- Normal-model VaR
- 16.4485 loss units
- Normal-model expected shortfall
- 20.6271 loss units
- Standard normal quantile
- 1.6449
- Standard normal tail-mean multiplier
- 2.0627
- Selected worst-tail probability
- 5 %
Declared normal loss law over one common horizon and unit. Positive loss, negative gain. Inputs are assumptions, not fitted data; no horizon scaling or regulatory-capital calculation is performed. The density chart uses standardized loss, not probability at a point.
The plotted z values run from −4 to 4 and include the selected quantile and tail-mean multiplier. The normal law has unbounded support. Curve height is density; probability is area over an interval. This is an illustrative example.
View the chart values
| Series | Standardized loss (z) | Density per z unit |
|---|---|---|
| Standard normal density | -4 | 0.00013383023 |
| Standard normal density | -3.875 | 0.00021893164 |
| Standard normal density | -3.75 | 0.00035259568 |
| Standard normal density | -3.625 | 0.00055906152 |
| Standard normal density | -3.5 | 0.0008726827 |
| Standard normal density | -3.375 | 0.0013411189 |
| Standard normal density | -3.25 | 0.0020290481 |
| Standard normal density | -3.125 | 0.003022258 |
| Standard normal density | -3 | 0.0044318484 |
| Standard normal density | -2.875 | 0.0063981203 |
| Standard normal density | -2.75 | 0.0090935625 |
| Standard normal density | -2.625 | 0.012724182 |
| Standard normal density | -2.5 | 0.0175283 |
| Standard normal density | -2.375 | 0.023771901 |
| Standard normal density | -2.25 | 0.031739652 |
| Standard normal density | -2.125 | 0.041720985 |
| Standard normal density | -2 | 0.053990967 |
| Standard normal density | -1.875 | 0.068786276 |
| Standard normal density | -1.75 | 0.086277319 |
| Standard normal density | -1.625 | 0.10653827 |
| Standard normal density | -1.5 | 0.1295176 |
| Standard normal density | -1.375 | 0.15501227 |
| Standard normal density | -1.25 | 0.18264909 |
| Standard normal density | -1.125 | 0.21187665 |
| Standard normal density | -1 | 0.24197072 |
| Standard normal density | -0.875 | 0.272055 |
| Standard normal density | -0.75 | 0.30113743 |
| Standard normal density | -0.625 | 0.32816097 |
| Standard normal density | -0.5 | 0.35206533 |
| Standard normal density | -0.375 | 0.37185509 |
| Standard normal density | -0.25 | 0.38666812 |
| Standard normal density | -0.125 | 0.39583769 |
| Standard normal density | 0 | 0.39894228 |
| Standard normal density | 0.125 | 0.39583769 |
| Standard normal density | 0.25 | 0.38666812 |
| Standard normal density | 0.375 | 0.37185509 |
| Standard normal density | 0.5 | 0.35206533 |
| Standard normal density | 0.625 | 0.32816097 |
| Standard normal density | 0.75 | 0.30113743 |
| Standard normal density | 0.875 | 0.272055 |
| Standard normal density | 1 | 0.24197072 |
| Standard normal density | 1.125 | 0.21187665 |
| Standard normal density | 1.25 | 0.18264909 |
| Standard normal density | 1.375 | 0.15501227 |
| Standard normal density | 1.5 | 0.1295176 |
| Standard normal density | 1.625 | 0.10653827 |
| Standard normal density | 1.6448536 | 0.10313564 |
| Standard normal density | 1.75 | 0.086277319 |
| Standard normal density | 1.875 | 0.068786276 |
| Standard normal density | 2 | 0.053990967 |
| Standard normal density | 2.0627128 | 0.047533023 |
| Standard normal density | 2.125 | 0.041720985 |
| Standard normal density | 2.25 | 0.031739652 |
| Standard normal density | 2.375 | 0.023771901 |
| Standard normal density | 2.5 | 0.0175283 |
| Standard normal density | 2.625 | 0.012724182 |
| Standard normal density | 2.75 | 0.0090935625 |
| Standard normal density | 2.875 | 0.0063981203 |
| Standard normal density | 3 | 0.0044318484 |
| Standard normal density | 3.125 | 0.003022258 |
| Standard normal density | 3.25 | 0.0020290481 |
| Standard normal density | 3.375 | 0.0013411189 |
| Standard normal density | 3.5 | 0.0008726827 |
| Standard normal density | 3.625 | 0.00055906152 |
| Standard normal density | 3.75 | 0.00035259568 |
| Standard normal density | 3.875 | 0.00021893164 |
| Standard normal density | 4 | 0.00013383023 |
On this page
Use a common loss unit and horizon
Mean loss and standard deviation must use the same unit and period. Positive mean loss means loss; negative mean loss means a gain location. Dispersion must be nonnegative. Confidence choices are 95%, 97.5% and 99%. The inputs declare assumptions rather than supplying a fitted data record.
Read the two coefficients
VaR uses the standard normal quantile; ES uses the upper-tail mean multiplier. Both are multiplied by loss standard deviation and shifted by mean loss. The output remains in the input loss unit and keeps its sign.
Inspect the original default
Mean loss 0, dispersion 10 and confidence 95% give VaR 16.448536 and ES 20.627128 loss units. The selected worst-tail probability is 5%.
Read the standardized density
The figure plots standard normal density against z from −4 to 4 and includes the chosen quantile and tail-mean coefficient coordinates. Probability is interval area, not curve height at a point. The normal law continues beyond the finite display window.
Handle the constant-loss limit
At zero dispersion, both summaries equal the entered mean. Standardization and a positive-scale normal density are undefined, so the figure is omitted.
Retain the assumption record
The calculator does not establish normality, estimate parameters, construct joint distributions, reprice positions, scale horizons or implement regulatory capital. Use the actual loss-model record and exposure basis when deciding which calculation fits the task.