What Is Negative Convexity?
Let me explain negative convexity to you directly: it happens when a bond's yield curve takes on a concave shape. As someone who's delved into bond mechanics, I can tell you that a bond's convexity is essentially the rate of change of its duration, calculated as the second derivative of the bond's price relative to its yield. You'll see this most often in mortgage bonds, which are typically negatively convex, and in callable bonds, especially when yields are low.
Key Takeaways on Negative Convexity
Here's what you need to grasp: negative convexity means the bond's price drops when interest rates do, creating that concave yield curve. I always recommend assessing a bond's convexity as a solid method to measure and manage your portfolio's exposure to market risk—it's straightforward and effective.
Understanding Negative Convexity
Think about a bond's duration: it shows how much the bond's price shifts with changes in interest rates. Convexity, on the other hand, illustrates how that duration itself changes as rates fluctuate. Normally, when rates drop, a bond's price goes up—but not with negative convexity. In those cases, the price actually decreases as rates fall.
Take a callable bond as an example. If interest rates drop, the issuer has more reason to call the bond back at par, so the bond's price won't rise as much as a non-callable one. In fact, it might even drop if the call becomes highly likely. That's exactly why the price-yield curve for a callable bond is concave, showing negative convexity. I see this pattern a lot in my analyses, and it's critical for you to recognize it in your investments.
Convexity Calculation Example
Duration alone isn't a perfect predictor of price changes, so we calculate convexity to refine those estimates. It's a key tool for risk management, helping you gauge and control market risk in your portfolio, ultimately making your price predictions more accurate.
The full formula for convexity is complex, but you can use this approximation: Convexity approximation = (P(+) + P(-) - 2 x P(0)) / (2 x P(0) x dy^2), where P(+) is the bond price with decreased rates, P(-) with increased rates, P(0) is the current price, and dy is the rate change in decimal form.
Applying the Convexity Approximation
- Suppose a bond is priced at $1,000. Drop rates by 1%, and the new price is $1,035. Increase by 1%, and it's $970.
- Plugging in: Convexity approximation = ($1,035 + $970 - 2 x $1,000) / (2 x $1,000 x 0.01^2) = $5 / $0.2 = 25.
Using Convexity Adjustments
To adjust for convexity when estimating price with duration, use: Convexity adjustment = convexity x 100 x (dy)^2. In our example: 25 x 100 x (0.01)^2 = 0.25.
Finally, for the full bond price change estimate: Bond price change = duration x yield change + convexity adjustment. This is how I approach these calculations to get reliable results—apply it directly to your scenarios.
