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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

Rt+1=σvt+1 ,vt+1  ~i.i.dN(0,1)

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.

VaRt+1a=-σt+1Φ-1(a)

where Φ is the standard normal cumulative density function. 

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

ESt+1α= Et[Rt+1 | Rt+1 < VaRt+1α ].

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

ESt+1a=σt+1aφ(Φ-1 (α))

where φ is the standard normal probability density function. Hint: From the properties of the normal distribution we know that E[Z | Z  A] =  φ(A) Φ(A  if Z is normally distributed

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