Lesson 2 of 8
Expected value
The probability-weighted average of every outcome. It is the right number to decide on and it is almost never the number that actually happens.
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
- Expected value multiplies each outcome by its probability and adds the results.
- It is a weighted average, so it usually equals no outcome that can actually occur.
- A positive expected value does not mean a good bet if losing is unrecoverable, because the arithmetic assumes you get to repeat it.
An expected value is the average outcome if a situation could be run many times. It is the cleanest tool in decision-making under uncertainty and it comes with a condition most people skip: that many times is available.
Three outcomes and the number none of them is
- Poor case, 25 percent60
- Central case, 55 percent100
- Good case, 20 percent175
- Expected value105
The arithmetic
Multiply each outcome by its probability. Add. That is the whole calculation.
Three scenarios for one holding
- Poor case: value 60, probability 0.25. Contribution 15.
- Central case: value 100, probability 0.55. Contribution 55.
- Good case: value 175, probability 0.20. Contribution 35.
- Check the probabilities add to 1.00. They do.
- Add the contributions: 105.
Step four is the one that gets skipped and it is the only structural check available. Probabilities that add to 0.9 or 1.1 make every number after them wrong by an unknown amount.
Why no outcome equals it
The expected value here is 105. The three possible values are 60, 100 and 175. None of them is 105, and none ever will be. That is not a flaw. An average of a set does not have to be a member of the set.
It matters because "the expected value is 105" gets read as a prediction of 105, which is the one thing the number never claims. What it claims is that if this exact situation occurred a thousand times, the mean result would be close to 105.
The condition nobody states
Expected value assumes the situation repeats and the results add up. Where a bad outcome removes your ability to continue, that assumption fails and the calculation stops describing your position.
This is why position size is a separate question from whether a holding is attractive, and why the two are conflated so often. A favourable expected value on a position large enough to end the exercise is not favourable in any sense that helps.
Where the probabilities come from
The hard part is never the multiplication. It is the three probabilities, and they are usually assigned by the same person who built the three scenarios, which tends to produce a central case that looks like the plan and a poor case that is not poor enough.
The defence is the reference class: before assigning a probability to this company's outcome, ask how often companies like it reached that outcome. That is the last lesson of this course.
Check yourself
Three outcomes, 60, 100 and 175, with probabilities 0.25, 0.55 and 0.20. What is the expected value?
- 111.67, the plain average of the three
- 105
- 100, the most likely outcome
Show the answer
Answer: B.
Weight each by its own probability: 15 plus 55 plus 35 is 105. The unweighted average and the most likely single outcome are different numbers answering different questions.Why is a positive expected value not sufficient to justify a position?
- Because expected values are usually miscalculated
- Because the arithmetic assumes the situation can be repeated, which a ruinous loss prevents
- Because probabilities cannot be known
Show the answer
Answer: B.
An expected value is the mean over many repetitions. An outcome that ends the ability to continue breaks the assumption the calculation rests on, however favourable the average is.Your three scenario probabilities add to 1.15. What does that tell you?
- That the good case is too likely
- That the calculation is invalid until they are fixed
- Nothing, as long as they are in the right order
Show the answer
Answer: B.
Probabilities over a complete set of outcomes must add to exactly one. Any expected value computed from weights that do not is wrong by an amount nobody can state.Sources
- Kahneman and Lovallo, 'Timid Choices and Bold Forecasts: A Cognitive Perspective on Risk Taking', Management Science, 1993, on the inside view and why scenarios built from a specific plan lean optimistic.
- Graham, The Intelligent Investor, revised edition 1973, chapter 20, on the margin of safety, which exists because a favourable average outcome is not a guaranteed one.
What this leaves out
- Every figure here is invented arithmetic, chosen so each step can be recomputed.
- An expected value is only as good as the probabilities put into it, and this lesson does not supply a method for producing those.
- Nothing here is advice about any security.