Explain in your own words William Sharpe's argument in The Arithmetic of Active Management.
When you compare funds, advisers or robo-advisors, you'll often hear that a provider's managers can beat the market after fees. If you're choosing between two providers, you need to know how much weight to give that claim. You don't need years of performance data to judge it. Arithmetic is enough. The economist William Sharpe set it out in 1991 in "The Arithmetic of Active Management", a short paper in the Financial Analysts Journal.
Arjun ran into this question while comparing robo-advisors. He had narrowed his shortlist from Lesson 4.4 until only one difference between two providers was left to settle. Provider A built its portfolios from index funds. Provider B used active funds and claimed its managers could beat the market after fees. Arjun wanted to know whether he could reasonably expect that, or whether every fund manager says it.
Sharpe's argument works for any market you choose, such as Singapore shares, world shares or US bonds. Take everyone who owns part of that market and put each of them into one of two groups.
A passive investor holds every security in the market in proportion to its size, the way an index fund does. If a company is worth 2% of the market (an illustrative figure), it makes up 2% of what a passive investor owns. An active investor is anyone who is not passive. Active investors hold more of some securities than the market does and less of others, and they leave some out completely, because they think they can choose better than the market average. Active fund managers are active investors. So are people choosing their own stocks, and so are hedge funds. Between them, passive and active investors own the entire market, and there is no third group.
Passive investors hold the market exactly, so before costs they earn exactly the market return. Suppose the market returns 8% before costs in one year (an invented example). Passive investors also earn 8% before costs.
Next, look at the active group as a whole. The average active dollar is the return across all active holdings, weighted by how much each active investor holds. The market is made up of the passive holdings plus the active holdings. If the market earned 8% and the passive part earned 8%, the active part must also have earned 8% before costs. Results inside the active group vary widely. One active investor might make 15% and another 1% (invented examples). Once each is weighted by the size of their holdings, the group averages 8%. This step holds every year and in every market, whatever the market does.
The two groups pay different costs. A passive fund only has to follow an index, so it trades rarely and charges low fees. An active fund pays for research and analysts, trades more often and usually charges higher fees.
Say passive investors pay 0.2% a year in costs and active investors pay 1.5% a year on average (invented examples). The average passive dollar keeps 8% minus 0.2% = 7.8%. The average active dollar keeps 8% minus 1.5% = 6.5%. After costs, the average active dollar earns less than the average passive dollar, and the shortfall is roughly the difference in costs.
This result doesn't depend on studying a particular period or a particular market. It follows from how passive and active are defined, so it holds in a year when the market rises 30% and in a year when it falls 30% (illustrative figures). In either year both groups earn the same before costs, active investors pay more, and the cost gap comes straight off the average active return. Lesson 7.1, "Why a 1% yearly gap becomes a large sum", works out how much that gap adds up to over time.
The full argument runs in five steps:
Choose a market and split all its holders into passive and active investors, who together own all of it. Before costs, passive investors earn the market return because they hold the market exactly. Before costs, the average active dollar also earns the market return, since the two groups together make up the market. Take off costs, which are low for passive investors and higher for active ones. After costs, the average active dollar trails the average passive dollar by roughly the cost difference.
The arithmetic has two limits. The first is that it describes averages. It doesn't say every active fund trails the market, and in any given year plenty of active funds beat the market after costs. The hard part is identifying the winners before the year starts. Last year's winning fund is not reliably next year's winner. A manager's track record is like any past return: it is not a forecast. If someone could pick the winners ahead of time, the arithmetic would not stop them.
This narrows what Arjun needs to ask about Provider B. The arithmetic already shows that active managers as a group can't beat the market on average after costs. What's left to answer is whether he, an adviser or a robo-advisor's managers can pick winners reliably enough to cover the extra cost. Lesson 6.2, "What the SPIVA reports measure and show", looks at how often active funds have actually beaten the market after costs, and whether the funds that won kept winning.
The second limit is that the argument depends on a strict definition of passive. In Sharpe's sense, passive means the whole market held at market weights. An index fund that tracks a narrow index, trades often or charges high fees doesn't meet that definition, and the arithmetic says less about it.
Arjun's friend Wei Ling holds an Asia equity fund through her bank, and the bank had made much the same claim about beating the market. Arjun explained Sharpe's point by shrinking the market to two people. One owns an index fund and the other picks shares, and between them they own the whole market. The market rises 8% (an invented example), so the index investor earns 8% before costs. Because the two of them own everything, the share-picker must also earn 8% before costs on average. After costs, the share-picker ends up behind unless they picked unusually well.
Wei Ling followed the reasoning, then asked how you would know whether your manager had picked well. The next two lessons take up her question. Before that, it is worth putting the arithmetic into your own words, because if you can explain it to a friend you will spot when a sales pitch skips over it.
Explain the arithmetic to a friend in a short paragraph. Use a simple market with only two investors in it.
Junxiong-WFG Organisation is an authorised representative of AIA Financial Advisers Private Limited (Reg. No. 201715016G).