Can Anyone Beat the Stock Market?

A rise of 50% multiplies the money by 1.5; a fall of 50% multiplies it by 0.5. Each year’s change is a growth factor: a change of r percent multiplies the money by $1 + \frac{r}{100}$, which is less than 1 for a fall. Two years in a row multiply:

$$1{,}00{,}000 \times 0.5 \times 1.5 = 1{,}00{,}000 \times 0.75 = 75{,}000.$$

This is the compound-interest idea of Grade 10, with one change: in a bank the rate is the same every year, and on a share it is different every year. Multiplication doesn’t care about order, so a fall then a rise lands in the same place as the rise then the fall.

The trap is that percentages up and percentages down are not opposites. To undo a fall, the price must rise by more than it fell, because it is rising from a lower floor. If a price falls by a fraction d, it is left at 1 − d of itself, and getting back needs a factor of $\frac{1}{1-d}$:

Fall	What’s left	Rise needed to get back
10%	0.90	11.1%
20%	0.80	25%
25%	0.75	33.3%
35%	0.65	53.8%
50%	0.50	100%
75%	0.25	300%
90%	0.10	900%

Small falls are nearly undone by equal rises, but big falls need enormous ones. A share that loses 90% has to multiply by ten just to get back.

Suman might say: “My average return was zero. +50% and −50% average to 0%.” He has added the two percentages and halved them. That is the arithmetic mean return, and here it says nothing useful: his money fell by a quarter.

What would tell the truth is the steady yearly rate that gives the same result. Two years of the same factor g must make 0.75, so $g^2 = 0.75$ and

$$g = \sqrt{0.75} \approx 0.866.$$

So Suman’s money did exactly what it would have done falling 13.4% every year. A steady factor found this way is the geometric mean of the factors, and the yearly rate it gives is called the compound annual growth rate. For money, it is the only average that matters.

Is it bad luck that up-and-down lost money, or does it always? Figure 42.1 answers with a picture. Rise by a fraction a, then fall by a (or the other way round): Rs. 1 becomes (1 + a)(1 − a). That is the area of a rectangle 1 + a wide and 1 − a tall. Slice off a strip a wide from its right end; the strip is a by 1 − a. Turn it and lay it along the top of what is left, a rectangle 1 wide and 1 − a tall. The two pieces fill a 1 × 1 square except for one small corner, a by a. So

$$(1 + a)(1 - a) = 1 - a^2,$$

always less than 1. With a = 0.5, the missing corner is 0.25: Suman’s quarter. With a = 0.2, it is 0.04: the 4% of the try box. The bigger the swings, the bigger the corner, and it grows as the square of the swing.

The more the years differ, the more the arithmetic mean flatters (question f proves it for any two years). This gap is often called volatility drag: for small swings it costs about half the square of the swing each year (0.2² ÷ 2 = 2%, against the exact 2.02% for a steady ±20%).

Now Suman’s second claim, that his share is “only going up”. People who study charts of prices look for trends, runs and shapes and use them to guess what comes next. Try it yourself: here is a share’s price for the last 40 trading days. Will it be up or down tomorrow?

Here is the secret: this share is made by tossing a coin. Each day, heads multiplies the price by 1.02 and tails by 0.98. The history you studied, with its runs, its climbs and its slides, came from coin tosses, and so did every day you guessed. No method can beat 50% on a fair coin in the long run, so your score is luck: after 20 guesses, 96 people in 100 score between 30% and 70% just by guessing (fewer than 6 right, or more than 14, happens about 2% of the time each).

A path made by adding a random step at each turn is a random walk. The coin share is one, and the surprise is how un-random it looks. In 250 coin tosses, about a year of trading days, the longest run of tosses landing the same way is about 8 on average, and a run of at least 7 turns up in 87% of years. A run of seven “up” days looks like a trend to anyone, but it means nothing about the eighth day.

