William Sharpe and John Bogle: the arithmetic of costs and the case for index funds
Two men made the argument that most investors are better off owning the whole market at a low cost. The logic, its limits, and how much of six famous portfolios is in index funds.
Who they were
William Sharpe (born 1934) is an economist who built on Markowitz's work. He developed the capital asset pricing model in the 1960s, invented the ratio that bears his name, which measures the return earned for each unit of risk, and shared the Nobel Prize in 1990. John Bogle (1929–2019) was a practitioner: he founded the Vanguard Group in 1975, and in 1976 launched the first index fund for ordinary investors.
They worked in different ways, but they arrived at the same message: for most people, the best strategy is to own the market itself and to pay as little as possible for it.
The core ideas
Sharpe's argument, set out in a short 1991 article called The Arithmetic of Active Management, is almost an accounting identity. Before costs, the average return of all investors who hold the market actively is equal to the return of the market, because together they are the market. Active investors collectively cannot beat it. After costs, they must trail it by the amount of those costs, and the passive investors who pay almost nothing must beat the average active investor.
Bogle added the practical side. Costs are the one thing that an investor can control, and they come out of the return every year, compounding against you over decades. He argued for broad, low-cost funds that simply hold what the market holds, and for a long horizon, and he repeated that investors who buy and sell often do worse than the funds they invest in.
How it works in practice
A broad index fund does not try to pick winners. It holds all the stocks in an index, weighted by size, and charges a small fee. The investor gets the return of the market minus a tiny cost, with a very high degree of diversification. A portfolio can be built from two or three such funds, such as a fund of US stocks, one of foreign stocks and one of bonds, and rebalanced once a year.
The argument does not say that no one can beat the market. It says that it is difficult to know in advance who will, that the winners of one period are often not the winners of the next, and that the costs of trying are certain while the benefits are not.
What six current portfolios show
The filings give a surprising confirmation. The largest hedge fund in the world, Bridgewater, holds about 30% of its reported stock portfolio in index and sector funds: the SPDR S&P 500 ETF alone is 16.3% and the iShares funds add 10.2% and 2.0%. Even Duquesne has funds and calls on funds among its largest positions, although calls are valued in the report at the underlying shares, not at the money paid for them.
Berkshire Hathaway reports no funds at all, but Buffett has said publicly many times, and written in his letters, that for most investors a low-cost fund of the S&P 500 is the right choice, and that this is what he has arranged for the money he leaves to his wife. The four concentrated stock pickers in the table, by contrast, hold no funds at all, and each of them is betting that selection beats the market after costs.
Six real portfolios side by side, from the latest filing of each investor (Form 13F, US-listed stocks only). Effective positions is the number of equal-sized positions that would be as concentrated as the real portfolio: one divided by the sum of the squared shares. Opened or closed is the average over the four quarters of half the combined share of positions that were opened or closed.
| Investor | Reported stocks | Positions | Effective positions | Largest position | Ten largest | Opened or closed per quarter |
|---|---|---|---|---|---|---|
| Warren Buffett | $299.3 billion | 26 | 8.1 | 22.0% | 91.0% | 1.0% |
| Ray Dalio | $24.4 billion | 990 | 23.9 | 16.3% | 42.1% | 5.8% |
| Bill Ackman | $13.7 billion | 10 | 6.9 | 17.6% | 100.0% | 7.0% |
| Stanley Druckenmiller | $5.2 billion | 94 | 22.2 | 16.6% | 46.3% | 23.8% |
| Seth Klarman | $5.4 billion | 23 | 13.4 | 16.5% | 74.9% | 9.5% |
| Mohnish Pabrai | $0.3 billion | 4 | 2.9 | 43.3% | 100.0% | 4.4% |
Where it is challenged
Critics make several points. Index funds are not risk-free: in a crash the whole market falls, and a person who is fully invested in 2008 or 2022 lost a lot. A market-weighted fund has most of its money in the biggest companies, so when a few large technology stocks dominate an index, the investor owns a concentrated portfolio without choosing it. Some argue that if everyone indexed, nobody would do the research that sets prices, although this point is far from reality: active trading is still a large part of the market.
The arithmetic also compares an average active investor with an average passive one. A particular manager can beat the market for years, and some do; the difficult part is to find them beforehand.
What to take from it
If you do not want to study companies, buy broad funds with very low fees, keep them for a long time and add to them regularly. Check the cost of everything you own, because a difference of one percent a year turns into a very large amount over thirty years. Do not confuse a good year with skill. And if you choose to be active, treat it as a smaller part of your money and be honest about the odds.
This is a description of an argument, not advice for any person's situation.
Where it meets our bots
Our reference bot is the Sharpe and Bogle argument in its purest form: it bought a fund of the 500 largest US companies once and does nothing else, and every other stock bot is measured against it. Many of the other bots are the active alternative, charging themselves a realistic trading cost on each trade, so you can watch in public whether the activity pays. The 60/40 bot is the classic two-fund portfolio that the index fund investors recommend.
All accounts are virtual and every trade is public. The guides below explain the rules.
Related strategy guides
Buy and hold the S&P 500: the yardstick for every strategy
The 60/40 portfolio: stocks and bonds
Sector rotation: the three strongest US sectors
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