Using Your Calculator to Find the Future Valueof a Single Sum with Annual Compounding

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Dr. Craig RuffDepartment of FinanceJ. Mack Robinson College of BusinessGeorgia State University

© 2014 Craig Ruff

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Solving with the Calculator - Formula

Suppose you have the following example: Calculate the future value of $100 in 5 years at an interest rateof 6%, compounded annually.

When using the calculator and the formula for the future value of a single sum with annual compounding:

you need to remember your calculator’s order of operations.

To do this one, you may need to add parentheses to ensure the calculator is doing what you want it to do, as in… “100 x ( 1.06 5 ) = “ and you will get an answer of 133.822.

Another way to do this (and without parentheses) is to first enter the 1.06 raised to the fifth and then multiply by 100, as in…

“1.06 5 x 100 = “ and you will get the answer of 133.822.

FVn PV (1 r)N

y x

y x

Recall ‘PEMDAS’ or ‘Please Excuse My Dear Aunt Sally’ from your school days?

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Using the time-value-of-money buttons on the calculator is generally a game of putting in four of the five pieces of information andasking the calculator to tell you the fifth.

It is helpful to set your calculator out to plenty of decimal places. To set your BA II Plus to 8 decimal places, hit 2ND decimal place (the ‘.’)9ENTER

Before starting, I also suggest making sure that your BA II Plus setting are:2ND I/Y = ‘1.0000000’2ND PMT = ‘END’ The buttons stand for:PV = present value FV = future valueI/Y = compound rate for the periodN = number of periods in the future (or the number of payments, if an annuity)PMT = this is for annuity payments; for this problem, we input a ‘0’ as we are dealing with a single sum.

Solving with the Calculator – TVM Buttons

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Using the time-value-of-money buttons, On the calculator, the future value of $100 in 5 years at an interest rate of 6%, compounded annually is solved as:

Buttons Numbers to Enter

PV -100FV ???? 133.822I 6N 5PMT 0

Solving with the Calculator – TVM Buttons

© 2014 Craig Ruff

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Using the time-value-of-money buttons, On the calculator, the future value of $100 in 5 years at an interest rate of 6%, compounded annually is solved as:

Buttons Numbers to Enter

PV -100FV ???? 133.822I 6N 5PMT 0

Solving with the Calculator – TVM Buttons

Be careful: Your calculator uses percent form (6).

However, the formula uses the decimal form (.06) as in:

$133.82=(1.06)*$100=FV 55

© 2014 Craig Ruff

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Using the time-value-of-money buttons, On the calculator,the future value of $100 in 5 years at an interest rate of 6%,compounded annually is solved as:

Buttons Numbers to Enter

PV -100FV ???? 133.822I 6N 5PMT 0

Notice that the PV was entered as -100. Hadyou entered a positive 100, you would havegotten an answer of -133.822. In terms ofdoing an easy problem like this one, entering apositive 100 is no big deal; you know that your-133.822 really represents an ending balance of133.822.

However, for some problems you willeventually see, the sign of what you enter intothe calculator is very important to the finalsolution.

Thus, it is healthy to get into the habit ofhaving money going out of your pocket (as in a deposit) be a negative and money coming inyour pocket (as in withdrawing the money from the bank) be a positive.

Solving with the Calculator – TVM Buttons

© 2014 Craig Ruff

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As a word of caution, using the calculator’s TVM buttons can make this topic seem deceptively easy. I suggestthat when you see this type of time-value question, you not think in robotic terms of “OK, now I push thisbutton and then this button, and…”

Instead, I would suggest that you first cast the question in words that make sense to you, such as: ‘Suppose Iplace $100 today in a bank account that pays 6% interest per year. At the end of five years, how much moneywill I have in the bank account?’

Also, it is often helpful to imagine the question graphically on a timeline. Here, we are “pushing” the moneyout to year 5. Or, we could say that we are moving the money forward in time from t=0 to t=5. Graphically, itwould look like:

t=0t=5

$1005$100(1.06)

Solving with the Calculator – TVM Buttons

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  A B1 Present Value -1002 Years 53 Annual Rate 6%4    5 Future Value ($133.82)

Excel has a lot of built in functions, including future value. The ‘0’ in the parenthesis indicates that the annuity payments arezero (as this is a single sum.)

