Let the random variable $X$ be defined as $X=0$ with probability $1-p$ and $X=c<0$ with probability $p$.
The Value at Risk for the confidence level $1-\alpha$ (with small $\alpha$) is defined as $V_{\alpha} = \sup \left\{ x\, |\, P( X \leq x ) \leq \alpha \right\}$.
Now if $p>\alpha$, I would say that $V_{\alpha}=c$, and if $p\leq \alpha$ then $V_{\alpha}=0$.
In Solvency II the Solvency Capital Requirement is defined als difference between the expected value and the Value at Risk. Now in case $p\leq \alpha$ the $SCR = pC - 0$ would be negative, which for me is nonsense, I would set $SCR= 0$ in this case.
Do you agree? Is there a special definition of the $VaR$ for dichtomic variables, or is $VaR$ not applicable in this case?
Thanks.
I do not think your problem is related to the variable being dichotomic. You would have the same result for any loss variable, where the VaR is larger than the expected value. Of course this is unusual, maybe even non-sensical, in real applications. Furthermore, it seems you might be confused by signs and/or definitions.
Definition: For SCR calculation VaR is taken of the centred P&L variable $\tilde{X}=X- E[X]$. This makes sense since the expected profit (or loss) is included in the valuation of the SII balance sheet i.e. ultimately in own funds.
Sign: The SCR requirement of a loss should be positive, i.e. if you have natural sign (P&L or losses are negative) $SCR = - VaR[\tilde{X}]$.
In your case $\tilde{X}=X- pc$ and for $p<\alpha$: $VaR[\tilde{X}] = 0 -pc$ so SCR is indeed equal to $pc$ and negative. To convince you this makes sense, see it as "compensation" for the deduction of the expected loss in 1. above.