# What is the formula for loan payoff with daily compounded interest and monthly payment?

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" Jun 7, 2015 at 16:31
• At 4% DAILY that debt more than doubles each month. There is no way to pay it back with given monthly payment. Jun 7, 2015 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. Jun 7, 2015 at 16:39
• The rate appears annual obviously. OP meant compounded daily. Jun 7, 2015 at 16:39
• @Ghanima Thanks for catching that, I did mean 4% annual but compounded daily. Jun 7, 2015 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). 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: Where PV = 21750, Pmt = 220, i = 0.0033387092772 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? Jun 7, 2015 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. Jun 7, 2015 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. Jun 7, 2015 at 17:12
• @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. Jun 7, 2015 at 17:38