What is Convexity Adjustment? Understand Here!

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Convexity Adjustment – Definition

This adjustment is done in reaction to a discrepancy between the forward and future interest rates, which must be applied to the former in order to arrive at the latter. Because of the non-linear relationship between bond prices and yields, this modification is required.

Convexity Adjustment Formula

CA=CV×100×(Δy)

where:

CV=Bond’s convexity

Δy=Change of yield

What Does the Convexity Adjustment Indicate?

The non-linear change in the price of output when the price or rate of underlying variable changes is referred to as convexity. The output price, on the other hand, is determined by the second derivative. Convexity refers to the second derivative of a bond’s price in relation to interest rates. Bond prices and interest rates fluctuate in opposite directions: when interest rates rise, bond prices fall, and vice versa. To put it another way, the connection between price and yield is convex rather than linear. The term of a bond can be used to quantify interest rate risk due to changes in the economy’s current interest rates.

Bond prices and interest rates fluctuate in opposite directions: when interest rates rise, bond prices fall, and vice versa. To put it another way, the connection between price and yield is convex rather than linear. The term of a bond can be used to quantify interest rate risk due to changes in the economy’s current interest rates.

The weighted average of the present value of coupon payments and principal repayment is known as duration. It calculates the percent change in a bond’s price for a slight change in the interest rate and is measured in years. Duration can be thought of as a tool for measuring the linear change in a nonlinear function.

The rate at which the tenor fluctuates along the yield graph is known as convexity. As a result, it’s the 1st derivative of the equation for the duration and the second derivative of the equation for the price-yield function or the function for changes in bond prices as interest rates vary.

In order to estimate a more accurate price for bigger changes in interest rates, a convexity adjustment takes into consideration the curvature of the price-yield relationship depicted in a yield curve. A convexity adjustment measure can be used to improve the estimate supplied by duration.

Convexity helps to approximate the change in price that is not recorded or explained by duration because the predicted price change using duration may not be correct for a big change in yield due to the convex shape of the yield curve.

In order to estimate a more accurate price for bigger changes in interest rates, a convexity adjustment takes into consideration the curvature of the price-yield relationship depicted in a yield curve. A convexity adjustment measure can be used to improve the estimate supplied by duration.

How to Use Convexity Adjustment?

Take a look at how convexity adjustment is used in this example:

AMD=−Duration×Change in Yield

where:

AMD=Annual modified duration

CA= 1/2 ×BC×Change in Yield

where:

CA=Convexity adjustment

BC=Bond’s convexity

Assume a bond with a 780 annual convexity and a 25.00 annual modified duration. The yield to maturity is 2.5 percent, with a 100 basis point (bps) increase expected:

AMD=−25×0.01=−0.25=−25%

Note that 100 basis points are equivalent to 1%.

CA= 1/2*780*0.01 = 0.039 = 3.9%

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

Annual Duration + CA= -25% + 3.9% = – 21.1%

Keep in mind that a rise in yield equals a drop in price and vice versa. When pricing bonds, interest rate swaps, and other derivatives, a convexity adjustment is frequently required. The unsymmetrical change in the price of a bond in respect to changes in interest rates or yields necessitates this adjustment.

In other words, the percentage gain in the bond price for a specific fall in rates or yields is always greater than the decline in the bond price for the same rate or yield increase. A bond’s convexity is influenced by a number of parameters, including the coupon rate, duration, maturity, and current price.

Wrapping Up

To accurately price securities in an arbitrage-free way, we must first understand how and why convexity adjustments occur. Furthermore, because there are so many different types of convexity adjustments that it is difficult to document them all, practitioners must be able to derive and modify such formulae on their own, as well as understand the advantages and disadvantages of various approaches, as well as the approximations and assumptions used.