4.9 Fractional Periods

Occasionally it may be necessary to deal with compound interest for a fraction of a period, for example taking money out of a savings account after two weeks.  In these cases, it is important to understand what the policy of the bank or lender is.

For a savings account that pays interest monthly you may still receive interest for the two weeks (at the end of the month) if they pay interest on the daily balance but pay it monthly.  In other words, some banks calculate interest based on the equivalent daily rate (j₃₆₅) , but pay monthly.  The interest rate paid depends on the total daily closing balance. Interest rate is applied to the entire balance, calculated daily, and paid monthly.

However, for GICs if you cash them in early you may get no interest or if it is a redeemable GIC you will get interest for the time that you had it.  So if you have a 3-year cashable GIC that pays interest annually and you cash it in after 10 months it is unclear whether they will use simple interest or fractional exponents.  We reached out to a bank to ask –  the head office of TD Bank has stated that they had no idea, they said the “computer calculates it”.

For such cases, unless otherwise stated, use the compound-interest formula:

[latex]FV = PV(1+i)^n[/latex]

with n having a fractional part. The following example justifies this procedure.

Example

Suppose that $1,000 was borrowed at an interest rate of 10% compounded annually, and originally was scheduled to be repaid after two years. Instead, it was decided to repay the loan after 1.5 years.

1. Both parties have agreed to have interest calculated for the partial period.  At that time, by the compound-interest formula, the amount to be repaid would be:
   [latex]FV=$1000(1.10)^{1.5}=$1,000(1.15369…)=$1,153.69[/latex].  Now suppose this amount were to be reinvested at the same rate for the remaining half year. In this case, the money accumulated would be:[latex]FV=$1,153.69(1.10)^{0.5}=$1,210.00[/latex], which is exactly what would have resulted from the original two ­year loan:[latex]FV =$1,000(1.10)^2=$1,210.00[/latex]
2. Both parties have not agreed to have interest calculated for the partial period.  In this case, we would round down, so $n = 1$, and we receive no interest for the final year.[latex]FV =$1,000(1.10)^1=$1,100.00[/latex].  In this case, we get a lot less interest!

In fact, part 1 gives us a general property of compound interest: If the balance is found at any time and reinvested at the same rate, nothing changes. You may find that some financial institutions prefer to deal with the fractional period by assuming that simple interest is paid for that portion. At one time this method was popular because of the difficulty of performing the calculations for the formula without calculators or computers. It also meant that the institutions would receive more money since, for a partial period, simple interest is slightly higher than compound interest.

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