Is investing a zero-sum game?

If you win at blackjack in a casino, you take someone else’s money. If you lose, someone else takes your money. There’s no wealth being created or destroyed, it’s just the same cash being redistributed between the players.

That’s a zero-sum game – more for me means less for you.

Other examples of zero-sum games include games such as poker, and other multiplayer games of chance, since the sum of the amounts won by some players equals the combined losses of the others. Games like chess and tennis, where there is one winner and one loser, are also zero-sum games.

Is investing a zero-sum game?

To understand whether investing is a zero-sum game, a useful place to start is a concept known as Sharpe’s Arithmetic of Active Management.

Sharpe’s arithmetic begins with the premise that the equity market is owned by two kinds of investors: a) market cap indexers, who own every equity in the market in proportion to its market capitalisation (known as “passive investors”), and, b) active investors, each of whom owns a portfolio of equities which they believe will outperform.

We can see that while each active manager will own a portfolio that diverges from market cap weights, combining all the portfolios of all the passive managers and all the active managers in aggregate is, by its very definition, the entire market. If you combine the portfolios of every single investor, you get the market.

It then follows that if the market represents the combined portfolios of all active and passive investors, then if the returns of the passive investor and the returns of the market are the same (which they always will be, before costs), then the returns for the active investors, in aggregate, must equal the returns of the passive investors, before costs.

As William Sharpe explained in his seminal 1991 note (found here – very short and definitely worth reading in full):

‘Each passive manager will obtain precisely the market return, before costs. From this, it follows (as the night from the day) that the return on the average actively managed dollar must equal the market return. Why? Because the market return must equal a weighted average of the returns on the passive and active segments of the market. If the first two returns are the same, the third must be also.’

Both the portfolio representing the aggregate of all active managers and the portfolio of index-trackers are the same capitalisation weighted ‘market’ portfolio. Therefore, the returns to active investors, on average, and the returns to passive investors must be the same, before costs.

For every 1% of outperformance against an index, there must be 1% of underperformance somewhere.

Using the FTSE 100 as an example, for every 1% of outperformance vs the FTSE there must be 1% of underperformance somewhere. That’s how we end up with ‘market performance’ – it’s the combined performance of all active and passive strategies in the market. So 1% of outperformance for me means 1% of underperformance for someone else. That’s zero sum.

However, that’s only true in a world with no transaction costs or management fees.

In reality, the higher fees charged for active investment management means that, on average, those paying for active management must underperform those paying lower fees for passive management. After all, they’re both receiving exactly the same return before fees, on average.

Once costs are factored in, passive investors must come out ahead of the average active investor, due to the active investors paying higher fees. This is one of the reasons why passive funds have so consistently outperformed active funds over the long run.

Source: Vanguard

The chart below shows how Sharpe’s arithmetic manifests itself in the real world. Historic performance has been consistent with the theory, presenting as a bell curve of both positive and negative manager excess returns, centred on a value of slightly below zero.

Source: Vanguard

Relative to an index then, active investing is only a zero-sum game in a fictional world with no transaction costs. In the real world, due to transaction fees and management fees, active management is a negative-sum game. And it’s more negative-sum the higher fees active investors as an aggregate are paying.

This is why Charles Ellis famously called active management a ‘loser’s game’, and one of the reasons I personally prefer to take a passive approach to investing.

It should be noted that passive management is also negative-sum, as passive investors will still be paying a fee for market performance. But given the fee for passive management is far lower than for active management (the fees are now so low to be almost negligible), a passive approach is much less negative-sum than active, and is approaching zero-sum.

Relative versus absolute

Active investing is negative-sum when viewed relative to an index. 1% of outperformance versus an index means there must be 1% of underperformance versus the index somewhere else.  

But even a fund which underperforms on a relative basis can still be generating value on an absolute basis for the investor. It just might not be as much as they could’ve generated by holding a passive fund, or one which outperformed the index.

If one fund generates a return of 15% versus the FTSE’s 10%, that means there will be 5% of underperformance somewhere else in the market. But all active investors, including those underperforming the FTSE, will still be generating positive returns.

Because the market rises over the long run, all investors are likely to generate positive returns if they have a long enough time horizon and are able to stay invested. The market creates wealth, which means that unlike in traditional zero-sum games where the amount of money being allocated between participants is fixed, the fact that markets rise over time means that wealth is being generated for all participants. The size of the pot is effectively increasing over time.

