# Unequal Loan Repayments

I am struggling with this question below.

Present Value = 100000

Interest =12.25% per annum

Number of months = 60

Payments =?

Payments increase by 15% per annum.

Calculate the PMT for all years.

• This looks like a homework problem in some course and not an issue of personal finance. – Dilip Sarwate Jul 28 '15 at 11:34
• The question might be homework, I agree, 100%. But the process has relevance to personal finance, and especially in light of Chris' answer, I'd vote to have it stay. – JTP - Apologise to Monica Jul 29 '15 at 14:20

A method of calculation is shown here:-

Investopedia - Calculating The Present And Future Value Of Annuities

Specifically, the section: Calculating the Present Value of an Ordinary Annuity.

The example shown calculates the present (or initial) value of a 5 year loan given annual payments of \$1,000 with an effective interest rate of 5%. By discounting the future payments according to the interest rate the present value is calculated to be \$4,329.48.

In your question the initial value is known and the payments are to be calculated.

``````p = initial loan value = 100,000
n = compounding periods per year = 12
r = effective annual interest rate = 12.25% = 0.1225
i = monthly interest rate = (1 + r)^(1/n) - 1 = 0.00967638
d = initial payment amount
``````

The present value of the loan is equal to the sum of the discounted payments for each year.

The payments for the first year are 1,688.56 increasing annually by 15% thereafter.

The calculation can also be generalised and converted into a formula for `d`.

``````y = number of years = 5
q = annual percentage increase in payments = 15% = 0.15
``````

Text form

``````d = ((1 + i)^(12 * y) * p *
(i * (2 + i) * (1 + i + i^2) * (2 + i * (2 + i)) *
(1 + i * (1 + i)^2 * (2 + i)) * (3 + i * (3 + i)) - q))/
((2 + i) * (1 + i + i^2) * (2 + i * (2 + i)) *
(1 + i * (1 + i)^2 * (2 + i)) * (3 + i * (3 + i)) *
((1 + i)^(12 y) - (1 + q)^y))
``````