It is common knowledge that if you want to make big money you must take big risks. If you invest in risky stocks, you will, on average, get more in return than if you invested in in less risky stocks. But is this really true?

Why would risk pay off? Consider two options. One option is to flip a fair coin. If it is heads the pay off is $20, otherwise nothing. The other option is to roll a fair die. If it is one the pay off is $60, otherwise nothing. Forgetting about non-linear utility of money, the expected value of the two choices are the same. So the rational choice is to go with the coin game because it is less risky. The only way to get people to play the die game is to increase the pay-off, say by raising it to $70. Presumably this is the reasoning that underlies the idea the risky stocks will pay-off more.

But if you have the option of playing these games multiple times, this argument breaks down. By rolling the die enough times you can reduce the variance of the pay-off below that of the coin game. No matter how little above $60 you make the pay-off of the die game, everyone will chose the die game, and play it with the same variance of the coin game. Markets would force the die game and the coin game to give the same pay-off.

I argue that stocks are more like playing multiple games than single games. If you can find many high-risk stocks that are not correlated then you can reduce the variance of the portfolio to that of a single low-risk stock. Since there are wealthy entities that are capable of making such portfolios with little friction, this means the pay-off of high-risk stocks will be driven to be the same as low-risk stocks.

Is there any empirical evidence that high-risk investments pay off more? My colleague has a book that analyses 58 investments strategies and calculated what the return would have been if one had used that strategy from 1954 to 1994. The authors also calculate the standard deviation in the investment strategies. The figure below plots this standard deviation (in percent) against the return on investment (in percent).

The linear correlation of the data is 1.8%. I don’t expect the return on investment to be linearly related to the standard deviation, but I don’t really see any relation whatsoever in the plot. Perhaps some statistician out there can tell me if this correlation is consistent with the hypothesis that the return on investment is unrelated to the standard deviation of the investment.

Does anyone have any evidence that riskier investments have higher returns?

Maybe what people mean is that if you want to make lots of money you must take risks; however taking risks does not mean you will make lots of money.