# Comparison of expected returns from index funds and single shares

I shall start by saying I know little about markets, investing or gambling, but I do know a little bit about probability. This question is prompted by comments to this question, such as "As a personal investor you have no real chance of beating the market." and "You have a chance of beating the market, but it's far more likely you'll lose a lot of money."

Let us compare 2 strategies:

1. Put \$10,000 in an S&P 500 linked index fund, keep for 1 year and cash out.
2. Buy \$10,000 of a single S&P 500 stock, randomly selected weighted by market capitalisation, keep for 1 year and cash out.

From a simplistic probabilistic view, I would expect both these strategies to have the same expected return, with the second having a greater variance than the first. If each company had an independent, identically distributed and symmetrical expected return you would expect both to have an equal chance of "winning". If companies have a left skewed expected return, in that you expect some complete failures and fewer to double in value, then you would expect the second to actually win more frequently, but to have a larger chance of losing all your money than of doubling it. A right skewed expected return would have the opposite result.

Another relevant observation is that people differ in their attitude to financial risk. Some people actively avoid risk, for example by hedging their investments and so accepting a lower expected return in exchange for lower variance. Others actively seek out risk, for example playing the lottery, gambling on horses or playing fruit machines. This is accepting a negative expected return for a very high variance, and gaining the "fun" of finding out the result.

Therefore one could conclude that it would be rational for some people, those who favour lower variance, to buy the S&P 500 index. Others, those who favour risk, would be rational in choosing the single stock. They get the same expected return but also gain the "fun" of seeing how their pick did against the market.

Is this analysis approximately correct, or is there something I am missing?

• By "S+P index", do you mean "S&P 500 index"?
– Flux
May 10, 2021 at 12:12
• Yes, I guess I mean S&P 500 index linked fund.
– Dave
May 10, 2021 at 12:16
• Have you tested this? Go back x years ago and do a couple of million trials and see what happens. Make sure X years is long enough to get great years, and poor years, and years where they moved sideways. May 10, 2021 at 12:17
• I have not tested this. It is based on theoretical probabilities, and I think is consistent with maths.
– Dave
May 10, 2021 at 12:19

From a simplistic probabilistic view, I would expect both these strategies to have the same expected return,

No. This would only be true if each individual stock in the S&P 500 has the same expected return, which is an overly simplistic assumption.

with the second having a greater variance than the first.

Correct. The standard deviation of the mean will always be smaller than the mean of the standard deviations.

If each company had an independent, identically distributed and symmetrical expected return

The statistical biggest piece you are missing in your model is the covariance between the stocks. Stocks in a portfolio are not independent random variables but too some extent "coupled" . The amount of coupling is quantified by the covariance. The coupling is large: more often than not the stocks move together but there is also a lot of relative movement. The assumption of "independence" is NOT justified. Not even close.

... would be rational in choosing the single stock.

That depends on your definition of rationality. A good read is Benjamin Graham's "The Intelligent Investor". He clearly delineates between "investing" and "speculation". The goal for investment is to make money, the goal for speculation is entertainment. Buying an individual stock as a personal is almost always "speculation". It becomes irrational if you buy individual stocks as an investment vehicle (and not being honest with yourself that it's speculation).

• "This would only be true if each individual stock in the S&P 500 has the same expected return" I do not think this is true. As long as your random choice of stock matches the weighting of the S+P then it should capture the variance in expected return, but will increase the variance of the single stock. "you are missing in your model is the covariance between the stocks" covariance will increase the variance of the return from the S+P index, not the expected return of either.
– Dave
May 10, 2021 at 12:06
• "... the goal for speculation is entertainment" — Where does Graham say that? In The Intelligent Investor chapter 1 Investment versus Speculation, Graham briefly mentions intelligent vs unintelligent speculation. In Security Analysis chapter 4 Distinctions between Investment and Speculation, Graham and Dodd carefully distinguish between intelligent and unintelligent speculation. The authors are not completely against speculation, only unintelligent speculation. They do, however, believe that intelligent speculation is unsuitable for most people, when investment is available.
– Flux
May 10, 2021 at 12:49

"They get the same expected return but also gain the "fun" of seeing how their pick did against the market."

Yikes. Be very, very careful. This is gambler-speak for "I am smart enough to make money but this will also become a form of entertainment".

The piece you are missing is the emotional baggage associated with making what can be significant financial decisions that end up poorly. As well as the fact that you may be downplaying exactly how large the difference in risk levels are between a single company and a diversified index. As well as the fact that if you lose all your money, that money will never again 'average back up' over time, because it's already gone.