People may be receiving different shares of the pot in the form of different levels of return, but the rising tide of positive market returns means all investors are highly likely to generate positive returns over the long term. The longer investors can stay invested, the more positive the returns are likely to be, and the more positive-sum investing becomes.

Fees will still act as a drag on returns, and reduce the degree to which markets are positive-sum (remember, fees are what caused active investing to be a negative-sum game, rather than just zero-sum). So the less investors as a whole pay in fees, again, the more positive-sum investing becomes.

Another simple way of demonstrating that investing is positive-sum is to think about what would happen if investing was zero-sum.

If the only way to make gains in the stock market was for someone else to take a loss, then the stock market wouldn’t be able to go up. For every person that made money, there’d have to be one that lost money. They would cancel each other out, and the market would remain flat. That’s clearly not the case, as we all know that stocks go up in the long run. So that mere fact that markets rise shows that investing can’t be zero-sum.

Even those investors who underperform on a relative basis are highly likely to generate wealth on an absolute basis over time.

On a relative basis, active investing is zero-sum (technically, negative-sum). But on an absolute basis, it’s not. Investing becomes more positive-sum the longer investors are able to stay invested, and the less they have to pay in fees.

Buyers and sellers

We’ve seen that markets are positive sum for those able to buy and hold investments for the long term. But where things can get a little confusing is at the transactional level.

When we’re thinking about markets in terms of buyers and sellers transacting in stocks/funds/bonds/etc, every buyer requires a seller. Some market participants are buying, some are selling, but at the end of the day because every buyer requires a seller, the transactions all net to zero.

At the transactional level, the market does appear to be zero-sum. If we exclude share issues, buybacks, and other corporate actions, then on any given day there are a fixed number of shares in existence. If I buy a share, that means there are now fewer shares of that company for you to buy. A fixed number of shares are being redistributed amongst the various buyers and sellers. That sounds pretty zero-sum to me (again, negative-sum after transaction costs).

But this ignores the motivations behind each transaction.

If I buy a share from you, you’ll use that cash for another purpose. You might buy a bond fund to better align your portfolio with your risk tolerance. You might leave it in cash to meet a future expense you have to pay. You might buy a share in a different company, or buy and index tracker fund. You might just want to spend the cash.

In all instances apart from you spending the cash, the proceeds from the sale are being reinvested into something with a different set of risk/return characteristics. The redistribution of stocks, funds, bonds, etc between the market participants is really just a redistribution of expected returns. By buying a share from you, I’m trading my cash, which has a low expected return, for a stock with a high expected return. If you use the proceeds from selling your stock to buy a bond fund, you’re trading a high expected return for a low expected return (and lower risk). I want to buy the share because I believe it enhances my portfolio’s risk/return characteristics. You’re buying a bond fund for exactly the same reason.

We’re both gaining value from the transaction, as it improves both our portfolios. We just have different objectives.  

If you don’t decide to reinvest the proceeds from your sale of the stock to me, and instead decide to spend it, then you’re also gaining value. You’re swapping the future value generated by the share’s future price appreciation for value today – in effect saying that the value you can create from spending the cash today is higher than the present value of the future cash created by owning the share.

Broadening this out to the whole market, although all transactions net to zero in aggregate, each transaction is creating value for both sides. No two investors are investing for the same reason, or hold the same portfolios. Each transaction creates value as it better aligns the two counterparty’s portfolios with their objectives.

So although the market is zero-sum at the transactional level in terms of the quantities of securities traded, each transaction creates value for both sides. Because both parties benefit from each trade, markets are also positive-sum at the transactional level.

Summary

• On a relative basis for actively managed investments, investing isn’t just zero-sum, it’s negative-sum due to the high fees paid.
• But on an absolute basis, investing is a positive-sum game. Even underperforming active funds can still generate value.
• The longer investors are able to stay invested, the more positive-sum investing becomes.
• Similarly, the less investors pay in fees, the more positive-sum investing becomes.
• As long as the market continues to rise over time, investing will remain a positive-sum game.
• Although all transactions require a buyer and a seller, because both parties gain value from the transaction this is also a positive-sum game.
• Investing is therefore a positive sum game both in the act of holding investments, and also in the act of transacting in them.