Solving the Equity Premium Puzzle, and Uncovering a Huge Flaw in Investment Theory.

The equity premium puzzle has troubled economists for over 30 years.  I’ve solved it, and in the process identified a major flaw with investment theory.

Put simply, the puzzle is:

“Equities (stocks) have provided a real return of about 7% for the last 100 years.  The risk-free rate (treasury bills) has provided about 1% of return.  The 6% difference between these two investments is very high.  Economic models imply this difference should be at most 1%.  We don’t know why the large difference exists.”

Graphically the puzzle looks like this:

Published in 1982, the original study (Mehra and Prescott) on the equity premium puzzle evaluated the returns from 1889-1978 of the S&P 500¹ and revealed the statistics above.  Since then, economists have evaluated this puzzle in different countries, different timelines, vs longer term bonds instead of t-bills, and for both arithmetic and geometric returns.  All studies continue to agree the puzzle still exists unexplained.  There is an acknowledgement that stocks are riskier and investors should be compensated for the risk.  But in economists’ opinion the risk to return ratio is beyond explaining.  Most believe investors are overly risk adverse because they don’t buy more and more equities to close this gap and match the economists’ models.  

Random Side Note

Jack Bogle, founder of Vanguard, invented the index mutual fund in 1976. It was a per-cursor to the Vangaurd 500 index fund that tracks the S&P 500.  We take index funds for granted now as they are everywhere, but before Mr. Bogle, index investing did not exist. 

A Mistaken Equivalency

Now I really don’t understand why the economic models say the gap should be around 1%, or what the models are really trying to accomplish.  I’ve looked at them briefly, but they just seem so overly complicated.  And, more importantly they are totally unnecessary to understand the puzzle once you fix the major apple to oranges comparison foundational to the puzzle.

Treasury bills (and bonds) are a single investment item.  An equity market index (SP500 for the original study and many others) is a portfolio of many investments, who’s composition changes all the time.  They are not the same thing and shouldn’t be compared as if they are!

The conclusion is akin to questioning why the world record for the men’s 400 meter race is 6 seconds slower than the men’s 4×100 meter relay.   One is a combination of participants, the other is a single performer.  You can’t really compare them.

I would re-characterize the original 1982 study’s conclusion as follows: 

Investing in the S&P 500 index from 1889 to 1978 clearly outperforms investing in treasury bills.  Even though index funds were only invented a few years before our study, we do not understand why people did not invest in this non-existent strategy more often.

Put this way, the original study’s conclusion is obviously ridiculous.  People did not invest in a market index strategy before 1975, they invested in individual stocks.  Until very recently, when passive investing gained a large following, people still generally invested in individual stocks.

“Stocks” and the “Stock Market Index” are not the same thing and never have been. One is an asset class, the other is a trading strategy of that asset class.  They don’t behave the same and don’t have the same properties, return, or standard deviation.   You can’t use one to replace the other.

The Equity Premium of Stocks

The prior studies peg the annual standard deviation of the S&P500 at around 20%.  From the little bit of historical research I have done, individual stocks possess a standard deviation of 30-35% — a 50% increase in risk over the market index.²  Shown graphically on our chart: 

That’s looks better, but we have one more step to solve the puzzle.  Most of the equity premium studies evaluated arithmetic returns (one study I found used geometric returns, but it still used the S&P 500).  From prior posts, I’ve shown that arithmetic returns by themselves are meaningless.  All compounding investments converge to the geometric average, the only returns investors actually experience.  Therefore, let’s convert our annual arithmetic equity premium of stocks into geometric returns using the formula:

Arithmetic Average of Premium – ½ x (standard deviation)² = Geometric Average of Premium

6%            –      ½   x   (0.33)²         =     0.55%

The geometric mean of individual stocks explains the puzzle!  The remaining half percent can be explained by pure randomness or if necessary by the 0-1% premium the economic models imply.  This is why the equity premium does not exist.³ The geometric average return of stocks converges to the risk free rate over time.

Far Reaching Ramifications Beyond the EPP.

It should be easy now to understand the enormous apple to oranges mistake in the original study.  The stock market and a stock market index are not interchangeable. When I explain this concept to people outside the financial industry, they get it instantly. Its seems obvious. A few in the financial community are starting to see this, but most don’t. One recent paper (Bessembinder, 2018) gets close by recognizing that 58 percent of common stocks do not keep pace with the returns of a one month treasury bill over time, but does not call out the false equivalency being made by the market index comparison.

The Equity Premium Puzzle has lasted for 37 years without anyone recognizing the market index doesn’t represent stocks. But this is understandable as the financial industry almost always makes this mistake. At least the equity premium’s only real purpose is to call investors foolish (they are not). Fund managers use the more foundational concepts of investing, built on the same the same mistake (CAPM, beta, factor models, stock for the long runs, etc) to justify the allocations of billions and billions of your dollars. When you get right down to it, it’s quite scary that an error this large has persisted and grown so influential.

But fear not, by the time I’m finished with this blog, you will recognize these false concepts and hold in their place a new, accurate perception of financial markets. We will re-invent the financial world together.

1-I’m not entirely sure how they truly took the S&P500 back to 1889, but that’s what the paper says. 

2-For anyone trying to reproduce the theory and validate my numbers, let me expand here.  The average standard deviation itself is actually meaningless since volatility drag is very convex.  20% standard deviation leads to 2% drag; 30% standard deviation leads to 4.5% drag;  40% standard deviation to 8% drag.  Increasing standard deviation grows the drag proportionally more.  This makes the actual math on this problem fairly difficult to calculate.  Therefore, I presented the concept very simply with an average to not muddy the waters with unnecessary complexity for the average reader.  One doesn’t need to understand volatility’s convexity to understand the concept that stocks and a stock market index are very different things. 

3-Update 5/7/19-  I read Ole Peters’ and Alex Adamou’s  evaluation of the puzzle. Their work is brilliant.  They mix the geometric average with a calculus called “stochastic efficiency” to address the puzzle.  I cross the geometric average with the returns of stocks vs. a market index.  Both of us have a left-over error that we feel can be explained through randomness.  When you combine our ideas and use the geometric average, stochastic efficiency, and the return of stocks, the puzzle converges right where Peters’ math says it should, to a small equity premium.  It’s intriguing, and I may change my mind and agree there is a small, but explainable premium, but I’m going to leave the post unchanged for posterity.  At a minimum my method explains most of the puzzle.  Furthermore, the post aims to convey the importance of the geometric average and highlight the foundational problem with financial studies- that a stock market index does not represent the stock market.  Introducing calculus would confuse that message.