My goal is to make a graph using this formula for all possible monthly payments.

Let's say...

Loans: 21750
Monthly Payment: 220
Daily Interest: 4.0% / 365

What formula would I use to get the total amount of money (or Principal) I'd use to pay off this loan at this rate?

  • keyword: "loan amortization" – keshlam Jun 7 '15 at 16:31
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    At 4% DAILY that debt more than doubles each month. There is no way to pay it back with given monthly payment. – Ghanima Jun 7 '15 at 16:38
  • @Ghanima excuse me but all internet rates in the market are quoted at nominal (annualized) basis. You divide 4% by 365 to get daily. – base64 Jun 7 '15 at 16:39
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    The rate appears annual obviously. OP meant compounded daily. – JoeTaxpayer Jun 7 '15 at 16:39
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    @Ghanima Thanks for catching that, I did mean 4% annual but compounded daily. – Vongdarakia Jun 7 '15 at 16:46

Since the compounding period and payment period differs (Compounded Daily vs Paid Monthly), you need to find the effective interest rate for one payment period (month).

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This means that each month you pay 0.33387092772% of the outstanding principal as interest.

Then use this formula to find the number of months:

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Where PV = 21750, Pmt = 220, i = 0.0033387092772

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That gives 120 Months.

Depending on the day count convention, (30/360 or 30.416/365 or Actual/Actual), the answer may differ slightly. Using Financial Calculator gives extremely similar answer.

The total cash paid in the entire course of the loan is 120 x $220 = $26,400

  • This is perfect! Thanks! Although I am quite curious to why ln is used. Can you explain the meaning of that? – Vongdarakia Jun 7 '15 at 17:04
  • @Vongdarakia It is "Natural Log". It is a button on most calculators. ln(y) gives the x in the equation 2.71828^x=y, where 2.71828 is a mathematical constant. – base64 Jun 7 '15 at 17:07
  • Ah, I see. Thanks! Also, 26,400 doesn't seem to be the exact amount. Would it make sense to calculate the total amount like this? total = total - (nMonths - wholeMonths) * monthlyPayment? i.e. 26400 - (120.184399 - 120) * 220 = 26359.43. Since the 0.184399 looks to be the overpayment. – Vongdarakia Jun 7 '15 at 17:12
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    @Vongdarakia I'd say it is reasonable but 120.18 is just an approximation. Sometimes there are 366 days in a year, sometimes a month is not 30 days, but 31 or 29 or 28. – base64 Jun 7 '15 at 17:38

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