# What is the relationship between interest rates and exponentials

### APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS This experimental study investigates subjects' ability to value correctly the exponential function underlying the interest-rate discounting relationship. The results. Interest rates and terminology were invented before the idea of compounding. Heck, loans were around in BC, before exponents, 0, or even the decimal point! So it's no wonder our .. That's a pretty big difference! Notice that e takes care. At an interest rate of 3% compounded annually means that at the end of each year, you earn 3% on the current value of your account. The interest is.

First, what does an interest rate mean? But both types of speed have a subtlety: Does driving 50 mph mean you must go a full hour? You could drive 15 minutes and go You get the idea. Interest rates are similar. Take a look at this: Connecting the dots gives us a trendline: Simple interest keeps the same trajectory: In other cases, our rate may change, like a skydiver: No need to hit a mosquito with the calculus sledgehammer just yet.

Basic Compound Interest Simple interest should make you squirm. Heck, we should use the golden eggs to fund research into cloning golden geese! Compound growth means your interest earns interest.

## A Visual Guide to Simple, Compound and Continuous Interest Rates

When you have a growing thing, which creates more growing things, which creates more growing things… your return adds up fast. Reinvesting our interest annually looks like this: After n years, this becomes: Compound growth is useful when: Interest can be reinvested, which is the case for most savings accounts. You want to predict a future value based on a growth trend. Most trends, like inflation, GDP growth, etc. Things get a bit crazy.

Years 4 to infinity: Left as an exercise for the reader. This is an interesting viewpoint. Breaking earnings into components helps understand e. Knowing more about e is a good thing because it shows up everywhere.

And besides, seeing old ideas in a new light is always fun. For one of us, at least. One more insight — take a look at our trajectory: With annually compounded interest, we get a new trajectory each year. We deposit our money, go to sleep, and wake up at the end of the year: You scream at your bank and get the rate adjusted. This process repeats forever — we seem to never learn. Compound Interest Revisited Why are we waiting so long? More importantly, after 6 months we have the same trajectory as when we started.

Imagine I took your money and returned it after 6 months. After all, interest is measured per year. Not per 6 months. Annual payouts are man-made artifacts, used to keep things simple.

How Interest Rates Affect the Market

But in reality, money should be earned all the time. We can pay interest after 6 months to reduce the gap: The key point is that our trajectory improved halfway through, and we earned For 1 year, the impact of rate r compounded n times is: Repeating this for t years multiplying t times gives: The more frequently we compound, the smaller the gap between earning interest and updating the trajectory.

Continuous growth is compound interest on steroids: We wait a millisecond, find our new sum, and go off in the new trajectory.

### Interest and Exponential Growth

Continuous growth keeps the trajectory perfectly in sync with your current amount. Another way of writing this is to write the balance at the end of the third month as. By now you should notice some common things in each end-of-month balance.

For one thing, the exponent is the same as the number of months that have passed. In this example the compounded is weekly, so the interest rate has to be converted to a weekly interest rate of. At the end of the week you will have This can also be written. At the end of the second week, you will have the you had at the beginning of the week plus the interest on the. At the end of the second week you will have This can also be written Another way of writing this is to write the balance at the end of the second week as At the end of the third week, you will have the you had at the beginning of the week plus the interest on the.

At the end of the third week you will have This can also be written Other way of writing this is to write the balance at the end of the third week as. By now you should notice some common things in each end-of-week balance. For one thing, the exponent is the same as the week. The annual percentage rate is the rate that you would receive if the interest was calculated at the end of the year. This means there was no compounding during the year.

From Example 2 above, we know that we can find the balance after t years as follows: We use 12 t because there are 12 months in every year. Take the natural logarithm of both sides of the above equation: Simplify the left side of the above equation: Divide both sides of the above equation by: If you multiply 0.