Positive vs Negative Convexity
Positive vs Negative Convexity
The simplest way to understand convexity is this: bonds with positive convexity are friends when rates move large; bonds with negative convexity are foes.
Key takeaways
- Positive convexity (ordinary bonds) means falling rates deliver outsized price gains; rising rates produce smaller losses than duration predicts.
- Negative convexity (callable bonds, MBS) means the bondholder loses optionality—gains are capped and losses are amplified.
- Most plain-vanilla bonds have positive convexity; the cost of negative convexity is compensated by higher yield.
- In volatile rate environments, positive convexity acts as a free insurance policy; negative convexity is an insurance policy you sold.
- Understanding which bonds you own—positively or negatively convex—is essential to managing interest rate risk.
Positive convexity: the typical bond
Most bonds you encounter—government bonds, investment-grade corporates, high-quality municipals—are positively convex. Their price-yield curve bends upward and to the left.
What positive convexity means for you: When rates fall 1%, the bond does not appreciate exactly 1% × duration; it appreciates more. When rates rise 1%, the bond does not depreciate exactly 1% × duration; it depreciates less. This asymmetry favors the bondholder.
Consider a 5-year bond yielding 4%, with positive convexity of 50. If yields fall 1% (to 3%):
- Duration predicts: price rises 5%.
- Convexity adds: ½ × 50 × (1%)² = 0.25% extra gain.
- Actual price rise: roughly 5.25%.
If yields rise 1% (to 5%):
- Duration predicts: price falls 5%.
- Convexity subtracts: ½ × 50 × (1%)² = 0.25% loss is offset.
- Actual price fall: roughly 4.75%.
Over time, in a volatile rate environment, these small asymmetries compound into meaningful outperformance. This is why positive convexity is valuable.
Historical example (2023–2024): In late 2023, Federal Reserve rhetoric shifted toward potential rate cuts. Long-duration bonds (10–30 years) with strong positive convexity posted exceptional returns because:
- Rates fell 1.5%–2%, so duration drove gains.
- The large rate moves amplified convexity gains.
- Investors reallocated from stocks to bonds, driving prices higher.
A 20-year Treasury with duration 18 and positive convexity 200+ delivered 18%+ gains in a few months—much more than duration alone would predict. The extra return came from convexity.
Negative convexity: the embedded option cost
Callable bonds and mortgage-backed securities are negatively convex. Their price-yield curve bends downward and to the right, working against the bondholder.
A callable bond: A corporation issues a 10-year bond at 5% coupon. Five years later, rates fall to 3%. The corporation exercises its call option, repays the bondholder £100 per bond, and refinances at 3%.
From the bondholder's perspective: you were anticipating a 5% coupon for 10 years. Now you receive only 5 years of coupons and your principal is returned early, forcing you to reinvest at 3%. You lost the expected gain from falling rates.
The price-yield curve bends downward because as rates fall, the call option becomes more likely to be exercised. The bondholder's upside is capped at the call price (usually par or a slight premium). You do not get the full price appreciation that a non-callable bond would enjoy. This is negative convexity.
A mortgage-backed security (MBS): A mortgage investor holds £1 million of 30-year mortgages yielding 4%. Homeowners are locked into 4% mortgages. If rates fall to 2%, homeowners refinance, prepaying the mortgage. The MBS investor receives principal early and must reinvest at 2%.
Again, falling rates cap the bondholder's gains. The MBS curve bends downward.
The price of negative convexity
If negative convexity is so bad, why do investors accept it? Answer: compensation via higher yield.
In May 2024, suppose:
- A 5-year non-callable Treasury yields 3.5% with positive convexity.
- A 5-year callable corporate bond yields 4.5% with negative convexity.
- A 5-year mortgage-backed security yields 4.0% with negative convexity.
The extra yield (100 basis points for the callable, 50 for the MBS) is the market's price for taking on negative convexity. Investors compare:
- Certainty of Treasury yield at 3.5% (positive convexity is a bonus).
- Higher yield of 4.5% on the callable (but capped gains if rates fall).
