Risk and uncertainty · RiskAtoms Academy

By the end you will be able to

1Tell measurable risk apart from uncertainty that no model can price.
2Describe three ways of putting a probability on an outcome: from the set-up, from data, or by judgement.
3Estimate a probability from records, and explain why few events or a changing world make it unreliable.
4Calculate the chance of at least one event over several years, and say how firms handle true uncertainty.

Why it matters

An insurer can tell you, quite precisely, how many of a million drivers will crash next year. Nobody could have told you in 2019 what a pandemic would cost in 2020. Both are “risk” in everyday speech. To a risk manager they are very different problems.

Core explanation

Known odds, estimated odds, and no odds at all

L2.1.01: What is risk? defined risk as uncertainty about outcomes that matter. To manage it, you would like to put a probability on each outcome. Sometimes you can. Sometimes you cannot, and pretending otherwise is one of the most dangerous mistakes in finance.

In 1921 the American economist Frank Knight drew the line that is still used today. Where the probabilities can be known or reliably estimated, he called it risk; here we say measurable risk. Where they cannot, he called it uncertainty. In his words, a measurable uncertainty “is so far different from an unmeasurable one that it is not in effect an uncertainty at all.”

Knight described three ways of getting a probability:

From the set-up

A probability known from the set-up itself, before any data: a fair die, a shuffled deck.

A die shows a six one time in six. A roulette wheel, a lottery draw.

From data

A probability estimated from how often something happened in many similar past cases.

Car crashes among a million drivers; deaths in a large group of 60-year-olds.

By judgement

A probability that is only an informed guess, because there are no similar cases to count.

Whether a new technology will succeed; the next pandemic; a country leaving a currency union.

Insurance, credit scoring and most of the risk models in this course live in the middle column: they count how often something happened in many similar past cases. That works when there are many cases and when the future resembles the past. It breaks down when events are rare, when the world changes, or when the past contains nothing like what is coming.

Sort itKnown odds, estimated odds, or uncertainty?

0 of 6 right

Where would each probability come from?

The chance that a fair coin lands heads twice in a row.
How many of 200,000 insured cars will be stolen next year.
The chance that a brand-new kind of cyber attack takes down a payment system.
The default rate next year on a pool of 50,000 prime credit-card accounts.
Whether a war breaks out that closes a major shipping route.
The chance of drawing an ace from a well-shuffled deck.

People treat unknown odds differently

Uncertainty does not just make numbers harder to find; it changes how people behave. Try this choice, devised by the economist Daniel Ellsberg in 1961.

ChooseTwo urns

You win £100 if the ball drawn is the colour you bet on. You choose the urn.

Round 1: you bet on red. Which urn do you draw from?

Ambiguity aversion matters in markets. When a crisis makes the odds unknowable, investors do not just demand a higher price for risk; many refuse to hold the asset at all, and markets that worked yesterday stop trading.

The formal version

A probability is a number from 0 (impossible) to 1 (certain). Estimated from data, it is a frequency (see M-C01: Probability basics):

\hat p = \frac{\text{number of times it happened}}{\text{number of cases observed}}

The hat on $\hat p$ says it is an estimate of the true $p$ . A common question is the chance of at least one event over $n$ years. If each year has chance $p$ and years are independent, work through the opposite:

\begin{aligned} P(\text{no event in a year}) &= 1 - p \\ P(\text{no event in } n \text{ years}) &= (1 - p)^n \\ P(\text{at least one in } n \text{ years}) &= 1 - (1 - p)^n \end{aligned}

A “1-in-100-year” flood ( $p = 0.01$ ) has a 26.0% chance of striking at least once during a 30-year mortgage. A 10% event has a 65.1% chance in 10 years.

The catch

Both formulas assume you know

p

and that it stays the same. Measurable risk is exactly the case where that holds well enough. Uncertainty is where it does not.

