# Calculating long intervals of Interest

Small interval calculations is fairly straight forward. But what about long intervals. If a loan's daycount is set to Actual/Actual and the duration has run into two different DayCount intervals, how abouts would you calculate the interest? Would you split it into two intervals?

A loan is accruing interest from December 15, 2015 to January 15 2016. Calculate the interest Accrued.

``````Principal is at 21,049.71
Interest is at 9.1%
Compounding semi-annually
``````

The total days passed was 31 days.

Do you calculate interest accrued from December 15 to December 31 then add it with January's accrued interest? Or do you calculate it in one whole interval.

Sample work:

``````Effective Annual Rate = (1 + 9.1/100/2)^2
= 1.09307025

Daily_Rate_365 = (1.09307025)^(16/365) = 1.003908571
Daily_Rate_366 = (1.09307025)^(15/366) = 1.00365381

Interest Accrued for first interval = (0.003908571) * 21049.71
= 82.274

Interest Accrued for second interval = (0.00365381) * (21049.71 + 82.274)
= 77.21

Total interest Accrued = 159.486
``````

This is incorrect, the interest accrued is 159.26 but I'm not sure how it's calculated.

Compounding semi-annually means 2 times in a year or every 6 months. Depending on various other terms, this at times means after 6 months of the loan period or it could mean once end of June and once end of December.

Assuming the reference is fixed for calendar year, then you need to calculate interest for December, add this back to principal and on this revised amount calculate interest for January.

• Hi @Dheer, I tried what you mentioned. I calculated december then added interest accrued and then tried to find second interval but it diidn't add up. Any thoughts? – user3276954 Oct 14 '15 at 14:07

First, as the OP did, find the effective interest rate, `r`. Then the monthly rate that produces the required result is based on an interval of 31/366.

``````p = 21049.71

i = 9.1/100 = 0.091

r = (1 + i/2)^2 - 1 = 0.0930703

m = (1 + r)^(31/366) - 1 = 0.00756592

p*m = 159.26
``````
• So the accrued interest from Dec doesn't go back into the principal amount to calculate the Jan interest? Think that is the step that was catching me up. This is a lot simpler that is for sure :) – Ross Oct 14 '15 at 15:28
• Hi Chris, thank you for the great example. What if the Payment date was changed to January 15, 2017. Now 397 days have passed. 366 of those days are in the leap year and 31 of those days weren't. – user3276954 Oct 14 '15 at 15:37
• For multi-year calculations they might use 365.25 days per year. – Chris Degnen Oct 14 '15 at 15:43