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Numbers Don’t Lie: Why Active Stock Management Can’t Win

There are two ways you can invest in stocks: active or passive stock management. If you’ve read this blog before, you know that I am a big fan of passive investing strategies. And there’s a lot of good peer-reviewed evidence that passive approaches beat active stock management.

However, huge numbers of investors still invest actively. And the reason why is because it just seems to make sense to us as humans that the more we study and work at something (in this case investing), the better the results…for these 5 reasons.

But when it comes to investing in the stock market (or bond market for that matter), the opposite is true. We are rewarded for the less that we do. Because the market is efficient.

And it’s not just me saying this. Even Steve Galbraith, a professor at Columbia University, says, “We recognize that the odds are against active managers.”

But it seems like we need even more definitive reason to prove to ourselves that passive investing is the way to go.

So let’s do just that. Courtesy of William Sharpe.

The mathematical proof that active stock management can’t win

William Sharpe is a pretty smart guy. In 1990 he won the Nobel Prize in Economics. And he followed that up in 1991 by writing an academic paper titled, The Arithmetic of Active Management.

active stock management
Don’t worry, no calculus is involved!

This paper provides an arithmetic proof that is theoretical but essentially shows that active management just can’t win. As you go through this with me, you may be taken back to your high school logic class and the law of modus tollens.

But stick with us…

Setting the stage

Sharpe asserts in his argument that active and passive stock management must be “sensible.”

This, in his proof, the following must be true:

  • Before expense (i.e. fees and taxes) returns on the average actively managed dollar will be the same as before expenses returns on the average passively managed dollar, and
  • After expenses returns on the average actively managed dollar will be the less as before expenses returns on the average passively managed dollar

Now, in his own words, Sharpe says, “These assertions will hold for any time period, Moreover, they depend only on the laws of addition, subtraction, multiplication, and division. Nothing else is required. To repeat, Properly measured, the average actively managed dollar must underperform the average passively managed dollar, net of costs. Empirical analyses that appear to refute this principle are guilty go improper measurement.

Emphasis is mine.

I’ll be honest. The first time I read this, it didn’t quite make total sense to me. How could we assume that average active and passive returns, before fees, were the same. After all, that’s the premise that the whole argument is based on.

But an example that Larry Swede provides in his book, The Quest for Alpha, illustrates things very clearly.

Here comes the math…

Again, the example and proof here is that active stock management, in aggregate, is a loser’s game.

In the stock market, there are only two types of investors: active investors and passive investors.

Let’s assume that:

  • 70% of investors are active and therefore 30% are passive
  • The market returns 10% i.e. a total stock market index fund earns 10% returns before expenses

The question then becomes, what before expenses rate of return, must active managers earn in this case?

And we can actually figure this out due to the simple mathematical concept that the sum of the parts must equal the whole.

In this case, the total stock market in sum (A) is equal to all of the active investors (B) plus all of the passive investors (C).

A = B + C

Now let’s factor in the rate of return before expenses, X. And we already know that the rate of return pre-expense for the total market and for passive investors is 10% in this case. The only variable we don’t know yet is the pre-expense return for active stock management.

So,

10% (100%) = X% (70%) + 10% (30%)

And we can solve this equation,

10% = X% (70%) + 3%
7% = X% (70%)
X = 10%

We did it!

We just proved the very first assumption that Sharpe makes in this proof. That before expense (i.e. fees and taxes) returns on the average actively managed dollar will be the same as before expenses returns on the average passively managed dollar.

With this proven, we already know the second point that after expenses returns for active stock management are lower must be true. Because active management incurs additional fees and taxes due to increased trading and management.

As Sharpe concludes, arguments in favor of active stock management can only be made by “assuming that the laws of arithmetic have been suspended for the convenience of [active managers].”

Mic drop.

But practically why is this the case?

With active stock management, there is a winner and a loser.

If one active investor wins because they got lucky and picked some better performing stocks, that means there must, be definition, be another active investor who underperformed.

And we know from other research that there is no evidence of active managers showing persistently over performing returns. So whoever is the winner one year will become the loser another year.

In contrast, passive investors just buy and hold index funds. And the stocks they buy (in the funds) must be sold by an active investors.

Thus, in aggregate, no one wins with active investing. Because returns are the same as passive investing but the fees and taxes result in overall lower returns.

And this is always the case

This is an important point.

You can do the same proof above with different numbers. Passive investors can equal 90% of all investors. Average market returns can be 5% or 25%.

Further, instead of the total stock market, you can run the same proof for small cap funds or other seemingly or purported inefficient investment types.

And you can run it during a bear market with active losses instead of returns. Both active and passive investors lose the same, pre expense. But passive investors lose less after expenses.

It always shows the same thing: active investing can’t win.

And if you can’t win, stop playing the active investing game.

Invest instead in passively managed, low cost, broadly diversified index funds based on your asset allocation. Then rebalance yearly.

And, in between, enjoy your life knowing you are on the path to financial freedom!

In the meantime, if you are looking for some help figuring out exactly how to structure your passive investments, these posts can help:

What do you think? Can active stock management win? If so, how? Would you invest your money actively after seeing this proof? Let me know in the comments below!

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    The Prudent Plastic Surgeon

    Jordan Frey MD, a plastic surgeon in Buffalo, NY, is one of the fastest-growing physician finance bloggers in the world. See how he went from financially clueless to increasing his net worth by $1M in 1 year and how you can do the same! Feel free to send Jordan a message at [email protected].

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