Convexity - It's Not That Complicated
Updated: Jan 4
Convexity is really just a measurement of change and it's typically good to have
Duration and Convexity
Convexity is probably the second most important concept in fixed income, just behind duration. It’s no coincidence that convexity is the second derivative of the price and yield relationship with duration being the first.
Duration and convexity go hand-in-hand. Duration is the change in price for a change in yield and is a good approximation for price movement for smaller changes in interest rate movements. As you have a larger changes in rates, the duration approximation gets further and further away from the actual price movement. This is because of convexity.
To get a better approximation for the change in price you will need to take the convexity effect into consideration. You can see in Figure 1, that if yields go down, duration alone will underestimate the price increase. Similarly, if yields go up duration alone will overestimate the price decline. You can also see that for small changes in yield, duration (the tangent line) will have a very good approximation for the change in price. If you want to more accurately reflect the price change for large changes in yield, you will need to add the convexity effect.
I want to make it clear, you will likely never actually calculate convexity. If you are in finance, you will almost certainly have software that will provide the convexity of a security for you. However, you must know how to correctly interpret convexity. I will provide the formulas for convexity as they should help you understand it, but I'll also provide context so you should get the logic behind it.
To calculate the price impact of convexity, you will first need to measure convexity and then apply the adjustment to the price. You can measure convexity using the formula in Figure 2.
Once you have the convexity measure, you will need to apply the convexity adjustment formula seen in Figure 3.
Once you have the convexity adjustment, you can then combine the duration and convexity adjustments for a much closer approximation of the change in the bond price for larger changes in yield.
Don't get too bogged down in the actual calculations here. Just remember the point here is that for larger changes in yield, duration alone is not a great estimate of price change and convexity is good. The intuition is this; if you have a bond that is positively convex, then this will add value as yields change. So when you analyze a security that has positive convexity, just know this will add value. The more convex the bond typically the better. If you remember only one formula here, then remember Figure 4. This will help you understand that convexity is additive to the change in price. The next section should help drive this point home.
Convexity as the Rate of Change of Duration
Similar to duration, there are multiple ways to think about convexity. Now that you know what convexity is, and that it adds value, you should also be aware that it is referred to as the rate of change of the rate of change. This makes sense since mathematically it’s the second derivative of the price yield relationship and duration is the first derivative.
Convexity is the rate of change of duration. It’s not uncommon to think of convexity as the accelerator for duration. That is, how fast duration is changing. Recall, duration is how much your price changes for movements in interest rate. So convexity is how fast your duration moves.
The chart in Figure 5 Illustrates how higher convexity can lead to larger price changes given large interest rate movements.
For a -300bps decline, the bond with more convexity has a ~+83% increase in price. The bond with low convexity (actually negative and I'll get into that) has a ~+1% increase in price. Now this is an extreme example, and the low convexity bond actually has negative convexity but the point is well illustrated.
This naturally leads us to negative convexity. Negative convexity basically means that as yields decline, the rate of price change, or duration, will also decrease. You will still get price appreciation, however, the increase in the rate of the appreciation will be declining. At some point, the increase in price will stop and you might actually even get a decline in price. Why does negative convexity happen?
Negative convexity is seen in bonds with imbedded call options. As the yields decline, the option becomes more in the money and the likelihood of the bond being called rapidly increases. Think about it this way, let’s say you issue a bond paying 5% and then rates drop to 2%. What would you do? You should call the bond! Why would you continue to pay 5% when you can call the bond, and issue new debt at 2%? You would save 3% by calling and issuing new debt (of course we’re ignoring all transaction costs).
Some of the most common negative convexity bonds are mortgage backed securities (“MBS”). (In the chart above, the security with negative convexity is an MBS 6% coupon security). This is because every MBS has an embedded call options since every home owner can call i.e., refinance their home whenever they want. Most home owners are sensitive to interest rates and will refinance if they can lower their monthly payment by calling their high rate mortgage and reissuing the debt at a lower interest rate. Because of this call option, the price of a negative convexity bond will cap out right around the call price of the security. It would be hard to pay $110.00 for a bond that could get called tomorrow at $100.00. That’s a 10% loss on one trade. It’s this reason why negative convexity caps the upside price appreciation when rates decline.
I hope this helps in your understanding of convexity. You should remember that convexity aids in explaining the price movement of bonds that duration will miss given larger interest rate movements. Convexity is also your friend as it is typically additive. However, if a bond has an embedded call option, convexity will be negative and cap your upside price gain at a level near the call price. If you have any questions or would like to any further information please reach out. Thanks for reading.