Question

A 30-year maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%. Use a financial calculator or spreadsheet to find the price of the bond if its yield to maturity falls to 7% or rises to 9%. What prices for the bond at these new yields would be predicted by the duration rule and the duration-with-convexity rule?

What is the percent error for each rule? What do you conclude about the accuracy of the two rules?

Short answer

Answer

Duration rule price at 7% and 9% respectively = $1605.28 and $1295.34; Percentage error - 0.94% and 0.98%

Duration with Convexity rule price at 7% and 9% respectively =$1619.23 and $1309.29’ Percentage error - 0.075% and 0.083 %

Step-by-step solution

Step by Step Solution Step 1: Calculation of the price of the bond

The price of the bond: (calculated using financial calculator)

for YTM 7% = $1620.45

for YTM 8% = $1450.31

for YTM 9% = $ 1308.21

Calculation of predicted price and % error using Duration rule at YTM 7%

Predicted price change = - Duration xΔy / 1 + y x P₀

= -11.54 x -0.01/ 1.08 x $1450.31

= $154.97

So predicted price = $154.97 + $1450.31 = $1605.28

Now we have calculated the actual price at 7% YTM = $1620.45

% error = $1620.45 - $1605.28 / $1605.28 = 0.94%

Calculation of predicted price and % error using Duration rule at YTM 9%

Predicted price change = - Duration xΔy / 1 + y x P₀

= -11.54 x + 0.01/ 1.08 x $1450.31

= - $154.97

So predicted price = - $154.97 + $1450.31 = $1295.34

Now we have calculated the actual price at 9% YTM = $1308.21

% error = $1308.21 - $1295.34 / $1308.21 = 0.98%

Calculation of predicted price and % error using Duration-with-Convexity rule at YTM 7%

Predicted price change =[(- Duration xΔy / 1 + y ) + (0.5 x Convexity x (Δy)²]x P₀

=[(- 11.54 x -0.01 / 1.08 ) + (0.5 x 192.4 x (-0.01)²]x $1450.31

= $ 168.92

So predicted price = $168.92 + $1450.31 = $1619.23

Now we have calculated the actual price at 7% YTM = $1620.45

% error = $1620.45 - $1619.23 / $1620.45 = 0.075%

Calculation of predicted price and % error using Duration-with-Convexity rule at YTM 9%

Predicted price change =[(- Duration xΔy / 1 + y ) + (0.5 x Convexity x (Δy)²]x P₀

=[(- 11.54 x + 0.01 / 1.08 ) + (0.5 x 192.4 x ( + 0.01)²]x $1450.31

= - $ 141.02

So predicted price = - $141.02 + $1450.31 = $1309.29

Now we have calculated the actual price at 9% YTM = $1308.21

% error = $1309.29 - $1308.21 / $1309.29 = 0.083%

Thus, it can be said that Duration with Convexity rule provides more accurate error percentage or actual price change.