Understanding Duration

Bond duration measures sensitivity to interest rate changes. The Macaulay duration represents the weighted average time until a bondholder receives the bond's cash flows. Modified duration divides Macaulay duration by one plus the yield per period. This approximation indicates how much a bond's price will move for a one percentage point change in yield. Investors use duration to compare interest rate risk across bonds.

Convexity Matters

Convexity captures the curvature in the price-yield relationship. A bond with higher convexity experiences larger price increases when yields fall and smaller price declines when yields rise, compared with a bond of lower convexity but equal duration. Combining duration and convexity gives a more accurate estimate of price changes for significant yield shifts.

How the Calculator Works

Enter the bond's face value, annual coupon rate, yield to maturity, years until maturity, and the number of coupon payments per year. The script discounts each coupon payment and the final redemption value using the yield per period. It then multiplies each present value by the time of payment to compute the numerator for duration and by the time squared for convexity. The formulas are:

Where is the present value of cash flow $t$, $n$ is the total number of payments, $y$ is the yield per period, and $\mathrm{Price}$ is the bond's present price. The calculator outputs Macaulay duration, modified duration, and convexity.

Practical Applications

Portfolio managers use duration and convexity to hedge interest rate exposure. For instance, a portfolio with high duration will lose more value when rates rise. By selecting bonds with specific duration and convexity characteristics, managers can match liabilities, immunize portfolios, or position for expected rate movements. Individual investors can also benefit from understanding these metrics when comparing bond funds or constructing fixed-income ladders.

Limitations

This calculator assumes constant yields and equal time periods between payments. It does not account for embedded options such as call features, which can alter effective duration. Additionally, for bonds with very long maturities or large coupon disparities, the approximations may be less precise. Nonetheless, it offers a useful starting point for analyzing interest rate risk.

Example

Suppose a bond has a face value of $1,000, a 5% annual coupon, a 4% yield to maturity, 10 years remaining, and pays semiannually. Entering these values reveals a Macaulay duration of about nine years and a convexity of roughly 85. Such information helps gauge the potential price change if yields move up or down by a certain amount.