Which is better? It depends on your forecast. If you expect rates to fall, the Treasury is more attractive (positive convexity pays off, and you get appreciated capital). If you expect rates to rise or stay flat, the callable might be worth it (the call is unlikely to be exercised, and you pocket the extra 100 basis points).
Comparing positive and negative convexity in different rate scenarios
Suppose you have £100 to invest and choose between:
Bond A (positive convexity): 5-year duration, yield 3.5%, convexity +50. Bond B (negative convexity): 5-year duration, yield 4.5%, convexity −30.
Scenario 1: Rates fall 2%
- Bond A: Duration gain = 10%, convexity gain = ½ × 50 × 4 = 1%, total = 11% return.
- Bond B: Duration gain = 10%, convexity loss = −½ × 30 × 4 = −0.6%, plus the call might be exercised, capping price appreciation. Total = 9%–10% return.
Bond A wins decisively.
Scenario 2: Rates rise 2%
- Bond A: Duration loss = −10%, convexity loss = −½ × 50 × 4 = −1%, total = −11% loss.
- Bond B: Duration loss = −10%, convexity gain = +½ × 30 × 4 = +0.6%, total = −9.4% loss. Plus, the call is unlikely to be exercised, so you keep the 4.5% yield.
Bond B loses less and pays more yield. Bond B is better in this scenario.
The tradeoff is evident: positive convexity rewards large favorable moves; negative convexity protects against large unfavorable moves (by capping losses) but at the cost of capped gains and extra compensation the issuer must pay.
Callable bonds: the call option explained
A callable bond is equivalent to owning a non-callable bond MINUS a call option on the bond (an option the issuer owns on you).
Bond Value = Non-Callable Bond Value − Call Option Value
The call option value is embedded in the higher yield of the callable. If a non-callable bond yields 3.5% and a callable yields 4.5%, the difference (100 basis points) is, roughly, the annual cost of the call option.
The call option becomes valuable (and likely to be exercised) when interest rates fall. As rates drop, the call option's value rises, and the bondholder's maximum price appreciation is capped. This is why callables have negative convexity.
Mortgage-backed securities: prepayment risk
MBS are negatively convex due to prepayment risk. When rates fall, homeowners refinance, and prepayments accelerate. The MBS investor's cash flow acceleration shortens the effective duration and limits price appreciation. The mathematics is complex, but the outcome is simple: falling rates = capped upside for the MBS holder.
This is particularly painful in an environment of rapidly falling rates. In March 2020, when yields plummeted due to Federal Reserve rate cuts, MBS did not rally as much as Treasuries, disappointing MBS investors. In late 2023, the same dynamic played out. MBS holders missed the rally because prepayments capped their gains.
Floaters and other convexity variations
Not all bonds fit neatly into "positive" or "negative" convexity:
- Floating-rate bonds have near-zero duration (because coupons reset), so convexity is minimal. They are stable in price but offer minimal capital appreciation if rates fall.
- Putable bonds (bonds with a put option owned by the bondholder) have positive convexity because the put protects you on the downside. You get both positive convexity from the bond itself plus the put option. This is rare and expensive, so putable bonds are uncommon.
- Convertible bonds (convertible to stock) have hybrid characteristics; convexity depends on the stock price and conversion features.
Why institutional investors obsess over convexity
Pension funds, insurance companies, and bond traders study convexity carefully because it directly affects returns in volatile environments. A pension fund managing billions in liabilities must understand whether its portfolio is long convexity (positioned to benefit from large moves) or short (vulnerable to large moves).
After 2022, when the Federal Reserve raised rates sharply (rates rose 4%+ in one year), convexity assumptions were tested. Many investors who had assumed "bonds are stable, low-return assets" discovered that duration and convexity had hidden them from large losses. The lesson: understand your convexity, or it will surprise you.
Decision tree: positive vs negative convexity
Next
Positive convexity in ordinary bonds is a gift: a free option that favors falling rates and stable returns. But negative convexity in callable bonds and mortgage-backed securities is deliberate and strategic. The issuers (corporations, mortgage servicers) own the options and exercise them when rates move against the bondholder. The next two articles dive deeper: one on callable bonds and their negative convexity, and one on mortgage-backed securities and how prepayment behavior amplifies losses in favorable rate environments.