TABLE OF CONTENTS In this chapter, we present a formal functional derivation of the VAR and CVAR equations for the linear case. We pay particular attention to the CVAR volatility. The derivation is done for in the continuous multivariate framework. This shows that CVAR uncertainties are present in the limit of an infinite-length Monte-Carlo (MC) simulation run. We indicate extensions for non-linear exposures (convexity) to VAR, as discussed in the last chapter. We end with a summary of the extension to multiple time steps [1].
To perform the calculations, we need the multivariate Gaussian probability distribution for one time step. The time difference of an underlying variable x ? is d t x ?( t)= x ?( t+ dt) ?x ?( t) at fixed time t. This is the return if x ? = ln r ?. To apply the formalism to the Stressed VAR, we would specify the vol ? ? of d t x ? as a fat-tail vol ? ?; FT, as discussed in Ch. 21. We would also specify the d t x ?, d t x ? correlation ? ?? as the stressed correlation ? ?? ( stressed), as described Ch. 23, 24.
The probability is then an integral over all possible values of each d t x ?, involving the measure d( d t x
