Bond Convexity Calculator
This bond convexity calculator works out the convexity of a coupon bond, along with its price and modified duration, so you can gauge how sharply its price responds to changes in yield. Duration tells you the straight-line sensitivity of a bond's price to yields, but the real price-to-yield relationship is a curve, not a line, and convexity measures that curvature. It matters because duration alone underestimates the price gain when yields fall and overestimates the loss when yields rise: convexity supplies the correction that closes the gap, and the correction grows with the size of the yield move. Enter the face value, the annual coupon rate, the market yield you are pricing at, the years to maturity, and how many coupon payments fall in a year, and the calculator builds the full schedule of cash flows, discounts them at the market yield to get the price, then computes the convexity and modified duration in annual terms. It reports convexity as a number, the price in dollars, and the modified and Macaulay durations in years. Higher convexity is generally prized by investors because it means a bond gains a little more when rates drop and loses a little less when they rise, for the same duration. Fixed-income analysts and students use these figures together to estimate price moves: price change is roughly minus duration times the yield change, plus a half times convexity times the yield change squared.
This 10-year bond, priced at $1,081.76, has a convexity of 75.47 and a modified duration of 7.92 years. For a 1% move in yields, convexity adds about 0.38% to the price change that duration alone predicts.
Assumes level coupons, a flat yield and settlement on a coupon date. Convexity is annualised by the payment frequency. Not financial advice.
How it works
The calculator builds every cash flow: a coupon of face value times coupon rate divided by payments per year at each period, plus the face value repaid at the final period. Each cash flow is discounted at the periodic yield, the market yield divided by payments per year, and the discounted values are added to get the price. Macaulay duration in periods is the time-weighted average of those discounted cash flows divided by the price, and dividing by payments per year puts it in years. Modified duration is Macaulay duration divided by one plus the periodic yield. Convexity in periods is the sum of each cash flow times its period times the next period, divided by price times one plus the periodic yield squared, and dividing by payments per year squared annualises it.
Worked example
Take a $1,000 bond with a 5 percent annual coupon paid semi-annually, a 4 percent market yield, and 10 years to maturity. That is 20 periods, each paying a $25 coupon, with the $1,000 face value returned at period 20, all discounted at a 2 percent periodic yield. Discounting and summing gives a price of about $1,081.76, above face value because the coupon beats the yield. The time-weighting produces a Macaulay duration of about 8.08 years and a modified duration of about 7.92 years, and the convexity works out to about 75.47. For a 1 percent yield move, the convexity term, one half times 75.47 times 0.01 squared, adds about 0.38 percent to the price estimate from duration alone.
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