Question

A 30-year maturity bond making annual coupon payments with a coupon rate of 12% has Macaulay’s duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.

a) What price would be predicted by the duration rule? You assume that yield to maturity falls to 7%.

b) What price would be predicted by the duration-with-convexity rule?

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

d) Repeat your analysis if the bond’s yield to maturity increases to 9%. Are your conclusions about the accuracy of the two rules consistent with parts (a)-(c)?

Please answer all parts to receive positive rating

Answer 1

Ans a) Bond Price at 8% yeild:

Bond price = coupon * (1 - (1+interest rate)^(-number of year))/interest rate + Fave value/(1+intrest rate)^n

= 120*(1 - (1.08)^(-30))/.08 + 1000/(1.08)^30

= $1450.31

at 7% yield

Bond price = 120*(1- 1.07^-30)/.07 + 1000/(1.07)^30

= $1620.45

Using the duration rule, assuming yield to maturity falls to 7%:

Predicted price change = –Duration × change in y/(1+y) * Price

= - 11.54 * -.01/1.08 * 1450.31

= $154.97

Therefore: Predicted price = $154.97 + $1,450.31 = $1,605.28

Ans b) Duration with convexity rule

Predicted price change = (–Duration × change in y/(1+y) + (1/2 * convexity * change in yield^2)) * price

= $168.92

Predicted price = $168.92 + $1,450.31 = $1,619.23

Ans c) percentage error using duration rule:

(1620.45 - 1605.28)/1620.45 = .94%

percentge error using duration with convexity rule

(1620.45 - 1619.23)/1620.45 = .075%

Duration with convexity rule is better because it has better accuracy.

d) bond price at yield of 9%

ond price = 120*(1- 1.09^-30)/.09+ 1000/(1.09)^30

= $1308.21

Using the duration rule, assuming yield to maturity ries to 9%:

Predicted price change = –Duration × change in y/(1+y) * Price

= - 11.54 * .01/1.08 * 1450.31

= -$154.97

Therefore: Predicted price = -$154.97 + $1450.31 = $1295.34

Duration with convexity rule

Predicted price change = (–Duration × change in y/(1+y) + (1/2 * convexity * change in yield^2)) * price

= -$141.02

Predicted price = -$141.02 + $1,450.31 = $1,309.29

Ans c) percentage error using duration rule:

(1308.21 - 1295.34)/1308.21 = .98%

percentge error using duration with convexity rule

(1308.21 - 1309.29)/1308.21 = .083%

The duration-with-convexity rule provides more accurate approximations to the actual change in price.