Lesson 3 of 8
Independence, and what breaks it
Multiplying two probabilities assumes they are unrelated. Two of Graham's tests are nearly independent in today's market and two others are not, measured.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- Two events are independent when knowing one tells you nothing about the other. Only then may their probabilities be multiplied.
- Screening criteria are usually not independent, and the error compounds with every extra condition.
- Whether two criteria are related is measurable on the market you are screening, and the answer is sometimes surprising.
Add a second condition to a screen and most people estimate the survivors by multiplying. That is correct only when the two conditions are unrelated, and whether they are is a fact about the market rather than an assumption you get to make.
Two pairs of screening tests, expected against measured
- P/E at most 15 and current ratio at least 2: expected127
- P/E at most 15 and current ratio at least 2: measured132
- P/E at most 15 and P/B at most 1.5: expected174
- P/E at most 15 and P/B at most 1.5: measured327
The multiplication rule, and its condition
If two events are independent, the chance of both is the product of the two chances. A fair coin twice gives a quarter. That is the easy case and it is the one everybody generalises from.
Independence means knowing the first outcome changes nothing about the second. Almost nothing in a set of company accounts satisfies that, because the ratios are computed from overlapping line items and driven by the same business conditions.
A pair that is nearly independent
Of the active US listings holding both figures, 1,484 in all, 315 have a price to earnings ratio at or below 15 and 597 have a current ratio at or above 2. Those are shares of 21.2 percent and 40.2 percent.
Multiply them and you would expect 127 companies to pass both. The measured count is 132. The multiplication is out by four percent, which for a rule of thumb over a whole market is close.
A pair that is not
Now take two tests that both look at price. Of the 1,908 listings holding both, 589 have a P/E at or below 15 and 565 have a price to book at or below 1.5.
Which direction the error runs
Positively related criteria make multiplication understate the survivors. The screen returns more names than expected, and those names are more alike than expected, because the same underlying condition satisfied both tests.
Negatively related criteria do the reverse. Ask for a high growth rate and a low multiple and the two conditions fight each other, so the product overstates the survivors and the screen comes back with almost nothing.
What this means for a portfolio
The dangerous version is not about counting. It is about what happens after the screen. Twelve holdings chosen by related criteria are not twelve independent positions. Whatever made them all pass the same test can stop being true for all of them at once, and the diversification the count suggests is not there.
The check is the one done above, and it takes one query: count how many pass each test, count how many pass both, and compare that with the product. Do it on the market you are actually screening rather than assuming.
Check yourself
When may two probabilities be multiplied together to get the chance of both?
- Whenever both are known
- Only when knowing one outcome tells you nothing about the other
- Only when both are below 50 percent
Show the answer
Answer: B.
That condition is what independence means. Multiplying related probabilities gives an answer that is wrong in a direction set by how they are related.Two screening criteria are positively related. What does multiplying their pass rates do?
- Understates how many companies pass both
- Overstates how many companies pass both
- Gives the right answer, since both are correct individually
Show the answer
Answer: A.
Companies that pass the first are more likely than average to pass the second, so the real overlap is larger than the product. The measured pair here was 327 against an expected 174.Why is a portfolio of twelve names from one screen not twelve independent positions?
- Because twelve is too few
- Because the condition that made them all pass can stop holding for all of them at once
- Because screens only find small companies
Show the answer
Answer: B.
Selection by a shared criterion imports the correlation that criterion carries. The count of holdings overstates the diversification when the holdings were chosen by the same test.Sources
- Counts computed by ValueScreener across the active US equity pool on 13 September 2026: 1,484 listings holding both a positive P/E and a positive current ratio, and 1,908 holding both a positive P/E and a positive P/B.
- Graham, The Intelligent Investor, revised edition 1973, chapter 14: the defensive investor's criteria, from which the three thresholds used here are taken.
What this leaves out
- These are counts on one date over the listings that hold both figures, not over the whole market. Listings missing either figure are excluded from both the pass rates and the overlap, which is the only treatment that keeps the arithmetic consistent.
- Listings are counted, not deduplicated companies, so a company with two listings can appear twice.
- Graham's own tests were stated in the context of his own era and his own definitions. They are used here as two well-known thresholds to measure dependence with, not as a screen anyone should run.
- Nothing here is advice about any security.