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Is active investing a zero sum game?

I have previously explained that investing [1] in equities does not need to be a zero sum game.

Now that doesn’t mean you’re guaranteed a positive return in any one year of investing – or even over many years.

Stock markets go up and down a lot in the short-term.

But even the long term – while a much more appropriate time frame to talk about when it comes to the expected returns [2] from shares – is unknown.

Perhaps we’ll go into some terrible 20-year economic depression, who knows? Or maybe shares will simply turn out to have been too highly rated and they’ll fall for a long time as they readjust to reality, similar to what happened in Japan [3].

The important point for today’s article is to understand there’s nothing inherent in investing in equities that means that for you to earn the market return, another investor must lose by the same amount.

If you’re not sure why – or if you don’t understand what ‘the market’ is, or the investing terms alpha and beta – then please read the previous article [1] so we’re all on the same page.

Market makers

Investing in a broad stock market to achieve the same return as the index (minus costs) is not a zero sum game.

However once you try to do better [4] than the market, it’s a different kettle of fish.

Specifically, active investing is a zero sum game.

The reason is obvious once you get it, but from experience [5] it can take some time to do so.

First off, we talk about ‘the market’. Key to understanding why active management must be a zero sum game is understanding what the market is.

When we talk about the market, we may mean different things.

We might mean the sum total of all shares listed in all stock markets around the world.

We might even include other kinds of assets, like bonds and commodities.

More commonly we might mean a specific market – such as the UK stock market – or a further subset of that market – such as the FTSE 100 index of the top 100 largest public companies in the UK.

Let’s focus on that FTSE 100 index for a moment.

Who owns this market?

Well, pension funds, hedge funds, mutual funds, and also individuals who buy and sell stocks.

How do they own the market?

They take a stake in three different ways1 [6]:

The first kind of investor is only aiming to get the market return – the beta – which is why they have invested via index funds. By definition, they track the market, and hold shares in proportion to the market’s ownership.

If some shares go up and others go down, then so will their holdings. They can do no better, but also no worse [11]. They will get the market return (less holding costs and any tracking error [12]).3 [13]

That leaves the non-passive investors, who may be invested in active funds or actually running the active funds, stock picking [4] their own portfolio of shares, or maybe owning a mix of passive and active funds.

For simplicity I’ll label them all as active investors from here, as it doesn’t matter exactly how they’re actively investing for this discussion.

Note: Sometimes people will say they are not trying to beat the market when non-index investing, but rather they are trying to be defensive, or ethical, or focused on income [14], or one of a gazillion other variations. That’s all fair enough – whatever floats your boat – but it amounts to the same thing from the market-beating zero sum game perspective.

Actively different

So what does it mean to be an active investor?

It means your shareholdings in a particular market are different to the market’s weightings.

But the ownership of that market is comprised entirely of…

Passive investors + active investors

…so if you want to take a different position to the market, then by definition you are going to have to do so by trading with other active investors – since passive investors just hold the market, and they ain’t giving you diddly squat if it distorts their market weightings.

An example might help.

Let’s say you own the market via a tracker fund, but also you’re keener on Tesco and so you want to own more than a market-weighting of its shares.

You therefore buy shares off an otherwise identical investor who is less keen on Tesco’s prospects than you are.

You take ownership of some Tesco shares off this investor, who now owns fewer Tesco shares. In place of the Tesco shares, let’s say they increase their holding of the market.

Now let’s imagine that was the only trading any active investors did in the FTSE 100 this year. (I think that’s what they call a quiet year!)


You own the market plus your extra Tesco shares.

The active investor you bought them off also owns the market, except she owns fewer Tesco shares.

In our hypothetical example all other investors in the world are passive investors.

So what happens next?

Well, the market delivers the market return.

So far so obvious.

However Tesco shares either return more than the market’s return, the same as the market, or less than the market.

That is why active investing is a zero sum game.

Note this has nothing to do with the discussion of whether some active investors can or will beat the market over the long-term [15].

In fact, it shows it’s trivial to create a market-beating fund over any one year – just hold the index plus shares in one company in one fund, and the index and a short position (i.e. bet against) the same company in the second fund.

One of the two funds is guaranteed to beat the market that year, before costs4 [16]. But sadly, the other is guaranteed to do worse.

Indeed, in aggregate – netting winners off against losers – active investors as a group earn the market return, just like passive investors.

They must do, since together they own the market.

What really does for investing in active funds overall is high costs [17].

