Expected Return, Risk, and Diversification

Posted on September 2, 2011 by


You’ve probably been told (over and over) that your portfolio should be diversified. You may have some sense of why that’s a good idea. Here’s a (slightly) mathematical analysis of the related concepts return, risk and diversification:

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Expected return

Let’s consider this in the context of a coin flipping game: Each time you play you can win

Each coin flip is an independent event.

the amount you bet if the coin turns up heads. Suppose you have a biased coin that turns up heads 51% of the time: For each bet you have a 51% chance of winning.

You have $1000 that you were willing risk in a series of bets. You are impatient, so you’d like to get your betting over with quickly. Does it make more sense to make a single bet with $1000 or to make one thousand small bets of $1 each? Let’s consider the expected return for each scenario. In the case of the single bet of $1000:

  • 49% chance you lose (-$1000)
  • 51% chance you win (+$1000)
  • Expected return = 49% * -$1000 + 51% * $1000 = $20

And for 1000 small bets:

  • For each $1 bet: 49% chance you lose (-$1)
  • 51% chance you win (+$1)
  • Expected outcome for each $1 bet is $0.02.
  • If we do this 1000 times in parallel, the total expected return is $20

The expected return is $20 in both cases, so why should we choose one one approach over the other? Why not just make the single bet? The answer lies in risk.


One way to measure risk is to consider the possibility of losing all of our investment. In the case where we make one thousand $1 bets the only way we can lose all of our money is if we lose every single one of these one thousand bets. The probability of that occurring is 0.49 * 0.49 * … * 0.49 (repeated 1000 times), or 0.49^1000.

  • Chance of losing all money with one thousand $1 bets: This number is so microscopically small that my calculator refuses to compute it.
  • Chance of losing all money in a single $1000 bet: 49%

The single bet is significantly more risky in comparison.  Almost infinitely more risky (if you believe my calculator).

Now let’s consider this in a way that’s a bit more like stock market investing: In the finance world “risk” is usually measured as the standard deviation of each number in a series. Assume we have 1000 opportunities to “bet”. In one case we will make 1000 $1 bets, and in the other case we’ll make one $1000 bet and 999 $0 bets. You can get mini loans to help you invest more, click to get loans.

The total amount we bet (“invest”) in both cases is $1000. So our series of returns in the single $1000 bet case look something like: +$1000, 0, 0, 0, 0, 0… Note that the leading $1000 there could have also been -$1000 (49% chance of it), but the standard deviation is not affected. In the case of betting making 1000 $1 bets our returns look like: +$1, +$1, -$1, +$1,… When we calculate standard deviations we find:

  • The standard deviation for 1000 $1 bets is $1.
  • The standard deviation for a single $1000 bet, and 999 $0 bets is $31.62.

Both measures of risk tell us that the many smaller bets approach is substantially less risky.

Implications for trading

The key lesson is that you can substantially reduce risk by “spreading your bets” across many small trades rather than a small number of large trades. But the stock market is a bit different from the coin flipping model we covered above. In coin flipping, each flip is an independent event, and the math above only works for independent events. Stock price moves are not independent: If Chevron goes down, Exxon is probably going down too.

This is the reason wealth managers recommend “spreading your bets” through diversification. Diversification means to place your bets in different sectors that are not highly correlated. In fact anti-correlated investments are the goal.

For further reading, here’s a blog entry on a related topic.

Where can I get these odds?

You probably think that nobody is going to play this game with you… You know, the game where you have a biased coin that comes up in your favor 51% of the time.

That’s not quite true. If you own a casino, people will line up to bet with you at even better odds. Casinos also leverage the diversification idea above (many small bets) by setting betting limits at their tables. If you want to join that game, buy a stock like WYNN.

Here’s an interesting paper on the statistics of coin tosses http://comptop.stanford.edu/u/preprints/heads.pdf

Disclosure: As of this writing, Tucker Balch does not have a position in WYNN.