Duration and convexity are two important measures of bond price sensitivity to changes in interest rates.
Duration represents the average time it takes to receive the bond’s cash flows. Mathematically, duration is the weighted average of the times over which the cash flows occur, where the weights are the present values of the cash flows. The formula for Macaulay duration is as follows:
$$D = \frac{\sum_{t=1}^{T} t\times CF_t}{P}$$
where D is the Macaulay duration of the bond, CFt is the cash flow at time t, T is the bond’s maturity, and P is the bond’s price.
Convexity measures how the duration of the bond changes as interest rates change. Specifically, it measures the curvature of the relationship between the bond price and the yield. Convexity can be defined as the rate of change of duration with respect to yield. The formula for convexity is as follows:
$$C = \frac{\sum_{t=1}^{T} \frac{t(t+1)}{2}\times CF_t}{P\times (1+y)^2}$$
where C is the convexity of the bond, CFt and T are as defined in the duration formula, and y is the yield to maturity of the bond.
Duration and convexity are important because they help bond investors and analysts to understand how a bond’s price is likely to react to changes in interest rates. Specifically, bonds with higher duration and convexity are more sensitive to changes in interest rates, while bonds with lower duration and convexity are less sensitive. This information can be used to construct bond portfolios that are more or less sensitive to interest rate changes, depending on the investor’s risk tolerance and investment objectives.
For example, suppose an investor is considering two bonds: Bond A has a duration of 5 years and convexity of 20, while Bond B has a duration of 10 years and convexity of 50. If interest rates rise by 1%, the price of Bond A would be expected to fall by approximately 5%, while the price of Bond B would be expected to fall by approximately 10%. This means that Bond B is more sensitive to interest rate changes than Bond A. If the investor is interested in preserving capital and avoiding losses due to interest rate changes, they may choose to invest in Bond A instead of Bond B. Conversely, if the investor is willing to take on more risk in exchange for potential higher returns, they may choose to invest in Bond B.