Alternative statistical distributions for estimating value-at-risk: theory and evidence

Abstract

A number of applications presume that asset returns are normally distributed, even though they are widely known to be skewed leptokurtic and fat-tailed and excess kurtosis. This leads to the underestimation or overestimation of the true value-at-risk (VaR). This study utilizes a composite trapezoid rule, a numerical integral method, for estimating quantiles on the skewed generalized t distribution (SGT) which permits returns innovation to flexibly treat skewness, leptokurtosis and fat tails. Daily spot prices of the thirteen stock indices in North America, Europe and Asia provide data for examining the one-day-ahead VaR forecasting performance of the GARCH model with normal, student’s t and SGT distributions. Empirical results indicate that the SGT provides a good fit to the empirical distribution of the log-returns followed by student’s t and normal distributions. Moreover, for all confidence levels, all models tend to underestimate real market risk. Furthermore, the GARCH-based model, with SGT distributional setting, generates the most conservative VaR forecasts followed by student’s t and normal distributions for a long position. Consequently, it appears reasonable to conclude that, from the viewpoint of accuracy, the influence of both skewness and fat-tails effects (SGT) is more important than only the effect of fat-tails (student’s t) on VaR estimates in stock markets for a long position.

Appendix 1: Derivation of the standardized SGT

The probability density function for the Skewed Generalized t (SGT) distribution introduced by Theodossiou (1998) is represented as follows:

$$ {\text{f}}\left( {{\text{x}}\left| {\sigma^{2} ,\lambda ,\kappa ,{\text{n}}} \right.} \right) = C^{\prime } \left\{ {1 + \frac{\kappa }{{{\text{n}} - 2}}\frac{{\left| {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x \sigma }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\sigma $}}} \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}({\text{x}})\lambda } \right]^{\kappa } \theta^{\prime \kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} $$

(16)

where $\theta^{\prime} = \frac{1}{{{\text{S}}(\lambda )}}\left( {\frac{\kappa }{{{\text{n}} - 2}}} \right)^{{\frac{1}{\kappa }}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{{ - \frac{1}{2}}} ,{\text{S}}(\lambda ) = \sqrt {1 + 3\lambda^{2} - 4{\text{A}}^{2} \lambda^{2} } ,$ ${\text{A}} = {\text{B}}\left( {\frac{2}{\kappa },\frac{{{\text{n}} - 1}}{\kappa }} \right){\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{1}{2}}} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{{ - \frac{1}{2}}} ,$ ${\text{C}}^{\prime } = \frac{\kappa }{2}\frac{{{\text{S}}(\lambda )}}{\sigma }{\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{3}{2}}}$.

The expected value of x is $\mu = {\text{E}}({\text{x}}) = 2\lambda {\text{A}}\sigma /{\text{S}}(\lambda )$ and the variance of x is $\text{var} ({\text{x}}) = {\text{E}}({\text{x}}^{2} ) - \mu^{2} = \sigma^{2}$. For simplicity, the derivation of parameters, C, θ and δ, for the standardized SGT distribution is accomplished using the transformed random variable ε = (x−μ)/σ, which has a mean of E(ε) = 0 and a variance of var(ε) = 1. The random x thus can be expressed as x = μ + σε and let $\delta = \mu /\sigma = 2\lambda {\text{A}}/{\text{S}}(\lambda )$, therefore ${\text{dx}}/{\text{d}}\varepsilon = \sigma ,$ ${\text{x}}/\sigma = \varepsilon + \mu /\sigma = \varepsilon + \delta$ and ${\text{sign}}({\text{x}}) = {\text{sign}}(\mu + \sigma \varepsilon ) = {\text{sign}}\left( {\sigma \left( {\varepsilon + \mu /\sigma } \right)} \right) = {\text{sign}}\left( {\varepsilon + \mu /\sigma } \right) = {\text{sign}}\left( {\varepsilon + \delta } \right)$.

