## Additional Detail on Present and Future Values

This section gives more detail on computing present and future values. It shows you how to compute more complex problems involving future and present values when there are multiple compounding periods and when the time duration of those problems are longer or are less than one year.

### Calculating Values for Fractional Time Periods

The value of money and the balance of the account may be different when considering fractional time periods.

#### LEARNING OBJECTIVE

• Calculate the future and present value of an account when a fraction of a compounding period has passed

#### KEY TAKEAWAYS

##### Key Points
• The balance of an account only changes when interest is paid. To find the balance, round the fractional time period down to the period when interest was last accrued.
• To find the $PV$ or $FV$, ignore when interest was last paid an use the fractional time period as the time period in the equation.
• The discount rate is really the cost of not having the money over time, so for $PV$/$FV$ calculations, it doesn't matter if the interest hasn't been added to the account yet.

##### Key Terms
• time period assumption: business profit or loses are measured on timely basis
• compounding period: The length of time between the points at which interest is paid.
• time value of money: the value of an asset accounting for a given amount of interest earned or inflation accrued over a given period

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 time 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 at June 1, 2014, there is a bit of a conundrum. The last time interest was actually paid was at 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 time 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 have 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 isn't exactly the interest rate that is earned on the money. It is the same as that number, but more broadly, 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 downto 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 6.9999th 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 can't be a fractional time period, so the balance of the account is always exactly the value of the money.