How far does a random walk wander? After n steps of size 1, it is typically about $\sqrt{n}$ steps from its start. With 250 daily steps of 1%, $\sqrt{250} \approx 16$: a coin share usually ends a year somewhere within about 16% either side of its start, and in 95 years out of 100 between about −27% and +33%.

Why real prices behave like coins

Why should real prices look like a coin? Suppose a pattern really did predict that a share would rise tomorrow. Then everyone who saw the pattern would buy it today, and their buying would push today’s price up until the rise was already in it. A pattern that can be seen tends to be used up by the people trading on it. What is left to move the price is news nobody knew yet, and news, by its nature, can’t be guessed from yesterday’s chart. This is the random walk hypothesis: past prices don’t tell you which way the next one goes.

Suman’s friend has been right eleven times in a row. Surely that can’t be luck?

Try a contest. 1,024 tipsters each say, every week, whether the market will go up or down, and each one decides by tossing a coin. Watch how many keep a perfect record.

Each week, about half of the perfect records end, so after w weeks about $1024 \div 2^w$ are left: 512, 256, 128, … and after 10 weeks, about 1. That last tipster has a record that looks like genius, ten right out of ten, a chance of 1 in 1,024 for any one tipster. But with 1,024 people tossing, someone is expected to get there, and the chance that at least one does is about 63%. And when you ask the champion for next week’s tip, the coin is still a coin: the chance is still 1 in 2.

You never hear about the 1,023 who missed. Groups that stopped being right go quiet; the one still right gets the followers. Judging by the survivors alone, while those who dropped out are invisible, is survivorship bias.

The same bias hides in tables of funds, which pool many people’s money and choose shares for it. Funds that do badly are often closed or merged into others and vanish from the lists of past results. S&P Dow Jones Indices, which keeps careful score in the United States, counts the funds that were closed too. Of 774 US funds investing in large companies at the start of 2005, only about a third, 255, still existed at the end of 2024. And over those 20 years, 92% of all of them did worse than the S&P 500, the simple list of 500 big companies they were trying to beat (as of the report for 2024).

So nobody can see tomorrow. Is there anything maths can do for an investor? Yes: it can show how to lose less to bad luck.

Imagine a coin company. Each year, a coin decides whether its share rises 60% or falls 40%. The arithmetic mean return is $\frac{60 - 40}{2} = 10\%$ a year, which sounds wonderful. But a good year and a bad year multiply to 1.6 × 0.6 = 0.96: the compound rate is $\sqrt{0.96} - 1 \approx -2\%$ a year. Over 20 years, a typical investor gets 10 good years and 10 bad, and Rs. 1,00,000 becomes $1{,}00{,}000 \times 0.96^{10} \approx$ Rs. 66,483. A few lucky ones become extremely rich, enough to make the average of everyone’s money 1.1²⁰ times the start, Rs. 6,72,750. The average investor is rich; the typical investor has lost a third. (Grades 6–8’s “What Does an Average Hide?” meets this gap from another side.)

Now hold two coin companies whose coins are tossed separately, with half the money in each, and split the money half and half again at the start of each year. Each year one of four things happens, each with chance 1 in 4: both rise (× 1.6), both fall (× 0.6), or one rises and one falls (half at 1.6 and half at 0.6, × 1.1 in all), the last in two ways. Over four typical years:

$$1.6 \times 0.6 \times 1.1 \times 1.1 = 1.1616,$$

a compound rate of about +3.8% a year instead of −2%. Nothing about either company changed. Spreading the money made the swings smaller, and the up-and-down loss of Figure 42.1 grows with the square of the swings. Spreading money over many investments is diversification.

With many companies, each on its own, the lines bunch together, and the typical investor gets close to the 10% a year that a single company only promised: Rs. 1,00,000 typically grows to Rs. 5 to 6 lakh with 10 companies and about Rs. 6.6 lakh with 100.

