What are the differences between compound, actuarial and simple interest?
Compound interest
Interest is said to be compounded when interest is added to the principal after each interest period (thus generating interest on interest).
FinFlow calculates the year fraction as calculated by the given day count convention and the effective rate based on the given compounding frequency.
Compound interest is calculated for both full and fractional periods using standard compound interest formulas.
Actuarial interest
Contrary to the compound interest method, actuarial interest calculates compound interest only on full compounding periods, if there is a fractional period, interest on that period is calculated as simple interest.
FinFlow first counts the number of full compounding periods from the end date back to the start date, it then counts any remaining odd days to determine the fractional period.
Simple interest is calculated for the fractional period.
Compound interest is calculated for the full periods on the sum of the principal and the simple interest for the fractional period.
Simple interest
For simple interest, interest is not added to the principal at the end of each period.
FinFlow first counts the number of full compounding periods from the end date back to the start date, it then counts any remaining odd days to determine the fractional period.
Simple interest is the sum of the simple interest for the full periods and the simple interest for the fractional period.
Example
A $100,000 deposit is made on January 6th, 2010. The interest rate is 12% and intervals are monthly. The day count is based on actual days (act/365 day count corresponding with a 365 day year length).
How much interest would be generated on June 10th, 2010 using the different compute methods?
Compound interest
Year fraction: 155 days/ 365 days = 0.424657534
Equivalent rate: 12 % compounded monthly = 12.6825 %
Interest: $100,000 * ((1 + 0.126825) ^ 0.424657534 - 1) = $5,201.33
Actuarial interest
Full periods: from June 10th back to January 10th = 5 months
Fractional period: from January 10th back to January 6th = 4 days
Simple interest for ractional period: $100,000 * 4/365 * 12 % = $131.51
New principal: $100,000 + $131.51 = $100,131.51
Compound interest for full periods: $100,131.51 * ((1 + 0.01) ^ 5 - 1) = $5,239.22
Simple interest
Full periods: from June 10th back to January 10th = 5 months
Fractional period: from January 10th back to January 6th = 4 days
Fractional period: $100,000 * 4/365 * 12 % = $131.51
Full periods: $100,000 * 5/12 * 12 % = $5,000.00
Simple interest: $131.51 + $5,000.00 = $5,131.51
Observations
As you can see in the previous example, when fractional periods come into play, the actuarial interest method generates significantly more interest than the actuarial method. This is probably why most financial institutions prefer this method for their loan/lease calculations, thus making it the "industry standard".
Please note that there is a cumulative effect: if there are more intervals with fractional periods, results start to differ significantly from those obtained with the standard compound interest formulas.
It may well be that for an actuarial interest loan that has been calculated without fractional periods, the actual amount of interest paid is significantly higher when fractional periods start to occur due to irregular real life payment intervals .
For instance, in the actuarial method, interest for two equal one month intervals from January 6th to March 6th on a $100,000 principal is $2,010.00. If a zero payment is introduced on February 5th, the interest for the same two month period rises to $2,029.70. Compare this to the compound interest method where interest doesn't change at all if a zero payment is inserted (use a 30/360 day count to mimic the actuarial method's month based counting).
Conclusion
If you want to perform accurate cash flow calculations (FinFlow's goal) make sure that you use the compound interest compute method to get mathematically correct results.
If you need to check loan/lease statements or want to set up loans and leases based on standard industry practices, use the actuarial or simple interest compute method.
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