Future Value, Single Amount

Calculating FV involves identifying PV, i (or r), and t (or n) and then plugging them into the compound or simple interest formula.

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 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 $5,000 loan for 8 years. The loan accrues interest at a rate of 3% per quarter. On January 1, 2015, you will take out another $5,000, 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, there are 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 start with the first loan. Its present value was $5,000 on June 1, 2014. It is possible to find the loan's value today and then in 2017, but since the value is the same in 2017, it is okay to just imagine it is 2014 today.

Next, we need to identify the interest rate. The problem says it is 3% per quarter, or 3% every three months. Since the problem does not say otherwise, we assume that the interest on this loan is compounded.

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 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 do not really care when the loan ends in this problem – we only care about the value of the loan on December 31, 2017.

\(FV = PV \cdot (1+i)^t\)

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=5,000, 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=$5,000, 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

\(FV = PV \cdot (1+rt)\)

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