10.2 The Time Value of Money and Interest
For each of the following questions, assume you have $1 and that interest on it will be paid in full, at the END of the stated period. What are the future values (FVs) given each of the following questions? In other words, how much money will you have at the relevant future points in time? (If you had more than $1, the answers would be the appropriate multiple thereof.)
As we go through the questions and calculations, observe how the outcomes, or solutions, change. Try to explain the reasons for the differences in the outcomes. Also, observe that the seemingly small differences in outcomes are really not as trivial as may seem at first glance. We are illustrating Future Values, in each question, of just one dollar of money that we have now – of Present Value. Suppose we were instead dealing with millions of dollars?
As we go through each question, we will, methodically and painstakingly, create a general symbolic formula, which may be employed for any similar problem. Insert the appropriate values into the formulae to solve the problems numerically. (Solutions follow.)
1. You will earn 5% interest, paid once a year, at the END of the year, for one year.
You have $1 now of Present Value (PV). In one year, you will receive your “principal” of $1 back plus interest at an annual rate of return (R) of 5%. A general Future Value (FV) formula will therefore be:
FV = PV (1 + R)
Insert the relevant data into the formula in order to solve for the Future Value.
As we go through this analysis, you will need to learn and remember the symbols or abbreviations.
2. Same as question #1, but R = 10%.
Here we will use the same formula as above, but you will insert a different rate for R. What is the Future Value? Why is the outcome different?
3. Same as question #2, i.e., R = 10%, but for two years (n years), rather than just one.
We will now have two years of compound interest; n = 2. Therefore, we apply the FV formula, slightly modified, a second time:
FV = PV (1 + R) (1 + R)
FV = PV (1 + R)n
Here, the exponent, “n,” stands for the number of years in which the money compounds.Once again, in this case, n = 2. What is the Future Value? Why is the outcome different than in the prior question?
4. 10% interest, twice a year, for one year.
Interest is always quoted as an annual rate, unless explicitly noted otherwise.
Our annual rate is still 10%, but we will receive half of it, i.e., R ÷ p = 0.10 ÷ 2 = 0.05, at the end of each half-year. The letter, “p” stands for the number of compounding periods per year; here p = 2.
FV = PV (1 + R/p)(1 + R/p)
Notice that, while n = 2, there are now two compounding periods per year, so the exponent must reflect that. Our exponent is therefore now: “n × p.”
Whenever p ≠ 1, we must make two adjustments to the formula: “R/p” in the rate part of the formula and “n × p” in the exponent. While in theory P can take on any integer value, it will actually be equal to 1, 2, 4, 12, or 365 for annually,