In the last lesson on Modern Portfolio Theory, I used an example involving coin flips that showed the results of a normal outcome of heads and tails for one coin and then for two coins in order to show the surprising benefits of diversification (lower risk combined with higher returns). More than one reader (but fewer than 1,000) objected to my calculation of returns, saying that the actual returns would have been exactly the same, regardless of the number of coins involved, had I calculated the average returns correctly. This would have been achieved by taking into account all the possible outcomes of the coin flips, each weighted by its probability of occurring, and the results, in the case of both one coin and two coins (and a billion coins, for that matter), would be an average gain of 25% per year, which is the simple result of having an equal chance of either making 100% (double) or losing 50% (half):

(100% – 50%) / 2 = 25%

This means the four-round experiment would have gained an average of 25% per round, so that the initial $100 would have, on average, grown to:

$100 x 1.25 x 1.25 x 1.25 x 1.25 = $244.14

At the risk of killing the interest of math haters, let me briefly prove this for one coin. If the experiment were repeated thousands of times, we would eventually see 16 different combinations of heads and tails showing up with equal frequency. Only one combination would have resulted in

four heads; four would have resulted in three heads; six would have resulted in two heads; four would have resulted in one head; and one would have resulted in no heads. These combinations are:

4 Heads = HHHH

3 Heads = HHHT, HHTH, HTHH, THHH

2 Heads = HHTT, HTHT, HTTH, THHT, THTH, TTHH

1 Heads = HTTT, THTT, TTHT, TTTH

0 Heads = TTTT

Starting with $100 each time, we’d end up with $1,600 if the bet doubled in all four rounds, $400 if it doubled three times and was cut in half once, $100 if it doubled two times and was cut in half two times, $25 if it doubled one time and was cut in half three times, and $6.25 if it was cut in half all four times. The average result of 16 rounds would have been:

1 x 1600 + 4 x 400 + 6 x 100 + 4 x 25 + 1 x 6.25 = $3,906.25

Since this represents 16 rounds, we can determine that the average result over time will be:

3,906.25 / 16 = $244.14

Anyone who wants to prove this for two coins (and 256 combinations) is welcome to do so and should return to this piece after finishing the calculations, and then reacquaint themselves with his or her family.

The point, of course, is that $244.14 in both cases is quite different from my conclusion that the experimenter would have ended up with just $100 for one coin and $156.25 for two coins. Where did I go wrong?

In math, there are three different ways to calculate the average. The method that takes into account all of the possible results and then calculates a weighted average of all of them is known as the mean. The method that ranks all of the outcomes from best to worst and then chooses the one in the middle is called the median. The method that identifies the frequency of all the different wealth outcomes and then chooses the one that occurs most frequently is known as the mode. The result I chose was both the median and the mode, but not the mean. So I win, 2 to 1.

Okay, okay, that’s not the way to decide this. The point of my explanation of Modern Portfolio Theory was to provide useful information to people who are trying to invest properly in the real world. I do not believe that the mean calculation, which results in a predicted gain of 25% per year, is of relevance to personal investors. There are three different reasons I believe this is so. Any one of them being correct is sufficient to make the mean inappropriate for planning investments. Now guess what the next three lessons are on.