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Table 2 Functional Relationship of VaR and tVaR

From: Economics of eBay’s buyer protection plan

Distribution

Uniform (a, b)

Gamma (1, b)

Lomax (a, b)

Normal (m, v)

\( \frac{tVar(w)}{VaR(w)} \)

\( \frac{b\left(2-w\right)+ aw}{2\left(b\left(1-w\right)+ aw\right)} \)

\( 1-\frac{1}{ \log (w)} \)

\( \frac{a}{\alpha -1}; a>1 \)

\( m+\sqrt{v}\frac{\ {f}_{\tilde{c}}\left[VaR\left(1-w\right)\right]}{1-w} \)

\( {f}_{\tilde{c}}\ is\ the\ Normal\ pdf \)

f(w) : tVaR[w] = VaR[f(w)]

\( \frac{w}{2} \)

\( \frac{w}{e} \)

\( w{\left(\frac{a}{\alpha -1}\right)}^a \)

\( m+\sqrt{v}\left(1-{f}_{\tilde{c}}\left[VaR\left(1-w\right)\right]\right) \)