# How to calculate the new price of a bond using duration rule and duration with convexity rule?

A bond with a 30 year maturity, par value of \$1000 and is 8% p.a. coupon is selling at an yield to maturity of 8% p.a. The modified duration of the the bond at its yield is 11.26%, and its convexity is 212.4. If the bond's yield increases from 8% to 10%, how to calculate the new price of the bond using the duration rule and how to compare this answer with one calculated using the duration with convexity rule.?

I know the formula for Modified Duration is -1/(1 + y) * Macaulay Duration and formula for convexity is (Modified Duration)^2 - ▲ Modified Duration / ▲ y.

But since Macaulay's duration is not given, I am unable to proceed in solving this problem. Any solution would be highly helpful.

• By inspection of the first sentence (no calculations), get the current price of the bond. Then from the Yield Change and the Modified Duration (both given) find the price change and hence the new price. – noob2 Sep 6 '20 at 15:18
• @noob2 Could you try to solve this? – Silent_bliss Sep 8 '20 at 6:05
• At a yield of 8% the price of the bond is 1000 (bond is at par). The duration is 11.25778 (not 11.26%, that's an error). At a yield of 10% a detailed calculation shows the price of the bond is 811.4617. That is the exact answer. The approx answer with the duration rule is 1000*(1-(0.10-0.08)*11.26) = 1000*(1-0.225156) = 774.844. This is below the true value, the estimate could be improved by adding the convexity. – noob2 Sep 8 '20 at 8:21
• At a yield of 8% the convexity is 212.4325. So the duration and convexity rule says there is a -0.225156 effect of duration (see above) and a 0.5*212.4325*(0.02)^2 = 0.042487 effect of convexity. The total price change is -0.182669 so the new price estimate is 817.3308, quite close to the actual price. – noob2 Sep 8 '20 at 8:49