Risk Forecast

A risk forecast states the chance of an adverse outcome rather than a central estimate of what will happen. “Demand will be 500” is a point forecast; “there is a 12% chance demand exceeds capacity” is a risk forecast. The difference is not one of emphasis — it is a different object, and one that a point forecast structurally cannot provide.

The reason is that a risk statement is about a region of a distribution. You get it by integrating the forecast distribution over the outcomes you care about, which requires having the distribution — see probabilistic forecasts. A point forecast is one summary of that distribution (“the point forecast is the mean of this distribution”), and a summary cannot be un-summarized.

Three shapes of question

Rows one and two are inverses of each other: fix the threshold and read off a probability, or fix the probability and read off a level. The second is usually easier to act on because it comes out in the units of the thing being planned. Row three is a different quantity and answers a question the first two do not — a 5% chance of breaching says nothing about how far past the line you would be. The financial version of row two is Value-at-Risk.

The tail assumption decides the answer

This is the warning that matters most, and it follows from how most intervals are produced: “most time series models produce normally distributed forecasts — that is, we assume that the distribution of possible future values follows a normal distribution.”

A risk forecast reads directly off the part of that distribution the assumption is least reliable about. Heavier-than-normal tails mean the true probability of an extreme outcome is larger than the model reports, and the model’s confidence is unchanged — so the number can be wrong by a large factor while looking precise. See normal distribution.

Two checks follow, and neither is optional for a number anyone plans against.

• Verify the frequency, not the label. Hold out periods and count how often the threshold was actually breached. A “5% risk” breached 15% of the time is a measurement you can act on.
• Prefer methods that do not assume the shape where the tail drives the decision — estimate the quantile directly, or resample. Hyndman and Athanasopoulos describe bootstrapping the residuals as one that “only assumes that the residuals are uncorrelated with constant variance.” Resampling also treats the residuals as interchangeable draws from one stable distribution, so when they are dependent or the tail shifts over time, use a block or model-based bootstrap that matches the data structure. See bootstrap and quantile regression.

Reporting one

A risk number is meaningless without three accompanying facts: the threshold, the horizon, and the population. “12% risk” could be per day or per quarter, for one region or all of them, against a soft target or a hard limit — and those readings differ by orders of magnitude.

Finally, keep the scope of the claim honest. A risk forecast quantifies what the model represents, and a change the model does not represent is outside it rather than at its edge — the same limitation that applies to any prediction interval. Scoring the whole distribution rather than one threshold gives a fuller picture of whether the model deserves trust in the tail: see CRPS.

Further reading

• Hyndman & Athanasopoulos, Forecasting: Principles and Practice (3rd ed.), §5.9 “Evaluating distributional forecast accuracy”
• §5.5 “Distributional forecasts and prediction intervals”