Future Value of a Single Sum with Annual Compounding Dr. Craig Ruff Department of Finance J. Mack Robinson College of Business Georgia State University © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding Let’s start with a very simple question…. Suppose you deposit $100 today (t=0) in a bank account paying 10%, compounded annually. How much money will you have at the end of one year (t=1)? The answer is easy; at the end of the first year (t=1) you would have your original $100 plus the $10 in interest earned for a total of $110: This is how much you have in the bank at the end of the first year (t=1). © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding While it may seem an odd exercise at this point, we will rewrite this simple equation: $100 + ($100*.1) = $110. First, factor out the $100 to get: $100*(1+.1) = $110 Next, raise the (1+.1) to the power of one (since raising anything to the power of one just equals itself) to get: Next, take away the ‘+’ sign to get: © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding Following the same pattern, by the end of second year (t=2), our $110 balance at the end of the first year would have grown to © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding Again, we can again rewrite this simple equation: $110 + ($110*.1) = $121 First, factor out the $110 to get: $110*(1+.1) = $121 Next, raise the (1+.1) to the power of one to get: Next, take away the ‘+’ sign to get: © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding This page is intentionally left blank. © 2014 Craig Ruff

Future Value of a Single Sum: Non-Annual Compounding . Future Value of a Single Sum: Non-Annual Compounding Time for the old switcheroo. Another try… © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding To understand the future-value formula, it is crucial to understand this simple substitution we are about to make. Start with the last equation for the account value at the end of two years (from the bottom of Slide 8): Going back to the bottom of Slide 5, recall that the $110 can be written as: So, we will simply substitute in the left-side of the equation from Slide 5 for the $110 in the equation from Slide 8 to get: And, remember that when we have a common base (here, 1.1), we can simply add the exponents (1+1=2) to rewrite this as: © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding And we can keep following this pattern…how much will the balance be at the end of the third year (t=3)? © 2014 Craig Ruff

Future Value of a Single Sum: Non-Annual Compounding . Future Value of a Single Sum: Non-Annual Compounding Time for the old switcheroo. Another try… © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding Again, we will do the same type of substitution: Following the same steps as Slides 5 and 8, we can rewrite the above equation as: Going back to the bottom of Slide 6, recall that the $121 can be written as: So, we will simply substitute in the left-side of the equation directly above for the $121 in the top equation to get: And, since we have a common base (1.1), we can add the exponents (2+1=3) to rewrite this as: © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding Based on the equations we have done, we can easily see the pattern: This leads to the general formula for the FUTURE VALUE OF A SINGLE SUM WITH ANNUAL COMPOUNDING: Future value at time N (measured in years) The amount of time (in years) you are compounding The amount you start with… the Present value The annual compound interest rate © 2014 Craig Ruff

(1) Future Value of a Single Sum: Annual Compounding This formula will work for any number of years (or amounts or annual compound rates): For instance, at the end of ten years, the bank balance will be: Or, at the end of 352 years, the bank balance will be: © 2014 Craig Ruff