Question

3. Suppose that a firm has borrowed $1000 in the current year at a 10% interest rate, with a commitment to repay the loan (principal and interest) in equal annual installments over the following five years. Calculate:

1. the amount of the annual repayment;
2. the stream of interest payments which can be entered in the tax calculation of the private benefit-cost analysis.

Answer:

(i) $263.80

(ii) year 1 = $100; year 2 = $83.62; year 3 = $65.60; year 4 = $45.78; year 5 = $23.98

I need to know the process to solve it.

Answer 1

Solution:

With loan of $1,000 and interes rate of 10% per annum, the firm has committed to repay the entire loan amount in 5 equal installments.

i) This is how we proceed: I won't use any shortcut formula, so that you understand better

The principal, P = $1,000, interest rate i = 0.1, let annual installment be X.

After first year, the firm has a total amount to be repaid = P*(1 + i)¹

= 1000*(1 + 0.1) = 1100

Since, the firm also pays back an installment amount, X, in year 2 the loan size becomes (1100 - X)

So, at the end of year 2, amount to be repaid = (1100-X)*(1 + 0.1)

= 1210 - 1.1*X

Going this way, by the end of 5th year, entire loan payment must be made.

Since again the installment will be made, by start of 3rd year, loan to be paid back = (1210 - 1.1*X) - X = 1210 - 2.1*X

So, by end of 3rd year, total amount to be paid back accumulates to (1210 - 2.1*X)*(1 + 0.1) = 1331 - 2.31*X

Again, since annual payment is made at the end of third year of $X, by start of 4th year, loan size = 1331 - 2.31*X - X

= 1331 - 3.31*X

By end of fourth year, loan to be paid (with interest as usual) = (1331 - 3.31*X)*(1 + 0.1) = 1464.1 - 3.641*X

By start of 5th year, payment to be made = 1464.1 - 3.641*X - X = 1464.1 - 4.641*X

Note that, as already mentioned, by end of 5th year, entire payment needs to be made, so the last installment must simply equal the total amount finally left to be paid

Total amount finally left to be paid, by end of 5th year = (1464.1 - 4.641*X)*(1 + 0.1) = 1610.51 - 5.1051*X

This must exact equal the last installment, which is X itself, so

1610.51 - 5.1051*X = X

Solving this, gives us the value of X, that is the value of annual repayment:

1610.51 = 5.1051*X + X

X = 1610.51/6.1051 = 263.797 = 263.80 (approximately)

Thus, the amount of annual repayment = $263.80

ii) Stream of interest payments which can be entered in tax calculation is simply the amount of interest payment firm has made every year. This equals the payment firm had in beginning of every year, to be paid (basically the loan size) multiplied with interest rate. Now, that we know X, putting its value directly, and using above part for loan sizes of different years, we shall find our answer.

In year 1, at beginning, loan size = $1,000

So interest stream in year 1 = 1000*0.1 = $100

In year 2, at beginning, loan size = 1100 - X = 1100 - 263.8 = $836.2

So, interest stream in year 2 = 836.2*0.1 = $83.62

In year 3, at beginning, loan size = 1210 - 2.1*X = 1210 - 2.1*263.8 = $656.02

So interest stream in year 3 = 656.02*0.1 = $65.60

In year 4, at beginning, loan size = 1331 - 3.31*X = 1331 - 3.31*263.8 = $457.822

So, interest stream in year 4 = 457.822*0.1 = $45.78

Finally, at beginning of year 5, loan size = 1464.1 - 4.641*X = 1464.1 - 4.641*263.8 = $239.8042

Interest stream in year 5 = 239.8042*0.1 = $23.98

Hope the solution makes it clear, and you find it helpful.