APR Calculation using Newton Raphson in php

Question

I am trying to calculate APR using Newton Raphson in PHP. I have modified the code from this question into php

Calculating annual percentage rate (need some help with inherited code)

However the formula in this question is based on an initial principal, recurring monthly payments and a total number of payments. I need my APR calculation to include an initial start up fee. This would be added to the first monthly payment meaning the monthly payments are no longer all the the same. I take it that this would alter the original formula and hence its derivative. I have no idea what to use now as the initial formula or how to modify this

Here is the php

$numPay = 12;
$payment = 875;
$amount = 10000;
$error = pow(10,-5);
$approx = 0.05/12; // let's start with a guess that the APR is 5% 
$prev_approx;

function f($x) {
    global $numPay;
    global $payment; 
    global $amount;
    global $error;



    return $amount * $x * (pow(1 + $x,$numPay)/(pow(1 + $x, $numPay) - 1)) - $payment;

}



function f_prime($x) {
    global $numPay;
    global $payment; 
    global $amount;
    global $error;

     return $amount * (pow(1 + $x,$numPay)/(-1 + pow(1 + $x,$numPay)) - $numPay * $x * pow(1 + $x,-1 + 2*$numPay)/pow(-1 + pow(1 + $x,$numPay),2) + $numPay * $x * pow(1 + $x,-1 + $numPay)/(-1 + pow(1 + $x,$numPay)));


}
echo f($approx) . "<br/>";

echo f_prime($approx) . "<br/>";
echo  "initial guess $approx" . "<br/>";

for ($k=0;$k<20; $k++) {
       $prev_approx = $approx;
       $approx = $prev_approx - (f($prev_approx)/f_prime($prev_approx));
       $diff = abs($approx-$prev_approx);
       echo "new guess $approx diff is $diff <br/>";
       if ($diff < $error) break;
}

$apr = round($approx * 12 * 10000 /100, 1); // this way we get APRs like 7.5% or 6.55%
echo "apr is $apr %";

Thanks in advance for any help

Answer 1

If it's just added to the first monthly payment as a constant, just .. well, add it to the first monthly term from the calculations. After all, the payment schedule and interest should be the same, it's just an added payment in the first term. It does not affect any other payments or the total amount to be paid back over time, as it's just a fee.