Understanding Diversification

In my quest to find different ways of explaining investing concepts, I have a new idea for explaining the value of diversification based on a hypothetical game show. It shows the trade-offs between concentrating your bets and spreading them out.

You’ve beaten the other contestants to make it to the prize round of the TV game show “No Guts—No Glory” to take a shot at winning up to a million dollars! There are five briefcases, each containing a different amount of money: $100, $1000, $10,000, $100,000, and $1,000,000. But they are in a randomly-selected order.

You can choose any number of briefcases from one to five, but none are opened until you’ve finished making your selections. The catch is that you get the average amount in the briefcases you pick, not the total. So, if you choose 3 briefcases that happen to contain $1000, $10,000, and $100,000, your prize is not the total of $111,000 but the average amount of $37,000. (Fun fact: no matter what the briefcase selection, the prize will be a whole number of dollars.)

The only way to win a full million dollars is to choose only one briefcase. But this is risky because the only way to end up with only $100 is also to choose only one briefcase. The average outcome is $222,220, but your odds of winning less than this amount are 80% if you choose only one briefcase. The following bar chart illustrates the possibilities:

Choosing only one briefcase seems too risky. Maybe it’s better to choose two. You can’t win a million dollars this way, but you could still win as much as $550,000. There are a total of 10 different outcomes. The average outcome is still $222,220, but the range of outcomes is narrower than if you choose only one briefcase. There’s still a 60% chance that you’ll end up with less than this average outcome. Here’s the bar chart of possibilities:

Choosing only 2 briefcases still seems awfully risky. Being left with around $50,000 or less would be hard to take when so much more was available. Selecting 3 briefcases could still win you $370,000. Again, the average outcome is $222,220. As long as one of the 3 briefcases holds the million dollars, the payoff is great. But things look much worse the other 40% of the time:

Let’s try 4 briefcases. This is starting to look much safer. There’s an 80% chance of winning at least a quarter of a million dollars. But that 20% chance of winning only $27,775 is scary:

Things get a lot simpler if you choose all 5 briefcases. You get exactly $222,220. It’s not a million but it’s nothing to sneeze at. For completeness, here’s the 5-briefcase bar chart:

We’ve talked ourselves all the way from a concentrated bet on one briefcase trying to win a million dollars all the way to the safe choice of a sure $222,220. This is analogous to the difference between picking a small number of stocks and owning (almost) all stocks through index ETFs or mutual funds. Assuming your stock choices are random, the expected outcome is the same, but the volatility is higher the fewer stocks you own. The only way to beat the stock market soundly is by making concentrated bets. Unfortunately, this is the way to lose badly to the market averages as well.

For those who think they don’t mind some extra volatility, there’s a wrinkle. As you compound your returns over the years, higher volatility skews your possible returns more and more. This means the odds of beating the market become progressively smaller. A small number of stock pickers make huge gains at the expense of the masses of other stock pickers whose returns lag the markets.