I analyze macroeconomic issues from a fundamental perspective, and I analyze market behavior from a technical perspective. Original macroeconomic analysis can be found here and both macro analysis and commentary can be found on my Caps blog. If you like or appreciate my analysis, please add yourself to my Following List

Saturday, September 19, 2009

Why e is the coolest number

Adapted from my original post here: Why e is the coolest number - June 18, 2008

Okay, this is obviously not going to be a blog about stocks or Elliott Wave counts. It will be about investing only in a very general and relatively abstract sense. But growth (and more importantly exponential growth) is why all of us are investing in the first place. And it is interesting to think about the fact that all exponential growth has its basis is one very cool number: e.

Why am I writing this? Who knows, binv271828 is a strange character [Note: My original Caps username is binv271828 and my new username in binve]. I really wanted to share why I like the number e so much that I did put it in my name and why it is related to investing.

Let me warn you now that there will be a lot of ideas, mostly math based and some very uninteresting except to those that really like math. Please feel free to skip, and I won’t be offended :). So, it should be abundantly clear to anybody who has read my blog posts that I am a fairly large nerd. I like math… a lot. Numbers are cool. But the relationships between numbers and how they describe physical phenomena are even more interesting.

e is a number that describes a whole class of relationships like this. But if you read a math textbook or looked up e on wikipedia you would have no idea how universally cool it is. So here is the dry definition: e, also called Euler’s number, is a transcendental number that is approximated by 2.71828182845904523536. …. Okay, who cares. So here is some more dry definition: the mathematical constant e is the unique real number such that the function e^x has the same value as the slope of the tangent line, for all values of x. … again, who cares!

Okay, lets see why e is so cool.

Everybody is familiar with compound interest. You begin with a starting amount of money, and then you earn interest. The next period you earn interest on the principal + interest from the first period, and this continues until you are rich!

So lets say you have an account where interest is calculated once a year. So the growth comes in yearly chunks. If you start with $1 and you get an interest rate of 100%, then at the end of the year you will have $2 (the interest earned on 1$ with a rate of 100% is 1$, and $1 + $1 = $2). Well, what if interest was calculated once every half year. Then that means after 6 months you will earn $0.50 (100% interest for half a year, or 50% earned on the $1) for a total of $1.50, then at the end of the next 6 months, you will earn interest on $1.50. This interest is 50% (for half a year) to give you $0.75. Add back to the $1.50, which gives you $2.25. Right on, so calculating in more intervals gives you more money. So now you have 2 payments instead of one step at the end of the year. Next imagine the interest was calculated once a month, or once a day or once an hour or once a minute or once a second or once a nanosecond…. What this does is to increase the number of steps, which makes your growth curve “smoother”. Eventually with an infinite number of steps in which your interest is calculated, your interest growth will represent a continuous curve.
That is an interesting relationship. And this relationship can be expressed as: (1 + 100%/n)^n where n is the number of steps taken. So lets list this relationship for an increasing number of steps:


And as you can see, the relationship begins to converge, and lo and behold, it’s e! So this is where e comes in, it is this idea of continuous growth.

What this actually is, is a limit. Okay, I am going to throw some calculus at you. e = limit as n goes to infinity of (1 + 1/n)^n. This is an exceptionally important relationship very useful in describing all kinds of phenomena, and has some very unique properties in relation to derivatives and integrals (more in a minute).

What is even more interesting is that if you start looking at any exponential relationship (interest calculated at interest rates other than 100%, population growth, cell division, bacteria replication, etc.) you can express it as function of e. Absolutely any exponential relationship at all. What this means is that every single continuous growth relationship in existence can be though of as a scaled version of e…. ! How cool is that!

So all of us are looking for continuous exponential growth in our portfolio returns :), e is always on our minds subconsciously.

Okay so that’s cool, so what’s up with all derivative and integral stuff? Because of the shape of this exponential curve and remembering the original dry wikipedia definition “the mathematical constant e is the unique real number such that the function e^x has the same value as the slope of the tangent line, for all values of x” an interesting property is discovered: The derivative d/dx (e^x) = e^x. This is very useful in casting functions for linearization.

Another concept as to why e useful is in the concept of imaginary numbers. It can be shown that e is actually a trig relationship (sines and cosines) in the imaginary domain. Now that is really abstract, but you can think of imaginary numbers as describing an oscillating signal or motion. Any motion that can be described as a magnitude and an angle or phase (such as a pendulum moving back and forth, a wing vibrating through the air, a cesium atom moving back and forth in an atomic clock) can be though of in terms of imaginary numbers, which then can be compactly represented in one number: e …!

e also has usefulness in integrals. Since e can represent imaginary numbers, it can represent any oscillatory signal. Any oscillating signal can decay (think of a bar door when you open it, rocking back and forth on its hinge until it eventually comes back to rest), stay stable or it can grow (if you are not familiar with the bridge “galloping gertie” check this out http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge and be sure to watch the video under the collapse section). So growing oscillating signals are bad for mechanical systems, like galloping gertie, and are also bad for electrical systems. Exponentially growing signals cannot be easily described since their integrals do not converge. So you cannot even analyze the effects of a system with a non-converging integral.

That is until you throw in some e! Since e is actually an oscillatory number, you can add a sufficiently large amount of negative e in order to force an integral to converge. This is the principle behind the LaPlace transform. Figuring out the size of negative e added can tell you something about the stability of a system, and is a very useful technique in controls.

e is just so cool!

Okay, okay enough geeking out. If you want to use e for some useful formulas for investing calculations, here are a few:

growth = e ^ (total rate * time)
annualized growth rate = e ( ln (total return multiple) / number years ) - 1
where ln is the natural logarithm (another cool relationship that is related to e).

If you also have a love of e, please feel free to share! If you have skipped everything in the middle and come down to the end, well I don’t blame you :)


For more reading on e, check out:
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