Substitute the above expressions and Eq. 16 into the following equation.

$$ \begin{aligned} {\text{f}}\left( \varepsilon \right) & = {\text{f}}\left( {\text{x}} \right)\left| {\frac{\text{dx}}{{{\text{d}}\varepsilon }}} \right| = {\text{C}}^{\prime } \left\{ {1 + \frac{\kappa }{{{\text{n}} - 2}}\frac{{\left| {{\raise0.7ex\hbox{${\text{x}}$} \!\mathord{\left/ {\vphantom {{\text{x}} \sigma }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\sigma $}}} \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}({\text{x}})\lambda } \right]^{\kappa } \theta^{\prime \kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \sigma \\ & = C^{\prime } \left\{ {1 + \frac{\kappa }{{{\text{n}} - 2}}\frac{{\left| {\varepsilon + \delta } \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}(\varepsilon + \delta )\lambda } \right]^{\kappa } \theta^{\prime \kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \sigma \\ & = \frac{\kappa }{2}\frac{{{\text{S}}(\lambda )}}{\sigma }{\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{3}{2}}} \left\{ {1 + \frac{\kappa }{{{\text{n}} - 2}}\frac{{\left| {\varepsilon + \delta } \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}(\varepsilon + \delta )\lambda } \right]^{\kappa } \theta^{\prime \kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \sigma \\ & = \frac{{\kappa {\text{S}}(\lambda )}}{2}{\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{3}{2}}} \left\{ {1 + \frac{\kappa }{{{\text{n}} - 2}}\frac{{\left| {\varepsilon + \delta } \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}(\varepsilon + \delta )\lambda } \right]^{\kappa } \theta^{\prime \kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \\ \end{aligned} $$

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where ${\text{C}}^{\prime } = \frac{\kappa }{2}\frac{{{\text{S}}(\lambda )}}{\sigma }{\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{3}{2}}} ,$ $\theta^{\prime } = \frac{1}{{{\text{S}}(\lambda )}}\left( {\frac{\kappa }{{{\text{n}} - 2}}} \right)^{{\frac{1}{\kappa }}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{{ - \frac{1}{2}}} ,$

Letting $\theta^{\kappa } = \frac{{({\text{n}} - 2)\theta^{\prime \kappa } }}{\kappa } = \left[ {\frac{1}{{{\text{S}}(\lambda )}}{\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{{ - \frac{1}{2}}} } \right]^{\kappa }$ and substituting the expression into Eq. 17 gives the equation which is the standardized SGT distribution.

$$ \begin{aligned} {\text{f}}\left( \varepsilon \right) & = \frac{{\kappa {\text{S}}(\lambda )}}{2}{\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{3}{2}}} \left\{ {1 + \frac{{\left| {\varepsilon + \delta } \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}(\varepsilon + \delta )\lambda } \right]^{\kappa } \theta^{\kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \\ & = {\text{C}}\left\{ {1 + \frac{{\left| {\varepsilon + \delta } \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}(\varepsilon + \delta )\lambda } \right]^{\kappa } \theta^{\kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \\ \end{aligned} $$

(18)

where $\theta = \frac{1}{{{\text{S}}(\lambda )}}{\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{{ - \frac{1}{2}}} ,{\text{S}}(\lambda ) = \sqrt {1 + 3\lambda^{2} - 4{\text{A}}^{2} \lambda^{2} } ,$ ${\text{A}} = {\text{B}}\left( {\frac{2}{\kappa },\frac{{{\text{n}} - 1}}{\kappa }} \right){\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{ - 0.5} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{ - 0.5} ,$ $\delta = \frac{{2\lambda {\text{A}}}}{{{\text{S}}(\lambda )}},$ ${\text{C}} = \frac{{\kappa {\text{S}}(\lambda )}}{2}{\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{{ - \frac{3}{2}}} = \frac{\kappa }{2\theta }{\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{ - 1} .$

