How to Calculate Convexity Adjustment in Bonds, with Formulas

What Is a Convexity Adjustment?

A convexity adjustment involves modifying a bond’s convexity based on the difference in forward and future interest rates. As its name suggests, convexity is non-linear. It is for this reason that adjustments to it must be made from time to time. A bond’s convexity measures how its duration changes as a result of changes in interest rates or time to maturity.

Understanding the Convexity Adjustment Formula


$\begin{aligned} &CA = CV \times 100 \times (\Delta y)^2 \\ &\textbf{where:} \\ &CV=\text{Bond’s convexity} \\ &\Delta y=\text{Change of yield} \\ \end{aligned}$​CA=CV×100×(Δy)2where:CV=Bond’s convexityΔy=Change of yield​

Implications of Convexity Adjustment for Bond Pricing

Convexity refers to the non-linear change in the price of an output given a change in the price or rate of an underlying variable. The price of the output, instead, depends on the second derivative. In reference to bonds, convexity is the second derivative of bond price with respect to interest rates.

Bond prices move inversely with interest rates—when interest rates rise, bond prices decline, and vice versa. To state this differently, the relationship between price and yield is not linear, but convex. To measure interest rate risk due to changes in the prevailing interest rates in the economy, the duration of the bond can be calculated.

Duration is the weighted average of the present value of coupon payments and principal repayment. It is measured in years and estimates the percent change in a bond’s price for a small change in the interest rate. One can think of duration as the tool that measures the linear change of an otherwise non-linear function.

Convexity is the rate that the duration changes along the yield curve. Thus, it’s the first derivative of the equation for the duration and the second derivative of the equation for the price-yield function or the function for change in bond prices following a change in interest rates.

Since duration might not accurately predict large price changes due to the yield curve’s convexity, using convexity can help estimate these price shifts.

A convexity adjustment considers the yield curve’s shape to better estimate price changes when interest rates shift significantly. We use convexity adjustments to enhance duration estimates.

Practical Example: Applying Convexity Adjustment to Bonds

Take a look at this example of how convexity adjustment is applied:

$\begin{aligned} &\text{AMD} = -\text{Duration} \times \text{Change in Yield} \\ &\textbf{where:} \\ &\text{AMD} = \text{Annual modified duration} \\ \end{aligned}$​AMD=−Duration×Change in Yieldwhere:AMD=Annual modified duration​

$\begin{aligned} &\text{CA} = \frac{ 1 }{ 2 } \times \text{BC} \times \text{Change in Yield} ^2 \\ &\textbf{where:} \\ &\text{CA} = \text{Convexity adjustment} \\ &\text{BC} = \text{Bond’s convexity} \\ \end{aligned}$​CA=21​×BC×Change in Yield2where:CA=Convexity adjustmentBC=Bond’s convexity​

Let’s say a bond’s annual convexity is 780 and its annual modified duration of 25. The yield to maturity is 2.5%, expected to rise by 100 basis points (bps):

$\text{AMD} = -25 \times 0.01 = -0.25 = -25\%$AMD=−25×0.01=−0.25=−25%

Note that 100 basis points is equivalent to 1%.

$\text{CA} = \frac{1}{2} \times 780 \times 0.01^2 = 0.039 = 3.9\%$CA=21​×780×0.012=0.039=3.9%

The estimated price change of the bond following a 100 bps increase in yield is:

$\text{Annual Duration} + \text{CA} = -25\% + 3.9\% = -21.1\%$Annual Duration+CA=−25%+3.9%=−21.1%

Keep in mind, when yield increases, prices fall, and vice versa. Adjusting for convexity is often needed when pricing bonds, interest rate swaps, and other derivatives. This adjustment is needed because bond prices change unevenly with interest rate shifts.

In other words, the percentage increase in the price of a bond for a defined decrease in rates or yields is always more than the decline in the bond price for the same increase in rates or yields. Several factors influence the convexity of a bond, including its coupon rate, duration, maturity, and current price.

The Bottom Line

Convexity adjustments modify bond yields to reflect expected future rates due to the non-linear nature of bond price changes. Convexity measures how a bond’s duration changes due to changes in interest rates or time to maturity. Duration measures the linear change in bond prices due to interest rate shifts, while convexity adjusts for non-linear changes.

Investors can use convexity adjustments to better estimate bond price changes when facing significant interest rate fluctuations. These adjustments enhance the accuracy of bond pricing models and help investors understand potential risks and rewards more clearly.