Wednesday, September 22, 2021

The Equity Premium Puzzle: A Suggested Solution

Most investors are free to invest in either stocks or bonds, so one would expect the return of both investments to be similar. It is an obvious argument, at least to an economist, but the conclusion does not appear to be true. In the U.S. over the past century, stocks have consistently outperformed bonds. This is the Equity Premium Puzzle.

A variety of explanations have been offered. The most obvious is that the return of stocks, both individual stocks and stock portfolios, is more uncertain than the return of bonds, at least over the short run, so risk averse investors would be willing to accept a lower return on bonds in exchange for lower risk. Attempts to estimate how risk averse investors would have to be to explain the observed premium, however, produce implausible answers: 

To quantify the level of risk aversion implied if these figures represented the expected outperformance of equities over bonds, investors would prefer a certain payoff of $51,300 to a 50/50 bet paying either $50,000 or $100,000.[1]

A variety of other explanations have been offered but none appears to be generally accepted.

I have another one.

Imagine a market where many investors are insider traders on a small scale, in an economic but not necessarily a legal sense. I know more than the market about the firm I work for. You know more than the market about a firm you have repeatedly had dealings with. He knows more than the market about a technology he has worked with extensively that will affect the performance of several firms.

Each person invests some of his money where he has an edge, buys a stock that is going to go up or sells short one that is going to go down. By the standard analysis of insider trading, he makes an above market return. Because he is risk averse he wants to diversify his investment, so part of his money goes into other investments. His marginal return is the return he can get without inside knowledge, his average return is a suitably weighted average of that and his insider return. Bond investments only have to do as well as his marginal investments, so the return on bonds is lower than the average return on stocks.

When I first came up with the idea, someone I described it to offered a proof that it could not be true, one that at the time seemed convincing. My argument implies that everyone, the investor without inside knowledge and the investor with inside knowledge making marginal investments,  is getting less than the average return. Why can’t he buy a random collection of stocks, in the limit a millionth of a percent of the stock of every company, and so guarantee himself the average return?

That proves too much, only that my explanation of the equity premium puzzle is inconsistent with rational behavior by investors but also that an insider cannot profit by his inside knowledge: If some insiders are getting an above average return, everyone else must be getting, on average, a below average return, and anyone who is could get an average return by buying stock at random.

The solution to the paradox is that it is not possible for every uninformed investor to end up with an equal share of every company on the market because some of those shares, the shares of companies particularly likely to go up, belong to insiders. That explains both how it is possible for insiders to make an above average return and why my explanation of the Equity Premium Puzzle is not inconsistent with investor rationality.

Whether it is the correct explanation, of course, I do not know.

Comments welcome.

[1] Mankiw, N. Gregory; Zeldes, Stephen P. (1991). "The Consumption of Stockholders and Nonstockholders". Journal of Financial Economics. 29 (1): 97–112.



Lawrence Kesteloot said...

"His marginal return is the return he can get without inside knowledge"

Why? If he has an extra dollar, why can't he invest it proportionally (say, 30% inside, 70% not)?

John said...

"The solution to the paradox is that it is not possible for every uninformed investor to end up with an equal share of every company on the market because some of those shares, the shares of companies particularly likely to go up, belong to insiders. "

I would update your assumption about equal shares to market capitalization weights. In this sense, insider information about a firm with a large market capitalization weight may matter more than one with a small market capitalization weight. Because of the large weight, however, it is harder for the marginal investor to buy up a significant amount of what is outstanding.

Eric Rasmusen said...

"When I first came up with the idea, someone I described it to offered a proof that it could not be true,..."
Let's call that the Dartboard Counterargument: an uninformed investor who chooses his stocks by throwing darts at a list will get the high average return---let's call it 8%, to be concrete.
You need to do more discussion of that. It's true that under your story, not ALL uninformed investors can do that. They can't invest equally in all stocks when the informed investors have locked up 30% of the shares, 90% of the shares of each "good" company.
But any individual uninformed investor could use his dartboard. The trick is that he has to buy-and-hold. If he invests in an index fund in 2000 and doesn't sell till 2021, he'll get the 8% return. That's because when the informed investor comes to him and says, "Will you sell your Good Company stock to me?", he will say No. On the other hand, the actively trading uninformed investor will say Yes, because the informed investor will offer what seems like a high price for Good Company.
The actively trading uninformed investor is "irrational" in this story, but we know that already, by the very fact that he is actively trading and paying brokerage fees. I don't see any way that sophisticated uninformed investors can take advantage of his irrationality, so it will persist so long as he has any money left.

A separate idea: Is active trading a good thing, because it takes moeny from foolish rich people who would otherwise waste it on foolish spending (that is, costly real good that would not give them or anybody else much utility) and transfers the money to clever rich people?

Eric Rasmusen said...

ps-- please do send me your semi-formal model, at or

David Friedman said...



Vincent said...

That is a great idea. However, I am not sure about the size of the effect. Estimates of (Jeng et al. (2003)) are around $0.10 per $10,000 in transactions for reported insider trading. Assuming that this estimate is reasonable and assuming an average holding period of a year, unreported levels of insider trading x associated excess returns will have to be a factor 10^3 higher in order to be meaningful. It would be very interesting (and sad) if that seems indeed to be the case.

Anonymous said...

Vincent, it's probably right that the size is very small. In the US, at least. I wonder what the equity premium is in countries with less regulation and disclosure requirements. In particular, in the US companies have to have annual reports with numbers, quarterly earnings, etc. and we have lots of analysts, so there's a lot of "private regulation". What about China, Korea, Italy? Maybe teh reported equity premium is 15% there. Or, more likely, people just don't invest in public stocks much, or buy American stocks if they do want to. Those are countries, also, where minority shareholders get ripped off by such things as conflict-of-interest transactions by the company.

On the other hand, there might well be massive unreported, legal, "insider trading" in the US. Hardly any of it would be by people classified as "insiders" in the law, or even subject to insider trading laws. It might be just that there are lots of people who acquire private info legally, on their own, by research such as counting truck shipments into a factory or being very good at reading accounting statements.

Vincent said...

Those are good points Anonymous. So there could be an argument that somehow there is a gap between "informed" and "uninformed" trading, which leads to a question about the efficiency of the market, which are likely different depending on the country. You are effectively arguing that the equity premium could, at least in part, be an outcome of stock market inefficiencies (in excess of any bond market inefficiency)?

But wouldn't such inefficiencies be easy to witness (through some big, consistent winners) and disappear over time (if cost of obtaining information is sufficiently low)?

Eric Rasmusen said...

They wouldn't be consistent winners necessarily. It might be people who stumble upon good info, say, about a supplier or a customer, or by watching a friend's husband go on lots of trips to a particular city.

The inefficiency would disappear-- in fact, that's how the informed guy makes money-- when the news becomes public and the stock price rises. But there's always new info coming up.

Note, too, that tho the cost of acuqiring info might be low for some people (the friend), it would be very high for other people (you and me). --Eric Rasmusen

Vincent said...

Thanks Eric, the idea that such information is distributed and in some way you have to be "lucky" makes sense. However, it seems hard to verify the level of impact of such information. It would be interesting if there is a way to measure the distribution and prevalence of such "informedness" (maybe looking at return distributions I am actually quite curious). (Trying to look at it a bit more deeply I notice that David Easly and Maureen O'Hara have written some interesting papers on the topic focussing on the firm level.)