What Is Convexity
If duration tells you 'how many percent the price moves for a 1-percentage-point rate change,' convexity explains why that calculation is slightly off. And this discrepancy usually works in the investor's favor.
Duration is a 'straight line,' reality is a 'curve'
If you graph the relationship between a bond's price and interest rates, you get not a straight line but a gently bending curve.
Duration 'approximates' this curve as a straight line at one point. So when rates move only a little it's fairly accurate, but when rates move a lot, an error appears between the actual curve and the straight line.
Convexity is the metric that captures the 'degree of bending (curvature)' of this curve. If duration is a first-order (linear) approximation, convexity can be seen as the second-order correction added on top of it.
Why it works in the investor's favor
An ordinary bond without options has 'positive (+) convexity.' This is good news for the investor.
Positive convexity means that for a rate move of the same size, the size of the price gain (when rates fall) is larger than the size of the price loss (when rates rise).
In other words, if you calculate using duration alone, the loss when rates rise comes out larger than reality (overestimated), and the gain when rates fall comes out smaller than reality (underestimated). In reality, there's a favorable asymmetry where losses are smaller and gains are larger.
By analogy, it has the quality of 'the brakes catching a little on the way down, and getting a boost of acceleration on the way up.'
This 'favorable asymmetry' applies to ordinary bonds without call options or early-redemption provisions. (Source: Raymond James, Breckinridge)
When it matters, and the exceptions
When rates move only slightly, duration alone is accurate enough. Convexity shows its true worth when rates move a lot (roughly 1 percentage point or more). At that point you need to add the convexity correction to estimate the price change properly.
That said, not all bonds have favorable positive convexity. Callable bonds that the issuer can repay early, or bonds bundled from mortgage loans (MBS), can have 'negative (-) convexity.'
Negative convexity is, conversely, an unfavorable quality where price appreciation is capped in a situation—falling rates—that should be good. So when looking at a bond, it's important to check 'whether this bond carries an option like early redemption.'
常见问题
Q. Do beginners really need to know how to calculate convexity?
You almost never need to calculate it yourself. Actual convexity figures are often already provided in bond information or fund materials. At the beginner stage, what matters is understanding the concept that 'duration is an approximation, an error appears when rates move a lot, and for ordinary bonds that error generally leans in the investor's favor.'
Q. If convexity is high, is it always a good bond?
Positive convexity is a favorable quality, but that alone doesn't make a bond good. A bond's appeal is determined by many factors besides convexity—the issuer's credit, yield, maturity, and so on. Also, a bond with negative convexity, like a callable bond, can actually be unfavorable, so it's right to treat convexity as just one of several factors to weigh.
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