Table 9 Probabilities of solvency in the case the risk capital is calculated without taking parameter uncertainty into account for \(n=50\) and \(n=100\) and different confidence levels, continuous distributions and methods of estimation, determined using a Monte-Carlo simulation with 10,000,000 simulations

From: Practical aspects of modelling parameter uncertainty for risk capital calculation

n Distribution Estimation \(95\%\) \(99\%\) \(99.5\%\)
method
True parameter
50 Gamma ML 94.28% 98.58% 99.20%
\(k=0.5\), \(\beta=1\)
ML 94.32% 98.64% 99.25%
\(k=2\), \(\beta=1\)
MM 93.99% 98.37% 99.03%
\(k=0.5\), \(\beta=1\)
MM 94.38% 98.59% 99.21%
\(k=2\), \(\beta=1\)
Normal/ ML* 94.33% 98.65% 99.26%
Lognormal
(two parameter)
  MM 94.32% 98.64% 99.25%
  \(\mu=0.1\), \(\sigma=0.1\)
Normal/ MM 94.32% 98.64% 99.25%
Lognormal \(\mu=1\), \(\sigma=0.1\)
  MM 93.45% 98.09% 98.84%
  \(\mu=1\), \(\sigma=1\)
Exponential ML* 94.54% 98.79% 99.35%
(one parameter)
Pareto ML* 94.34% 98.69% 99.30%
  (two parameter)
100 Gamma ML 94.64% 98.80% 99.36%
\(k=0.5\), \(\beta=1\)
ML 94.66% 98.83% 99.38%
\(k=2\), \(\beta=1\)
MM 94.48% 98.68% 99.27%
\(k=0.5\), \(\beta=1\)
MM 94.64% 98.80% 99.36%
\(k=2\), \(\beta=1\)
Normal/ ML* 94.67% 98.83% 99.39%
Lognormal
(two parameter)
  MM 94.66% 98.83% 99.38%
  \(\mu=0.1\), \(\sigma=0.1\)
Normal/ MM 94.66% 98.83% 99.38%
Lognormal \(\mu=1\), \(\sigma=0.1\)
  MM 94.10% 98.48% 99.13%
  \(\mu=1\), \(\sigma=1\)
Exponential ML* 94.74% 98.89% 99.42%
(one parameter)
Pareto ML* 94.67% 98.85% 99.40%
  (two parameter)