Bond Convexity Calculator

Bond convexity is a measure of the curvature in the relationship between a bond's price and its yield. It captures the non-linear effect of interest rate changes on bond prices that duration alone cannot fully explain. A higher convexity means the bond's price is more sensitive to large interest rate movements, and bonds with greater convexity are generally more valuable to investors.

Duration measures the linear (first-order) sensitivity of a bond's price to changes in interest rates — it estimates how much the price changes for a 1% move in yield. Convexity captures the second-order (non-linear) effect, improving the accuracy of price change estimates for larger yield movements. Together, duration and convexity give a more complete picture of interest rate risk.

Bond convexity is calculated by summing, for each cash flow period t, the quantity [CF_t × t × (t+1)] / [(1 + periodic_yield)^(t+2)], all divided by the bond price times the square of the number of periods per year. This calculator uses the exact cash flow method to derive convexity alongside Macaulay and modified duration.

A bond with higher convexity experiences larger price increases when yields fall and smaller price decreases when yields rise, compared to a bond with lower convexity and the same duration. Positive convexity is generally desirable for investors as it provides a cushion against interest rate risk. Bonds with embedded options (like callable bonds) can exhibit negative convexity under certain conditions.

Modified convexity is calculated using expected cash flows that do not change with interest rates, making it suitable for straight (option-free) bonds. Effective convexity accounts for the fact that cash flows may change when interest rates change, which is important for bonds with embedded options such as callable or putable bonds. For plain vanilla bonds, both measures produce the same result.

Macaulay duration is the weighted average time (in years) until a bond's cash flows are received, where the weights are the present values of each cash flow relative to the bond's total price. It tells you the effective time horizon of a bond and is the foundation for computing modified duration and convexity.

The approximate price change for a given yield shift (Δy) is: ΔP ≈ –(Modified Duration × ΔP × Δy) + 0.5 × Convexity × Price × (Δy)². The duration component provides a linear estimate, while the convexity term corrects for the curvature, giving a more accurate estimate especially for large interest rate moves.

More frequent coupon payments mean cash flows are returned to the investor sooner, which shortens the effective duration and generally lowers convexity. A semi-annual bond will have a different convexity than an otherwise identical annual-coupon bond because the timing and present value weighting of cash flows differ. This calculator lets you select annual, semi-annual, quarterly, or monthly frequencies to reflect your bond's actual structure.