## Saturday, August 17, 2013

### Time to Double Your Money!

Last year, I did a post about a number called e. This number is around 2.71828, and has many applications to calculus and finance. In the other post, I discussed its application to finding compound interest. Click here to see that post.

You might remember that the formula for finding the amount of money you have when your interest gets compounded continuously is:

Pert

P = amount of money originally deposited
r = interest rate
t = time (years)

This is a prettier formula, but a more practical formula, which also has variable n for the number of times the money was compounded in the year is:

P(1 + r/n)nt

For instance, if you deposited 1000 dollars in the bank, and your money got compounded every year with a 3% interest rate, after 30 years, you would have:

P = 1000
r = 3% = .03
t = 30
n = 1

P(1 + r/n)nt
1000(1 + .03/1)1(30)
1000(1 + .03)30
1000(1.03)30
1000(2.42726)
2427.26

So, after thirty years, you would have about \$2427.26, which means you were able to more than double your money! This might be a little surprising that it is possible for your money to double if you leave it alone for long enough.

You might be wondering how long it will take to double. How many years does it have to sit there? In other words, what value of t makes that equation equal to 2P? We deposited P, so to make it double, we must get 2P.

2P = P(1 + r/n)nt

In our scenario, we deposited \$1000 with an annually compounding interest rate of 3%. So, plug all of this in and we get:

2P = P(1 + r/n)nt
2(1000) = 1000(1 + .03/1)1t
2000 = 1000(1.03)t
2000/1000 = (1000(1.03)t)/1000
2 = (1.03)t
ln(2) = ln((1.03)t)
ln(2) = t • ln(1.03)
ln(2)/ln(1.03) = t
24 ≈ t

Note: this computation required something called logarithms. They look weird, but are very easy to understand. I explained them in my post on Benford's Law. Click here to read it.

So, it will take 24 years to double. So, let's set a rule to this - figure out a simple formula where you can figure out how long it will take for your interest to double, assuming it is compounded annually.

2P = P(1 + r/1)1t
2P/P = (P(1 + r)t)/P
2 = (1 + r)t
ln(2) = ln((1 + r)t)
ln(2) = t • ln(1 + r)
ln(2)/ln(1 + r) = t

This is as simplified as the equation will get without using calculus. And this doesn't look very simple anyways. But, let's look at a graph of it. It is easier to see as a picture than a messy jumble of logarithms.

This is a logarithmic function. However, it looks very much like a rational function, or a function that is the quotient of two polynomials (for example, 1/x is a rational function). So, let's try to find a rational function that fits this blue curve.

You can play around with it on your graphing calculator if you want, but I will just tell you that the function that fits it best is 72/(100r). Here are the two graphs:

As you can see, the two graphs are practically touching. In fact, all the way up through 0.5, they are very close together, as you can see here:

Since no bank on this planet offers 50% interest rates (if anyone has heard otherwise, please contact me), the 72/(100r) should be a good approximation for any of our purposes. Even to see the difference in the ones and tens, I had to set the graph below two. So, they are very close together.

This 72/(100r) equation can look even better. You might remember that r is currently in decimal form. To turn it into a percentage, you must move the decimal over twice, or multiply it by 100. So, dividing the interest percentage into 72 will give the same approximation.

Let's try it out on the original example. We said that it was a 3% interest rate. 72 ÷ 3 = 24, and we did conclude that it would take about 24 years. For a 6% interest rate, it would take about 72 ÷ 6 = 12 years. For a 4.5% interest rate, it would take about 72 ÷ 4.5 = 16 years. Couldn't be easier!

This rule is normally called the Law of 72. I find it very cool because you don't even need to know the amount of money you deposited to figure out this time. It doesn't matter. You don't even need to convert the interest rate into a decimal. I think this is a really intriguing and practical formula.