Future Value, Single Amount

Since the number of periods ($n$ or $t$) is one, $F V=P V(1+i)$, where $i$ is the interest rate.

LEARNING OBJECTIVE

• Calculate the future value of a single-period investment

KEY TAKEAWAYS

Key Points

• Single-period investments use a specified way of calculating future and present value.
• Single-period investments take place over one period (usually one year).
• In a single-period investment, you only need to know two of the three variables $PV$, $FV$, and $i$. The number of periods is implied as one since it is a single-period.

Key Terms

• Multi-period investment: An investment that takes place over more than one periods.
• Periods ($t$ or $n$): Units of time. Usually one year.
• Single-period investment: An investment that takes place over one period, usually one year.

EXAMPLE

• What is the value of a single-period, $100 investment at a 5% interest rate? $PV=100$ and $i=5 \%$ (or $.05$) so $\mathrm{FV}=100(1+.05)$. $\mathrm{FV}=100(1.05)$ $\mathrm{FV}=$105$.

The amount of time between the present and future is called the number of periods. A period is a general block of time. Usually, a period is one year. The number of periods can be represented as either $t$ or $n$.

Suppose you're making an investment, such as depositing your money in a bank. If you plan on leaving the money there for one year, you're making a single-period investment. Any investment for more than one year is called a multi-period investment.

Let's go through an example of a single-period investment. As you know, if you know three of the following four values, you can solve for the fourth:

1. Present Value ($PV$)
2. Future Value ($FV$)
3. Interest Rate ($i$ or $r$) [Note: for all formulas, express interest in it's decimal form, not as a whole number. 7% is .07, 12% is .12, and so on. ]
4. Number of Periods ($t$ or $n$)

In a single-period, there is only one formula you need to know: $F V=P V(1+i)$. The full formulas, which we will be addressing later, are as follows:

Compound interest: $\mathrm{FV}=\mathrm{PV} \cdot(1+\mathrm{i})^{\mathrm{t}}$.

Simple interest: $\mathrm{FV}=\mathrm{PV} \cdot(1+\mathrm{rt})$.

We will address these later, but note that when $t=1$ both formulas become $\mathrm{FV}=\mathrm{PV} \cdot(1+\mathrm{i})$.

For example, suppose you deposit $100 into a bank account that pays 3% interest. What is the balance in your account after one year?

In this case, your $PV$ is $100 and your interest is 3%. You want to know the value of your investment in the future, so you're solving for $FV$. Since this is a single-period investment, $t$ (or $n$) is 1. Plugging the numbers into the formula, you get $F V=100(1+.03)$ so $\mathrm{FV}=100(1.03)$ so $F V=103$. Your balance will be $103 in one year.

Source: Boundless
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