Suppose that I have a loan value x and interest rate r. The simple interest is then x*(1+r). If I take out a loan compounded annually and paid monthly for 12 months the amount at the end of the year would be the same. Why is it then that loan calculators give a different value? For example, if I were to borrow 100 at 10% I would owe 110 in simple interest and so 110/12 each month, yet using an online calculator I get that I owe 5.48 in interest for a total of 105.48. Why the discrepancy?

EDIT: Since it's due to differences in principal due, which one is actually the APR of the loan?

  • 1
    When you make a monthly payment, part of that is a repayment of principal amount, and part is interest. So you really have not borrowed $100 for a whole year, and so you don't really owe $10 in interest. – Dilip Sarwate Nov 26 '14 at 1:51
  • You state "compounded annually and paid monthly." This is never actually the case. Most loans either accrue interest daily, and payments are credited as of the day they are received, or are monthly, with interest charged each month, and payment date not really affecting balance so long as it doesn't run past due. – JoeTaxpayer Nov 26 '14 at 2:40

Most online calculators use monthly compounded interest rate (I created a few).

What this means is that it assumes you pay each month (as with usual payments), this has the effect that the principal amount becomes less and less through the course of the year and the interest becomes less and less.

The interest per month on 10% is 10%/12 = 0.833%

The Interest = Principal * Interest rate

Monthly Payment

 Principal   Interest rate   Interest  Payment 
 $100,00     0,833%          $0,83   $8,79 
 $92,04      0,833%          $0,77   $8,79 
 $84,02      0,833%          $0,70   $8,79 
 $75,93      0,833%          $0,63   $8,79 
 $67,77      0,833%          $0,56   $8,79 
 $59,55      0,833%          $0,50   $8,79 
 $51,25      0,833%          $0,43   $8,79 
 $42,89      0,833%          $0,36   $8,79 
 $34,46      0,833%          $0,29   $8,79 
 $25,96      0,833%          $0,22   $8,79 
 $17,38      0,833%          $0,14   $8,79 
 $8,74       0,833%          $0,07   $8,79 
 $0,02       0,833%          $0,00  

Note the decrease in interest

The sum of the Interest is $5.5, which accumulates to a total payment of $105.5

Yearly Payment

For this example and for sake of argument i'm continuing with monthly compounding interest

 $100,00    0,833%   $0,83   $-   
 $100,83    0,833%   $0,84   $-   
 $101,67    0,833%   $0,85   $-   
 $102,52    0,833%   $0,85   $-   
 $103,38    0,833%   $0,86   $-   
 $104,24    0,833%   $0,87   $-   
 $105,11    0,833%   $0,88   $-   
 $105,98    0,833%   $0,88   $-   
 $106,86    0,833%   $0,89   $-   
 $107,75    0,833%   $0,90   $-   
 $108,65    0,833%   $0,91   $-   
 $109,56    0,833%   $0,91   $110,47 
 $0,00  0,833%   $0,00  

Note the increase in interest

The sum of the Interest is $10.0, which accumulates to a total payment of $110

Conclusion

This is not really what happens with yearly or simple interest, it's more like a "boom" 10% added each year! But this should give you a feeling of how it works.

I hope this made sense.

It should be noted that banks and lenders usually have there own set of formulas they use to calculate the interest rate (usually something close to compounding daily), thus you will not know what you will pay until they tell you.

A mathematical expression for a loan can be made by setting the sum of the discounted future payments equal to the present value.

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where

pv is the initial (present) value
p  is the regular payment amount
n  is the number of periods
r  is the periodic interest rate

The closed form can be found by induction from the exponential summation.

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or

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(also given by this loan formula in a slightly different form.)

This formula can be used for your calculation.

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The twelve payments amount to 105.499

  • +1 for the math. The r is not directly solvable, correct? i.e. solving for r is iterative, even in the guts of the spreadsheet or calculator. – JoeTaxpayer Nov 26 '14 at 15:05
  • Hi Joe, that's correct. – Chris Degnen Nov 26 '14 at 15:17

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