Problem 16:13 A 30-year maturity bond making annual coupon payments with a coupo
ID: 2643908 • Letter: P
Question
Problem 16:13
A 30-year maturity bond making annual coupon payments with a coupon rate of 9.5% has duration of 13.53 years and convexity of 258.03. The bond currently sells at a yield to maturity of 6%.
A 30-year maturity bond making annual coupon payments with a coupon rate of 9.5% has duration of 13.53 years and convexity of 258.03. The bond currently sells at a yield to maturity of 6%.
What is the percentage error for each rule? (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places. Omit the "%" sign in your response.) Duration Rule Duration-with-convexity Rule Percent error for 5% YTM % % Percent error for 7% YTM % %
Explanation / Answer
Formula:
YTM = {Annual Coupon + [(Par Value - Market Price) / Number of Years to Maturity]} / [(Par Value + Market Price) / 2]
% Error for Duration Rule = (Market Value - Predicted New Price) / Market Value
where,
Predicted New Price = Predicted Price Change + P0
Predicted Price Change = (-Duration) * (change in YTM) / (1+YTM) * P0
% Error for Duration-with-Convexity-Rule = (Market Value - Predicted New Price) / Market Value
where,
Predicted New Price = Predicted Price Change + P0
Predicted Price Change = [(-Duration)*(change in YTM) / (1+YTM)] + [0.5*Convexity*(change in YTM)2] * P0
Solution:
Let, Market Value = x.
Assume, Par Value = $1,000
6% = {95 + [(100 - x) /30]} / [(100+x)/2]
6% = {95 + 3.33 - 0.033x} / [50 + 0.50x]
6 + 0.03x = 98.33 - 0.33x
0.63x = 92.33
x = 1457.89
Market Value = P0= $1457.89
Percent Error for 5% YTM-
Predicted Price Change = (-13.53) * (-0.01) / (1+6%) * 1,457.89 = 186.09
Predicted New Price = 186.09 + 1,457.89 = 1,643.98
% Error for Duration Rule = 0.13 = 13% (increase)
Predicted Price Change = [(-13.53) * (-0.01) / (1+6%)] + [(0.5) * (258.03) * (-0.01)2] * 1,457.89 = 204.90
Predicted New Price = 204.90 + 1,457.89 = 1,662.79
% Error for Duration-with-Convexity-Rule = 0.14 = 14% (increase)
Percent Error for 7% YTM-
Predicted Price Change = (-13.53) * (0.01) / (1+6%) * 1,457.89 = (-186.09)
Predicted New Price = (-186.09) + 1,457.89 = 1,271.81
% Error for Duration Rule = 0.13 = 13% (decrease)
Predicted Price Change = [(-13.53) * (0.01) / (1+6%)] + [(0.5) * (258.03) * (-0.01)2] * 1,457.89 = (-167.28)
Predicted New Price = (-167.28) + 1,457.89 = 1,290.61
% Error for Duration-with-Convexity-Rule = 0.11 = 11% (increase)