In my piece on Modern Portfolio Theory, I used an example involving coin flips that showed the results of a normal outcome of heads and tails for 1 coin and then for 2 coins, to show the surprising benefits of diversification (lower risk combined with higher returns). More than 1 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 1 coin and 2 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 4-round experiment would have gained an average of 25% per round, so that the initial $1,000 would have, on average, grown to:
$1,000 x 1.25 x 1.25 x 1.25 x 1.25 = $2,441.41
At the risk of killing the interest of math haters, let me briefly prove this for 1 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 1 combination would have resulted in 4 heads, 4 would have resulted in 3 heads, 6 would have resulted in 2 heads, 4 would have resulted in 1 head, and 1 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 $1,000 each time, we’d end up with $16,000 if the bet doubled in all 4 rounds, $4,000 if it doubled 3 times and was cut in half once, $1,000 if it doubled 2 times and was cut in half two times, $250 if it doubled 1 time and was cut in half three times, and $62.50 if it was cut in half all 4 times. The average result of 16 rounds would have been:
1 x 16,000 + 4 x 4,000 + 6 x 1,000 + 4 x 250 + 1 x 62.50 = $39,062.50
Since this represents 16 rounds, we can determine that the average result over time will be:
39,062.50 / 16 = $2,441.41
Anyone who wants to prove this for 2 coins (and 256 combinations) is welcome to do so, and should return to this piece after finishing the calculations and watching President Edwards' Inauguration Address.
The point, of course, is that $2,441.41 in both cases is quite different from my conclusion that the experimenter would have ended up with just $1,000 for 1 coin and $1,562.50 for 2 coins. Where did I go wrong?
In math, there are 3 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 3 different reasons I believe this is so. Any one of them being correct is sufficient to make the mean inappropriate for planning investments:
(1) Equity returns revert toward long-run averages
The book that first popularized Modern Portfolio Theory was Burton Malkiel’s classic, A Random Walk Down Wall Street. In the first edition, published in 1973, Malkiel made the strong assertion that neither fundamental analysis of a business nor technical analysis of price movements would produce superior returns, and that the future behavior of markets was unpredictable from past behavior. After convincing most of the academic world of this, he then went on to modify his own position in subsequent editions of the book (the current 9th edition was published in early 2007). He has, for quite a long time, admitted that there is some evidence of market behavior inconsistent with a complete random walk. The strongest evidence, in his view (and mine), based on the work of several other researchers as well as his own careful examination and confirmed repeatedly in academic studies of both the US and foreign markets, is that returns above long-term historical averages over a period of a few years (between 3 and 5, depending on the study) are less likely than usual to be followed with similar or better returns in the following few years, and that below-average returns over a few years are substantially less likely than usual to be followed by similar or worse returns over the next few. While the studies vary as to how strong they find the effect, it is rare to find an academic paper on the topic which contradicts this finding.
The much better average performance of the coin flip than the normal results I cited depends enormously on the extremely positive result that comes from winning all 4 rounds (that single $16,000 round in the above example contributed nearly half the $39,062.50 total wealth that resulted from playing the game 16 times). For an investor with a time frame over 5 years, though, the effect of reversion will be to reduce the likelihood of that type of result (as well as its horrible opposite, of course). The good news is that stock market investing appears to be less risky for anyone with an investment time frame of more than 5 years than the short-term volatility of stocks would imply, and SUBSTANTIALLY less risky for someone with a time frame of 10 or more years. The bad news is that the real life investment version of the coin flipping game with the 25% per round average expected return is not going to provide an average return of 25% per round to the player, even if the experiment can be repeated thousands of times.
(2) Your life is only one experiment
To be fair, I must note that a few theorists have offered a plausible case against reversion based on chaos theory, arguing that “off-the-chart” results occur in real life that normal statistics claim are impossible (for instance, based on the calculated daily volatility of stocks, there is simply no way the US stock market could have dropped more than 20% in a single day. It still did so on October 19, 1987). Thus, it is possible that the real life equivalent of a coin flipping test doesn't produce 4 consecutive heads as often as expected, but under a unique condition will get locked in and produce heads 100 times in a row. Although this doesn't really offer an argument for the mean, since chaos makes the average impossible to compute, it does at least allow that it might not be upwardly biased, as I've suggested (chaos theory also makes diversification even more important, as chaotic events are far more likely to be disasters than jackpots).
