x=0, y=3, z=1

In case Y=0, use (Y+1). In case Z=0, use (Z+1). X=0, Y=3, Z=1 There are three financial securities: A, B, and C. Each of them has a chance of default (in other words, net return = - 100%) equal to Y%. The event that one of the three security will default is independent of the event that any other security will default. The security A would have a net return of r=0% in 50% of cases, while in left (50-Y)% cases the net return is expected to be Y%. Suppose you buys $1000 worth of security A, which we will refer to as portfolio P1. Suppose that you buy the second portfolio, Pz, for $1000. This portfolio consists of two securities, B and C. If neither B nor C default, this portfolio will have a net return of r=0%. If exactly one of B or C default, it will have a net return of r = -Y0%. If both B and C default, it will have a net return of r = -100%. A) Calculate the Value-at-Risk at the a = Z% critical level for portfolios P, and Pz. B) Calculate the Value-at-Risk at the a = 2% critical level for the joint portfolio P, + Pz. C) Is the Value-at-Risk sub-additive in this example. Show your reasoning. Why do you want your risk measure to be sub-additive? In what case is this property more desirable, in A) or B)? D) Calculate the expected shortfall at the a = 2% critical level for portfolio P2. Is this risk measure sub-additive