Bond Convexity Calculator

Bond convexity measures the non-linear relationship between a bond's price and its yield, acting as a second-order derivative approximation. While modified or effective duration provides a linear estimate of price change for small yield shifts, convexity captures the bending of the price-yield curve. Effective convexity is computed using the starting bond price, the coupon parameters, and the resulting prices when yields move up and down by a differential (Δy). The standard mathematical formulation for effective convexity is expressed as: EffConvexity = (P_+ + P_- - 2P_0) / (2 * P_0 * (Δy)^2), where P_0 is the initial bond price, P_+ is the price when yield decreases, P_- is the price when yield increases, and Δy is the yield differential.

Consider a corporate bond with a par value of $1,000, an annual coupon rate of 6% paid semi-annually (frequency = 2), and 5 years remaining until maturity. The current yield to maturity (YTM) is 5.5%, resulting in a baseline bond price (P_0) of $1,022.30. To find the effective convexity, we apply a yield differential (Δy) of 50 basis points (0.0050). First, calculate the up-price when the yield drops to 5.0%, yielding $1,043.76. Next, calculate the down-price when the yield rises to 6.0%, yielding $1,001.38. Substituting these values into the effective convexity formula: ($1,043.76 + $1,001.38 - 2 * $1,022.30) / (2 * $1,022.30 * (0.0050)^2), which simplifies to ($2,045.14 - $2,044.60) / (0.051115) = approximately 10.56. This positive convexity indicates that bond prices rise at an accelerating rate when yields fall.

When analyzing fixed-income assets, always combine duration and convexity for a comprehensive risk profile, because duration alone underestimates price gains when rates drop. Ensure that your yield differential input is small enough to capture local curvature accurately, typically ranging between 10 to 50 basis points. Finally, remember that bonds with embedded options, such as callable or puttable bonds, can exhibit negative convexity, making effective convexity an essential metric for accurate risk management.