Calculating diminishing interest amount

Question

The agent says he will loan an amount of 24,660 dollars and let me pay 414 dollars monthly + interest of 1% per Month on the diminishing balance.

How do I calculate the time by which I should be paid up?

What is the total interest I pay in this scenario ?

Answer 1

The easy way to do this is with an excel spreadsheet.

Put 24,660 in the first row. and put 24,660*(1.01) - 414 in the row below, and spread it down until the number goes negative.

The other way to do it is to derive a formula.

The NPV of future cashflows is:

$414 \sum_\limits{i=1}^n (1.01)^{-i} = 24,660\\ 414 \frac {1-(1.01)^{-n}}{0.01} = 24,660\\ n = -\frac {\log(1-\frac {24,660}{414}0.01)}{\log 1.01}$

Answer 2

By C program,

#include<stdio.h>

int main()
{
    double capital = 24660;
    double interest = 0;
    double juice;
    int months = 0;
    while (capital > 0) {
        capital -= 414;
        juice = capital / 100;
        capital += juice;
        interest += juice;
        months++;
    }
    printf("Balance  %.2f\n", capital);
    printf("Interest %.2f\n", interest);
    printf("Months   %d\n",   months);
}

It will take you 90 months but you will have overpaid by $190 in the last month, and added about half the loan in interest.

Balance  -189.84
Interest 12410.16
Months   90

Answer 3

Assuming that interest accrues immediately after the loan is made, and that the first payment is made after one month, then after one month you owe $$24660\times1.01-414$$

Similarly, after 2 months you owe $$(24660\times 1.01-414)\times1.01-414$$ $$=24660\times1.01^2-414(1+1.01)$$

After $n$ months you owe nothing, so you will have $$24660\times1.01^n-414(1+1.01+1.01^2+...)=0$$

So, using the formula for the sum of a geometric series, , we have to solve for $n$ the equation $$24660\times1.01^n=414\times\frac{1.01^n-1}{1.01-1}$$

This rearranges to give $$1.01^n=\frac{41400}{41400-24660}\implies n=90.99997...$$

So it looks like you make 90 payments of 414 and your last payment is 409.89 plus interest for one month, so 413.99.