Lesson 22 of 25 · 7 min
Uncertainty that has no distribution to put on it
When the only input is a guess at the probability, the output range is just the range of the guess.
Every lesson to this point assumed a distribution existed, even a badly estimated one. Some quantities do not have one, and treating them as though they did is a specific and recognisable error rather than a matter of degree.
A risk has a distribution that can be argued about: there is a sample, a mechanism, or both, and two people disagreeing about it can bring evidence. An uncertainty has neither. The revision of a haircut on a particular facility is an uncertainty in this sense. It is a discretionary act by one party, it depends on that party's own funding conditions and internal policy, it is not observed across a population you belong to, and the count of past revisions on your own facility is zero or one.
Watch what happens when a number is demanded anyway. Assume a 2 percent annual chance of a revision: over three years the probability is 1 − 0.98³ = 5.88 percent. Assume 10 percent instead: 1 − 0.90³ = 27.10 percent. A factor of 4.61 separates the two, and the entire factor is the gap between two numbers that were invented. The arithmetic was faithful. It converted the guess, added nothing, and returned it with more decimal places than it went in with.
So the useful response is not a better prior. It is a different question, and lesson 20 supplies it. The frontier says that a 12.49 point haircut revision breaches that facility on a flat market. That statement contains no probability, cannot be improved by one, and does not change if the true likelihood turns out to be 1 percent or 30 percent a year.
Where a probability cannot be estimated, a question that does not require one is not a compromise. It is the only version of the question that has an answer.
Key terms
- Uncertainty
- A quantity with no sample and no mechanism from which to derive a distribution. Distinct from a risk, where the distribution exists and is merely estimated badly.
- Assumed prior
- A probability supplied by the analyst rather than by evidence. Everything downstream of it is a restatement of it, however elaborate the arithmetic in between.
- Probability-free statement
- A conclusion that holds whatever the likelihood turns out to be, such as which combination of conditions breaches a threshold.
Worked example
What an invented probability produces
A discretionary haircut revision over three years. Two analysts, each assuming a different annual probability. Illustrative.
- First analyst's assumption
- 2% a year
- Second analyst's assumption
- 10% a year
- Ratio of the two answers
- 4.61×
- Evidence separating the two assumptions
- None available
- What the arithmetic contributed
- Compounding, correctly
- The lesson 20 statement instead
- A 12.49 point revision breaches on a flat market
Three-year probability 1 − 0.98³ = 5.88 percent.
Three-year probability 1 − 0.90³ = 27.10 percent.
The two answers differ by a factor of 4.61 and the difference is entirely upstream of the model. The frontier statement is unaffected by either assumption.
Check yourself
Two analysts model the chance of a haircut revision over three years. One assumes 2 percent a year and gets 5.88 percent; the other assumes 10 percent and gets 27.10 percent. Which is closer to right?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.