Calculator Inputs
Example Data Table
| Observation | Loss Amount |
|---|---|
| 1 | 1,200 |
| 2 | 1,400 |
| 3 | 1,500 |
| 4 | 1,700 |
| 5 | 1,900 |
| 6 | 2,300 |
| 7 | 2,800 |
| 8 | 3,200 |
| 9 | 4,100 |
| 10 | 4,900 |
| 11 | 6,100 |
| 12 | 7,600 |
Formula Used
This tool applies a generalized maximum likelihood approach to fitted loss severity models. When a threshold is supplied, the likelihood is adjusted for left truncation.
General truncated likelihood:
L(θ) = ∏ [ f(xᵢ | θ) / (1 - F(T | θ)) ] for all xᵢ ≥ T.
Log-likelihood:
ln L(θ) = Σ ln f(xᵢ | θ) - n ln(1 - F(T | θ))
Supported Models
Normal: f(x) = (1 / σ√(2π)) exp(-(x-μ)² / (2σ²))
Lognormal: f(x) = [1 / (xσ√(2π))] exp(-(ln x-μ)² / (2σ²))
Exponential: f(x) = λ exp(-λx)
Risk Outputs
Value at Risk: the selected quantile of the fitted severity model.
Expected Shortfall: the conditional mean beyond the selected VaR cutoff.
AIC and BIC: penalized fit measures for comparing alternative distributions.
KS Statistic: the largest gap between empirical and fitted truncated CDF values.
How to Use This Calculator
- Select a severity model that matches your loss pattern.
- Paste observed loss amounts into the data box.
- Optionally enter a threshold for left-truncated samples.
- Choose a confidence level for VaR and expected shortfall.
- Set decimal precision, then click Estimate Parameters.
- Review fitted parameters, likelihood diagnostics, and tail estimates.
- Inspect the histogram and fitted density overlay.
- Export the results as CSV or PDF for reporting.
Frequently Asked Questions
1. What does this estimator calculate?
It estimates distribution parameters from observed loss data. The tool also reports likelihood diagnostics, model comparison scores, and tail risk measures such as VaR and expected shortfall.
2. When should I use a threshold?
Use a threshold when your dataset only contains losses above a reporting floor, deductible, or operational capture limit. The likelihood is then adjusted for left truncation.
3. Which distribution should I choose?
Lognormal often suits skewed severity data. Normal can fit symmetric losses. Exponential is useful for simple thin-tail decay. Compare AIC, BIC, and visual fit together.
4. Why are nonpositive values removed sometimes?
Lognormal and exponential models only support positive values. Zero or negative amounts break those density formulas, so the calculator removes them before fitting.
5. What do AIC and BIC mean here?
They balance goodness of fit against model complexity. Lower values usually indicate a better tradeoff when comparing candidate distributions on the same filtered dataset.
6. Why is expected shortfall larger than VaR?
VaR marks a tail cutoff. Expected shortfall averages the losses beyond that cutoff, so it is usually equal to or larger than VaR.
7. Does the chart reflect truncation?
Yes. When a threshold is used, the plotted curve is the fitted truncated density, not the unadjusted full distribution density.
8. Can I export the results for audit work?
Yes. The CSV export is useful for spreadsheets, while the PDF export creates a clean report table for review, archiving, or presentation.