# Financial Math I: Rule of 72 and other fast math

When making financial decisions, it’s important to understand how financial math works. This is the first of a short series where I hope to cover some financial math basics.

Rule of 72

For starters, when we are faced with an opportunity, it’s good to be able to quickly assess if it’s worth spending more time investigating. We often call these “back of the envelope estimates”. Without a spreadsheet can we get a ballpark estimate of the value of something?

One of the most well known mathematical devices is the “Rule of 72”. The formula provides a way to see in what time frame and with what interest we can double our money.

Interest x Years = 72

For an example, imagine we had an account growing at 10% annual interest rate. Take the 10, multiply if by 7.2, and you get 72. This means that in 7.2 years (approximately), our money would double.

You can flip things around. If we wanted our money to double in 10 years, we would need at least a 7.2% interest rate.

If you wanted something to double in six years, you would need 12%. 3 years? 24%.

To generalize, any situation that involves the compound interest effect works here. It’s also important to recognize this is only an approximation, and the further you get away from the middle, the less accurate it is. Rule of 72 may say that 2% takes 36 years to double, but this is much less accurate than 7.2% and 10 years. (BTW, you can swap the two such that 7.2 years and 10%.)

This type of quick math can help you. If you hear some investor advertise that there is no risk in doubling your money in six years. In your head, you can quickly deduce he is suggesting 12% growth. Sorry, but that is 3x what people average when using mutual funds. In real estate, this is definitely doable, but it requires either a lot of cash to mitigate the risk, or it requires taking on debt. None of this is risk free.

Gains and Losses

I’m always hearing things like “The Dow is up 5%” or “The NASDAQ is down 10%”. People seem to think that growth and losses are arithmetic, i.e. pluses and minues. They aren’t. They are multiplicative.

If you had \$1000, to grow it by 10% means you multiply by 1.1, another way of saying (1 + 10%) or (1 + .10). \$1000 x 1.1 = \$1100.

To reduce by 25% mean you multiply by 0.75, another way of saying (1 – 25%) or (1-.25). \$1000 x 0.75 = \$750.

Why is this important to understand? Because combining multiple gains and losses together involves multiplying all of the factors together, not adding them. When you have a 10% gain following by a 25% loss, what is the result? You might think 10% – 25% = -15%, but that isn’t right.

Instead, let’s take what we did earlier (1.1 and 0.75) and multiply them together. 1.1 x 0.75 = 0.825, or -17.5%. That loss (-17.5%) is actually greater than you would get if you just subtracted one from the other other (-15%).

It’s important to know that getting 10% each year for three years doesn’t add up to 30% gain. Instead, it’s 1.1 x 1.1 x 1.1 = 1.331 or 33.1%. This is power of compounding. But when we suffer losses, the effect is equally drastic. And people often don’t realize just how hard losses can be on overall performance.

Understanding this basic tenet of gains and losses is critical to building spreadsheets that evaluate things that are more complex. And the more you use this mechanism, the more you notice how people like to quote “Your investment will average x%!” You will be able to size something up and ask, “Hmm, if I multiple by 1.xx every year, do I really get what they just advertised?”

That’s enough for starters. Tune in for future posts about more financial math.

## 2 thoughts on “Financial Math I: Rule of 72 and other fast math”

1. makgabo says:

Guys help me out here…. Financial math :
R3802 invested at 15% PA for 5 years on simple interest.

HELP!! HELP!!! HELP!!!! HELP!

1. Greg Turnquist says:

I don’t follow all your shorthand. Could you expand a little bit?