An Interactive Applet powered by Sage and MathJax.

(By Prof. Gregory V. Bard)

On this page, we're going to explore the question of which is stronger (i.e. which will bring you more money): 6% compounded monthly, or 8% simple interest.

At first, it might not be at all obvious. Particularly if you've not had
much opportunity to explore simple and compound interest. However, maybe if
you have some familiarity, you might know that compound interest is really
quite a powerful instrument. In fact, Albert Einstein was once asked what
the strongest force in the universe was, and people were expecting answers
such as gravitation, or the nuclear forces that keep an atom together.
Einstein replied that compound interest was the strongest force in the universe!
So we can conclude that perhaps compound interest will *eventually*
out-perform simple interest. However, the question is a bit more complicated
than it first appears.

It turns out that for very short intervals of time, compound interest has very little advantage over simple interest. On the other hand, for very long periods of time, compound interest will win by a huge factor. We're going to try to figure out which is stronger for any particular length of time. To do that, we should first figure out when 6% compounded monthly and 8% simple interest are equal.

Click the big button below to launch the interactive applet. For now, do not allow the applet to display the graph. (In other words, do not change the "no" next to "Show the graph" into a "yes" until we reach the next part of this lesson.) You'll see two sliders below. One governs principal, and the other one governs time. Leave the principal alone now, and try moving the slider back and forth to change the value of time. Can you find out at what time 8% simple interest and 6% compounded monthly have the same return?

**Note:** When you drag the slider back and forth, the applet won't recompute
the answers until you let go of the mouse button. There's also a 1-3 second delay,
depending on the speed of your internet connection.

Once you've explored the question numerically a bit---even if you didn't find the answer from Part One---you can now try exploring it graphically. Move the time slider to some arbitrary value like t=45. Keep the principal at $\$$ 5000. Now, next to the words "Show the Graph?" is a choice between "no" and "yes." Change that to "yes."

This graph has several pieces to it:- The graph for simple interest is salmon colored. (Hint: "salmon" and "simple interest" both start with an "s.") As you can see, it is a line.
- The graph for compound interest is chrome colored. (Hint: "chrome" and "compound interest" both start with a "c.") As you can see, it is curved. As it turns out, it is an exponential curve, but you cannot see that from the graph alone.
- The time chosen by the slider is the vertical tan line. (Hint: "tan" and "time" both start with a "t.")
- As you slide the time slider back and forth, the vertical line should move to keep up with you. The salmon-colored graph and the chrome-colored graph report the amount of simple interest and compound interest respectively.
- The yellow line coming out from the intersection of the line for time and the graph for simple interest is there to help you read the graph. You can see how many dollars simple interest returns by looking at the y-axis.
- Likewise, the yellow line coming out from the intersection of the line for time and the graph for compound interest is there so that you can see how many dollars compound interest returns. Just look at the y-axis.
- The place where the two graphs cross is where simple interest and compound interest are equal. Try to move the time-slider to that spot. The two yellow lines should merge at that time, as both compound interest and simple interest agree on how much money to return.

I don't know about you, but graphically, I was able to get the ball-park time-value to have the amounts close together. After that, I had to rely on the numerical values of the amounts to find the time where they are closest. I was able to find a point in time where they differ only by a mere $\$$ 2.33. The graph was useful for me, but this is an excellent example of how a graph is only a picture---it is not enough, alone, to answer the big questions.

Now, repeat the experiment for principal values of $\$$ 2500, $\$$ 7500, and $\$$ 10,000. How is the value of time (when the simple interest and the compound interest are as close together as possible) changing as compared to above, where you used $\$$ 5000?

Now here are some questions that you should be able to answer based on your work above.

- For a principal of $\$$ 5000, at how many months will 8% simple interest and 6% compounded monthly come out the same dollar value? (Or as close as possible to the same?)
- How does the answer to the above question change when the principal is $\$$ 2500? $\$$ 7500? Or $\$$ 10,000?
- For what lengths of time will 8% simple interest return more money?
- For what lengths of time will 6% compounded monthly return more money?
- There are no right or wrong answers for this one: How useful was the graph in helping you determine the answers to these questions? How useful was the numerical output?

Last modified on January 15th, 2013, but with trivial modifications on June 17, 2013.