Convexity
March 4, 2025
What Is Convexity?
Convexity is a mathematical concept in fixed income portfolio management that is used to compare a bond’s upside price potential with its downside risk. It builds on the modified duration, which is a measure that examines the bond’s price sensitivity to changes in interest rates, but however has one caveat – it assumes that bond prices and interest rates have constant linear relationship, which in reality is not the case. Convexity aims to resolve this by accounting for any effect that interest rates could potentially have on the bond’s duration and therefore measures the rate of change in duration (or the rate of change to the rate of change in bond’s price).
Key Learning Points
- Convexity is a measure that similarly to modified duration estimates the bond’s sensitivity to changes in interest rates
- However, while modified duration assumes linear relationship between prices and yield, convexity lies on the assumption that this relationship is curvature
- There are two types of convexity – positive where duration rises and yields fall, and negative in which duration increases as yields increase
- Although it can also be used for smaller changes in interest rates, convexity is generally more useful for larger changes
- In normal market environment, higher coupons would result in lower bond convexity
Understanding Convexity
Although modified duration is a useful tool for estimating the bond’s interest rate risk, the prediction is not entirely accurate since it assumes a linear relationship (i.e. moving in a straight line) between bond prices and yields. As shown on the chart below, in reality this relationship is convex, and convexity helps explain the difference between predicted and actual bond prices.
This difference arises from the fact that bond prices react differently to small and large changes in interest rates.
For example, when yields fall bond prices rise, but due to convexity the price does increase more steeply than a linear calculation using the modified duration would estimate. On the other hand, when yields rise bond prices fall, but not as steeply as predicted by a linear model.
By refining the modified calculation investors get a closer estimate to the actual price, which helps them reduce losses when yields rise and enhance gains when yield fall.
Drivers for Convexity
Different bonds have different levels of convexity and this is influenced by a few key factors such as the bond’s time to maturity and coupon size. We look into this below:
As a result, the modified duration provides a fairly accurate estimation when there is a small change in yields and the prediction error will be relatively insignificant. Nevertheless, the model does not account for the curvature in larger yield changes, and this is where convexity becomes particularly useful.
Convexity Formula
To calculate the approximate convexity of a bond we need use the below formula:
Where:
(P+) – the bond’s price if yield increase
(P−) – the bond’s price if yield decrease
(P0) – the current bond price
Δy – the change in yield
Example of Convexity
Below are some details about a bond:
- Matures in 5 years
- Has a face value of £1000
- Coupon is 5% per annum
- Yield to Maturity (YTM) is 4%
- Current price is £1038.2
First, we make the following assumptions:
- (P+) – If yield decreases by 0.5%, the bond price rises to £1,057.14
- (P−) – If yield increases by 0.5%, the bond price drops to £1,020.07
Then we use the above formula to calculate the bond’s convexity:
Higher degree of convexity means that the bond is less affected by interest rate volatility relative to low convexity bonds.
Convexity and Risk
One risk management strategy that is popular among financial institutions (and insurers in particular) to mitigate the potential risks that can arise from convexity is convexity hedging. Instruments such as mortgage-backed securities (MBS) are known for having negative convexity, due to their embedded prepayment option, and can fall more rapidly when interest rates rise and achieve lower return when rates fall.
This typically involves taking positions in financial instruments with negative correlations (i.e. move in the opposite direction) with the convexity of the assets or liabilities being hedged. This is typically used to achieve optimal interest rate risk management and better capital adequacy.
Negative and Positive Convexity
Generally speaking, there are two types of convexity:
- Positive – it means that as interest rates fall, bond prices increase at an accelerating rate, and conversely when interest rates rise, bond prices decrease at a decelerating rate. This asymmetry would typically benefit investors by providing greater returns when yields drop and smaller losses when yields rise.
- Negative – some bonds (specifically those with embedded options such as callable bonds) can show negative convexity in certain yield environments. For example, when interest rates fall, issuers are more likely to call the bond, capping its price appreciation. Therefore, these bonds could experience smaller price returns when yields fall and steeper price falls when yields rise.
As a rule of thumb, non-callable bonds would normally have positive convexity, while many bonds that can be redeemed prior to maturity (callable bonds, i.e. those that have an embedded option) should have negative convexity.
Conclusion
Overall, while modified duration is among the most popular measures to gauge the bond’s price sensitivity to interest rates changes, it is less accurate when accounting for larger changes. This is where convexity becomes more useful in what is essentially providing a refined and more accurate calculation of the modified duration. Convexity is influenced by several factors such as coupon size and maturity and is widely used as a risk management tool and/or to enhance potential returns.