Recreate Excel RATE function using Newton's Method - php

Question

I'm working on converting a mortgage calculator in PHP, but I don't necessarily need a PHP solution. I'm looking for the logic needed to replicate the Excel RATE function. I've found a solution which uses bisection, and if worse comes to worse, I use that.
I know someone out there in the interwebs world has knowledge of such a function, so I'd love to have an easy answer instead of creating a solution from scratch.
References:
http://office.microsoft.com/en-us/excel-help/rate-HP005209232.aspx
http://en.wikipedia.org/wiki/Newton%27s_method
Thanks

Answer 1

Implementation of the MS Excel RATE() function using the secant method (a finite difference approximation of Newton's method) taken from PHPExcel:
define('FINANCIAL_MAX_ITERATIONS', 128);
define('FINANCIAL_PRECISION', 1.0e-08);
function RATE($nper, $pmt, $pv, $fv = 0.0, $type = 0, $guess = 0.1) {
$rate = $guess;
if (abs($rate) < FINANCIAL_PRECISION) {
$y = $pv * (1 + $nper * $rate) + $pmt * (1 + $rate * $type) * $nper + $fv;
} else {
$f = exp($nper * log(1 + $rate));
$y = $pv * $f + $pmt * (1 / $rate + $type) * ($f - 1) + $fv;
}
$y0 = $pv + $pmt * $nper + $fv;
$y1 = $pv * $f + $pmt * (1 / $rate + $type) * ($f - 1) + $fv;
// find root by secant method
$i = $x0 = 0.0;
$x1 = $rate;
while ((abs($y0 - $y1) > FINANCIAL_PRECISION) && ($i < FINANCIAL_MAX_ITERATIONS)) {
$rate = ($y1 * $x0 - $y0 * $x1) / ($y1 - $y0);
$x0 = $x1;
$x1 = $rate;
if (abs($rate) < FINANCIAL_PRECISION) {
$y = $pv * (1 + $nper * $rate) + $pmt * (1 + $rate * $type) * $nper + $fv;
} else {
$f = exp($nper * log(1 + $rate));
$y = $pv * $f + $pmt * (1 / $rate + $type) * ($f - 1) + $fv;
}
$y0 = $y1;
$y1 = $y;
++$i;
}
return $rate;
} // function RATE()

Answer 2

I tried to use the code above, but the results simply aren´t the same as Excel (or Google Spreadsheet).
I dont know if you need to implement this function yet, but in any case, I looked at how this algorithm was built and even though I was not able to access the excel source code (or the google worksheet) I found that this is not a simple calculation. About this math, more can be read here:
https://brownmath.com/bsci/loan.htm#Eq8
The function, in PHP, may be something like this:
function rate($nprest, $vlrparc, $vp, $guess = 0.25) {
$maxit = 100;
$precision = 14;
$guess = round($guess,$precision);
for ($i=0 ; $i<$maxit ; $i++) {
$divdnd = $vlrparc - ( $vlrparc * (pow(1 + $guess , -$nprest)) ) - ($vp * $guess);
$divisor = $nprest * $vlrparc * pow(1 + $guess , (-$nprest - 1)) - $vp;
$newguess = $guess - ( $divdnd / $divisor );
$newguess = round($newguess, $precision);
if ($newguess == $guess) {
return $newguess;
} else {
$guess = $newguess;
}
}
return null;
}

Answer 3

For Laravel use the same function but you remove define
define('FINANCIAL_MAX_ITERATIONS', 128);
define('FINANCIAL_PRECISION', 1.0e-08);
and financial_max_iterations = 20; -> same excel
The code is:
function RATE($nper, $pmt, $pv, $fv = 0.0, $type = 0, $guess = 0.1) {
$financial_max_iterations = 20;
$financial_precision = 0.00000008;
$rate = $guess;
if (abs($rate) < $financial_precision) {
$y = $pv * (1 + $nper * $rate) + $pmt * (1 + $rate * $type) * $nper + $fv;
} else {
$f = exp($nper * log(1 + $rate));
$y = $pv * $f + $pmt * (1 / $rate + $type) * ($f - 1) + $fv;
}
$y0 = $pv + $pmt * $nper + $fv;
$y1 = $pv * $f + $pmt * (1 / $rate + $type) * ($f - 1) + $fv;
// find root by secant method
$i = $x0 = 0.0;
$x1 = $rate;
while ((abs($y0 - $y1) > $financial_precision) && ($i < $financial_max_iterations)) {
$rate = ($y1 * $x0 - $y0 * $x1) / ($y1 - $y0);
$x0 = $x1;
$x1 = $rate;
if (abs($rate) < $financial_precision) {
$y = $pv * (1 + $nper * $rate) + $pmt * (1 + $rate * $type) * $nper + $fv;
} else {
$f = exp($nper * log(1 + $rate));
$y = $pv * $f + $pmt * (1 / $rate + $type) * ($f - 1) + $fv;
}
$y0 = $y1;
$y1 = $y;
++$i;
}
return $rate;
}
it worked for me

