## Present Value and Single Amount

This section discusses how to calculate the present value of a future single-period payment, the return on a multi-period investment over time, and what real-world costs to the investor comprise an investment’s interest rate. It also addresses what a period is in terms of present value calculations and distinguishes between the formula for present value with simple interest and compound interest.

### Single-Period Investment

When considering a single-period investment, n is one, so the PV is simply FV divided by 1+i.

#### LEARNING OBJECTIVE

• Calculate the present value of a future, single-period payment

#### KEY TAKEAWAYS

##### Key Points
• A single period investment has the number of periods ($n$ or $t$) equal to one.
• For both simple and compound interest, the $PV$ is $FV$ divided by $1+i$.
• The time value of money framework says that money in the future is not worth as much as money in the present.

##### Key Terms
• period: The length of time during which interest accrues.
• interest rate: The percentage of an amount of money charged for its use per some period of time. It can also be thought of as the cost of not having money for one period, or the amount paid on an investment per year.

The time value of money framework says that money in the future is not worth as much as money in the present. Investors would prefer to have the money today because then they are able to spend it, save it, or invest it right now instead of having to wait to be able to use it.

The difference between what the money is worth today and what it will be worth at a point in the future can be quantified. The value of the money today is called the present value ($PV$), and the value of the money in the future is called the future value ($FV$).There is also a name for the cost of not having the money today: the interest rate or discount rate ($i$ or $r$). For example, if the interest rate is 3% per year, it means that you would be willing to pay 3% of the money to have it one year sooner. The amount of time is also represented by a variable: the number of periods ($n$). One period could be any length of time, such as one day, one month, or one year, but it must be clearly defined, consistent with the time units in the interest rate, and constant throughout your calculations.

$F V=P V(1+i)^{n}$

$FV$ of a single payment: The $FV$ is related to the $PV$ by being $i%$ more each period.

All of these variables are related through an equation that helps you find the $PV$ of a single amount of money. That is, it tells you what a single payment is worth today, but not what a series of payments is worth today (that will come later). relates all of the variables together. In order to find the $PV$, you must know the $FV$, $i$, and $n$.

When considering a single-period investment, $n$ is, by definition, one. That means that the $PV$ is simply $FV$ divided by $1+i$. There is a cost to not having the money for one year, which is what the interest rate represents. Therefore, the $PV$ is $i%$ less than the $FV$.

Source: Boundless