# How to quantify the effect of compounding interval on total interest accrued

I am struggling with calculating the total interest to be paid on a loan, when given Principal, APR, Compounds per Year, and term.

Mostly, I get confused when the term is not the same unit as the compounds per year. So, my math doesn't come close. This problem is based on my own analysis of these relationships. Perhaps, I am just running in a circular reference and getting confused....

For example:

``````Principal = \$18,500
APR = 2.800%
Compounds/yr = 365 (daily)
term = 60 months

Total interest paid over lifetime of loan is.... ?
First, do i have to calculate payment? Can payment be unknown?
``````

Next example:

``````Principal = \$18,500
APR = 2.800%
Compounds/yr = 12 (monthly)
term = 60 months
``````

My goal is to compare the solutions of these two problems and understand how the compounding period has an effect on the total interest accrued over the life of the loan.

• Is that nominal APR or effective APR? See this for the difference. – Lawrence Jan 5 at 16:16
• Is this homework for which you need the exact answer, or your own research, where Close Enough is Good Enough? – RonJohn Jan 5 at 16:50
• We aren't finance professionals. – RonJohn Jan 6 at 0:59
• Out of curiosity, why did you think we are finance professionals? (Though I'm sure that some of the regulars here can answer the question.) – RonJohn Jan 6 at 15:52
• @RonJohn some of us are finance professionals – MD-Tech Jan 23 at 14:36

It's not clear in the first example if a payment is also made daily or if just the interest is compounded daily. For the simple case, lets assume that they are the same.

The formula for the payment amount for a loan is:

``````PV  *  r
---------
1-(1+r)^-n
``````

in your example, n=5*365 (5 years), r is 2.8%/365 = 0.000077, and PV is 18,500. So the result is:

``````18,500 * 0.000077
-----------------       =   10.8635
1-(1.000077)^(-1825)
``````

So the total amount paid is `10.8635 * 5 * 365 = 19,826`. Since 18,500 of that is principal, the amount of interest paid is 1,326.

The second example follows the same way but with r=2.8%/12 and n=60:

``````18,500 * .002333
-----------------       =   330.7758
1-(1.002333)^(-60)
``````

So the total amount paid is `330.7758 * 5 * 12 = 19,846`. Since 18,500 of that is principal, the amount of interest paid is 1,346.

As you can see, the compounding frequency doesn't make a huge difference in this case.

When the compounding frequency is different than the payment frequency then things get a little more complicated. For instance, if the interest compounds daily but payments are made monthly, then you need to calculate how much interest is in each monthly payment. Since months have different number of days, the most accurate way to compute it is to accrue interest each day and then apply

One way to estimate it is to calculate an average equivalent interest rate for a month. We can compute the equivalent interest rate for an average month by compounding interest by 365/30 or 30.4 days:

``````rm = 1-(1+rd)^30.4
``````

So in your first example, the equivalent monthly rate would be

``````1 - (1+(2.8%/365)^30.4) = .2335%
``````

Plugging that into the monthly payment formula yields

``````  18,500 * .002335
-----------------       =   330.7955
1-(1.002335)^(-60)
``````

For a total interest amount of `330.7955*60 - 18,500 = 1,347`