The origins of interest

The concept of interest is very old. It was first mentioned in texts from the Sumerian Third Dynasty of Ur (circa 2100 BC to 2000 BC).

In Sumerian the word for interest is mash, which was also the word for calves. The first loans were probably loans of cattle, and when the cattle was returned to the lender the calves were the 'interest'. For loans of grain and loans of silver, the 'interest' was to be returned as a percentage of the principal.

Evidence of compound interest calculations dates back to the Old Babylonian period (circa 2000 BC to 1600 BC).

The Babylonians and Assyrians who later ruled Mesopotamia had basic rates of interest for loans of money of 1/5th to 1/4th (20 to 25 %), for loans of grain it was 1/3rd (33.33 %). For overdue periods it could be as high as 100 %!

The cuneiform sign for the Sumerian word mash is the symbol in the icon for InterestThing.

Simple interest

For simple interest, interest is calculated on the original principal over the term at the nominal annual rate.

The formula for simple interest is:

• accumulated value = principal x (1 + rt)
• where r is the nominal annual rate, and t is the term in years

Example: three months of simple interest at a nominal annual rate of 12 % on a 100.000 deposit.

• term is 3 / 12 or 0.25 years
• rate times term is 12 % x 0.25 = 3 %
• accumulated value = 100.000 x (1 + 0.03) = 103.000
• accrued interest = 100.000 x 3% = 3.000

Compound interest

For compound interest, interest is added to the principal after each period and taken into account when calculating the next period.

The formula for compound interest is:

• accumulated value = principal x (1 + r) to the power of t
• where r is the periodic rate or the nominal annual rate divided by the number of periods per year, and t is the number of interest periods

Example: three months of compound interest at a nominal annual rate of 12 % on a 100.000 deposit

• the periodic rate for one month is 12 % / 12 = 1 %
• the number of periods or t = 3
• accumulated value = 100.000 x (1 + 1%) ^3 = 103.030,10
• accrued interest = 100.000 x ((1 + 1%) ^3 - 1) = 3.030,10

As you can see, compound interest produces more interest than simple interest.
When intervals are shorter, more compounding occurs and more interest is generated.

Interest calculation issues

When computers appeared, calculating interest became very easy, however, results vary widely depending on what program is used to calculate them.

Even though the formulas for simple and compound interest are not complicated, there are many ways to actually compute interest, depending on how non-standard situations are handled.

The following paragraphs describe some of the issues and how they are handled in InterestThing.

1. Using the theoretical or the practical method

The formulas for simple and compound interest were developed on the assumption that periods are represented by integers. A first problem arises when there are fractional periods, for example when calculating with monthly intervals and the time interval is 3 months and 10 days, which is roughly 3,33 periods.

Theoretically, using 3,33 for the number of periods is OK as computers can easily calculate with non-whole exponents.

Using fractional periods is known as the exact or theoretical method.

When InterestThing calculates interest based on a year fraction it uses the exact method. The number of periods is the year fraction as calculated by the selected day count convention.

To calculate compound interest, an equivalent annual rate is calculated based on the nominal annual rate and the compounding frequency.

Odd days are taken into account in the year fraction calculation, which means that interest for odd days is also compounded.

As the exact method involves more complex calculations, in practice compound interest is used for full periods and simple interest for the odd days.

This method is called the approximate or practical method.

When InterestThing calculates interest based on periods and days, it uses the practical method and a standard year divided in a number of equal periods.

The daily rate that is used for odd days depends on the days fraction method, which can be any of the provided day count conventions.

The days fraction can have a denominator that is either fixed or variable depending on the stub period and the days fraction method

Examples:

• 30/360 based daycounts always use 360 as the denominator
• Act/365 Fixed always uses 365
• Act/Act ISDA sometimes uses 365 or 366, depending on whether the stub period falls in a leap year, when the stub period contains both days in a normal year and a leap year, it uses both 365 and 366 as denominators
• Act/Act ICMA multiplies the stub period length by the period frequency as the denominator, e.g. for quarterly periods it could use period lengths of 89, 90, 91 and 92 days resulting in denominators 356, 360, 364 and 368 respectively

2. Implicit capitalization of interest

When just calculating interest between a start and an end date, the balance is only adjusted on the end date.

When calculating interest from date to date over a series of dates, for example when amortizing a loan, it matters how accrued interest is handled.

Most programs just calculate the accrued interest from date to date, calculate the principal and update the balance.

This is OK for regular periods when payments coïncide with the compounding frequency, but when there are multiple payments inside a period, updating the balance within an interest period is in fact influencing the interest over the next sub-period, as that calculation uses the updated and not the original balance.

For example, when payments are weekly but compounding is set to monthly, you would end up with four or five payments per month, resulting in four to five capitalisations per period.

This way you would end up with results as if you were compounding weekly instead of monthly.

This is what we call implicit capitalization or compounding.

Amortization applications that recognize this problem deal with it in various ways:

2.1 Adjust rates

Using effective and equivalent rates to adjust for differing payment and compounding frequencies for a given nominal annual rate could be a solution as long as the payments are regular.

However, this will fall apart for irregular payments, resulting in unpredicable results.

2.2 Don't allow lower payment frequencies

Some applications simply don't allow users to create series of payments that have intervals that are smaller than interest periods or the compounding frequency.

However, when irregular payments are added, some applications accept them and additional capitalization compounding occurs for each irregular payment.

2.3 Add capitalization dates

Some applications provide for simple interest to be capitalized only at periodic capitalization dates.

When examining the actual values in such an amortization table you may find that capitalization still occurs after each payment and additionally on the periodic capitalization dates.

We think that instead of solving the problem, it makes it worse.

2.4 Provide total control

InterestThing's approach is to let you set compute options that define when and how capitalization is applied.

When selecting simple interest as the Interest type, InterestThing lets you choose whether interest is capitalized or not.

• Choosing no just calculates simple interest as described above, using the method selected for Date intervals.
• For date intervals based on year fractions, choosing yes results in simple interest that is capitalized from date to date.
• For date intervals based on periods and days, choosing yes adds extra options:
  
• to capitalize interest on the event date, i.e. from date to date.
• to capitalize interest on capitalization dates as defined by a reference date.

If you absolutely need to calculate with both capitalization on each payment date and on a periodic capitalization date as some applications do, don't use reference dates but set capitalization to occur on the item date and add a series of zero amount payments to the cashflow to force a capitalization on those dates.

3. Capitalized simple interest vs. compound interest

When using reference dates, capitalized simple interest is not the same as compound interest!

When the time interval is equal to or less than one period, results are the same for capitalized simple interest and compound interest. Capitalization or compounding (if any) will occur only once.

When the time interval includes more than one period, capitalized simple interest will only capitalize on the last capitalization date, whereas compound interest will compound at the end of each compounding period.

4. Interest on interest

Appendix J to part 1026 of the Truth in lending Act (Regulation Z) determines two methods to calculate the APR of closed-End Credit transactions.

Under the actuarial method the balance is always updated even if the payment is insufficient to cover the accrued interest. This can lead to interest being charged on interest, for instance when the first payment is made after a longer time interval or when there are multiple skipped payments

Under the U.S. Rule method, if the payment is insufficient to cover the accrued interest, the interest is kept pending and added to the total interest due and the adjustment of the balance is postponed until the total of the payments covers the total interest due.
Using this method interest is never charged on interest.

InterestThing uses the U.S. Rule method when the US Rule checkbox is checked, otherwise it uses the actuarial method when calculating capitalized simple interest.