Part II · Chapter 3 · Risk, without the score · lesson 10

The only expected-loss figure in this subject, and why it is here

15 min

Builds on Write down what your middle band actually means

This is the only expected-loss figure in the whole subject, and it is here to be taken apart rather than used.

Nine lessons of this track have refused to multiply two bands together. Now one model multiplies a frequency by a cost and produces a number in money, with units, that you could put in a slide. It belongs here for one reason: it is the respectable version of the arithmetic this track rejects, and because it is respectable, nobody checks it.

A frequency band and a cost per event. Band three means once a year. Call the event a quarter of a million. The answer is a quarter of a million a year, and it looks like a fact.

Before you touch anything: the frequency band is the softest input in the model. Nobody counted. It is a word somebody picked in a workshop. Move it one notch — the smallest move a five-point scale permits — and commit to what happens.

Predict

Frequency band three to band four. The smallest change your own definitions allow. How far does the annual figure move?

Hardly at all. One band out of five is a small change.Roughly double, which is what one step up usually means.By around a factor of ten, because the bands are decades apart.Impossible to say without knowing the cost per event.

Loss expectancy

Hypothetical model, not a forecast

Multiplies a frequency band by a cost per event, and then reports what that product does when the frequency moves by one band — the smallest change the definitions allow. It answers how far the figure moves when its softest input moves a notch, and how that compares with the whole declared range on the cost. It does not answer what the loss will be, and nothing computed this way can: an average is the one summary of a tailed loss distribution guaranteed to be wrong.

It assumes

The frequency band is a word somebody chose, never a rate anybody measured.
Every event is assumed to cost the same, which no incident has ever done.
The cost range is a plausible span, not a distribution with a shape.
Nothing is discounted, so a loss next month and a loss in twenty years weigh the same.
The band and the cost are treated as independent, which real incident years are not.
The figure is an average, and the loss it averages has a tail the average cannot see.

It ignores

The shape of the loss distribution, which is the omission that matters most here. Real losses have a tail: a great many small ones and a rare enormous one. The mean of that sits above almost every outcome and below the one that ends the company, so it describes neither, and an average is the single summary of a tailed distribution guaranteed to be wrong. Every figure this model reports is that average
That likelihood multiplied by impact — the risk score this subject refuses — is the same mistake as this arithmetic, with the units taken out. Putting real units back in makes the product look like money rather than like a made-up number, and it is the same made-up number
Where the frequency band came from. Nobody counted. It is a word somebody chose in a workshop, and the model treats it as though it had been measured, which is precisely the pretence the one-notch ratio is here to break
That the bands are not evenly spaced, so a notch at the top of the scale is worth a third of a notch at the bottom. The arithmetic is happy to multiply by either
Correlation between frequency and cost, which is almost always positive — the year with six incidents is the year the response was already overwhelmed, so the sixth one costs more than the first, not the same
Ruin. A loss larger than the organisation can absorb is not a point on an expected-value line, it is the end of the line, and an expected annual figure cannot see the difference. That is what ruin-probability is for
The cost of moving the band. This model never divides a loss figure by the price of a control, because that quotient is where a number like this gets used to justify a decision already taken
Insurance, deductibles, transfer and the limits on all three, which change who absorbs the loss without changing the loss
Losses absent from the frequency band altogether, because the band was fitted to the incidents somebody remembered and the ones that have not happened yet are not in it
That this figure, once it is in a slide, outlives every caveat attached to it — which is the reason it is the only one in this subject

Leave the money alone. Walk the frequency band from one to five and find the band where moving up one notch changes the answer least. There is a band that is not the top one.

Frequency band3

1 once in a century, 2 once in ten years, 3 once a year, 4 once a month, 5 once a week.

Cost of one event250,000

The central estimate, which is the figure the expected annual loss is built from.

Cost of one event — low estimate100,000

The cheapest the event could plausibly be. Clamped to the central estimate if it sits above it.

Cost of one event — high estimate1,000,000

The dearest the event could plausibly be, excluding the tail nobody priced.

