Question

An investor has two bonds in her portfolio, Bond C and Bond Z. Each bond matures in 4 years, has a face value of $1,000, and has a yield to maturity of 8.6%. Bond C pays a 11% annual coupon, while Bond Z is a zero coupon bond.

a. Assuming that the yield to maturity of each bond remains at 8.6% over the next 4 years, calculate the price of the bonds at each of the following years to maturity.
b. Plot the time path of prices for each bond.

Answer 1

a)

YTM of a coupon paying bond:

YTM = {coupon + [(face value - market value)/n]} / [(face value + market value)/2]

YTM of a zero coupon bond:

YTM = (face value / market value)⁻¹/ⁿ - 1

Bond C:

bond price at 4 years to maturity

0.086 = {110 + [(1,000 - MV)/4]} / [(1,000 + MV)/2]

0.086 x [(1,000 + MV)/2] = 110 + [(1,000 - MV)/4]

0.086 x (500 + 0.5MV) = 110 + 250 - 0.25MV

43 + 0.043MV = 360 - 0.25MV

0.293MV = 317

MV = 317 / 0.293 = $1,081.91

bond price at 3 years to maturity

0.086 = {110 + [(1,000 - MV)/3]} / [(1,000 + MV)/2]

0.086 x [(1,000 + MV)/2] = 110 + [(1,000 - MV)/3]

0.086 x (500 + 0.5MV) = 110 + 333.33 - 0.333MV

43 + 0.043MV = 443.33 - 0.333MV

0.376MV = 400.33

MV = 400.33 / 0.376 = $1,064.71

bond price at 2 years to maturity

0.086 = {110 + [(1,000 - MV)/2]} / [(1,000 + MV)/2]

0.086 x [(1,000 + MV)/2] = 110 + [(1,000 - MV)/2]

0.086 x (500 + 0.5MV) = 110 + 500 - 0.5MV

43 + 0.043MV = 610 - 0.5MV

0.543MV = 567

MV = 567 / 0.543 = $1,044.20

bond price at 1 year to maturity

0.086 = {110 + [(1,000 - MV)/1]} / [(1,000 + MV)/2]

0.086 x [(1,000 + MV)/2] = 110 + (1,000 - MV)

0.086 x (500 + 0.5MV) = 110 + 1,000 - MV

43 + 0.043MV = 1,110 - MV

1.043MV = 1,067

MV = 1,067 / 1.043 = $1,023.01

Bond Z:

bond price at 4 years to maturity

0.086 = (1,000 / MV)⁻¹/⁴ - 1

1.086 = (1,000 / MV)⁻¹/⁴

⁻¹/⁴√1.086 = ⁻¹/⁴√(1,000 / MV)⁻¹/⁴

0.7189 = 1,000 / MV

MV = 1,000 / 0.7189 = $1,390.97

bond price at 3 years to maturity

0.086 = (1,000 / MV)⁻¹/³ - 1

1.086 = (1,000 / MV)⁻¹/³

⁻¹/³√1.086 = ⁻¹/³√(1,000 / MV)⁻¹/³

0.7807 = 1,000 / MV

MV = 1,000 / 0.7807 = $1,280.82

bond price at 2 years to maturity

0.086 = (1,000 / MV)⁻¹/² - 1

1.086 = (1,000 / MV)⁻¹/²

⁻¹/²√1.086 = ⁻¹/²√(1,000 / MV)⁻¹/²

0.8479 = 1,000 / MV

MV = 1,000 / 0.8479 = $1,179.40

bond price at 1 year to maturity

0.086 = (1,000 / MV)⁻¹ - 1

1.086 = (1,000 / MV)⁻¹

⁻¹√1.086 = ⁻¹√(1,000 / MV)⁻¹

0.9208 = 1,000 / MV

MV = 1,000 / 0.9208 = $1,086

b) I do not have any drawing tools here, but the prices are:

years to maturity price of bond C price of bond Z

4 $1,081.91 $1,390.97

3 $1,064.71 $1,280.82

2 $1,044.20 $1,179.40

1 $1,023.01 $1,086