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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Multi-period investments take place over more than one period (usually multiple years). They can either accrue simple or compound interest.

LEARNING OBJECTIVE

• Calculate the future value of a multi-period investment with simple and complex interest rates

KEY TAKEAWAYS

Key Points

• Investments that accrue simple interest have interest paid based on the amount of the principal, not the balance in the account.
• Investments that accrue compound interest have interest paid on the balance of the account. This means that interest is paid on interest earned in previous periods.
• Simple interest increases the balance linearly, while compound interest increases it exponentially.

Key Terms

• accrue: To add, or grow.
• principal: The money originally invested or loaned, on which basis interest and returns are calculated.

There are two primary ways of determining how much an investment will be worth in the future if the time frame is more than one period.

The first concept of accruing (or earning) interest is called "simple interest". Simple interest means that you earn interest only on the principal. Your total balance will go up each period, because you earn interest each period, but the interest is paid only on the amount you originally borrowed/deposited. Simple interest is expressed through the formula in.

$F V=P V \cdot(1+r t)$

Simple Interest Formula: Simple interest is when interest is only paid on the amount you originally invested (the principal). You don’t earn interest on interest you previously earned.

Suppose you make a deposit of $100 in the bank and earn 5% interest per year. After one year, you earn 5% interest, or $5, bringing your total balance to $105. One more year passes, and it's time to accrue more interest. Since simple interest is paid only on your principal ($100), you earn 5% of $100, not 5% of $105. That means you earn another $5 in the second year, and will earn $5 for every year of the investment. In simple interest, you earn interest based on the original deposit amount, not the account balance.

The second way of accruing interest is called "compound interest". In this case, interest is paid at the end of each period based on the balance in the account. In simple interest, it is only how much the principal is that matters. In compound interest, it is what the balance is that matters. Compound interest is named as such because the interest compounds: Interest is paid on interest. The formula for compound interest is.

$F V=P V \cdot(1+i)^{t}$

Compound Interest: Interest is paid at the total amount in the account, which may include interest earned in previous periods.

Suppose you make the same $100 deposit into a bank account that pays 5%, but this time, the interest is compounded. After the first year, you will again have $105. At the end of the second year, you also earn 5%, but it's 5% of your balance, or $105. You earn $5.25 in interest in the second year, bringing your balance to $110.25. In the third year, you earn interest of 5% of your balance, or $110.25. You earn $5.51 in interest bringing your total to $115.76.

Compare compound interest to simple interest. Simple interest earns you 5% of your principal each year, or $5 a year. Your balance will go up linearly each year. Compound interest earns you $5 in the first year, $5.25 in the second, a little more in the third, and so on. Your balance will go up exponentially.

Simple interest is rarely used compared to compound interest, but it's good to know both types.

The Future Value can be calculated by knowing the present value, interest rate, and number of periods, and plugging them into an equation.

LEARNING OBJECTIVE

• Distinguish between calculating future value with simple interest and with compound interest

KEY TAKEAWAYS

Key Points

• The future value is the value of a given amount of money at a certain point in the future if it earns a rate of interest.
• The future value of a present value is calculated by plugging the present value, interest rate, and number of periods into one of two equations.
• Unless otherwise noted, it is safe to assume that interest compounds and is not simple interest.

Key Terms

• compound interest: Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest.

When calculating a future value ($FV$), you are calculating how much a given amount of money today will be worth some time in the future. In order to calculate the $FV$, the other three variables (present value, interest rate, and number of periods) must be known. Recall that the interest rate is represented by either $r$ or $i$, and the number of periods is represented by either $t$ or $n$. It is also important to remember that the interest rate and the periods must be in the same units. That is, if the interest rate is 5% per year, one period is one year. However, if the interest rate is 5% per month, $t$ or $n$ must reflect the number of periods in terms of months.

Example 1

What is the $FV$ of a $500, 10-year loan with 7% annual interest?

In this case, the $PV$ is $500, $t$ is 10 years, and $i$ is 7% per year. The next step is to plug these numbers into an equation. But recall that there are two different formulas for the two different types of interest, simple interest and compound interest . If the problem doesn't specify how the interest is accrued, assume it is compound interest, at least for business problems.

$F V=P V \cdot(1+i)^{t}$

Compound Interest: Interest is paid at the total amount in the account, which may include interest earned in previous periods.

