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The Problem and the Future Value Formula
Let's say we have $100 and the interest rate is 5%. What will the value of the investment be at the end of the first year? The formula for the future value of this lump sum investment is as follows:
FV1 = PV + INT or PV(1 + I)ⁿ
The first part of this equation -- FV₁ = PV + INT -- reads, "the future value (FV) at the end of one year -- the subscript ᵢ -- equals the present value plus the added value at the specified interest rate.
The next formula presents this in a form that is easier to calculate the value added by the accrued interest -- PV(1 + I)ⁿ -- which reads, "the present value (PV) times (1 + I)ⁿ, where l represents the interest rate and the superscript ⁿ is the number of compounding periods.
Now let's use the example from above. In one year, your $100 lump sum investment earning 5% interest per year will equal:
FV = $100(1 + 0.05) = $105
In this instance, you do not see a superscript (n) for compounding periods because at this point you're solving for the first year only.
If you wanted to determine the value at the end of two years, your calculation would look like this:
FV = $100(1 + 0.05)² = $110.25
You can solve this, which is a compound interest problem, by calculating the value at the end of the first year, then multiplying the outcome by the same 5 percent rate for the second year:
FV = [$100(1+0.05)] + [$105(1+0.05)] = $110.25
You can continue this process to find the future value of the investment for any number of compounding periods.
Spelling out this process way -- manually performing each year's added value from interest, then using that value to make similar calculations for each following year -- makes it clear how we're arriving at the result, but it's time-consuming. Solving for a future value 20 years in the future means repeating the math 20 times. There are faster and better ways of accomplishing this, one of them being the use of a financial calculator.
Future Value of an Investment Using a Financial Calculator
The formula for finding the future value of an investment on a financial calculator is:
FVN = PV(1 + I)ⁿ
Although it doesn't quite look like it, this is the same formula we used when we did the calculations manually.
Incidentally, you can use this formula with any calculator that has an exponential function key -- most do. However, using a financial calculator is better because it has dedicated keys corresponding to each of the four variables we'll be using, speeding up the process and minimizing the possibility of error. Here are the keys you will punch:
Punch N and 2 (for 2 years)
Punch I/YR and 5 (for the interest rate of 5%)
Punch PV and -105 (for the amount of money we are calculating interest on in year 2)
Punch PMT and PMT (there are no payments beyond the first one)
Punch FV, which returns the answer of $110.25
The advantage of the financial calculator over the manual method is obvious. With the calculator, it's no more difficult or time-consuming to solve for a future value 20 years from now than to solve for a single year. Another time-saving method uses a spreadsheet.
Future Value of a Lump Sum Investment Using a Spreadsheet
Spreadsheets, such as Microsoft Excel, are well-suited for calculating time value of money problems. The function that we use for future value of an investment or a lump sum on an Excel spreadsheet is:
To use the function in the worksheet, click on "Formulas" in the title bar, then click on "Financial." You will then see a list of functions. Click on FV. That will open the Formula Builder box where you'll see five boxes labeled rate, nper, pmt, pv and type. If you want to determine the future value at the end of two years, fill out the boxes as follows:
rate (interest rate) = .05
nper (total number of payment periods = 2
pmt (repeated payments, in this case none) = 0
pv (present value, expressed as a negative number) = -100
type (this refers to the timing of subsequent payments) = Since there are none, enter 0
Earlier versions of Excel require that you click on Calculate to see the result. Later versions present the result automatically.
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