But even if equity prices didn’t revert toward long-run averages, the mean return wouldn’t be very useful in the real world. Since the average return, once again, depends heavily on the occasional jackpot result, a single human being cannot consider the mean return to mean much. In a stadium containing Bill Gates and 49,999 paupers, the weighted average person is a millionaire, but if you were to ask anyone the wealth of the average person in that stadium, they’d say the average person was broke. When a game has a variety of possible outcomes, but can only be played once, the result that falls in the middle of all the others or the result that occurs most often is a far more useful guess than the weighted average of all the possibilities. This result was $1,000 in the 1 coin game I cited above (in the 16 rounds, the 8th best and 8th worst results were grouped among the 6 that ended up at $1,000, so it was the median, and no other outcome occurred as often as 6 times, so it was the mode).
Of course, there can be safety reasons not to choose the investment with the higher median (or mode), when the weaker possibilities of that investment are substantially worse than the weaker possible results of the investment with the lower median return. Going back to the Modern Portfolio Theory piece, however, you will be reminded that the more diversified games always had BOTH a higher average return AND lower risk. In a single human life, playing a game with critical life consequences, the possibility of winning the lottery just can’t be taken seriously in estimating returns.
(3) Wealth has a diminishing marginal utility
This, in my view, is the most important argument, and the one I stress with my clients when helping them plan their future. Even if the first two reasons didn’t apply, an investment strategy designed to serve real lives mustn’t overlook this one: $2 is not twice as valuable as $1.
People often forget the difference between price and value. The price system in a free market simply provides the information to buyers and sellers as to the exchange rate between different goods and services that will avoid both shortages and surpluses. On the other hand, a voluntary exchange between a seller and buyer is not an exchange of “equal value”. Clearly, the seller wouldn’t provide the good or service unless they preferred the money, and the buyer wouldn’t exchange the money unless they preferred the good or service. Both parties must believe they are receiving something of more value to them than what they are relinquishing, or the exchange wouldn’t take place (with the exception of coerced exchanges in which one of the parties to the exchange is a violent criminal or government official).
Why is this important? Well, since we each have limited resources, we will typically look to exchange them for what we value most highly. We must have valued them more highly than alternatives we didn’t choose, or else we would have chosen the alternatives. So when we get more resources, and can also acquire the alternatives, we must be acquiring things of less value to us with those additional funds. This is the diminishing marginal utility of wealth.
There is nothing all that surprising in this observation. The caveman first uses his time to acquire food. With a little more time, he can find food and shelter. With more, he can acquire food, shelter, and clothing. After that, food, shelter, clothing, and car insurance from GEICO. And so forth. Each thing added is less valuable to him, even though it may be more expensive to acquire.
Since money is nothing more than a medium of exchange, its value is in what can be acquired with it (even when money is a useful commodity, such as gold, its value as a medium of exchange eventually dwarfs its value as a commodity for most people). Since the things you acquire with money will be, at least unconsciously, affected by how you rank the value of your options, twice as much money will always be less than twice as valuable to you.
Up to a point, the things you need to acquire are so critical that the diminishing value of wealth isn’t obvious. After basic needs are met, however, most people will acknowledge that much of their spending adds comparatively little to their happiness. Some studies actually suggest that wealth beyond basic needs adds next to nothing to personal happiness and my own experience with clients is that, once basic financial independence is achieved, their satisfaction with life depends primarily on the quality of their personal relationships and not on their surplus wealth. We can debate the magnitude of the dropoff in value, but not its existence. Achieving your financial goal from investing is valuable, exceeding it much less so.
Thus, average return calculations that depend heavily on the occasional jackpot massively overstate the value of a strategy. Less diversified strategies always do.
Since this piece is already too long, let me just note that there are ways to raise the median result through methods other than diversification, and that I don't necessarily endorse such methods, especially when they increase the probability of a horrible outcome (for example, going without insurance against catastrophic events increases your wealth most of the time, but reduces it substantially if the catastrophe occurs). We are ONLY talking about the benefits of diversification in this piece. And in such cases, the median and mode, not the mean, are the most meaningful averages.
