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

Answer:

The amount Chris pay for his bond = $109.44

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

∴

$100=\dfrac{3}{Yield}$

Yield=$\dfrac{100}{3}$

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:  $PV= \dfrac{A*[ (1+r)^n -1]}{[(1+r)^n * r] }$

$PV= \dfrac{3*[ (1+0.024)^{20} -1]}{[(1+0.024)^{20} *0.024 ] }$

$PV= \dfrac{3*[ (1.024)^{20} -1]}{[(1.024)^{20} *0.024 ] }$

$PV= \dfrac{3*[1.606938044 -1]}{[1.606938044 *0.024 ] }$

$PV= \dfrac{3*[0.606938044]}{[0.03856651306 ] }$

$PV= \dfrac{1.820814132}{0.03856651306 }$

PV = 47.21

The PV of the face value = $\dfrac{100}{(1+r)^n}$

The PV of the face value =  $\dfrac{100}{(1+0.024)^{20}}$

The PV of the face value = $\dfrac{100}{(1.024)^{20}}$

The PV of the face value = $\dfrac{100}{1.606938044}$

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