Is There Actually An Equity Premium Puzzle?

The equity risk premium, or ERP, is defined as the excess return you get for investing in stocks over the risk-free rate. Simply, it’s the premium return you earn in exchange for dealing with path. The fact that you might experience a 20% drawdown every few years (with U.S. equity markets currently sitting on all-time highs it’s hard to believe that just 1 year ago the SP500 had a 20% drawdown). I admit this “no pain, no gain” explanation sounds a bit weird.

Student: Hey prof, why do I get paid extra for buying stocks instead of t-bills?

Master: Because if you weren’t offered a discounted price to buy stocks you wouldn’t. Duh.

Proof by induction can be unsatisfying. To be fair, my use of the word proof is straining its English definition. Instead, it’s typical to hear ERP referred to in the context of a puzzle since some economists with calculators decided that this roughly 6% historical premium has been excessive compared to what they would expect even risk-averse investors to demand.

Enter the Witch

But what if I told you that there is actually no ERP and therefore no puzzle. Well, you’d accuse me of heresy since I’m directly contradicting widely accepted financial orthodoxy. After all, I’m ignoring the fact that equities have in fact outperformed t-bills by a wide margin.

Let’s look at that assertion again — equities have outperformed t-bills by a wide margin.

Well, what do we mean by equities? Single stocks or indexes? This is where I let the witch take over. The heretic, BreakingTheMarket who states:

The Equity Premium Puzzle has lasted for 37 years without anyone recognizing the market index doesn’t represent stocks.

Mistaken Equivalency

Turns out the existence of an ERP depends on your definition of equities and an index of equities is just not just equities. It’s a strategy. An index is a rule-based weighting that rebalances intermittently. The difference cannot be overstated. Why?

“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 math makes it clear.

When you compare the geometric return of stocks not a stock index you do not find an ERP!

How This Ties Together With What We Have Learned In The Past

As you digest this, there should hopefully be a comforting reinforcement of past ideas, namely:

• When we deal with multiplicative processes, like returns that compound wealth, we care about geometric or logreturns not arithmetic returns because of the “volatility drain”. (Link)
• Portfolio components are not perfectly correlated so when we rebalance, we capture a premium geometric return. (Link)
• The imperfectly correlated aspect of a portfolio contributes to what Fernholz called the excess growth component that diversification earns when you are in logreturn space. (Link).

If we presume stock index volatility is only 17% (as opposed to the 33% for single stocks), we can use napkin math to make additional observations.

• Index ERP is closer to 6% – .5 * (.17^2) = 4.56%…the extra 4% represents Fernholz’s “excess growth rate”. This is why some pros refer to diversification as the only “free lunch” in investing.
• The average cross-correlation of stocks in an index can be approximated by the ratio of index variance to average weighted stock variance. Using our estimates (.17^2) / (.33^2) = .27 which is in the ballpark of where long term average SP500 index correlations have realized (although option folks know how spikey that number can be, especially on short measures).

Summing Up

ERP doesn’t exist if you look at stock; only stock indexes!

• Researchers commonly mistake equivalency between a single asset and a portfolio:
  • 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!

A Final Note

I chat with BreakingtheMarket on Twitter and follow his discussions with quants. So much of the merit of Twitter, and the internet in general, is the beauty of being able to learn and engage in conversations with talented, curious people whom you may not have found otherwise. Breaking the Market is not in finance. He’s an engineer with a strong math background who approached markets with a “beginner’s mind”. I don’t think it’s an accident that two of my favorite finance writers on the internet are from scientifically minded people from a different field. I think the best finance blog is PhilosophicalEconomics.com which is penned by another finance outsider, the pseudonymous Jesse Livermore. Jesse did his first interview this year and it’s worth checking out, along with his widely influential writing. (Link to interview with my notes)

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