ModelRisk example · General techniques
Operational Risk Capital (FFT)
How much capital covers a year of operational losses at 99.9%? Forty-five loss cells, one FFT.
From the model’s saved run
The capital charge
Exact from the FFT, against 10,000 simulated years.
| Measure | Exact | FFT | FFT, sampled | Cell by cell |
|---|---|---|---|---|
| Mean (expected loss) | 8,290 | 8,373 | 8,373 | 8,296 |
| Standard deviation | 583 | 588 | 580 | 583 |
| 99th percentile | 9,876 | 9,826 | 9,787 | |
| Required percentile (99.9%) | 10,705 | 10,514 | 10,683 | |
| Capital charge (FFT percentile − FFT mean) | 2,332 |
S-curve (cumulative probability) of Sampled from the FFT, Cell by cell, 99.9th percentile, FFT from the saved simulation.
The business question
A bank models its operational losses in 45 cells: three risk types, three processes and five sub-processes. In each, loss events arrive as a Poisson process (from 0.8 to 47 a year) and each event costs a Lognormal amount. Basel II sets capital at the 99.9th percentile of the year's total loss, less the expected loss: a one-in-a-thousand year.
Simulation estimates that percentile from a handful of extreme samples; VoseAggregateMultiFFT computes the total's distribution numerically from all 45 cells and gives it directly.
- Who it is for
- Operational risk and capital teams, and anyone who needs a far-tail percentile of a total built from many frequency–severity cells.
- Main outputs
- The expected annual loss · the 99th and 99.9th percentiles of the year's total, exactly by FFT · the capital charge (99.9th percentile minus expected loss) · the same by risk type and the diversification benefit · the total sampled from the FFT and simulated cell by cell.
How the model works
Four sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The percentile for the capital charge (99.9%).
Cells sheet
Each cell's expected number of events and loss per event (yellow), as a Poisson and a Lognormal object; its expected loss; and one simulated year (VoseAggregateMC).
Capital sheet
VoseAggregateMultiFFT combines all 45 cells: with U it returns the percentile, without it a sample of the total. The same for each risk type's 15 cells. Capital is the percentile less the FFT distribution's mean. The FFT's mean is checked against the exact one.
Results sheet
The capital charge, the capital by risk type and the diversification benefit, and the total two ways.
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The results in detail
Capital by risk type
Each type on its own, and the whole institution (FFT).
Histogram of Total loss from the saved simulation.
| Risk type | Percentile | Capital |
|---|---|---|
| Type 1 | 4,520 | 1,608 |
| Type 2 | 2,759 | 757 |
| Type 3 | 5,301 | 1,842 |
| Sum of the three | 4,208 | |
| Whole institution | 2,332 | |
| Diversification benefit | 45% |
How to read these results
- The institution expects about 8,290 of operational losses a year from about 635 events. In a one-in-a-thousand year the total reaches about 10,600 (an independent fine-grid calculation gives 10,595), so the capital charge, the loss above the expected, is about 2,300.
- VoseAggregateMultiFFT computes the total's distribution from all 45 cells and gives the 99.9th percentile directly, with no sampling noise. A 10,000-sample simulation estimates it from its 10 worst years only.
- Check an FFT against the exact mean: here the FFT's discretisation shifts the whole total up by about 1% (its mean is 8,373 against 8,290), so its 99.9th percentile is about 110 too high. Measuring the capital from the FFT's own mean cancels most of the shift (2,333 against 2,305). From the exact mean it would keep it (2,416).
- Held risk type by risk type, the three charges add up to nearly twice the institution's. Independent types rarely have their worst year together; that is the diversification benefit. Losses that move together (a common cause) would shrink it.
- A few cells with a large standard deviation (Type 1 B2, Type 3 B5, Type 3 C1) drive the tail far more than the frequent small losses.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
Capital standard 1 input
| Assumption | Unit | Value |
|---|---|---|
| Percentile for the capital chargeBasel II: a one-in-a-thousand year | 99.9% |
Units and conventions
- Capital charge
- The total's percentile at the required level less the expected loss (the unexpected loss).
- Diversification benefit
- 1 − capital for the whole ÷ the sum of the capitals of the three risk types.
Key assumptions and limitations
- Independence
- Cells are independent: events in one say nothing about another. Correlation between cells would raise the capital and shrink the diversification benefit.
- Distributions
- Poisson frequencies and Lognormal severities with the stated means and standard deviations.
- Data
- The 45 cells and the 99.9% are those of ModelRisk's earlier Operational Risk example.
How to use it
- Replace the cells on the Cells sheet with your own (keep each risk type's rows together) and set the percentile on the Inputs sheet.
- Read the capital at once on the Capital sheet; run a simulation (10,000 samples with a fixed seed are set) to compare with sampling.
- Use VoseAggregateMultiFFT's percentile for far-tail measures; a simulation cannot estimate a 99.9th percentile well without very many samples.
ModelRisk functions used
- VosePoissonObject, VoseLognormalObject
- Each cell's frequency and severity as objects.
- VoseAggregateMultiFFT
- The total of several frequency–severity pairs by Fast Fourier Transform: a percentile (with U) or a sample.
- VoseAggregateMC
- One cell's simulated annual loss: a number of events, then one loss each, added up.
- VoseSimMean, VoseSimStdev, VoseSimPercentile
- Statistics of the simulation.
Further reading
- Frequency-severity modelling (Vose guide)
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: operational risk and the loss distribution approach.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.