Trapped in a Negative-Sum Game?

The debate between active and passive investment management [1] is one of the most well-trodden discussions in the field of finance. Active management advocates argue that skilled fund managers can outperform market indices through expert analysis and trading strategies. However, a growing body of empirical research challenges this premise. Studies consistently reveal that passive strategies, such as index funds, offer superior long-term returns when considering the effects of fees, transaction costs, and market inefficiencies.

This article delves deeper into the mechanisms causing that make active management less effective, focusing on the economic and statistical evidence that suggests active strategies often underperform after accounting for the associated costs. In doing so, the paper highlights key concepts such as management fees, zero-sum and negative-sum games, and the role of luck in outperformance. We also highlight the lack of persistence in active managers’ ability to sustain outperformance (alpha) over extended periods.

Active Management: Theory and Reality

Active management is predicated on the belief that fund managers can consistently identify mispriced securities, leverage market inefficiencies , and time market movements to beat the broader market. In theory, this outperformance would justify the higher fees and transaction costs that are characteristic of actively managed funds. However, this assumption overlooks several key factors:

— High Management Fees: Active funds generally charge higher management fees (0.5% to 2% annually) compared to passive funds (0.1% - 0.5%), significantly eroding returns. Additionally,

— Transaction Costs: Active managers trade frequently if they perceive mispricings or to capitalize on short-term trends. Each trade incurs costs like commissions and the bid-ask spreads. The costs due to higher turnover rates of active funds (0.5% - 1% per year) further eat into potential returns.  

— Market Efficiency: According to the efficient market hypothesis (EMH), markets are informationally efficient, meaning that prices already reflect all available information. In such an environment, outperforming the market consistently becomes exceedingly difficult for active managers[2]. Even assuming that active managers possess superior insight, their ability to capture mispricing is limited by transaction costs and competition from other market participants.

The Negative Sum Game Argument

In the world of active management, the term "negative sum game" refers to the concept that due to the high fees and transaction costs, even if some active managers outperform the market, the overall collective performance of active managers will be lower than the market’s return. The net aggregate returns of active managers are lower than those generated by passive strategies. The aggregate gross return of active managers must equal the market return. Taking fees and costs leads to an overall underperformance.

This argument is not new; over 30 years ago, Sharpe [3] , in “The Arithmetic of Active Management”, stated that active management is a zero-sum game for gross returns and a negative sum game for net returns. In aggregate, all investors are invested in the whole market. Passive investors will earn the market return. Active managers invest in subsets of the market, with the aim of picking the winners. But in aggregate, active managers will also hold the market: If an active manager is underweight a stock, there must be another who is overweight. Since active managers as group must hold the market, in aggregate, they will also earn gross market returns. Each particular active manager’s gain will be another’s loss. If you take active managers’ fees into consideration, active managers’ net aggregate returns will be lower than passive managers, hence the negative sum game.

A Simplified Illustration

To illustrate a negative-sum game, consider a simplified market example involving only two assets, Alpha and Beta, with the following initial characteristics:

Asset	Initial Market Value ($M)	Initial Market Weight (%)	Annual Performance (%)
Alpha	50	50	20
Beta	50	50	0

The total market value is $100M, and the initial market weights of Alpha and Beta are 50% each. The example explores how passive and active investors perform over the course of a year.

Market Performance After One Year

The total market return is determined by the weighted performance of Alpha and Beta. After one year:

• Alpha’s new value: $50M × (1 + 20%) = $60M

• Beta’s new value: $50M × (1 + 0%) = $50M

• Total market value: $60M + $50M = $110M

• Market return: 110/100−1=10%

Passive Investor's Performance

Passive investors, who hold assets in proportion to the market weights (50% in Alpha and 50% in Beta), earn exactly the market return of 10%. For example, a passive investor with $10 million would see their portfolio grow to $11 million.

Active Investors' Performance

Active investors, on the other hand, allocate their funds differently in an attempt to outperform. Let’s assume there are two active investors:

1. Investor A allocates 70% of their portfolio to Alpha and 30% to Beta.

2. Investor B allocates 30% to Alpha and 70% to Beta.

After one year:

• Investor A’s return: 70%×20%+30%×0%=14%

• Investor B’s return: 30%×20%+70%×0%=6%

The Zero-Sum Game

While Investor A outperforms the market, Investor B underperforms. The average return for the two active investors is: (14%+6%)/2=10%. This matches the market return, demonstrating that active investing is a zero-sum game before costs. For active investors who outperforms, another group underperforms by an equal amount.

The Negative-Sum Game

Now, consider the effect of fees. Assume both active investors pay a 1% annual fee for management and trading costs.

• Investor A’s net return: 14%−1%=13%

• Investor B’s net return: 6%−1%=5%

The average net return for the two active investors is: (13%+5%)/2=9% This is lower than the 10% return achieved by passive investors. The inclusion of fees shifts active investing from a zero-sum game to a negative-sum game. While some active investors may still outperform after fees, the collective performance of all active participants falls short of the market.

This example illustrates the inherent challenges of active investing and underscores the advantages of cost-efficient passive strategies for capturing market returns over time.

Empirical Evidence

Such a line of thinking isn't popular among active managers and financial institutions selling their various products. They will all claim that due to their sophistication and significant resources, they are able to beat the market. Some investors too, would feel better thinking they have access to that special manager.

No matter the argument, it has to stand the scrutiny of empirical validation. The SPIVA reports, published by S&P Dow Jones Indices, provide semiannual analyses comparing the performance of actively managed funds to their respective benchmark indices across various regions and asset classes. These reports offer valuable insights into the effectiveness of active management strategies. They regularly find that the United States, approximately 85% of active managers fail to outperform their benchmark indices over a 10-year period. The performance numbers are similar for global funds.

Morningstar finds similar results, 85% of active funds failed to outperform the S&P500 over a 10-year period. It was found that active funds underperformed passive funds by an average of 0.5% per year for the same period. After factoring in fees and transaction costs, the underperformance increased by 1% to 2% annually.

Conclusion

In a Zero-Sum Game, the total gains and losses of all active investors balance out to match the market’s return before fees. Passive investors, by holding a market-weighted portfolio, earn the full market return without the variability of active bets.

In a Negative-Sum Game, fees are added and the collective returns of active investors fall below the market return, making active management a losing proposition for the majority over time. Passive investors, with lower fees, retain the full market return, making them more likely to outperform active investors in aggregate. 

[1] Passive management refers to an investment strategy that aims to match the performance of a market index (like the S&P 500) rather than outperform it. It involves minimal buying and selling of securities, typically through index funds or exchange-traded funds (ETFs), and focuses on long-term growth with lower fees.

In contrast, active management involves a more hands-on approach, where portfolio managers actively make decisions about buying and selling investments with the goal of outperforming the market. This strategy requires more frequent trading and often incurs higher fees due to the active involvement of fund managers.

Passive management seeks to replicate the market’s performance, while active management aims to beat the market, though with higher costs and more risk.

[2] Arguably, active managers may have the potential to outperform in emerging markets because these are seen as less efficient due to factors such as lower liquidity, less transparency, higher volatility, political instability, and underdeveloped financial systems. But, like developed markets, the zero or and negative-sum games arguments remain valid and the ability to consistently outperform remains a debated issue.

[3] Sharpe, W. F. (1991). “The Arithmetic of Active Management,”, Financial Analyst Journal 47, 7-9.