Investment Fractals & the Break-Even Point for Opportunity Cost

Give us your money, NOW!

Often times I will see an image encouraging people to invest earlier than later. Here is the first picture I found on a Google Image search for “investment 401k.”  These graphs emphasize that time is a key factor for investing: The sooner you invest, the faster you compound your returns.  (And, also the sooner you give an investor your money.)

This concept returned to my thoughts after playing with some of my personal financial goals.  I saw the same pattern, in a visual graph, no matter how I adjusted the dependent variables, such as time, initial investment, annuity investments, and interest rates.

Without perspective, investment curves have similar exponential patterns.

This second image has an initial investment of 100 over 30 periods, in the upper left hand corner.  In the same image, the other two graphs display an independent, 20% increase in time, from 30 to 36, and also in the initial investment, from 100 to 120. This is when I realized that the first graph always has at least two scenarios to show perspective. Otherwise, the pattern is a fractal. “Zooming in” or “zooming out” of the time periods will render the same image (unless there is perspective). Natural fractals are seen on the shoreline, mountains, or even man-made buildings. Often times something of known dimensions, like a person, is included in the picture to give the observer perspective on the relative size.

Unless you overlap the lines onto one graph, the latter rendering with graphs on different axes is a cumbersome display of the importance of time in investing.  This is why perspective is added on the investment literature I have seen so far, like in the first graph.

Displays that if an additional 20% is made at 4 time units later, the same future value is compounded.

My next curiosity is to quantify how important time is in investing.  Using the same basis, how much time can I lose if I invest 20% more of the initial amount, for a total of 120, at a later time?  By balancing a future value formula, for 100% of a present value and then for 120% of a present value, the answer is 3.74 time periods, at a 5% interest rate.  This graph should visually confirm that.  This is intuitive.  It is simply catching up on the opportunity cost of compounding interest.

This lost opportunity cost in time can be generalized, as a function of how what percentage of money can offset.  In my example above, the offset would be 0.20, and the interest would be 0.05.


As the real rebel that I am, I plotted time as the independent variable, even though in the formula above I have time, “n,” as the dependent variable.

Looking at catching up, more generally.

I used my perspective learning, and plotted two different interest rates.  I feel like the plot summarizes opportunity cost of investing well.  It is also interesting to note that money can be doubled in 15 years, based on median historical rates.  I feel like this calculation is just a generalization from the ideas of net present value and internal rates of return calculations, which makes it that much better of an adventure in missing the point!


2 Responses so far »

  1. 1

    Allen Taylor said,

    Nice writing. You are on my RSS reader now so I can read more from you down the road.

    Allen Taylor

  2. 2

    […] I compared time and interest rate dependencies in the exponential function, I started wondering more about relative interest return rates.  I believe applying […]

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