The fundamental law: skill times the root of independent bets

Warren Buffett holds a handful of companies and says wide diversification is only for people who do not know what they are doing. Renaissance Technologies holds hundreds of positions, each held briefly, and has done at least as well. Romero and Balch open their chapter on the fundamental law with exactly this contrast, because one short formula explains how both can be right. The earlier post on Sharpe, Sortino and the information ratio defined the quantity the law predicts; this post is about what drives it. The numbers come from the book’s coin-flipping example and from the blog’s five tickers, 2020 to 2025.

Two ways to be good

Romero and Balch paraphrase Grinold’s law in one line before they define anything: performance equals skill times the square root of breadth. Skill is how well a manager turns information about a stock into a correct prediction of its return. Breadth is the number of investment decisions the manager makes in a year. Buffett has a great deal of the first and very little of the second: he studies a few companies deeply and cannot do that for the four thousand on the exchange. A quantitative fund has a modest amount of the first applied thousands of times a day. The law says the two routes lead to the same place, and the square root says which route pays faster.

To double performance at the same skill, you need four times as many opportunities. Doubling the number of decisions improves the result by only about 41 percent.

That is the book’s explanation for why Buffett does not simply buy more companies. Working twice as hard on breadth would buy him 41 percent, and only if his skill held up on companies outside what he calls his circle of competence, which it probably would not. For someone whose skill is already among the highest, raising it further pays better than spreading it thinner.

The coin: same reward, a thirtieth of the risk

The book makes breadth concrete with a coin. Each flip is an even-money bet: call it right and you win your stake, call it wrong and you lose it. A fair coin pays nothing on average however often you flip. Now let the coin land heads 51 percent of the time, and let that small bias stand for skill. One dollar on heads is worth $\good{0.51 \times 1 + 0.49 \times (-1)} = \good{\usd 0.02}$ on average, two cents a flip. You have a thousand dollars. Do you bet it all on one flip, or one dollar on each of a thousand flips?

Table 1 compares the two, and the expected profit is identical: two cents times a thousand dollars either way. The risk is not. One flip is a coin toss on your whole stake, so the profit swings by almost the full thousand. Spread over a thousand flips, the lucky and unlucky ones cancel most of each other out and the swing is about thirty-two dollars. Dividing reward by risk, the thousand small bets are $\key{31.6}$ times better, and that number is no accident: it is the square root of a thousand.

the reward-to-risk ratioexpected profitstandard deviation of the profit

Add n independent bets and the expected profit adds up n times, but the random swings partly cancel, so the standard deviation grows only as the square root of n. The ratio of the two therefore grows as the square root of n (Figure 1). The book notes that with a risk-free rate of zero this ratio is a Sharpe ratio, and this particular version of it is the information ratio.

The three names in the law

With the coin in hand, the book introduces the terms, and warns that their names are not well chosen but are too established to change.

The information ratio is the measure of performance: excess return per period divided by the standard deviation of that excess return. Excess here means the part due to the manager rather than the market. Using the capital asset pricing model from the beta post, a stock’s return is beta times the market’s return plus alpha, and the alpha is the part the market does not explain. So the ratio the book uses is the mean of alpha over its standard deviation. A hedging strategy that removes the market risk leaves exactly this, and the earlier post has a worked example.

The information coefficient is the measure of skill: the correlation between a manager’s predictions and what actually happened. A perfect forecaster scores 1, a perfectly wrong one −1, and a random guesser 0. It only has to be positive on average. Many individual calls can be wrong and the coefficient still above zero, which is why the 51 percent coin counts as skill.

Breadth is the number of trading opportunities over time, and the book shows two ways to raise it: hold more positions at once, or turn the same positions over more often. The first is how a managed portfolio adds breadth, the second how an active trading strategy does.

IR, the information ratioIC, the information coefficient

Reading the formula left to right: performance equals skill times the square root of breadth, the same sentence as the book’s opening paraphrase with the names filled in. Table 2 has a few worked values.

The highlighted row is the Buffett-versus-quant lesson in numbers: four times the breadth at the same skill doubles the ratio, while the row below it gets back to the first row’s 0.50 with double that row’s skill and a quarter of its breadth. The last two rows push it further: a skill of 0.15 applied ten times a year is worth less than a skill of 0.03 applied a thousand times. These are arithmetic on the formula, not measured funds.

Why “independent” is the whole catch

The book defines breadth as the count of trading opportunities, and its coin example only works because every flip is a fresh draw: the cancelling of lucky and unlucky bets that shrank the risk by the square root needs the bets to be unrelated. In a portfolio they rarely are. Table 3 has the correlation of daily returns between the blog’s five tickers over 2020 to 2025.

Goldman Sachs and the index move together three days in four, in the loose sense that their correlation is $\risk{0.75}$; gold moves on its own. Five long positions in these are not five coin flips. On a day the market falls, SPY, NVDA, GS and XOM mostly fall together, so a bet on each is largely the same bet made four times, and only gold adds a genuinely separate draw. The law counts independent decisions, and a hundred positions that all ride one sector view are one decision. The same applies over time: re-deciding the same position every morning on the same information is not a new bet, so turnover only adds breadth when each trade rests on something new.

Breadth is the number of independent bets, not the number of positions or trades. Correlated positions shrink it, and a strategy that looks broad can have a breadth of one.

The book’s other caveat is the trade-off with skill: the broader the portfolio, the less expertise can go into each holding, and since alpha is assumed to come from investment-specific knowledge, breadth bought by spreading attention thinner can lower the information coefficient at the same time. The law holds skill fixed while breadth rises; a manager rarely can.

Traps

Table 4 has the beliefs this post set out to correct.

The code

The snippet reproduces the coin table and the worked values of the law; Table 5 names its variables.

import numpy as np

def coin(n, p_heads=0.51, stake=1000.0):
    bet = stake / n
    mean_one = bet * (p_heads - (1 - p_heads))   # win bet with p, lose bet with 1 - p
    var_one = bet**2 - mean_one**2               # E[x^2] - E[x]^2; x^2 is always bet^2
    return n * mean_one, np.sqrt(n * var_one)    # means add; variances add, so std grows as sqrt(n)

for n in (1, 1000):
    m, s = coin(n)
    print(f"{n:>4} bets: expected ${m:.2f}, std ${s:.2f}, ratio {m / s:.4f}")

for ic, br in ((0.05, 100), (0.05, 400), (0.10, 25), (0.15, 10), (0.03, 1000)):
    print(f"IC {ic:.2f} x sqrt({br:>4}) = IR {ic * np.sqrt(br):.2f}")

It prints the one-bet line as expected $20.00, standard deviation$999.80, ratio 0.0200, the thousand-bet line as $20.00,$31.62, ratio 0.6326, then the five rows of the law’s table. No data is loaded: the whole post is arithmetic.

Summary

Table 6 collects the ideas, with the number behind each.

References

• Romero and Balch, What Hedge Funds Really Do (Business Expert Press, 2014). Chapter 9 for the Buffett and Renaissance contrast, the one-line paraphrase of the law and its “four times as many opportunities” consequence, the two ways to add breadth, the 51 percent coin with its $20 expected profit and the “more than 30 times” reward-to-risk comparison, the definitions of the information ratio (as mean alpha over its standard deviation), the information coefficient and breadth, the full law, and the trade-off between breadth and the manager’s circle of competence. The chapter names Grinold and Kahn, Active Portfolio Management, as the source of the law.