Live model
Separate threshold chance, average loss, and percentile reserve
Risk planning often mixes three incompatible summaries. This page calculates the inclusive chance of reaching a critical count, the mean consequence across all periods, and the whole-number event reserve required at a selected percentile.
| Risk state | Event count | Consequence marker | Probability / CDF | Decision role |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
P(breach)=1-F(m-1;lambda); E[L]=lambdac; qp=min{q:F(q;lambda)>=p}; reserve=qpc
The threshold is inclusive. The percentile count is the smallest whole outcome whose cumulative probability reaches the selected percentile. Expected excess-event units use a Poisson tail identity rather than truncating an arbitrary table.
Use the calculator
Five steps for a threshold-risk record
- Define one risk period. Align lambda with the month, project, voyage, or other governance window.
- Set the critical count before calculation. State whether the threshold itself is a breach; this page treats it as inclusive.
- Enter one consistent consequence basis. Use currency, downtime value, or another additive scale consistently.
- Choose a percentile tied to policy. Do not select 95% or 99% merely because it is conventional.
- Compare all three outputs. Use breach probability for control likelihood, expected loss for average budgeting, and quantile reserve for a percentile commitment.
Five foundations
Frequency risk under a Poisson model
1. A breach is a tail event
For m=5, outcomes 5, 6, 7, ... all breach. Computing only P(X=5) understates trigger frequency.
2. Expected loss is a mean
lambdaxconsequence averages all periods, including event-free and severe periods. It is not the loss at a stated confidence level.
3. A count quantile is discrete
The percentile usually jumps past the target. The selected count is the first integer whose CDF meets or exceeds policy.
4. Severity is held constant
This model isolates event frequency. Real loss events often have a distribution of severities, producing a compound process.
5. Excess burden measures depth
Breach probability says how often; expected excess-event units add how far into the critical region the count travels on average.
Calculation anatomy
Symbols and default risk substitution
| Symbol | Meaning | Unit / domain |
|---|---|---|
| lambda | Expected loss-event count | events per period |
| m | Inclusive critical threshold | whole events, at least 1 |
| c | Consequence per event | currency/event |
| p | Planning percentile | 0.50-0.9999 |
| qp | Smallest count with CDF >= p | whole events |
| L | Linear period loss | currency |
Defaults substitute P(X>=5)=1-F(4;2.4), E[L]=2.4x$8,000, and search whole counts until F(q;2.4)>=0.95. The prior count must remain below 0.95, which the live reconciliation verifies.
Deep analysis
Three lenses for governance
Control trigger
Use breach probability to compare prevention options that shift lambda. It answers frequency, not the budget needed after breach.
Expected-loss budget
The mean is additive across independent portfolios and useful for long-run funding, but it can sit below a high-confidence reserve.
Percentile reserve
The quantile funds a stated fraction of count outcomes under the model. Multiplying by fixed severity is only valid when loss is linear in count.
Decision cases
Operating and boundary examples
Data-center outage incidents
With 2.4 outages expected per year and five as the escalation threshold, leaders compare the annual trigger probability with a 95th-percentile incident reserve. The expected budget remains a separate line.
Zero-rate control claim
At lambda=0, breach and expected loss are zero and every percentile count is zero. Treat this as a boundary calculation; a claimed real-world zero rate still needs evidence and uncertainty analysis.
Terms
Risk vocabulary
- Breach probability
- Probability of meeting or exceeding the critical count.
- Expected loss
- Probability-weighted mean consequence across periods.
- Quantile
- Smallest count whose CDF reaches a selected probability.
- Reserve count
- Whole-number count at the planning percentile.
- Expected excess
- Average count depth beyond the pre-threshold baseline.
- Frequency-severity model
- Model combining random event counts with random impact sizes.
FAQ
Questions for Poisson threshold risk
Why can expected loss be below reserve?
One is an average; the other targets a high count percentile.
Is the threshold inclusive?
Yes, m itself is a breach.
How is reserve count selected?
It is the first whole count whose CDF meets the percentile.
Must consequence be money?
No, but use one consistent additive value scale.
What if severity varies?
Use a compound frequency-severity model.
Can this replace safety or capital review?
No. It omits controls, dependence, severity tails, and regulatory rules.
Limits and evidence
Risk-model boundaries
- Event frequency is Poisson with fixed lambda for one defined period.
- Consequence is constant, additive, and independent of event count.
- The percentile reserve covers count variation only, not severity or parameter uncertainty.
- Common-cause events and contagion can make upper tails materially heavier.
- No legal, insurance, safety, or regulatory sufficiency conclusion is provided.
Sources and related tools