Loans

Explanation

A loan, as we are using the term here, can be understood by combining thinking about an ordinary annuity and thinking about compound interest.

Basically, when you take out a loan, you are going to pay it back with regular payments. That works quite a bit like an ordinary annuity; your group of payments are going to have a future value when they are done. That future value will be all the money that you pay plus the interest you earn with your payments over the time of the loan.

How much interest will you earn on your payments? Well, that would be the amount of money the bank would have earned if the bank would have invested the loaned sum at the loan rate over the term of the loan. In fact, the bank did make an investment. They invested in you.

So, a loan is like an ordinary annuity in that it has regular payments, except you do not collect the future value of that annuity. You give it to the bank.

The future value of the 'annuity' (loan) is equal to the value of a compound interest investment over the loan's time period at the same rate.

So, the bank gives you an amount of money and asks you to pay it back at regular intervals. Your payments act like an annuity and have a future value. As the bank sees it, that future value is equal to the value of an compound interest investment (an investment that the bank has made in you) compounded at the rate and payment period over the term of the loan.

By equating the future value of an annuity with the future value of a compound interest investment, both with the same interest rate and payment schedule, we get the following mathematics:

Formula symbols

A	Amount to be loaned
R	Amount of payment
r	Annual percentage rate (APR)
k	Interest periods per year
i = r/k	Percentage rate per interest period
n	Number of payments

The following equation relates the above quantities:

A = R((1 - (1 + i)^-n) / i)

Below is a calculator for the amount of money to be loaned.

Enter values for the above formula:

(Example: For r enter 5.0% as 0.05, etc.)

Click the following 'Code' button to see the actual JavaScript code that executes in the above calculator. In that way, if you wish, you can see exactly how this calculator arrives at its result.

Here's the code that runs when you click the above 'Calculate' button. It is shown here as an aid for your study and understanding.

You can change the code, if you like, and then click the following 'Reevaluate code' button. The program will then work as per your changes when you click the 'Calculate' button on the above calculator. Of course, your changes, especially random changes, can introduce errors, miscalculations, and browser crashes. If you need to get things back to their original condition, just reload this page using your browser's reload button.

Again, the intention here is to conveniently show you the inner workings of this program so that you understand how the program arrives at the result. Click the 'Code' button again to close this section.

window.calculate = function()
{
	var A = 0.0;
	var R = 0.0;
	var r = 0.0;
	var k = 0.0;
	var n = 0.0;
	var i = 0.0;

	R = parseFloat(document.inputForm.R.value);
	r = parseFloat(document.inputForm.r.value);
	k = parseFloat(document.inputForm.k.value);
	n = parseFloat(document.inputForm.n.value);
	
	i = r / k;

	A = R * ((1 - Math.pow((1 + i), -n)) / i);

	alert("A, amount to be loaned = " + A);
};