Particularly, the SGT distribution generates the student’ t distribution for λ = 0 and κ = 2. Using the recurrence identities of Beta function, ${\text{B}}\left( {{\text{a}},{\text{b}}} \right) = {\text{B}}\left( {{\text{b}},{\text{a}}} \right)$, ${\text{B}}\left( {{\text{a}} + 1,{\text{b}}} \right) = {\text{a}} \cdot {\text{B}}\left( {{\text{a}},{\text{b}}} \right)/\left( {{\text{a}} + {\text{b}}} \right),$ and ${\text{B}}\left( {{\text{a}} - 1,{\text{b}}} \right) = \frac{{{\text{a}} + {\text{b}} - 1}}{{{\text{a}} - 1}} \cdot {\text{B}}\left( {{\text{a}},{\text{b}}} \right),$ we can obtain ${\text{B}}\left( {\frac{3}{2},\frac{{{\text{n}} - 2}}{2}} \right) = \frac{1}{{{\text{n}} - 2}}{\text{B}}\left( {\frac{1}{2},\frac{\text{n}}{2}} \right)$.

Therefore $\delta = 2\lambda {\text{A}}/{\text{S}}(\lambda ) = 0,{\text{S}}(\lambda ) = \sqrt {1 + 3\lambda^{2} - 4{\text{A}}^{2} \lambda^{2} } = 1,$ and

$$ \begin{aligned} \theta & = \frac{1}{{{\text{S}}(\lambda )}}{\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{3}{\kappa },\frac{{{\text{n}} - 2}}{\kappa }} \right)^{{ - \frac{1}{2}}} \\ & = {\text{B}}\left( {\frac{1}{2},\frac{\text{n}}{2}} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{3}{2},\frac{{{\text{n}} - 2}}{2}} \right)^{{ - \frac{1}{2}}} = {\text{B}}\left( {\frac{1}{2},\frac{\text{n}}{2}} \right)^{\frac{1}{2}} {\text{B}}\left( {\frac{1}{2},\frac{\text{n}}{2}} \right)^{{ - \frac{1}{2}}} \sqrt {{\text{n}} - 2} = \sqrt {{\text{n}} - 2} \\ {\text{C}} & = \frac{\kappa }{2\theta }{\text{B}}\left( {\frac{1}{\kappa },\frac{\text{n}}{\kappa }} \right)^{ - 1} = \frac{2}{{2\sqrt {{\text{n}} - 2} }}{\text{B}}\left( {\frac{1}{2},\frac{\text{n}}{2}} \right)^{ - 1} = \frac{1}{{\sqrt {{\text{n}} - 2} }}\frac{{\Upgamma \left( {\frac{{{\text{n}} + 1}}{2}} \right)}}{{\Upgamma \left( \frac{1}{2} \right)\Upgamma \left( {\frac{\text{n}}{2}} \right)}} = \frac{{\Upgamma \left( {\frac{{{\text{n}} + 1}}{2}} \right)}}{{\sqrt {\pi \left( {{\text{n}} - 2} \right)} \Upgamma \left( {\frac{\text{n}}{2}} \right)}}. \\ \end{aligned} $$

The probability density function for the standardized student’ t distribution thus can be represented as follows.

$$ \begin{aligned} {\text{f}}\left( \varepsilon \right) & = {\text{C}}\left\{ {1 + \frac{{\left| {\varepsilon + \delta } \right|^{\kappa } }}{{\left[ {1 + {\text{sign}}(\varepsilon + \delta )\lambda } \right]^{\kappa } \theta^{\kappa } }}} \right\}^{{ - \frac{{{\text{n}} + 1}}{\kappa }}} \\ & = \frac{{\Upgamma \left( {0.5({\text{n}} + 1)} \right)}}{{\Upgamma \left( {0.5{\text{n}}} \right)\sqrt {\pi \left( {{\text{n}} - 2} \right)} }}\left\{ {1 + \frac{{\varepsilon^{2} }}{{{\text{n}} - 2}}} \right\}^{{ - \frac{{{\text{n}} + 1}}{2}}} \\ \end{aligned} $$

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