One of the great concerns I have when I share my views on asset allocation in public forums is that people have a tendency to underestimate the critical importance of diversification. I am often identified, not unreasonably, as advocating a 100% equity allocation for most people. This doesn’t, however, mean that someone who put all their money into Enron in 2000 was following my advice. For that matter, anyone who has all their money in the Vanguard Total Stock Market Index Fund is NOT following my advice, and is not nearly diversified enough, even though this fund attempts to duplicate the performance of an investment in all of the shares of all U.S. companies that are publicly traded.
Most people have no idea how diversification works (which is why most people have no business trying to do-it-themselves when it comes to investing). They believe that diversification means “not putting all your eggs in one basket,” which is true enough, but the analogy is a sloppy one. If you have two eggs to transport, and a 50% chance of dropping a basket, putting them in a single basket means that you will lose no eggs half the time, and both eggs the other half. The average loss is 1 egg per trip, since (0 lost +2 lost) / 2 = 1 lost per attempt. If you use two baskets, transporting only one egg at a time, you will need to make two trips in your attempt to move both eggs. You’ll have two safe trips and lose no eggs a quarter of the time, lose one egg from a dropped first trip a quarter of the time, lose one egg from a dropped second trip a quarter of the time, and lose 2 eggs from having two failed trips a quarter of the time. The average loss is STILL 1 egg per trip, since (0 lost + 1 lost + 1 lost + 2 lost) / 4 attempts = 1 lost per attempt. You HAVE cut the probability of a 2-egg loss from 50% in the first case to 25% in the second case, so not putting all your eggs in one basket did, in fact, reduce the possibility of a complete disaster, but didn’t reduce the AVERAGE result, which remained a loss of 1 egg per attempt.
That is not how diversification of a portfolio works. In the early 1950s, Harry Markowitz noted that diversification reduced both the probability of maximum loss AND the average expected loss, while increasing the average expected gain (this is a remarkable free lunch, if you think about it).Assume you are playing a double-or-half coin-flipping game, and that you bet your entire stake on each round (I didn’t develop this example but cannot, at present, recall the person from whom I am now stealing it: I will modify my wiki to credit the source if some kind reader can provide it to me. Update: a reader believes that my source was probably William Bernstein's THE INTELLIGENT ASSET ALLOCATOR). Starting with $1000 and playing the game for 4 rounds, I would expect to double my money on two occasions and cut it in half the other two occasions, and will, on average, be back to even. The order doesn’t matter, but assume head-tail-head-tail (so that I win the first and third round and lose the second and fourth). The money starts at $1000. After round 1, it doubles to $2000. After round 2, the $2000 is cut in half to $1000. Rounds 3 and 4 have the same effects as 1 and 2, so we’re back to $1000 at the end of the game.
Now, let’s diversify into two coins, so that in each round, 50% of the money is being bet on Coin A and 50% on Coin B. Over the course of 4 rounds, the average expectation is for one round in which both coins come up heads, one round in which both coins come up tails, one round in which only Coin A is heads, and one round in which only Coin B is heads. Assuming it occurs in that order (again, it doesn’t matter), we have an interesting result. The starting $1000 is bet $500 on Coin A and $500 on Coin B, and since both come up heads, both $500 bets double to $1000, and I have $2000 at the end of round 1. Round 2 starts with the $2000, with equal $1000 bets on Coin A and Coin B, and since both came up tails, each $1000 is cut in half to $500, and the total is $1000. Notice that the results are the same as in the first example, in which only one coin was involved, since in Rounds 1 and 2, both coins had the same result. Where it gets interesting is in Round 3 and 4. The $1000, split between Coin A and B bets of $500 each, results in the Coin A bet doubling to $1000, and the Coin B bet cutting in half to $250, and the net result is that I now have $1250. In Round 4, the $1250 is split into equal bets of $625, with Coin A, the loser, being cut to $312.50, and Coin B, the winner, doubling to $1250, so that I now have $1,562.50!
