Lesson 9 of 9
Scenarios and expected value
Replace one value with a bear, base and bull case, each with a probability, and read the skew between what you lose when wrong and what you make when right.
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
- A single value hides how confident it is, where three or four scenarios with probabilities show the range and the odds instead.
- Expected value weights each scenario's value by its probability and adds them, which is not the same number as the base case alone.
- The distance between the price and each scenario, not just the expected value, is what a margin of safety actually has to cover.
One number implies certainty a valuation rarely has. Naming a bear, a base and a bull case, each with a probability, states the uncertainty instead of hiding it, and it lets you read what you lose if you are wrong against what you make if you are right.
Three scenarios, one price
- Bear case (25%)40
- Base case (50%)100
- Bull case (25%)160
Current price: 80
The arithmetic
0.25 x 40 + 0.50 x 100 + 0.25 x 160 is 100, against a price of 80. The base case alone was also 100 here, which will not usually happen. A skewed set of scenarios, where the bear case sits closer to the base case than the bull case does, moves the expected value away from the base case, in a direction the base case alone never shows.
Make the bear case 10, a near-total loss, and the expected value falls to 92.5 while the base case stays 100.
Expected value is not the whole answer
Two positions can share the same expected value and be completely different investments. One where the bear case loses a little and the bull case gains a little is a mild bet. One where the bear case is a near-total loss and the bull case is a large gain is a wild one, and averaging away the difference is the mistake this exercise exists to prevent. Marks's point in The Most Important Thing is that risk is not the same thing as the variability of a quoted price: it is the chance of a genuinely bad outcome, which a single expected-value number can bury.
What you lose against what you make
Set the current price against each scenario rather than only against the average. At a price of 80, the bear case here loses 50 percent, the base case gains 25 percent, and the bull case gains 100 percent. That asymmetry, not the single expected figure of 100, is what a margin of safety has to survive. Klarman's point in chapter 8 of Margin of Safety is that business value is a range rather than a point. A scenario table is that range made explicit instead of left as a feeling.
Solve for the odds the price implies
Run the arithmetic backwards and a price tells you the odds the market is assigning. Rappaport and Mauboussin's method in Expectations Investing is to solve for the expectation embedded in a quoted price rather than only comparing your own forecast with it. Simplify to just the bear and bull values above, 40 and 160, and ask what probability of the bear case makes their weighted average equal the price of 80: two thirds. The price also carries a charge for risk, so treat that two thirds as a rough reading. That is a sentence about market sentiment, built from the same two numbers, that you can then agree or disagree with directly.
Check yourself
A share has a 30% chance of being worth 40, 50% of 100 and 20% of 160. What is its expected value?
- 100, the base case
- 94
- 100, the plain average of the three
Show the answer
Answer: B.
0.30 x 40 + 0.50 x 100 + 0.20 x 160 = 12 + 50 + 32 = 94. The weights pull it below both the base case and the plain average.Two positions have the same expected value. Why might they still be very different investments?
- Expected value is never a real number
- The spread between the bear and bull cases can differ even when the weighted average is the same
- Probabilities must always be equal across scenarios
Show the answer
Answer: B.
A mild bet and a wild bet can share an expected value while differing sharply in how bad the bad case is and how good the good case is. The expected value alone does not show that.What does Klarman's 'range' argument in Margin of Safety add to an expected-value calculation?
- That only the average matters
- That business value is a spread of outcomes, not a single point, and a margin has to survive the worst of them
- That probabilities should never be estimated
Show the answer
Answer: B.
Klarman treats value as a range and advises leaning towards its lower values, so the margin is judged against the bad cases, not the average.Sources
- Marks, The Most Important Thing, 2011, on risk as the chance of a bad outcome rather than the variability of a quote.
- Rappaport and Mauboussin, Expectations Investing, 2001, on solving for the expectations already built into a price.
- Klarman, Margin of Safety, 1991, chapter 8, on business value as a range rather than a point.
What this leaves out
- The probabilities and values in the worked example are round invented inputs, chosen so the arithmetic can be checked by hand.
- A probability attached to a scenario is a judgement, not a measurement. Two careful readers can assign different odds to the same three cases.
- Expected value is a summary, not a substitute for looking at the worst case directly.
- Nothing here is advice about any security.