Why active investing is worse than a zero sum game

Imagine the two fund managers who took different positions in my Tesco example charged 1% for their services.

Let’s also suppose that all that furious Tesco share trading cost them 0.5% in expenses. (Heck, it’s an expensive business in this hypothetical world of low volume trading…)

This means the total cost of their active investing is 1.5%.

In contrast, an investor in the passive fund might pay say 0.25% in annual charges.

Before taking into account these costs:

After costs:

Schoolboy algebra tells us that combined, the active funds earn…

… since the +alpha and -alpha cancel out.

Therefore in aggregate the passive investors beat the aggregate active investors, because of the latter’s much higher charges [18].

So in practice, active management is worse than a zero sum game.

The zero sum game writ large

While the example I gave above is obviously as simple as it can be, exactly the same principles hold true on the grand scale of real-world markets.

The fact is it must hold – ‘alpha’ cannot be magicked out of thin air.

The only place an active manager can go to get more or fewer shares than are held by the market is by dealing with other active investors in that market. (Because the passive investors by definition hold the market).

And then you have to subtract those higher costs.

This is summarized by Vanguard [19] in this handy graphic:

Click the link to Vanguard in the paragraph above for full-sized image. [20]

Click the link to Vanguard in the paragraph above for a full-sized image and PDF.

In this graph, the green vertical line above the ‘0’ on the x-axis is the return from the market.

You can see that there’s a bell curve of returns on either side of this market return.

This indicates how all the money that beats the market is offset by money that loses to the market.

Finally, there are costs, which are indicated by the gap left to the dotted line.

This has the affect of moving the bell curve of returns to the left – lowering average returns – as indicated by the curve in red.

Note that particular active funds do not necessarily deliver negative returns just because it’s a zero sum game. As a group they get the return from the market on average, before adding or subtracting alpha and costs. (They must get the market return as a group, because summed up active funds are the market, alongside passive investors).

So even though beating the market is a zero sum game, it’s not like poker, where a winner-takes-all.

The ‘beta’ delivered by the market goes to active managers as well as passive investors.

It is the ‘alpha-chasing’ part of their business that is a zero sum game.

And as shown above, it is a costly game since their higher costs mean [17] that in aggregate, investors in active funds see lower returns than passive investors.

Is there always a winner for every loser?

Like other writers, I have often written that “for every winner there must be a loser” when it comes to active fund management’s attempts at beating the market.

This is convenient shorthand, but it is not strictly accurate.

It’s not that one active investing individual or active manager must lose for every one that wins.

It’s that the total amount of winning money (the total excess return over the market, the alpha) must be offset by the same amount of losing money.

The Vanguard article cited above puts it as follows:

The concept of a zero-sum game starts with the understanding that at any one time, the holdings of all investors in a particular market make up that market.

As a result, for every invested pound that outperforms the total market over a given period, there must by definition be another pound that underperforms.

Another way of stating this is that the asset-weighted performance of all investors, both positive and negative, will equal the overall performance of the market.

Incidentally, this is a big reason reason why many active investors poo-poo the idea of index investing.

“I’ve done fine with my funds,” they say. “I have made £12,343”.

They have no idea [21] what the market return would have been if they’d just invested passively. And even if they did beat the market then they do not really understand that they risked [22] losing to the market in order to do better.

I’m not saying don’t actively invest if you know what and why you’re doing it, and you also appreciate the nailed-on headwind from higher costs and the other potential downsides.

It’s a free world. Heck, I’m a stock picker myself [23].

But time and time again people who have a dangerously small amount of knowledge about investing – that masquerades as a deep understanding – will tell you:

“Why bother with index funds when the Something Or Other Fund did 22% last year and the market only did 12%.”

The statement might be factually correct, but there’s typically an iceberg of ignorance and misconception beneath it.

We’ve written over one million words on Monevator trying to address these misunderstandings.

We’ll keep at it!

But if for some reason you want a second opinion, you might try reading Noble prize winners French and Fama’s thoughts on the zero sum game [24]:

Suppose we define a passive investor as anyone whose portfolio of U.S. equities is the cap-weight market portfolio.

Likewise, define an active investor as anyone whose portfolio of U.S. equities is the not the cap-weight market portfolio.

It is nevertheless true that the aggregate portfolio of active investors (with each investor’s portfolio weighted by that investor’s share of the total value of the U.S. equities held by active investors) has to be the market portfolio.

