Calculating APR using binary chop

Hi.

This is similar to a binary search algorithm, which you can find good examples on for .NET.

Remember, if you are dealing with continuous compound interest, then:
{future value} = {present value} * e^({annual interest rate}*{number of years})

Therefore, you can rearrange the formula to solve for different values, such as the present value or interest rate given the other values.  

e.g., {present value} = {future value} / e^({annual interest rate}*{number of years})
{annual interest rate} = ln({future value}/{present value}) / {number of years}

Where ln() is the natural logarithm.  

Similarly, you can do the same for periodic compound interest:
{future value} = {present value} * (1 + {annual interest rate}/{periods per year})^({periods per year}*{number of years})

Regarding the pseudo code, here is my interpretation.

(1) obtain known values of above formula, including the amount of future values and the specific time period.
(2) initialize minimum and maximum values of i.
(3) repeat binary chop algorithm where you try "i" at the midpoint of min/max.  I read this as you pass "i" to present value function with the values of advances/installments.  If you have the correct APR, the present value of the amount paid out should equal the present value of the amount paid in if I understand correctly.  If installments are higher, you move to upper half of "i"; if lower, the bottom half.

You iterate until the difference between the min/max approaches zero:
Do While ((max - min) < 0.0000001)
    ...
Loop


Does that help?

This might help lead you in the right direction.  I'll see if I have any other examples later today.
http://www.laprbass.com/RAY_irr.php

<?php // RAY_irr.php
error_reporting(E_ALL);


// COMPUTE (IRR) INTERNAL RATE OF RETURN - ITERATIVE METHOD, ANNUAL


// SOME TEST DATA
$cashflow = array
( -100.0   // INITIAL INVESTMENT
, 5.0      // ANNUAL INTEREST OR DIVIDEND PAYMENTS
, 5.0
, 5.0
, 5.0
, 5.0
, 110.0    // CASH OUT
)
;

// TEST THE FUNCTION AND SHOW THE WORK PRODUCT
$irr = irr($cashflow);
echo " CASH: ";print_r($cashflow);
echo " IRR: $irr";


// DETERMINE INTERNAL RATE OF RETURN FROM AN ARRAY OF ANNUAL PAYMENTS
function irr($cashflow)
{
    // DETERMINE IF GROSS CASHFLOW IS POSITIVE
    $cash = round(array_sum($cashflow),2);
    if ($cash == 0.0) return 0.0;

    if ($cash >  0.0) $iterator =  0.0001;
    if ($cash <  0.0) $iterator = -0.0001;
    $interest = 0.0;
    $npv      = 1.0;
    $years    = count($cashflow) - 1;

    while ( (round(abs($npv),4) > 0.0) )
    {
        $interest = round($interest + $iterator,4);
        $year     = 0;
        $npv      = 0.0;

        while ($year <= $years)
        {
            $yearly = $cashflow[$year];
            // DENOMINATOR IS (1 + r)**t
            $denominator = pow((1+$interest), $year);
            $denominator = round($denominator,6);
            $npv = $npv + ( $yearly / $denominator );
            $year++;
        }
        if  (($iterator > 0.0) && ($npv < 0.0) ) break;
        if  (($iterator < 0.0) && ($npv > 0.0) ) break;
    }
    return round((($interest-$iterator) * 100.0),2);
} // END FUNCTION irr

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Thank you for your replies. I have been off over christmas but will look at this information today and hopefully close the question

Hello, on the first answer posted by JohnRay, what is the last input option 'cash out'

How would i modify the second answer for something that wasn't an exact number of years e.g. 30 months?

Thanks

@Ray: you also may want to chime in on explanation of NPV/IRR.
https://www.experts-exchange.com/questions/28327419/Calculating-APR.html

If i test run a flat rate loan of 7500 at 6% interest paid back over 1 year I ran it with this input based on 12 repayments of 662.50

$cashflow = array
( -7500.0   // INITIAL INVESTMENT
, 7950
)
;

Have i ran it right?

I posted this in the other question, but it has merit here, too.
http://en.wikipedia.org/wiki/Internal_rate_of_return

The period of the calculations is infinitely adjustable, as are the inflows and outflows of cash.  I just did it in years because it was convenient to the test data.

what is the last input option 'cash out'

When analyzing multiple discounted cash flows, your investment is typically treated as a negative value and the returns of interest, dividend yields, etc., are treated as positive values.  In the example, the "cash out" value would represent liquidation of the investment at a future date.  This would occur if a bond matured and the principal was returned or a rental property is sold for cash.

Have i ran it right?

Sort of.  There is a lot of deep background involved in making these calculations.  The cash flows are not usually as hard-and-fast as you see in these theoretical examples, and interest on loans usually does not compound annually; there are different compounding periods for different financial instruments.  Mortgages, for example, are typically compounded continuously.

Thanks for your replies. The person who wants APR calulcated does want a flat rate of interest and not compound. I should have mentioned that in my post!