Calculate APR from loan, monthly payments, and amount of months

Question

For some reason calculating APR is very difficult. A lot of people ask, but not a lot of great answers. I'm working on a project, where I need to calculate APR. I have already calculated what the loan is, how much the user has to pay each month, and I have the amount of months. I found a formula online, but it doesn't make a lot of sense.

function calculateAPR(loan, repayments, months) {
    var p = 1;
    var tmp = 1;
    var a = p;
    var b = 0;
    while (Math.abs(tmp) > 0.0001) {
        p = (a - b) / 2 + b;
        tmp = (loan / repayments) - (1 - Math.pow(1 + p, -months)) / p;
        if (tmp > 0) {
            a = p;
        } else {
            b = p;
        }
    }
    var apr = Math.pow((1 + p), 12) - 1;
    return apr;
}

Math.pow()'s second argument is what you want to power by an amount. Basically 1+p^12 on the third last line. Math.abs() returns the absolute value (-10 becomes 10). If I input this:

loan = 80000
repayments = 7366
months = 12

I get an APR of 20.56%. That seems correct according to the site I got the formula from, but when I input the numbers into other websites, I get no where near the same result.

Can someone explain how this formula works? Why is there a loop? Is it because of compound interest? Is there a better formula? I'm looking for any kind of help, because I can't seem to find answers that solve this programmably. Thanks.

Answer 1

The loop is designed to progressively narrow the interval in which the correct apr resides. At the outset, the program assumes (arbitrarily) that the correct monthly apr, $r$ say, is between $0$ and $1$ (i.e., between $0\text{%}$ and $100\text{%}$).

The variables $a,b$ denotes the bounds on $r$, where $b \le r \le a$.

As the program progresses, the values of $a$ and $b$ are adjusted, as follows . . .

The variable $p$ is set to the middle of the known range (i.e., $p = b + (a-b)/2$).

The value $p$ is then tested to see whether it's too low or too high.

If too low, $b$ is set to $p$, and $a$ is left as is.

If too high, $a$ is set to $p$, and $b$ is left as is.

After each pass through the loop, the value of $a - b$ is half its previous value.

When $a-b$ is less than $0.0001$ (an arbitrary tolerance), the loop terminates.

Then the value of $p$ at the end is the desired $r$, which is then adjusted from a monthly rate to an annual rate in the standard way.