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jonacpa, Certified Public Accountant (CPA)

Category: Finance

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Loan Repayment Plan

Customer Question

For a loan with a single principle of P dollars, what is the future worth of a repayment plan that pays interest when earned and reduces the balance by P/n at the end of each of the n interest periods (interest rate is i per interest period)

There is a formula for this. It is FV = (i * (n+1)/2 * P) + P where P, n, and i have the definitions given in your post.

For example, P=50,000, n = 10, and i = .08, then FV = (.08 *(10+1) / 2 * 50000) + 50000 = (.88 / 2 * 50000) + 50000 = 22000 + 50000 = 72000. This can be proven by stepping through the calculations for each period. Interest at end of first period = 4000, therefore payment = 50000/10 + 4000 = 9000, etc. Add it all up at the end and you will prove the answer.

This assumes that the payments are not reinvested - that would complicate things significantly.

If you need further information or assistance, please let me know.

Reply to jonacpa's Post: ok here is the complete question:

Two repayment plans for a loan with a single principle of P dollars are being considered.
One has a single payment at the end of n interest periods.
The second one pays interest when earned and reduces the balance by P/n at the end of each of the n interest periods.

If the interest rate is i per interest period,show that the future worths of the two repayment plans are identical. The solution must be in terms of arbitrary values of P, i, and n.

HINT: The second repayment plan is the difference of two types of series.

I think there must be a mistake in there somewhere. If the one plan has a single payment at the end, then interest will continue on the full amount of principal for the entire n periods. The other plan reduces principal each of the n periods and, therefore, cannot possibly have the same FV unless there are other factors that are being considered.

Please make sure that the problem is stated correctly and if there are any other variables or circumstances that are to be considered.

Reply to jonacpa's Post: Thank you Jon i think I got it, it asks about equivalency in time not in actual amount in money anyway I handed in my test this morning that problem had me staying up all night. here is what i handed in:

The first plan is single principle single payment. In this plan no payment is made until the end of the nth period when the load is completely repaid with a single payment. The formula for this kind of plan is:

Fwp = P(F/P, i%, n) = P * (1+i) n

The second plan is a declining gradient over time. The factors can not be used directly for a declining gradient. Instead, we will subtract an increasing gradient from an assumed uniform series of payments.

The uniform series of payments A is:
(P/n)+Pi

And the increasing gradient is:
G =(P/n)i

Thus:
A' = P/n + Pi - (P/n)i (A/G, i%,n)
= P/n + Pi - (P/n)i *(1/i - n/(1+i)^n -1 )
Then we can find the future worth using the uniform series compound amount.
F = A' (F/A', i%, n) =