In other words, playing the double-or-half game with one honest coin will, over time make no money, but playing the game with two honest coins will, over time, cause the player to make lots of money (an average of 56.25% every 4 rounds, or about 11.8% a round). Of course, it has the same benefit of the eggs in the baskets, in that the possibility of a maximum loss (in the double-or-half game, that would be a 50% loss) is reduced from 1 out of 2 rounds to 1 out of 4 rounds, but the AVERAGE result improves dramatically, from breakeven to nice profits.
[Note: This example assumes that the result of flipping coins was 2 heads and 2 tails and that the 2 coins came up on the same side as each other half the time and on opposite sides the other half the time. I chose that because it is the most normal result, making the example easier to follow (I hope). It is certainly true that many tests will result in a larger or smaller number of coins coming up heads, or that the two coins could have had identical sides more or less often, and the results will be better or worse in such cases, but the results from splitting the bet between two coins will still be better than the results from an identical head-tail outcome for one coin (except when the two coins are identical in each round, in which case it is just like one coin). Readers with a mathematical bent are encouraged to convince themselves that 3 heads and 1 tail will cause 1 coin to rise to $4,000 and 2 coins to rise to $6,250 (unless the tails outcome is in the same round for both coins), and that 1 head and 3 tails will result in a drop to $250 for 1 coin but to only $390.63 for 2 coins (again, except when the head outcome is in the same round for the 2 coins). Since it is impossible for the results to differ if both come up heads every time or tails every time, you'll end up with $16,000 and $62.50, respectively, either way. But you will never find a case where the 2 coin result is worse than 1 coin. Never.
As to the valid point brought up by some statisticians that this paradox doesn't exist if we use the "mean" return (I used the "median" return), I will cover in a later piece (to which I'll link from here) why I believe it is not relevant to the situation an individual investor faces. I will also discuss the differences between the mean, median, and mode, for those who are NOT mathematically inclined.]
This, by the way, is the fundamental insight of what is called Modern Portfolio Theory, and what won Harry Markowitz the Nobel Prize in Economics. It is also of tremendous practical value. Let’s look at an investment example:
Assume you had $10,000 to invest at the beginning of 1972, and put all of it into an index fund that was intended to mirror the performance of the Standard & Poor’s 500 Stock Average (an index of 500 of the largest U.S. companies). As of the end of 2006, it would be worth $427,300, a return of 11.3% per year. Along the way, though, you would have had to endure quite a bit of volatility. Measured once a year, you would have suffered a cumulative 37% loss during the 1973-4 bear market and then, coincidentally, would have suffered another 37% drop during the 2000-2002 bear market (measured on a daily basis, the top to bottom would have been even worse, but anyone who measures their results on a daily basis needs to see a psychiatrist).
Alternatively, let’s say you could have put that $10,000 into an index fund that mirrored the performance of the Dow Jones AIG Commodity Index (an index of 20 different energy, agricultural, and metal futures contracts). Your January 1, 1972 investment would have grown to $363,600, a 10.8% annual return. Your worst performance would have been a 30% drop in 1997-8.
Hopefully, you know what’s coming. Assume you had split that $10,000 between these two options, and had evened up the amounts at the end of each year, maintaining an equal allocation throughout the 35 years. For the 5 minutes of attention you would have had to devote each year, you would have been rewarded with a portfolio of $508,200, an 11.9% annual return, better than the return of either stocks or commodities, and your worst cumulative performance would have been a one-year drop in 2001 of 15%, half as much as the worst loss in commodities and LESS than half the worst loss in stocks.
Read the last sentence again to make sure you realize what happened. By splitting your money between two investments that had their good and bad years at different times, you made much more money than EITHER investment individually and took much less risk in doing so.
Markowitz figured this out in 1951. Most investors still don’t know it. A portfolio of investments will ALWAYS be less volatile than the average volatility of the individual investments and ALWAYS have a higher return than the average return of the individual investments (with the trivial exception of investments that have the exact same return in every single period).
The trick in real life is to find investments that have similar returns to each other but that have them in different years, and that’s a heck of a lot easier than trying to pick stocks that beat the market. You won’t win a Nobel Prize by applying this insight, but you may make enough to buy one.
