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3. a assume daily returns that are normally distributed with constant...

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

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

${\mathbit{R}}_{\mathbf{t}\mathbf{+}\mathbf{1}}\mathbf{=}\mathbit{\sigma }{\mathbit{v}}_{\mathbf{t}\mathbf{+}\mathbf{1}}$ ,

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.

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

where Φ is the standard normal cumulative density function.

(b) The expected shortfall at the critical level α% and 1-day horizon can be defined as

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

where φ is the standard normal probability density function. Hint: From the properties of the normal distribution we know that if Z is normally distributed

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