A 30-year maturity bond making annual coupon payments with a coupon rate of 16.3

ID: 2762019 • Letter: A

Question

A 30-year maturity bond making annual coupon payments with a coupon rate of 16.3% has duration of 10.54 years and convexity of 161.2. The bond currently sells at a yield to maturity of 9%. a. Find the price of the bond if its yield to maturity falls to 8% or rises to 10%. (Do not round intermediate calculations. Round your answers to 2 decimal places.) YTM Price 8% $ 10% $ b. What prices for the bond at these new yields would be predicted by the duration rule and the duration-with-convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.) YTM Duration Rule Duration-with- Convexity Rule 8% $ $ 10% $ $ c. What is the percent error for each rule? (Do not round intermediate calculations. Round "Duration Rule" to 2 decimal places and "Duration-with-Convexity Rule" to 3 decimal places.) Percent Error YTM Duration Rule Duration-with- Convexity Rule 8% % % 10% % % d. What do you conclude about the accuracy of the two rules? The duration-with-convexity rule provides more accurate approximations to the actual change in price. The duration rule provides more accurate approximations to the actual change in price.

Explanation / Answer

Present value of annuity factor (PVAF)= {1 – (1+r)-n}/r

PVAF at 8% for 30 years = (1 – 1.08-30)/0.08 = 11.2578

PVAF at 9% for 30 years = (1 – 1.09-30)/0.09 = 10.2737

PVAF at 10% for 30 years = (1 – 1.10-30)/0.10 = 9.4269

Present value factor (PVF) = 1 / (1+r)n

PVF at 8% for 30 years = 1/1.0830 = 0.0994

PVF at 9% for 30 years = 1/1.0930 = 0.0754

PVF at 10% for 30 years = 1/1.1030 = 0.0573

Price of bond = present value of annuity of coupon payments + present value of face value

YTM = 8%

YTM = 9%

YTM = 10%

Face value

$1,000.00

Coupon rate

16.30%

Annual coupon payment

$163.00

PVAF

11.2578

10.2737

9.4269

Present value of coupon payments

$1,835.02

$1,674.61

$1,536.58

PVF

0.0994

0.0754

0.0573

Present value of face value

$99.40

$75.40

$57.30

Price of bond

$1,934.42

$1,750.01

$1,593.88

Using Duration rule

Predicted price change = {-duration/(1+r)} * Change in r * Price of bond at r

Predicted price change at 8% = (-10.54/1.09) * (-0.01) * $1,750.01 = $169.22

Predicted price at YTM 8% = $1,750.01 + $169.22 = $1,919.23

Predicted price change at 10% = (-10.54/1.09) * (0.01) * $1,750.01 = -$169.22

Predicted price at YTM 10% = $1,750.01 - $169.22 = $1,580.79

Using Duration with Convexity rule

Predicted price change = {(-Duration/1+r)*Change in r} + {0.05 * Convexity * (Change in r)2} * Price of bond at r

Predicted price change at YTM 8% = {[(-10.54/1.09)*(-0.01)] + [0.05 * 161.20 * (-0.01)2]} * $1,750.01 = $170.63

Predicted price at YTM 8% = $1,750.01 + $170.63 = $1,920.64

Predicted price change at YTM 10% = {[(-10.54/1.09)*(0.01)] + [0.05 * 161.20 * (0.01)2]} * $1,750.01 = $167.81

Predicted price at YTM 10% = $1,750.01 - $167.81 = $1,582.20

Duration Rule

Percent error at YTM 8% = ($1,919.23 - $1,934.42)/$1,934.42 = -0.79%

Percent error at YTM 10% = ($1,580.79 - $1,593.88)/$1,593.88 = -0.82%

Duration with Convexity rule

Percent error at YTM 8% = ($1,920.64 - $1,934.42)/$1,934.42 = -0.71%

Percent error at YTM 10% = ($1,582.20 - $1,593.88)/$1,593.88 = -0.73%

The duration-with-convexity rule provides more accurate approximations to the actual change in price. In this example, the percentage error using convexity with duration is less than the error using only duration to estimate the price change.