Times the figure moves if the frequency goes up one band: 12
Expected annual loss one band up: 3,000,000
Expected annual loss at this band, which is one notch from the figure above: 250,000
Expected annual loss one band down: 25,000
Times the figure moves if the frequency goes down one band: 10
The same band, costed at the low estimate: 100,000
The same band, costed at the high estimate: 1,000,000
Times the figure moves across the whole declared cost range: 10
One notch of frequency moves the figure further than the entire cost range: 1
One band up, costed at the low estimate: 1,200,000
One band up at its cheapest still exceeds this band at its dearest: 1
Events a year this band declares: 1
Years between events at this band: 1
Cost of one event, as used: 250,000
Times the annual figure that one actual event costs: 1
Chance of at least one event in a year, at a constant arrival rate: 63.212%
Chance the year costs nothing at all: 36.788%
The single most likely annual loss is nothing: 0
Largest step between adjacent bands on this scale: 12
Smallest step between adjacent bands on this scale: 4.33
Times more a notch is worth at the steepest part of the scale than the shallowest: 2.77
Frequency band: 1 once a century, 5 once a week: 3
Already at the top band, so there is no notch above to move to: 0
Already at the bottom band, so there is no notch below: 0
The low or high estimate contradicted the central one and was clamped to it: 0

If every event is cheap
At the central cost
If every event is dear

Three lines climbing across five evenly drawn bands, from thousands at band one to tens of millions at band five — four orders of magnitude of travel on an axis a register prints as five identical cells.
Frequency band	If every event is cheap	At the central cost	If every event is dear
1	1,000	2,500	10,000
2	10,000	25,000	100,000
3	100,000	250,000	1,000,000
4	1,200,000	3,000,000	12,000,000
5	5,200,000	13,000,000	52,000,000

Not there yet — keep moving the controls.

Two things the model says about itself

One notch beats the whole argument about the cost. At the opening values a single band is worth twelve times. The entire declared cost range — from the cheapest the event could plausibly be to the dearest — is worth ten. So the input nobody measured moves the answer further than the input the meeting was about, and it does it with the smallest move the scale allows. The two flags in the summary are there to say so out loud.

The band ladder is not evenly spaced either. Once a century to once a decade is ten. Decade to year is ten. Year to month is twelve. Month to week is a little over four. A notch is worth between four and twelve times depending on where you were standing, and the challenge above makes you find the cheap one.

Read that second point again with the first nine lessons in mind. This is the same complaint. Two ordinals multiplied give a number on a scale whose steps are uneven — and here the uneven steps have climbed inside a model that produces money, where the unevenness is invisible because the output has units on it.

That is why this lesson exists. The product of two bands is obviously made up; anybody can see there are no units. A figure in pounds is not obviously made up, and it is made up in exactly the same place.

Check

You have the model's output in front of you. Which sentence would you put in front of somebody who has to make a decision?

We expect to lose about a quarter of a million a year to this.Between a hundred thousand and a million a year, depending on the cost.Our answer is governed by a band nobody counted, and one notch moves it by twelve.The figure is unreliable, so we should not use numbers here at all.

The omission that matters most, which is the tail

Everything above treats the cost of an event as a point with a range around it. Real losses are not shaped like that. There are a great many small ones and a rare enormous one, and the average of that distribution sits above almost every year you will actually have and below the one that ends the company.

Which means the average describes neither. It is the one summary of a tailed loss guaranteed to be wrong, and every figure this model reports is that average. That is the condition under which this entire lesson stops being useful, and it is the normal condition for the risks anybody worries about.

The honest use of the model is therefore narrow. It can tell you which of your inputs your answer is hostage to. It cannot tell you what you will lose, and no arithmetic of this family can, because the number that matters is the one in the tail and the tail is the part nobody priced.

Why it is dividing the loss by the cost of a control that does the damageOptional

The next thing anybody does with a figure like this is divide it by the price of a fix. Two hundred and fifty thousand of expected loss, forty thousand of control, and the ratio makes the decision look settled.

This model will not do that division, and the refusal is deliberate. The numerator moves by twelve when its softest input moves one notch, and the denominator is a quote from a vendor that is accurate to the pound. Dividing a number with an order of magnitude of slack by a number with none produces a quotient that reads as precise and inherits all of the slack.

In practice the quotient is almost never used to make a decision. It is used to justify one already taken, by whoever chose the band.

Worth remembering

One frequency band is worth around a factor of ten, not a fifth of anything.
The cost cancels, so the one-notch ratio holds whatever an event costs.
The input nobody measured moves the answer further than the cost range.
The band ladder's own steps run ten, ten, twelve and about four.
Money units hide the uneven scale that a unitless product shows plainly.
The honest output is a sensitivity statement, never a forecast.
An average is the one summary of a tailed loss guaranteed to be wrong.
Dividing an expected loss by a control price launders the slack.