Question

Suppose that a loan of amount L is being repaid by n installments of R at the end of each period. Denote by B. the outstanding loan balance immediately after the tth payment has been made, t=0,1,2,...,n. Then Bo = L, Bn=0, and for t=1,2,...,n, Bi's satisfy the following recurrence relation: B+1= B.(1+i) - R, where i is the interest rate per period. (a) By using (*) to form the sum SBS 1+i)'-s, t=1,2,...,n, (**) show that B = L(1 + i)' – R], t=1,2,..., n. (b) Interpret (**). (c) By using the result of (a), show that L(1 + i)" R =- ST

Answer 1

Part A:

Assume k = 1+i

Where i is the interest rate per period.

Initial Loan Amount = L = Initial Outstanding Amount = B0, So -
B0 = L

For next period, interest will be charged on B0. Also at the end of the next period R(installment amount) will be repaid.
So outstanding amount B1 is given by -
B1 = B0*k – R

Extending the argument in the previous period to next Period
B2 = B1*k – R = (B0*k – R)*k – R = B0*k^2 – R(1+k)

Similarly,
B3 = B2*k – R = (B0*k^2 – R(1+k))*k – R = B0*k^3 – R(1+k+k^2)

..

..

..

Bt = B0*k^t – R(1+k+k^2+k^3+……+k^(t-1))

      = B0*k^t – R(k^t – 1)/(k – 1)

Replacing k with (1+i)

Bt = B0*(1 + i)^n – R((1 + i)^t – 1))/i

If ((1+i)^t - 1)/i = s|t
and, B0 = L

Bt = L*(1+i)^n - R*s|t ........................(1)

Another way to derive the above equation is as follows -

If no repayment is done, total outstanding after t period would be = L*(1+i)^n. ---------------(A)

But some R amount is being repaid every period.
so Interest saved on installment of the first period = R*(1+i)^t
interest saved on installment of the 2nd period = R*(1+i)^t-1
and so on...

Total interest saved = _s=0 ∑ ^t-1(R*(1+i)^(t-s))
= R((1+i)^t - 1)/(1+i - 1)
= R((1+i)^t - 1)/i ---------------------(B)

So Loan outstanding = total outstanding if no interest paid(A) - Interest saved due to repayment(B)
= L*(1+i)^t - R((1+i)^t - 1)/i
= L*(1+i)^t - R*s|t

Part B:

Bt is nothing but loan outstanding at the end of period t. Outstanding amount at the end of period t if no repayment would have been made is L*(1+i)^t. Interest saved due to repayments = R*s|t.

Hence total outstanding at end of period t is loan in absence of repayment minus savings due to interest payment.

Part C:

if this is a n period loan, outstanding after nth period should be 0

Bn = L*(1+i)^n - R*s|n = 0
R*s|n = L*(1+i)^n
R = L*(1+i)^n/s|n