Risk Measures (VaR / TVaR)
Value-at-Risk and Tail Value-at-Risk from a simulated or fitted loss distribution, with capital allocation views.
n = 10000, mean = 1,793, sd = 2,373
VaR / TVaR by level
| Level | VaR | TVaR |
|---|---|---|
| 80.0% | 2,529 | 5,035 |
| 90.0% | 3,951 | 6,966 |
| 95.0% | 5,585 | 9,243 |
| 99.0% | 10,977 | 16,793 |
| 99.5% | 14,205 | 21,053 |
Premium principles
| Principle | Parameters | Premium |
|---|---|---|
| Expected value + loading | E[X](1+θ) | 1,972 |
| Standard deviation | E[X] + k·σ | 2,979 |
| Variance | E[X] + k·σ² | 1,796 |
| Wang transform | λ = 0.2 | 2,188 |
| Proportional hazard (PH) | κ = 1.2 | 2,257 |
| Esscher | h = 0.0002 | 69,078 |
Euler TVaR allocation (two correlated lines)
Total portfolio TVaR0.95 = 12,220 (Line A + Line B via gaussian copula, ρ = 0.50).
Assumptions & limitations
- Empirical VaR/TVaR are computed from order statistics of the sample or simulation; larger samples give more stable tail estimates.
- Premium principle parameters (θ, k, λ, κ, h) are user-supplied loadings, not calibrated to a target return.
- The Euler allocation demo simulates two lognormal lines joined by a Gaussian copula purely for illustration of the additive TVaR-allocation property.