# Loan repayment calculations when interest compounding frequency does not match repayment frquency

Is there a formula for calculating loan repayments where interest is compounded daily, but repayments are made only monthly, for instance?

I would like to be able to calculate the repayment amount for any combination of repayment and compounding frequencies.

Cheers!

Yes, you must first convert your daily interest rate to a monthly interest rate:

Use the following formula:

$$(1+i^{(365)})^{365}=(1+i^{(12)})^{12}=(1+i)^1$$

Where $i^{(m)}$ is your m-thly effective interest rate

Then, you can use annuity formulas to determine your m-thly payment.

You can also determine the "daily payment" that is equivalent to the monthly payment but generally the method above is better when working with numbers

Let

• $P$ = the principal amount
• $n$ = the number of compounding periods that fit into each repayment period = e.g., 30 days in a month, in your case
• $r$ = the interest rate over the compounding period = a daily interest rate in your case, expressed as a decimal, e.g. 0.1% expressed as 0.001

The principal outstanding at the end of the repayment period is

$$P\left( 1+ r \right)^n$$

So the interest paid is the outstanding principal less the original principal

$$P\left( 1+ r \right)^n - P$$