Question

Chris purchased a 10 year 100 par value bond where 6% coupons are paid semiannually. Cheryl purchased a 100 par value bond where 6% coupons are paid semiannually. There is no maturity date or redemption value for Cheryl’s bond. Cheryl paid $100 for her bond. The yield for Chris’s bond is 80% of the yield for Cheryl’s bond. How much did Chris pay for his bond?

Answer 1

The amount Chris pay for his bond = $109.44

Step-by-step explanation:

Given that:

Chris purchased a 10 year 100 par value bond where 6% coupons are paid semiannually. Cheryl purchased a 100 par value bond where 6% coupons are paid semiannually.

The Price of the Cheryl's bond is 6% given that it is purchased at at par value where 6% coupons are paid.

Suppose The yield for Chris’s bond is 80% of the yield for Cheryl’s bond.

Then:

Price of the Cheryl's bond = Present Value of the coupon in perpetuity

∴

`100=(3)/(Yield)`

Yield=
`(100)/(3)`

Yield =0.03

Yield = 3%

The Yield of Chris = 0.8 × 3

The Yield of Chris = 2.4% semiannual

However;

Present Value of the coupons is:
`PV= (A*[ (1+r)^n -1])/([(1+r)^n * r] )`

`PV= (3*[ (1+0.024)^(20) -1])/([(1+0.024)^(20) *0.024 ] )`

`PV= (3*[ (1.024)^(20) -1])/([(1.024)^(20) *0.024 ] )`

`PV= (3*[1.606938044 -1])/([1.606938044 *0.024 ] )`

`PV= (3*[0.606938044])/([0.03856651306 ] )`

`PV= (1.820814132)/(0.03856651306 )`

PV = 47.21

The PV of the face value =
`(100)/((1+r)^n)`

The PV of the face value =
`(100)/((1+0.024)^(20))`

The PV of the face value =
`(100)/((1.024)^(20))`

The PV of the face value =
`(100)/(1.606938044)`

The PV of the face value = 62.230

Finally:

The amount Chris pay for his bond = PV of the coupons + PV of the face value

The amount Chris pay for his bond = 47.21 + 62.230

The amount Chris pay for his bond = $109.44