Since the aggregate portfolio of all investors (active plus passive) is the market portfolio and the aggregate for all passive investors is the market portfolio,the aggregate for all active investors must be the market portfolio.

All this is obvious. It is just the arithmetic of the fact that all U.S. equities are always held by investors.

(That is why we call it equilibrium accounting.)

In short, total actively invested money cannot beat the market because it is the market, together with neutral passive funds.

Remember that the next time someone says “Stock picking has the edge in a bear market” or similar nonsense.

It’s mathematically impossible.

Some stock pickers might have the edge. If they do better, then an equivalent amount of actively invested money will do worse.

To repeat myself, this doesn’t mean Warren Buffett [25] isn’t Warren Buffett.

Some tiny number of managers might have skill (I personally think some do) or luck, and they deliver years and years of market-beating returns.

If you have a way of finding these needles in a haystack, good luck to you [15].

But be sure you understand that their excess returns – their alpha – must come at the expense of the poor schmucks in active funds that are doing worse.

What if I invest actively for some other reason?

Most private investors who invest in active funds do so out of ignorance, and the rest because they want to beat the market.

However there are other reasons for investing in active funds.

For instance, you may value lower volatility in your portfolio, and for some reason you don’t want to achieve that by holding more bonds [26].

Or you may be trying to invest more ethically, and so you don’t want to exactly mirror the market for that reason.

These are personal choices, and if they have value to you then you can argue you’re not participating in a zero sum game.

Remember in part one I said that a zero sum game was one where everyone valued their ‘slice of cake’ equally?

Well, in this case you’re valuing certain shares for more than their market beating return potential. So arguably, you have stepped out of the mathematics of the zero sum game.

True, I think this discussion is investing’s equivalent of existential philosophy! But I mention it for completeness.

It is specifically active investing to beat the market that is a zero sum game.

Wait, I invest for dividends – who am I hurting?

I’ll also just mention this briefly, as it’s a very frequent retort from British stock pickers with their affinity for buying and holding shares for dividend income [27].

They’ll typically say something like:

“Sure, but I don’t get involved in all that share trading shenanigans. I’m not a gambler, I just buy and hold shares and pick up the income.

Who is losing out here when I’m not even trading?”

There are two parts to understanding why you’re still playing in a zero sum game (leaving aside the caveat just mentioned above).

Firstly, while we talk about ‘the market return’ mostly in terms of indices and share prices, we should really think of it as capital gains or losses plus dividends.

That is, we should think about the total return from the market.

If you own a bunch of high-yield shares [28] instead of (or in addition to) the market, then by definition you own a different weighting of shares to the market.

For you to have this different weighting, some active investors somewhere else (or as discussed above, ‘some money’) must have an underweight position in the same shares.

It doesn’t matter whether you trade the shares after you take this position:

It’s still a zero sum game in terms of beating the market, although by not trading [29] you do reduce the cost of playing it.

Again, this is not to say you should or shouldn’t invest in a portfolio [14] of dividend paying shares.

Each to their own, and I’m rather a fan of income investing myself.

You may beat the market return over the long-term. You may do worse.

But from that perspective it’s still a zero sum game.

Similarly, people say “who cares what the market does, I am just concerned about my own finances.”

Agreed, a very healthy attitude to have towards income investing [30]. But it doesn’t have anything to do with whether active investing is a zero sum game.

I don’t care less about the physics of fluid dynamics when I take a shower, but that doesn’t mean the laws of physics don’t apply when I lather up.

If you’re buying different shares to the index for their dividend income5 [31], then you’re taking a position versus the market, and you’re playing in the zero sum game.

That’s not bad or good – if you know what you’re doing. It’s a calculated risk.

But as Warren Buffett has said:

“If you’ve been playing poker for half an hour and you still don’t know who the patsy is, you’re the patsy.”

Don’t be an ignorant patsy.

  1. I’m ignoring shorting here, although it’s really just a special category of temporary ownership via active investing. [ [34]]
  2. So called synthetic ETFs [35] may track an index without owning the shares, but they can be set aside for the purposes of this article. [ [36]]
  3. Someone pedantic may point out in the comments that tracker funds may move prices when they are forced to buy or sell shares when they enter or leave the market. This is true, but the impact is irrelevant in the context of this discussion. [ [37]]
  4. One will do so just so long as the company returns more or less than the market, as opposed to exactly the same return [ [38]]
  5. Remember that you could sell some shares after they rise to generate an income instead. [ [39]]