bonds have negative convexity. The curve bends downward and to the right, working against the bondholder. This happens when the bond has embedded options, specifically a call option.
Callable bonds are issued by corporations or governments with the right to repay early. Suppose a company issued a 10-year bond at 5% when rates were high. Five years later, rates fall to 3%. The company calls the bond, repays principal, and refinances at the cheaper 3% rate. The bondholder had locked in 5%, but loses that advantage.
From the bondholder's perspective, this is terrible. They expected the bond to appreciate as rates fell (positive convexity), but the call caps the gain. The issuer (the company) benefits from falling rates; the bondholder does not. This is negative convexity.
Mortgage-backed securities (MBS) are similarly negatively convex. When rates fall, homeowners refinance their mortgages, prepaying the MBS principal early. The MBS investor loses the higher cash flows they expected, and the principal is returned at a time when reinvestment rates are low. Again, gains from falling rates are capped.
For the bondholder, negative convexity is a cost. Why accept it? Usually because the issuer compensates with a higher yield. A callable corporate bond yields 5.5% instead of 5% to compensate for the call option. An MBS yields 4.2% instead of 3.8%. Investors trade away convexity in exchange for higher current yield.
Measuring convexity: the formula
Convexity is calculated mathematically using the second derivative of the bond price with respect to yield. The formula is complex, but the intuition is simple: it measures how much the curve bends.
For practical purposes, you do not need to compute convexity yourself. Bond traders and portfolio managers use pricing software that outputs convexity automatically. A typical bond might have convexity of 40–100 (the units are "basis points per basis point squared," which is a bit awkward, but the number itself gives you a sense of how much curvature there is).
The key insight: higher convexity is better (for the bondholder) if it is positive, worse if it is negative. A positively convex bond with convexity = 80 is preferable to a similar bond with convexity = 40. A negatively convex bond with convexity = −60 is less attractive than a similar bond with convexity = −20.
Convexity and risk
In volatile interest rate environments, convexity becomes crucial. Consider two bonds with the same duration but different convexity.
Bond A: 5-year duration, positive convexity = 60. Bond B: 5-year duration, negative convexity = −40.
If rates stay flat, both perform identically (same duration = same yield = same return). If rates move 0.5%, the difference is small (duration dominates). If rates move 2%, convexity starts to matter:
- Bond A: duration loss/gain is partially offset by convexity gain → net loss/gain is small.
- Bond B: duration loss/gain is amplified by convexity loss → net loss/gain is large.
Over a 5-year holding period, Bond A is less risky because convexity protects against large moves. Bond B is riskier because you lose in both directions of large moves.
This is why pension funds and insurance companies scrutinize convexity. They often avoid negative-convexity bonds (or hedge the embedded options) to reduce portfolio risk.
Convexity and duration interaction
Convexity and duration work together:
- Duration tells you how much a 1% yield move affects the bond price (first-order effect).
- Convexity tells you how much additional price change occurs due to the curvature (second-order effect).
The full approximation for a bond price change is:
ΔPrice ≈ −Duration × ΔYield + ½ × Convexity × (ΔYield)²
For a 1% yield move, the convexity term contributes ½ × Convexity. For a 2% move, it contributes ½ × Convexity × 4 = 2 × Convexity, which is much more significant.
This is why convexity matters most when rates are volatile. In a stable, low-volatility environment (like 2017–2019), convexity is a nice-to-have. In a volatile environment (like 2022, when rates swung 3%+), convexity is crucial to your total return.
Comparing bonds by duration and convexity
When comparing two bonds, smart investors look at both metrics:
- Same duration, higher positive convexity: Choose it. You get the same sensitivity to rate moves but better protection on the downside and more upside on the favorable side.
- Same duration, one has negative convexity: Be cautious. You pay for the option embedded in the bond through either a lower yield or a hidden cost in the form of capped upside.
- Higher duration, higher convexity: Often a win. The longer duration means more sensitivity to rates, but the higher convexity cushions you. This is why investors sometimes prefer longer bonds even though rates could rise.
The role of coupon rates
Interestingly, coupon rate affects convexity. A high-coupon bond has lower convexity than a low-coupon bond with the same duration. This is because high-coupon bonds return capital more quickly (through larger coupons), reducing the curvature.
A zero-coupon bond (which pays nothing until maturity) has maximum convexity for its duration. A deeply discounted bond (trading far below par) has higher convexity than a premium bond (trading above par).
Flowchart: Understanding convexity
Next
Convexity is a universal property of all bonds, but it becomes especially pronounced and strategically important when bonds carry embedded options. The next article examines positive convexity in detail—why ordinary bonds enjoy it and how to exploit it. After that, we turn to negative convexity in callable bonds and mortgages, understanding why investors accept these unfavorable characteristics in exchange for higher yields.