Model Uncertainty in a Holistic Perspective

Abstract

This paper focuses on model uncertainty within a holistic perspective. The latter is characterized by a consistent approach to risk measurement by combining stochastic, economic, operational and regulatory elements. This paper is a plea to account for model uncertainties on the level of consequences and not at the level of risk factors. This has important implications for validation, auditing and is of use testing of internal models. In line with risk management approaches, uncertainties have to managed. The starting point for this process is the identification and measurement of uncertainties. To achieve this goal further specific criteria for validity and resilience, are introduced in this paper. Examples from real world internal models highlight the practical relevance of the introduced concepts. A concluding section summarizes the main insights.

Appendix

Gaussian Distribution

Non parametric estimator

For the non parametric case we use the estimator

$$\begin{aligned} \epsilon _{np}=\hat{\sigma }*\sqrt{\frac{\alpha *(1-\alpha )}{n*f^{2}(\Phi ^{-1}(\alpha ))}}*\Phi ^{-1}\left( 1-\frac{\vartheta }{2}\right) , \end{aligned}$$

(54)

with $\hat{\sigma }$ the estimated standard deviation, $\alpha$ the quantile for which the standard error is calculated, n the sample size, f the density of the standard normal distribution, $\Phi ^{-1}$ the quantile function of the standard normal distribution and $\vartheta$ the confidence level.

Parametric estimator

For the parametric case we use the estimator

$$\begin{aligned} \epsilon _{p}=\frac{1}{\sqrt{n}}*\Phi ^{-1}\left( 1-\frac{\vartheta }{2}\right) *\hat{\sigma }*\sqrt{1+\frac{(\Phi ^{-1}(\alpha ))^{2}}{2}}, \end{aligned}$$

(55)

with the same notation as before.

GEV Distribution

The density of the GEV distribution is given by

$$\begin{aligned} f(x;\mu ,\sigma ,\xi ) = \frac{1}{\sigma }\left[ 1+\xi \left( \frac{x-\mu }{\sigma }\right) \right] ^{(-1/\xi )-1} \exp \left\{ -\left[ 1+\xi \left( \frac{x-\mu }{\sigma }\right) \right] ^{-1/\xi }\right\} \end{aligned}$$

for $1+\xi (x-\mu )/\sigma >0$, where $\mu \in \mathbb {R}$ is the location parameter, $\sigma > 0$ is the scale parameter and $\xi \in \mathbb R$ denotes the shape parameter.

Non parametric estimator

For the non parametric case we use the estimator

$$\begin{aligned} \epsilon _{np}=\sqrt{\frac{\alpha *(1-\alpha )}{n*\bar{f}^{2}(\bar{F}^{-1}(\alpha ))}}*\Phi ^{-1}\left( 1-\frac{\vartheta }{2}\right) , \end{aligned}$$

(56)

with the same notations as in (54) and $\bar{f}=f(x;\hat{\mu },\hat{\sigma },\hat{\xi })=f(x;-8530.60,739.99,-0.116)$ the density of the estimated GEV distribution $\bar{F}$. The parameters were estimated with the Log-likelihood function using the method of Nelder and Mead to determine the maximum of the function.

Parametric estimator

For the parametric case we use the estimator

$$\begin{aligned} \epsilon _{p}=\sqrt{\frac{1}{n}*\bar{a}^{-1}*(I(\hat{\Theta }))^{-1}*\bar{a}}*\Phi ^{-1}\left( 1-\frac{\vartheta }{2}\right) , \end{aligned}$$

(57)

with

$$\begin{aligned} \bar{a}=\begin{pmatrix}1\\ \frac{1}{\hat{\xi }}*((-\log {(\alpha )})^{-\hat{\xi }}-1)\\ \hat{\sigma }*\left[ -\frac{1}{\hat{\xi }^{2}}*((-\log {(\alpha )})^{-\hat{\xi }}-1)-\log {(-\log {(\alpha )})}*(-\log {(\alpha )})^{-\hat{\xi }}\right] \end{pmatrix}, \end{aligned}$$

(58)

where $I(\hat{\Theta })$ denotes the observed Fisher information matrix. For different sample sizes we get the following results for the 0.05 % quantile and 95 % confidence level:

BURR Distribution

The density of the BURR distribution is given by

$$\begin{aligned} f(x; a,b,q)= & {} \frac{aq (x-c)^{a-1}}{ b^a \left[ 1 + \left( \frac{x-c}{b}\right) ^a\right] ^{1+q}},\,\,\, \text{ for } x>c. \end{aligned}$$

In addition to the shape parameters $a>0$ and $q>0$ and the scaling parameter $b>0$ we introduce a location parameter $c \in \mathbb {R}$ to relax the property that the density lives on the positive half line.

Non parametric estimator

For the non parametric case we use the estimator

$$\begin{aligned} \epsilon _{np}=\sqrt{\frac{\alpha *(1-\alpha )}{n*\tilde{f}^{2}(\tilde{F}^{-1}(\alpha ))}}*\Phi ^{-1}\left( 1-\frac{\vartheta }{2}\right) , \end{aligned}$$

(59)

with the same notations as in (54) and $\tilde{f}=f(x;\hat{a},\hat{b},\hat{q})=f(x;14.24,6912.24,1.28)$ the density of the estimated BURR distribution $\tilde{F}$. The parameters were estimated with the Log-likelihood function using the method of Nelder and Mead to determine the maximum of the function. The parameter c was estimated with 1.33*min(data).

Parametric Estimator

For the parametric case we use the estimator

$$\begin{aligned} \epsilon _{p}=\sqrt{\frac{1}{n}*\bar{E}^{-1}*(I(\hat{\Theta }))^{-1}*\bar{E}}*\Phi ^{-1}\left( 1-\frac{\vartheta }{2}\right) , \end{aligned}$$

(60)

with

$$\begin{aligned} \bar{E}=\begin{pmatrix}-\frac{\hat{b}}{\hat{a}^{2}}*\log {(\beta )}*\beta ^{\frac{1}{\hat{a}}}\\ \beta ^{\frac{1}{\hat{a}}}\\ \frac{\hat{b}}{\hat{a}}*\frac{\log {(1-\alpha )}}{\hat{q}^{2}}*(1+\beta )*\beta ^{\frac{1}{\hat{a}}-1}\end{pmatrix} \end{aligned}$$

(61)

and $\beta =(1-\alpha )^{-\frac{1}{\hat{q}}}-1$ and the same notations as before.