Question

There is a bond on a company's books with an original term of 10 years that was purchased for a premium at its issuance, just over 2 years ago. The bond pays semi-annual interest. With the receipt of the latest coupon, the corresponding amount for amortization of the premium was $665.68. Exactly one year ago, the amount for amortization of the premium was $607.26. Based on the relation between subsequent amounts for amortization of the principal, what was the original value of the premium?

Answer 1

Premium or discount is amortized on a straight line basis, based on the number of interest payments. Hence, the difference in the amount for amortization between two successive interest payments = Original amount / n where n is the total number of interest payments of the bond since beginning.

Within a year, two interest payments would have been made. Hence the gap between the amount for amortization = 2 x Original amount / n

n = number of semiannual payments in 10 years = 2 x 10 = 20

Hence, 665.68 - 607.26 = 58.42 = 2 x Original amount / 20 =

Hence, original value of the premium = 58.42 x 10 = $ 584.2

Answer 2

From the question, we know that:

n = (10)(2) = 20

4 coupon payments have already happened, with the 2nd payment containing a premium amortization of 607.26, and the 4th containing a premium amortization of 665.68.

Using the formula for amortization of loan repayment: K is the coupon payment every half a year.

Principal repaid at time t = (K)((1/1+i)^(n-t+1))

The coupon payment is also represented by Fr, with F being the face value, and r being the coupon rate.

Therefore Principal repaid at time t = (Fr)((1/1+i)^(n-t+1))

From the relation of the 2 premium amortizations, we know that:

665.68 = 607.26 ((1+i)^2)

From the above equation, we get that i is approximately 0.047.

Therefore, each coupon payment is 607.26(1.047^(20-2+1)) = 1453.24.

Then using the prospective method for loan repayment (the premium is essentially the outstanding balance at time 0), we have:

Premium = 1453.24(annuity present value factor: n = 20, i = 0.047)

             =  18580.76