Bond Convexity Adjustment: Definition, Calculation, and Applications

A convexity adjustment is a modification applied to a forward interest rate or yield to account for the expected future interest rate or yield. It addresses the non-linear relationship between bond prices and yields by adjusting for differences between forward and future interest rates. Let’s delve deeper into the formula, significance, and practical applications of convexity adjustment in finance.

Convexity adjustment is essential in bond markets due to the non-linear nature of bond price-yield relationships. It involves modifying a bond’s convexity based on discrepancies between forward and future interest rates. Since convexity is inherently non-linear, adjustments become necessary to ensure accurate pricing and risk management in bond investments.

The formula for convexity adjustment is:
CA = CV × 100 × (Δy)^2
Where:
CA = Convexity adjustment
CV = Bond’s convexity
Δy = Change of yield

Convexity adjustment accounts for the curvature in bond price-yield relationships, providing a more accurate estimation of bond prices, especially for larger changes in interest rates. Unlike duration, which measures linear changes in bond prices, convexity captures non-linear variations, offering insights into the impact of interest rate changes on bond values.

Consider a bond with an annual convexity of 780 and an annual modified duration of 25.00. If the yield to maturity increases by 100 basis points, the convexity adjustment can be calculated to estimate the bond’s price change accurately. For instance:
CA = 1/2 × BC × (Change in Yield)^2
CA = 1/2 × 780 × (0.01)^2
CA = 0.039 or 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%

Convexity adjustment is crucial in finance, particularly in bond markets, because it helps in accurately estimating the impact of interest rate changes on bond prices. By accounting for the non-linear relationship between bond prices and yields, convexity adjustment enhances pricing accuracy and risk management in bond investments.

Several factors influence the convexity of a bond, including its coupon rate, duration, maturity, and current price. Generally, bonds with longer maturities and lower coupon rates tend to have higher convexity, indicating greater price sensitivity to changes in interest rates.