Albert Einstein called compound interest the eighth wonder of the world. We want to teach students the value of investing early and steadily for their future. In our Foundations in Personal Finance: High School Edition chapter 3 we go over the story of Jack and Blake and the power of compound growth. Many interested students will search online for investment calculators and "crunch the numbers" to see if Jack really was able to make his early contributions grow to more than a million more than Blake while putting in way less money. Maybe you have too. Do your numbers match the numbers in the graph? Maybe, or maybe not.
When discussing this with your students, first, let's remember the power of compound growth in examples like this are just that: examples. Real world results will vary. Let's walk through how we came up with our final numbers (and you can too!)
Let's dig into the math. There are a lot of little factors that make a difference in the calculations including the frequency of compounding, rounding, monthly vs annual contributions, etc.
Compound Growth Vs. Annuity Formula
In the Foundations lesson we talk about the Future Value (FV) of an investment compounded over time. The formula for compound interest takes the present value (PV) of your investment and calculates how it will grow over time. It does not factor in on-going contributions. There is a slightly different formula, referred to as “Future Value of Annuity Formula” which includes regular contributions (annuity in this context just means an investment with regular scheduled contributions).
The standard compound growth formula for calculating a Future Value (FV) from our text is:
FV = PV × (1 + r/m)^mt
The ordinary annuity formula for calculating a Future Value (FV) looks like this:
FV = P × [ (1 + r)^n – 1 ] / r
Where: P is the periodic payment, r is the interest rate in decimal form, n is the number of payments.
Remember the interest rate r must reflect the frequency at which the compounding occurs. So an 11% annual interest rate must be converted to a monthly interest rate expressed as a decimal (0.11/12).
Any online investing calculator, like the Ramsey Compound Interest Calculator will use the annuity formula or the basic compound growth formula based on the contributions frequency, the starting balance, adjusts the annual interest rate to monthly, etc. It does all the heavy lifting for you.
Calculating Jack's Growth
For Jack, we have to make two calculations: the compounding of his investment for the first nine years while he is contributing using the annuity growth formula and a second calculation for the 38 years it grows using the compound interest formula. We know that Jack starts investing when he is 21 and invests $200 for a total of $21,600 for 108 months (9 years x 12 months = 108 months). Using our Ramsey Compound Interest Calculator we put in a $0 starting balance, contributing $200 a month for 9 years, at 11% annual interest, compounding monthly, we get a result of $36,635. That becomes our starting balance for the next calculation.
Here's where it gets tricky, and your calculations might be off a little. We say that Blake starts investing and Jack stops when they are both 30. They compare their final values when they are 67. So that's 38 years of growth. But how far into their 67th year do they stop to compare? You see where it gets tricky. If you just punch in 38 years in your investment calculator, you should get a result of around $2.35 million for Jack. If you put in 39 years, you should get a result of $2.62 million. One number is a little under and one is a little over the graph in the curriculum. So, the actual time would be more accurately calculated at about 38 and a half years.
Again, this is just an example. Whether you use 38 years, 39 years, daily, monthly or annual compounding in your calculations, Blake never catches up. That's the lesson we want students to learn.
Start early and invest consistently over time to capture the power of compound growth.
Want to do it by hand? Remember the formula from your text:
We hope this helps you in your classroom discussions.