Future Value, Single Amount

Read this section that discusses four separate but related concepts. They include: (1) multi-period investment, (2) approaches to calculating future value, and (3) single-period investment. How are these topics used in the business world? Applying these concepts is helpful when comparing alternative investments and when scarce capital resources are available. Often in a business setting, limited capital resources are available. Therefore, deciding which investment is best depends on comparing which investments will bring the highest returns to the business.

Single-Period Investment

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