Now the twist. In a real market, companies don’t toss separate coins. When the whole market falls, as NEPSE did in 2021–22, nearly every share falls with it. Switch the widget to partly together: each company now follows a coin shared by all of them half the time. More companies still help, but only up to a point: even with 100, about a tenth of the investors lose money, and the typical result stays near Rs. 3.8 lakh. Diversification removes each company’s own luck. It cannot remove the luck everyone shares, the market risk.

One more number every investor meets is the only one they control: costs. As of 2026, the commission in Nepal is 0.36% on a trade between Rs. 2,501 and Rs. 50,000, SEBON adds 0.015%, and selling costs a fixed Rs. 25 more for the depository. Buy Rs. 20,000 of shares and sell them at the same price: Rs. 75 to buy and Rs. 100 to sell, Rs. 175, or 0.875% of the money, before any tax on profit.

That seems small. But costs compound, just as gains do, and they never have a bad year. Someone who buys and sells their whole Rs. 20,000 once a month keeps $(1 - 0.00875)^{12} \approx 0.90$ of it a year from costs alone: they must beat the market by about 11% a year (1 ÷ 0.90 ≈ 1.11) just to stand still against someone who bought once and waited. Trade every week and the factor is $0.99125^{52} \approx 0.63$: over a third gone in a year.

Yearly fees do the same thing slowly. A fund that charges a fee each year takes it from the growth. Suppose the shares themselves grow 10% a year for 30 years (our own example, not a promise):

Kept each year after fees	Rs. 1,00,000 after 30 years
10% (no fee)	Rs. 17,44,940
9% (a 1% fee)	Rs. 13,26,768
8% (a 2% fee)	Rs. 10,06,266

A “small” 1% fee takes almost a quarter of the final amount (1.09³⁰ ÷ 1.10³⁰ = 0.76), and 2% takes over 40%. The fee is charged on everything, the gains of earlier years included, so its share of the end grows every year.

Why the average investor can’t beat the market

The economist William Sharpe put the last piece in place in 1991, in a short paper called “The Arithmetic of Active Management”, with nothing but adding up. Split all the money in a market into two kinds: passive money, which simply owns a little of every share in proportion (and so earns exactly the market’s return), and active money, which picks and chooses, hoping to beat it. Together the two own the whole market, so together they earn its return. Take away the passive part, which earns exactly that, and the active part, taken all together, must earn it too: for every active rupee that beat the market, other active rupees did worse by the same amount.

That is before costs. Active money trades more and pays more for advice, so Sharpe’s two statements are: before costs, “the return on the average actively managed dollar will equal the return on the average passively managed dollar”, and after costs, it “will be less”. It isn’t a fact about markets that might change one day; it is arithmetic.

In any year many people do; with enough people trying, some must, by luck alone. What the maths on this page says is how to read their claims:

• A record means little until you know how many tried, and how many dropped out of sight.
• A chart’s patterns predict nothing by themselves: coin tosses make them too.
• Up and down loses, by the square of the swing, so the arithmetic mean flatters.
• Spreading money cuts each company’s own luck, though not the market’s.
• Costs are certain and compound, and on average everyone’s picking earns the market’s return before costs, and less after.

The practical side, what a share is, how an account works, the IPO lottery, what trades cost, and why never to borrow for shares, is in Personal Finance 101 on this site.

• What Is a Game Worth on Average? Expected value: why a lottery ticket is worth less than it costs, and what “on average” promises.
• What Is e, and Why Does Growth Keep Finding It? Growth compounded more and more often, and the number it settles on.
• Can 1,000 People Speak for 3 Crore? Why spread shrinks like $\frac{1}{\sqrt{n}}$, the same square root as the random walk’s.
• What Does an Average Hide? (Grades 6–8) When the average and the typical are far apart.
• How Far Does a Wandering Ant Get? (Computing, Going Further 6–8) Random walks, simulated and measured.
• Grade 10 Math: Compound interest Growth factors with the same rate every year.