Try it

Here is a century of flood records for an illustrative town. Estimate this year's chance of a flood from the most recent years. Start in the stable world and slide from 10 years to 100. Then switch to the changing world, where the chance quadrupled 30 years ago, and do it again.

Try itHow good is the record?

Probability calculator →

The world behind the data

Years of records used (most recent)40 years

A century of records, oldest at top left, one square a year, twenty to a row. Filled squares are flood years; bright squares are the years you are using.

Estimated chance of a flood this year

10.0%

4 floods ÷ 40 years

True chance this year10.0%

Error0.0 pts

Spot on, this time.

In the stable world, more data usually helps, though even 100 years can be a few points out. In the changing world, the long record is the problem: it averages in decades that no longer apply. Risk becomes uncertainty when you cannot tell whether the past still describes the future.

Worked example

Pricing flood cover from a short record

An insurer covers 1,000 homes in a riverside town. A flood damages them all at once, with claims of about £50,000 a home. Its records show 4 floods in 40 years. Amounts in pounds; illustrative.

From the record to a price, and its limits

Step 1 of 5

1
Estimate the probability
4 ÷ 40 = 0.10, a 10.0% chance a year.

Regulatory lens

Capital rules are built on measurable risk. Solvency II, for example, asks EU insurers to hold enough capital to survive a 1-in-200-year loss over one year. Supervisors know that models cannot capture everything, so they also require stress tests and scenario analysis for events the data cannot price, climate change among them. L2.5.09: Stress testing and L2.7.05: Solvency I to Solvency II return to both.

Sparks

SparkWhy uncertainty pays

Knight was really writing about profit. He argued that measurable risks can be insured or pooled away, so they cannot be a lasting source of profit. What is left for the entrepreneur is the reward for facing true uncertainty, which nobody can insure.

Spark“We simply do not know”

John Maynard Keynes made the same point in 1937, listing the prospect of a European war and the rate of interest twenty years ahead as examples: “About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know.”

Key takeaways

Measurable risk has probabilities that can be known or reliably estimated; uncertainty, in Knight's sense, does not.
Probabilities come from the set-up (dice), from data (insurance records), or from judgement when there are no similar cases.
Estimates from data are shaky when events are few, and misleading when the world has changed.
The chance of at least one event in n independent years is 1 − (1 − p)^n: rare events become likely over long periods.
Firms handle uncertainty with limits, buffers and scenarios that work whatever the odds turn out to be.

Reaction

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Glossary

Probability: A number between 0 and 1 (or 0% and 100%) saying how likely an outcome is.
Measurable risk: Risk where the probabilities of the outcomes are known or can be reliably estimated.
Uncertainty (Knightian): In Knight's sense: a situation where no reliable probabilities exist, because the event is new, rare or the world has changed.
A priori probability: A probability known from the set-up itself, before any data: a fair die, a shuffled deck.
Statistical probability: A probability estimated from how often something happened in many similar past cases.
Judgement (estimate): A probability that is only an informed guess, because there are no similar cases to count.
Ambiguity aversion: A preference for known odds over unknown odds, even when the unknown odds could be better.
Independent events: Two events are independent if one happening does not change the chance of the other.

Deep Dives & Origins

Deep Dives

Knight, F. H., Risk, Uncertainty and Profit, Houghton Mifflin, 1921, chapters 1 and 7. Intermediate, a classic.
Kay, J. and King, M., Radical Uncertainty: Decision-Making Beyond the Numbers, Bridge Street Press, 2020. Beginner, narrative.

Origins

Knight, F. H., Risk, Uncertainty and Profit, Houghton Mifflin, 1921.
Keynes, J. M., “The General Theory of Employment”, Quarterly Journal of Economics 51(2), 1937.
Ellsberg, D., “Risk, Ambiguity, and the Savage Axioms”, Quarterly Journal of Economics 75(4), 1961.
Directive 2009/138/EC (Solvency II), Article 101: the Solvency Capital Requirement calibrated to a 99.5% value at risk over one year.

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