$F V=P V \cdot(1+r t)$

Simple Interest Formula: Simple interest is when interest is only paid on the amount you originally invested (the principal). You don’t earn interest on interest you previously earned.

So from the formula, we see that $F V=P V(1+i)^{t}$ so $\mathrm{FV}=500(1+.07)^{10}$. Therefore, $\mathrm{FV}=\$ 983.58$.

In practical terms, you just calculated how much your loan will be in 10 years. This assumes that you don't need to make any payments during the 10 years, and that the interest compounds. Unless the problem states otherwise, it is safe to make these assumptions - you will be told if there are payments during the 10 year period or if it is simple interest.

Example 2

Suppose we want to again find the future value of a $500, 10-year loan, but with an interest rate of 1% per month. In order to get our total number of periods ($t$), we would multiply 12 months by 10 years, which equals 120 periods. Therefore:

$\mathrm{FV}=500(1+.01)^{120}$

$\mathrm{FV}=\$ 1,650.19$

Calculating $FV$ is a matter of identifying $PV$, $i$ (or $r$), and $t$ (or $n$), and then plugging them into the compound or simple interest formula.

LEARNING OBJECTIVE

• Describe the difference between compounding interest and simple interest

KEY TAKEAWAYS

Key Points

• The "present" can be moved based on whatever makes the problem easiest. Just remember that moving the date of the present also changes the number of periods until the future for the $FV$.
• To find $FV$, you must first identify $PV$, the interest rate, and the number of periods from the present to the future.
• The interest rate and the number of periods must have consistent units. If one period is one year, the interest rate must be X% per year, and vis versa.

Key Terms

• quarter: A period of three consecutive months (1/4 of a year).

The method of calculating future value for a single amount is relatively straightforward; it's just a matter of plugging numbers into an equation. The tough part is correctly identifying what information needs to be plugged in.

As previously discussed, there are four things that you need to know in order to find the $FV$:

1. How does the interest accrue? Is it simple or compounding interest?
2. Present Value
3. Interest Rate
4. Number of periods

Let's take one complex problem as an example:

On June 1, 2014, you will take out a $5000 loan for 8-years. The loan accrues interest at a rate of 3% per quarter. On January 1, 2015, you will take out another $5000, eight-year loan, with this one accruing 5% interest per year. The loan accrues interest on the principal only. What is the total future value of your loans on December 31, 2017?

First, the question is really two questions: What is the value of the first loan in 2017, and what is the value of the second in 2017? Once both values are found, simply add them together.

Let's talk about the first loan first. The present value is $5,000 on June 1, 2014. It is possible to find the value of the loan today, and then find it's value in 2017, but since the value is the same in 2017, it's okay to just imagine it is 2014 today. Next, we need to identify the interest rate. The problem says it's 3% per quarter, or 3% every three months. Since the problem doesn't say otherwise, we assume that the interest on this loan is compounded. That means we will use the formula in . Finally, we need to identify the number of periods. There are two and a half years between the inception of the loan and when we need the $FV$. But recall that the interest rate and periods must be in the same units. That means that the interest must either be converted to % per year, or one period must be one quarter. Let's take one period to be one quarter. That means there are 10 periods. Please note that we don't really care when the loan ends in this problem–we only care about the value of the loan on December 31, 2017.

$F V=P V \cdot(1+i)^{t}$

Compound Interest: Interest is paid at the total amount in the account, which may include interest earned in previous periods.

Next, we simply plug the numbers into . $PV=5000$, $i=.03$, and $t=10$. That gives us a $FV$ of $6,719.58.

Now let's find the value of the second loan at December 31, 2017. Again, $PV=$5000$, but this time, pretend it is January 1, 2015. This time, the interest is 5% per year and it is explicitly stated to be simple interest. That means we use the formula in . January 31, 2017 is exactly two years from the January 1, 2015 and since the interest is measured per year, we can set $t=2$ years.

When we plug all of those numbers into , we find that $FV=$5,500.00$

$F V=P V \cdot(1+r t)$

Simple Interest Formula: Simple interest is when interest is only paid on the amount you originally invested (the principal). You don’t earn interest on interest you previously earned.

Since the problem asks for the total $FV$ of the loans, we add $6,719.58 to $5,500.00, and get a total value of $12,219.58.