The problem of implicit capitalization
When capitalizing on item dates, capitalization occurs at the frequency of the items in the cash flow, not at the period frequency set in Compute options.
The period frequency determines how the interval between two dates is broken down into periods and days fractions, i.e. to measure time — if the interval between items is shorter than the period frequency, the interval consists of a days fraction, otherwise it contains a periods fraction plus any odd days fraction.
When items are regular, capitalization is consistent and predictable. When items are irregular, capitalization varies with the shorter or longer intervals between items. In both cases, actual capitalization is dictated by the cash flow rather than the interest periods setting — this is what we call implicit capitalization.
To help you figure out whether the interest periods frequency in Compute options corresponds with the actual cash flow intervals, InterestThing calculates the dominant interval and displays it in the Summary tab. Compare this value to the period frequency to see if there is a significant difference.
To avoid implicit capitalization, you can use the period anchors setting to specify that capitalization should only occur on a reference date.
What are reference dates?
The base reference date (or anchor date) can be either the first date in the table or a custom date, for example the first day of the month or the date of the first payment.
Depending on the reference date set in Compute options, InterestThing generates a series of capitalisation dates so that it can calculate for each interval between two dates whether capitalization should occur or not.
How the interval between two dates relates to the reference date grid determines what happens with accrued interest:
- when the interval does not contain a capitalisation date it treats accrued interest as pending interest and adds it to the total interest due.
- when the interval contains one or more capitalisation dates, pending interest is capitalised together with any accrued interest up to the last capitalisation date. Interest generated after the last capitalisation date goes to pending again.
Depending on where the start and end dates fall relative to the reference date grid, combinations of leading stubs, full periods, trailing stubs and floating stubs are possible. An overview of possible combinations can be found under method 3 on the Date interval methods page.
Reference dates and pending interest
When capitalizing on item dates, the interest due column in the schedule will likely remain empty, unless you have rate changes.
When capitalizing on reference dates, extra columns for capitalized/compounded and pending interest appear. The pending interest column will contain values unless the cash flow is completely regular and reference dates correspond with item dates.
As capitalization is only meant to happen on reference dates it is perfectly normal to have pending interest:
- when item dates do not correspond with reference dates, for instance when payments are on the 15th of a month and reference dates are the 1st of a month
- when item intervals are shorter than reference periods, for instance when payments are weekly and reference dates are monthly: pending interest accumulates over multiple weeks, capitalization occurs on the monthly reference date and any interest from a trailing stub is kept pending again.
Reference dates and US Rule
When US Rule is active and capitalization occurs on reference dates, the two mechanisms interact in a way that can produce unexpected results.
Under US Rule, a payment is first tested against accrued and pending interest before any remainder is applied to the balance. If the payment does not cover the interest, the shortfall is kept pending and the balance is not reduced.
1. Capitalization frequency equals payment frequency
When capitalization occurs on every payment date (for instance monthly payments with monthly reference dates), every payment goes through this test. If a long first interval has generated more pending interest than payments can cover, the balance stays frozen until a payment is finally large enough to clear both accrued and pending interest. Only then does capitalization occur and the balance start to decrease.
2. Capitalization frequency higher than payment frequency
When the capitalization frequency is higher than the payment frequency (for instance weekly capitalization with monthly payments), capitalization can only occur when a payment is made. Between payments there may be multiple reference dates, but this does only trigger a single capitalization, so the higher capitalization frequency has no effect in practice.
In the example above, weekly reference dates with monthly payments produces a similar schedule to monthly reference dates with monthly payments. Capitalization occurs monthly, driven by the payment dates, not by the weekly reference date grid.
With capitalized simple interest, multiple reference dates within a single payment interval trigger only one capitalization. This is fundamentally different from compound interest, where each reference date would capitalize separately and thus generate more interest.
3. Capitalization frequency lower than payment frequency
When capitalization occurs less frequently than payments (for instance monthly payments with quarterly reference dates), some payment intervals contain no reference date. Those payments are not subject to the US Rule test and are applied directly to the balance. This means the balance starts decreasing earlier, even while pending interest is still accumulating between capitalization dates.
Paradoxically, less frequent capitalization can lead to earlier and faster balance reduction than more frequent capitalization, because more payments bypass the US Rule test and go directly to principal. For those payments, accrued interest is added to pending interest to be taken into account at the next capitalization date.
In extreme cases, when capitalization is so infrequent that payments on capitalization dates never cover the accumulated pending interest, no capitalization ever occurs. The balance decreases through non-capitalization payments and eventually turns positive, at which point it generates positive interest that reduces the interest due.
Meanwhile, pending interest decreases slowly through capitalization date payments. The two converge from opposite sides until they offset each other at the end of the schedule. The result is mathematically correct but the loan does not amortize in the traditional sense.
InterestThing informs you when this situation is detected by displaying US Rule: no capitalization occurred in the status bar. Click the warning icon for details.
Whether capitalization eventually occurs depends on three interacting factors: the pending interest built up during the (long) first interval, the capitalization frequency, and the payment size relative to ongoing accrual between capitalization dates. Because these interact, the only reliable way to determine the outcome is to examine the schedule.
Simple interest and pending interest
With pure simple interest, interest is never capitalized by definition — all payments go directly to the balance while interest accumulates separately in the interest due column. This produces the same convergence behaviour as the extreme US Rule case, but for a different reason.
As payments reduce the balance, it eventually turns positive. At that point the positive balance generates positive interest that reduces the interest due, while interest due decreases from its peak. The two converge from opposite sides until they offset each other at the end of the schedule.
This is the expected behaviour for pure simple interest with regular payments — not an error, but a consequence of interest never being capitalized. For irregular cash flows, the total due column shows the outstanding amount at any point in the schedule without needing the solver.