Question

ABC Corp. wants to issue a 20 year $1000 bond with semi-annual coupons at a nominal annual coupon rate of 5%. The bond is callable at the end of n1 years, n2 years, and 20 years with a redemption amount of $1100. You are given the following requirements:

(i) The ABC Corp. wants to offer an annual effective yield rate no more than i = 5%. (ii) The price P of the bond should satisfy $930 ≤ P ≤ $1070.

(iii) There are at least 5 years between any two callable dates. Answer the following questions.

1. (a) List all possible pair values of (n1,n2) satisfying the above three requirements.

2. (b) For each pair, write down an equation satisfied by the highest semi-annual effective yield rate that a bondholder can earn.

Answer 1

Given,

Face Value (FV)	$ 1000
Coupon rate	5%
Time Period (years)	20
n (pmt frequency per year)	2
PMT (payments per period)	$ 25

Bond Price (PV) = ?

The bond calling periods need to be separated by at least 5 years. Hence, n2 <= 15 years and consequently n1<= 10 years.

Annual Effective Yield rate = (1+YTM)^2 - 1

For AEY to be less than 5%, the price of the bond must be greater than the Par Value. Therefore, PV of bond > $1000.

1. Price of Bond = $1067, n1 = 10 years, n2 = 15 years

Bond Price (PV)	-1067	-1067	-1067
Face Value (FV)	1000	1100	1100
Coupon rate	5%	5%	5%
Time Period	20	15	10
n (pmt frequency per year)	2	2	2
PMT	25	25	25
YTM	2.24%	2.41%	2.46%
Annual Effective Yield	4.54%	4.89%	4.99%

At calling option after 10 years and 15 years of issuing the bond, price of the bond must be greater than or equal to $1067 for the Annual Effective Yield rate to be less than 5%.

2. Price of bond = $1069, n1 = 9 years, n2 = 14 years or 15 years

Bond Price (PV)	-1069	-1069	-1069
Face Value (FV)	1000	1100	1100
Coupon rate	5%	5%	5%
Time Period	20	15	9
n (pmt frequency per year)	2	2	2
PMT	25	25	25
YTM	2.24%	2.41%	2.47%
Annual Effective Yield	4.52%	4.87%	4.998%

If the price of the bond is kept equal to $1069, for the Annual Effective Yield rate to be less than 5%, n1 = 9 years and n2 = either 14 years or 15 years.

Therefore, the following three combinations are possible for (n1, n2) : (9, 14) , (9, 15) , (10, 15) years.