Tycoon Tommy – Follow Up

A follow up to yesterday’s quasi-riddle, on Tycoon Tommy, who despite apparently high average returns, lost money over a 20 year period.

As noted by the commenters, Tommy fell victim to a mathematical illusion. Here’s the full set of returns I postulated when setting up the problem:

Tommy Carl 30/70 Blend
Begin  $     10,000  $     10,000  $       10,000
1 14%  $     11,400 5%  $     10,500  $       10,770
2 41%  $     16,074 5%  $     11,025  $       12,472
3 3%  $     16,556 5%  $     11,576  $       13,020
4 -36%  $     10,596 5%  $     12,155  $       12,070
5 114%  $     22,675 5%  $     12,763  $       16,620
6 -37%  $     14,286 5%  $     13,401  $       15,357
7 15%  $     16,428 5%  $     14,071  $       16,586
8 2%  $     16,757 5%  $     14,775  $       17,266
9 59%  $     26,643 5%  $     15,513  $       20,926
10 -49%  $     13,588 5%  $     16,289  $       18,582
11 6%  $     14,403 5%  $     17,103  $       19,567
12 26%  $     18,148 5%  $     17,959  $       21,778
13 6%  $     19,237 5%  $     18,856  $       22,933
14 -41%  $     11,350 5%  $     19,799  $       20,914
15 3%  $     11,690 5%  $     20,789  $       21,835
16 102%  $     23,615 5%  $     21,829  $       29,280
17 -65%  $      8,265 5%  $     22,920  $       24,596
18 73%  $     14,299 5%  $     24,066  $       30,843
19 -38%  $      8,865 5%  $     25,270  $       28,406
20 12%  $      9,929 5%  $     26,533  $       30,423

Specifically, while the arithmetic average of Tommy’s returns was high (10.5%), the geometric average was slightly negative (-.04%, if my math is right).

In a nutshell, taking an initial sum, then adding 20% to it, then subtracting 20% from THAT result, does not result in your original value, but rather less (i.e. 100 * (1+.20) * (1-.20) = 96). It doesn’t even matter whether the negative result comes before or after the positive result.

The geometric mean is basically the Nth root of the total growth. So, for the simple +20%, -20%, above, with a final result of 96, the geometric mean can be computed as:

((final total/initial total) ^ (1/num periods)) – 1
or
((96/100) ^ (1/2)) – 1 = -2.02%

If there is no deviation in the annual results, the arithmetic mean will be the same as the geometric mean (i.e. as in Carl’s steady 5% return).

The wider the deviation in annual results, the greater the difference between the geometric mean and arithmetic mean. Even when all the returns are positive. For instance, two years of +10%, then +90%, gives you (100 * 1.1 * 1.9) = 209, with a geometric mean of 44.6%, which is noticeable less than two years of +50%, then +50% (100 * 1.5 * 1.5) = 225, geometric mean, of course, of 50%.

You’ll note in the table above a third column, showing the results of a 30/70 blend – with 30% of the money invested in Tommy’s fund and 70% in Carl’s fund, and with the money rebalanced each year to restore that 30/70 balance. Interestingly, the final result here is noticeably better than either of the 100% approaches. This goes against many people’s expectations that you should invest as much as possible in whatever has the highest return.

In fact, it is usually possible to get a better result by splitting money between multiple investment options, then periodically rebalancing the returns. A full explanation of this phenomenon, and how to apply it in the real world, is beyond the scope of this article, but, if interested, I’d strongly encourage you to do on-line searches for the terms ‘Efficient Frontier’ and ‘Mean Variance Optimization’, and/or read up on this site. You actually get a double bonus by this strategy – higher long term returns (geometric returns, the ones that count), and lower volatility/deviation.

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