You will be able to explain William Sharpe's argument in The Arithmetic of Active Management in your own words.
Arjun's robo-advisor shortlist from lesson 4.4 came down to one difference he could not settle. Provider A used index funds. Provider B used active funds and said its managers could beat the market after fees. Wei Ling's bank had said much the same about her Asia equity fund. Arjun wanted to know whether that was a reasonable thing to expect, or something every fund manager says.
There is an answer, and it does not need any data. It needs only arithmetic, set out in 1991 by the economist William Sharpe in a short paper called The Arithmetic of Active Management, published in the Financial Analysts Journal. This lesson walks through his argument.
Pick any market: Singapore shares, world shares, US bonds. Everyone who holds that market can be put in 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 makes up 2% of the market's value, it makes up 2% of a passive investor's holdings.
An active investor is anyone else. They hold some securities more heavily than the market does and some less, or leave some out, because they think they can pick better than the market average. Active fund managers, stock-picking individuals and hedge funds are all active investors in this sense.
Between them, the two groups own the entire market. There is no one else.
Now take one year. Suppose the market returns 8% before costs, an example figure.
Passive investors hold the market exactly, so before costs they earn exactly the market return: 8%.
The market as a whole earned 8%, and passive investors, holding a slice of it in exact proportion, earned 8%. The rest of the market is held by active investors, so what they hold, taken all together, must also have earned 8%. Some active investors earned 15% and some earned 1%, but the average active dollar, weighted by how much each holds, earned 8%. It has to, because the active and passive holdings add up to the market.
That is the first step of the argument, and it holds every year, in every market, whatever happens.
Costs are different for the two groups. Passive funds charge low fees and trade rarely, because they only need to follow an index. Active funds pay for research and analysts, trade more often, and usually charge higher fees.
Suppose passive investors pay 0.2% a year in costs and active investors, on average, pay 1.5%, both example figures. After costs, the average passive dollar earns 8% minus 0.2%, which is 7.8%. The average active dollar earns 8% minus 1.5%, which is 6.5%.
So after costs, the average actively managed dollar must earn less than the average passively managed dollar, by roughly the difference in costs. This is not a finding from any one period or market. It follows from the definitions. It would be true in a year when shares rose 30% and in a year when they fell 30%, because both groups earn the same before costs and the active group pays more.
This is the second step, and it is why lesson 7.1, Why a 1% yearly gap becomes a large sum, matters so much: the cost gap comes straight off the average active return, year after year.
Here is what the arithmetic does not say. It does not say every active fund trails the market. Around the average, some active funds do better and some do worse, sometimes by a lot. In any given year, plenty of active funds beat the market after costs.
The hard part is knowing in advance which ones. A fund that beat the market last year is not reliably the one that beats it next year, and a manager's past results, like any past returns, are not a forecast. If you could pick the winners in advance, the arithmetic would not stop you. The question is whether you, or your adviser, or your robo-advisor's managers, can do that reliably enough to cover the extra cost. Lesson 6.2, What the SPIVA reports measure and show, looks at how often active funds have managed it in practice, and whether the winners stayed winners.
The argument also rests on an exact definition of passive: holding the whole market at market weights. An index fund that tracks a narrow index, trades often, or charges high fees is not passive in Sharpe's sense, and the arithmetic says less about it.
When Arjun put this to Wei Ling, he used two investors and a simple market. If the market holds two investors, one holding an index fund and one picking shares, and the market rises 8%, the index investor earns 8% before costs. Together they own the whole market, so the stock-picker must also earn 8% before costs, on average. After costs, the stock-picker earns less, unless they picked unusually well.
Wei Ling's reply was the right question: so how do you know whether yours picked well? That is where the next two lessons go. 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.
Write a short paragraph explaining the arithmetic to a friend, using a simple market made of two investors.
Junxiong-WFG Organisation is an authorised representative of AIA Financial Advisers Private Limited (Reg. No. 201715016G).