Additional Detail on Present and Future Values

The value of money and the account balance may differ when considering fractional periods.

Up to this point, we have implicitly assumed that the number of periods in question matches to a multiple of the compounding period. That means that the point in the future is also a point where interest accrues. But what happens if we are dealing with fractional periods?

Compounding periods can be any length of time, and the length of the period affects the rate at which interest accrues.

Compounding Interest The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies.

Suppose the compounding period is one year, starting January 1, 2012. If the problem asks you to find the value on June 1, 2014, there is a bit of a conundrum. The last time interest was actually paid was on January 1, 2014, but the time value of money theory clearly suggests that it should be worth more in June than in January.

In the case of fractional periods, the devil is in the details. The question could ask for the future value, present value, etc., or it could ask for the future balance, which has different answers.

Future/Present Value

If the problem asks for the future value (FV) or present value (PV), it doesn't really matter that you are dealing with a fractional time period. You can plug in a fractional time period to the appropriate equation to find the FV or PV. The reasoning behind this is that the interest rate in the equation is not exactly the interest rate that is earned on the money. It is the same as that number, but more broadly, it is the cost of not having the money for a time period. Since there is still a cost to not having the money for that fraction of a compounding period, the FV still rises.

Account Balance

The question could alternatively ask for the balance of the account. In this case, you need to find the amount of money that is actually in the account, so you round the number of periods down to the nearest whole number (assuming one period is the same as a compounding period; if not, round down to the nearest compounding period). Even if interest compounds every period, and you are asked to find the balance at the 6.9999^th period, you need to round down to 6. The last time the account actually accrued interest was at period 6; the interest for period 7 has not yet been paid.

If the account accrues interest continuously, there is no problem: there cannot be a fractional time period, so the balance of the account is always exactly the value of the money.