Dogbert and log points

(I promise, this is my last math-nerd post on log points.)

Last week's Dilbert cartoon, though facetious, essentially captures how I intend to grow my money for retirement. It also helps illustrate how log points can help you compute compounding in your head.

In the cartoon, Dogbert wants people to multiply their money by a factor 10,000. Because of the long timeframe, compounding will dominate his calculations, so he can't simply say "1,000,000%/5% = 200,000 years" because he will be very far off. However, he can still do the calculation in his head if he uses log points.

We start with three rules of thumb that need to be memorized. Think of these as the "Rule of 72" on steroids:

• Rule 1: a 1% increase is about 1 log point
• Rule 2: a doubling is about 70 log points (which should be reminiscent of the Rule of 72)
• Rule 3: a 20× increase is about 300 log points

Adding log points multiplies the gains; so, for example, a gain of 370 log points equals 300+70, which gets you a 20× increase followed by a 2× increase, giving a 40× increase overall. Likewise, 230 log points equals 300-70, which gets you a 20× increase followed by a 2× decrease, giving a 10× increase overall.

If 10× is 230 log points, then 10,000 (which is 10×10×10×10) means 230+230+230+230 = 920 points. If our investment gains 5% per year (which is about 5 log points), then we need 920/5 = 184 years. This is a much better guess than 200,000 years, since the right answer is 188.7 years.

(Our estimates are still off by a bit, because all of the rules above are rounded, so each one you apply can introduce a 1% error. We applied four of them here (four 230s) so we should expect a total error of up to 4%. Rule 2 introduces the most error: rather than 70 log points, the correct value would be about 69.3, which I think you'll agree is a much more awkward number.)

You don't have to get all 920 log points from interest though: you could also get some of them by increasing your initial investment. If you start with 10× as much, you get the first 230 log points immediately, leaving you to earn the other 690 of them via interest, which would take about 690/5 = 138 years (which is close to the actual 141.5 years).

However, I have no plan to invest $100 at the beginning and leave it for 184 years. I'm much more likely to invest every month until I retire. How do we compute this with log points?

Suppose I invest $1/month for 10 years. Some of those $1 installments would be invested for the whole 10 years, while others would be invested in the last month or two and would spend very little time invested; overall, each dollar spends an average of 5 years invested.

This gives us our fourth rule of thumb (which, though unrelated to log points, is still very useful in log point calculations):

• Rule 4: Periodic equal payments at equal interest are (practically) equivalent to one lump sum invested at the same rate for half as long
  

So, applying this rule to the Dogbert scenario: instead of investing $100 for 184 years, I could divide that $100 into equal (tiny) payments over twice as long (368 years) and still end up with a million.

Thanks to these four rules, I did these calculations entirely in my head, using a calculator only to confirm my results.