Answer 4

TL;DR: Here's a SQL Server version. It doesn't work for some values, and the PHP code above will probably fail for the same values.
LONG ANSWER: I needed a RATE function for SQL Server. Using the PHPExcel answer above, and using https://charlottecredittechnology.blogspot.com/2013/05/sql-2008-excel-like-rate-function-part.html I wrote a SQL Server scalar function:
ALTER function [dbo].[Rate](
#nper integer, #pmt float, #pv float, #fv float, #type bit = 0, #guess float = 0.1
) returns numeric(38,10) as
/*
Calculate the effective interest rate of a sequence of regular payments.
*/
begin
declare #returns numeric(38,10) = 0;
if #type is null set #type = 0;
declare #i integer;
declare #rate float = #guess;
declare #FINANCIAL_MAX_ITERATIONS integer = 100;
declare #FINANCIAL_PRECISION float = 0.0000001;
declare #y float, #y0 float, #y1 float, #f float, #x0 float, #x1 float;
set #rate = #guess;
if Abs(#rate) < #FINANCIAL_PRECISION
begin
set #f = 0;
set #y = #pv * (1+#nper*#rate) + #pmt * (1+#rate*#type) * #nper + #fv;
end
else
begin
set #f = Exp(#nper * Log(1+#rate));
set #y = #pv * #f + #pmt * (1/#rate + #type) * (#f-1) + #fv;
end;
set #y0 = #pv + #pmt * #nper + #fv;
set #y1 = #pv * #f + #pmt * (1/#rate + #type) * (#f-1) + #fv;
-- Newton secant method.
set #i = 0;
set #x0 = 0;
set #x1 = #rate;
while Abs(#y0-#y1) > #FINANCIAL_PRECISION and #i < #FINANCIAL_MAX_ITERATIONS
begin
set #rate = (#y1 * #x0 - #y0 * #x1) / (#y1-#y0);
set #x0 = #x1;
set #x1 = #rate;
if Abs(#rate) < #FINANCIAL_PRECISION
begin
set #y = #pv * (1+#nper*#rate) + #pmt * (1+#rate*#type) * #nper + #fv;
end
else
begin
set #f = Exp(#nper * Log(1+#rate));
set #y = #pv * #f + #pmt * (1/#rate + #type) * (#f-1) + #fv;
end;
set #y0 = #y1;
set #y1 = #y;
set #i = #i + 1;
end;
return Convert(numeric(38,10), #rate);
end;
Unfortunately it does not always work. Here's the results of some tests I put together and checked using Excel:
-- (1) OK
select dbo.RATE(4*12, -200, 8000, 0, default, default) * 12 -- SQL formula
0.0924 (9.24%) -- SQL result
=RATE(4*12, -200, 8000, 0) * 12 -- Excel formula
9.24% -- Excel result
-- (2) OK
select dbo.RATE(12, -1000, 12000, 0, default, default) * 12 -- SQL formula
0 (0%) -- SQL result
=RATE(12, -1000, 12000, 0) * 12 -- Excel formula
0% -- Excel result
-- (3) OK
select dbo.RATE(30, -400, 4000, 0, 1, default) -- SQL formula
0.10496 (10.496%) -- SQL result
=RATE(30, -400, 4000, 0, 1) -- Excel formula
10.4964% -- Excel result
-- (4) OK
select dbo.RATE(120, 28.1, -2400, 0, default, default) -- SQL formula
0.0059905810 (0.599%) -- SQL result
=RATE(120, 28.1, -2400, 0) -- Excel formula
0.5991% -- Excel result
-- (5) OK
select dbo.RATE(10, -1000, 10000, -10000, default, default) -- SQL formula
0.1 (10%) -- SQL result
=RATE(10, -1000, 10000, -10000) -- Excel formula
10% -- Excel result
-- (6) WRONG ANSWER (unless you set #guess to 0.01)
select dbo.RATE(475, -1022.93, 272779.21, 0, default, default) -- SQL formula
0 -- SQL result
=RATE(475, -1022.93, 272779.21, 0, 0) -- Excel formula
0.2716% -- Excel result
-- (7) ERROR
select dbo.RATE(252, -29002.85, 2500000, 0, default, default) -- SQL formula
invalid floating point operation -- SQL result
=RATE(252, -29002.85, 2500000, 0) -- Excel formula
1.0833% -- Excel result
-- (8) OK
select dbo.RATE(24, -46.14, 1000, 0, default, default) -- SQL formula
0.0083244385 (0.83244%) -- SQL result
=RATE(24, -46.14, 1000, 0) -- Excel formula
0.8324% -- Excel result
Tests (7) and (8) were taken from RATE Function from EXCEL in Swift providing different results and look for the answer using the Newton-Raphson method.