Checking Interest Formulas
Why we need them
How to think about Interest
For these calculations the variables will be:
Principal: $12,000, (12000)
Interest Rate: 12%, (0.12)
Interest for 1 year: $1440, (1440)
Number of test periods: nper = 1, 1 year
nper = 2, 6 months
nper = 3, 4 months
nper = 4, 3 months
nper = 6, 2 months
We start with a loan for one year
	Principal	Interest	
	$12,000		$1,440
We borrowed $12,000 and had the use of the full value of $12,000 for a full year.
We also had the FULL use of the interest as it accumulated during the year.
Next, we will borrow the same amount but will pay interest twice a year.
    Principal         Interest      Lost use
                                    of interest						
    $12,000           $720.00	
                      $720.00          $43.20
Total Interest paid:  $1440.00      
Total Interest plus lost interest:  $1,483.20
The lost interest is the interest on $720 (720*0.06) paid half way through the year. Had we not lost the use of $720, by paying it, we would have had the use of that $43.20. So we really paid, and lost the use of a total interest of $1,483.20. So that is not an interest rate of 12%. It is higher. Although we also lost the use of the $720 payment, we did not pay interest on the $720 payment.
Now we will use the interest factor calculated by using the formula ((1+(12/100))^(1/2))-1 = 0.05830052443 
     Principal         Interest      Lost use
                                     of interest						

     $12,000           $699.61	
                       $699.61        $40.79
Total Interest:       $1399.21	    
Total Interest plus lost interest:  $1440.00
Note: The calculations were made with 11 places of decimal precision but rounding may lead to odd results. For instance 699.61 + 699.61 really is 1399.21. The lost use of interest is the interest on $699.61 ($699.61*0.05830052443) paid half way through the year. Had we not lost the use of $699.61 we would have had the use of that $40.79. So we really paid, or lost the use of, a total interest of $1,440.00. So that IS an interest rate of 12%. That explains why the formulas should be used to determine effective interest rates. That was for two payments a year. Lets try 3 payments, every four months, in a year.
Now we will use the interest factor calculated by using the formula ((1+(12/100))^(1/3))-1 = 0.03849882037 
     Principal         Interest      Lost use     Lost use
                                     of interest  of interest				
     $12,000           $461.99			
                       $461.99       $17.79
                       $461.99       $35.57            $0.68     
Total Interest:       $1385.96              
Total Interest plus lost interest: $1439.32         $1440.00
Now we nave 2 "Lost use of Interest" columns. There is lost interest on the lost interest.
Now lets try 4 payments, every three months, in a year.
Now we will use the interest factor calculated by using the formula ((1+(12/100))^(1/4))-1 = 0.02873734472 
     Principal   Interest      Lost use      Lost use     Lost use			
     $12,000                   of interest   of interest  of interest
                 $344.8481			
                 $344.8481       $9.9100
                 $344.8481      $19.8200     $0.2848
                 $344.8481      $29.7301     $0.8544       $0.0082
Total Interest: $1379.3925
Total Interest + lost int:    $1438.8526   1439.9918    $1440.0000
It is only by thinking of the use of the "lost use of interest" that we can see that without using the formulas to determine the "effective" interest rate, comparison of rates is, in most cases meaningless. Comparing the cost of a mortgage at 6% with the cost of another mortgage at 6.5% is pointless unless we know that the effective rate is quoted in both cases. Comparing the cost of a mortgage at 6% with the cost of credit card interest at 18%, is not meaningful even if both are quoted using effective interest rates. A choice may be made without such niceties.