Question: a assume daily returns that are normally distributed with constant...

(a) Assume daily returns that are normally distributed with constant mean and variance, i.e. they are given by

${\mathit{R}}_{\mathbf{t}\mathbf{+}\mathbf{1}}\mathbf{=}\mathit{\sigma }{\mathit{v}}_{\mathbf{t}\mathbf{+}\mathbf{1}}$ ,${\mathit{v}}_{\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{ }\mathbf{ }}\stackrel{\mathbf{i}\mathbf{.}\mathbf{i}\mathbf{.}\mathbf{d}}{\mathbf{~}}\mathit{N}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$

where the time increment t + 1 is 1 day. Derive the following formula for the Value-at-Risk at the α% (VaR) critical level and 1-day horizon.

$\mathit{V}\mathit{a}{\mathit{R}}_{\mathbf{t}\mathbf{+}\mathbf{1}}^{\mathbf{a}}\mathbf{=}\mathbf{-}{\mathit{\sigma }}_{\mathbf{t}\mathbf{+}\mathbf{1}}{\mathit{\Phi }}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathit{a}\mathbf{\right)}$

where Φ is the standard normal cumulative density function. 

(b) The expected shortfall$E{S}_{t+1}^{\alpha }$ at the critical level α% and 1-day horizon can be defined as

${\mathbf{ES}}_{\mathbf{t}\mathbf{+}\mathbf{1}}^{\mathbf{\alpha }}\mathbf{=}\mathbf{ }\mathbf{-}{\mathbf{E}}_{\mathbf{t}}\mathbf{\left[}{\mathbf{R}}_{\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{ }}\mathbf{\right|}\mathbf{ }{\mathbf{R}}_{\mathbf{t}\mathbf{+}\mathbf{1}}\mathbf{ }\mathbf{<}\mathbf{ }\mathbf{-}{\mathbf{VaR}}_{\mathbf{t}\mathbf{+}\mathbf{1}}^{\mathbf{\alpha }}\mathbf{ }\mathbf{]}\mathbf{.}$

Using the V aR formula from part (a) derive the following formula for the 1-day expected shortfall at critical level α 

$\mathit{E}{\mathit{S}}_{\mathbf{t}\mathbf{+}\mathbf{1}}^{\mathbf{a}}\mathbf{=}\frac{{\mathbf{\sigma }}_{\mathbf{t}\mathbf{+}\mathbf{1}}}{\mathbf{a}}\mathit{\phi }\mathbf{(}{\mathit{\Phi }}^{\mathbf{-}\mathbf{1}}\mathbf{ }\mathbf{(}\mathit{\alpha }\mathbf{\left)}\mathbf{\right)}$

where φ is the standard normal probability density function. Hint: From the properties of the normal distribution we know that $\mathit{E}\mathbf{[}\mathit{Z}\mathbf{ }\mathbf{|}\mathbf{ }\mathit{Z}\mathbf{ }\mathbf{\le }\mathbf{ }\mathit{A}\mathbf{]}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{-}\mathbf{ }\frac{\mathbf{\phi }\mathbf{\left(}\mathbf{A}\mathbf{\right)}}{\mathbf{ }\mathbf{\Phi }\mathbf{(}\mathbf{A}}\mathbf{ }$ if Z is normally distributed