=FV(B3,B2,0,B1)

Solving with Excel

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Examples

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Let’s say Bill Gates is worth $56 billion. Suppose Bill takes his entire $56 billion and deposits it in the Commerce Bank of BeverlyHills at a rate of 2%, compounded annually.

How much interest will Bill earn in the first 10 minutes?

I have used a PV of -$5.6 billion, since my calculator doesn’t handle $56 billion. That is no problem. I just need to remember tomultiply my answer by 10. (Again, I made this a negative number to indicate money going out of Bill’s pocket to make the deposit.)

Since the rate is 2%, compounded annually,, I simply plug in ‘2’.

The tricky part is telling the calculator how long. There are 525,600 minutes in a year (365 * 24 *60). Thus, N is equal to10/525600 (=.00001903).

PV -5,600,000,000

FV ??? 5,600,002,110

IY 2

PMT 0

N 0.00001903

Example: Future Value of a Single Sum: Annual Compounding

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To see how much Bill has made in the first ten minutes, directly subtractthe original 5,600,000,000 from this number. The answer you get onyour calculator is $2,109.867. Notice that this number is not exactly the2,110 that the results to the left would suggest. That is because, due tospace display constraints, your calculator (on the left) is showing you aslightly rounded number.

But underneath the necessary ‘rounding’ by your calculator, yourcalculator is really carrying the true number of 5,600,002,109.867 (andalso with lots more decimal places than the three I use). Once yousubtract the initial 5,600,000,000, your calculator can show you theaccurate number of $2,109.867.

We are not done yet…. Remember that this answer is only based on $5.6billion. Bill has $56 billion. So, we need to multiply this answer by 10 toget the answer. Bill would make $21,098.67 in interest in the first tenminutes.

PV -5,600,000,000FV ???? 5,600,002,110IY 2

-5,600,000,000

PMT 0 = 2,109.867N 0.0000190

Example: Future Value of a Single Sum: Annual Compounding

© 2014 Craig Ruff

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When doing these problems, do not round your inputs.

For instance, when entering the value for N, it may be tempting to simply use .000019 instead of .00001903. Theproblem is that the error associated with the rounding on an input gets magnified in the problem and can really throwoff the final answer.

But we actually can take this one step further: using .00001903 is also rounding. The correct number is0.0000190258751902588. Yes, that looks like a ridiculous number to enter into a calculator.

The point is that you don’t have to enter the number since the number should have never left your calculator in the firstplace. As soon as you calculate a number, enter it right away as the TVM button value . In this case, for instance, youwould enter something like 10 / ( 365 * 24 * 60 ) = and as soon as you hit the equal sign then hit the N button. Your calculator will place the correct (non-rounded) number into N.

If you calculate a number that you are not yet ready to use, then use your STO and RCL buttons to hold onto the number until you are ready to use it. On the BA II, the storage buttons will hold 9 numbers simultaneously.

PV -5,600,000,000FV ???? 5,600,002,110IY 2PMT 0N 0.00001903

Example: Future Value of a Single Sum: Annual Compounding

© 2014 Craig Ruff

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As another example, we can turn the basic future value calculation around and solve for other variables. For instance, suppose you invest $1,000 today in a savings account and 10 years later you have $2,000 in your account. What annual compound return have you earned over this period? Using your TVM buttons…

PV -1000

FV 2000

IY ???? 7.1773

PMT 0

N 10

Notice that the signs here are very important. The negative on the $1,000 signals money coming out of the pocket and the positive on the $2,000 signals money coming back in the pocket.

Indeed, if you were to enter both numbers as positives (or both as negatives), your calculator would ‘blow up.’

Example: Future Value of a Single Sum: Annual Compounding

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Or, as another example, we can ask something along the lines of ... Suppose you invest your money in an account paying 5%, compounded annually. How long will it take to triple your money? Using your TVM buttons…

PV -100

FV 300

IY 5

PMT 0

N ???? 22.517

Again, the signs on the FV and PV are veryimportant.

Notice also that it would not matter had weused -1 and +3 or -10 and +30, etc., as long asthe FV was triple the value of the PV.

Example: Future Value of a Single Sum: Annual Compounding

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