Write down the mathematical definition of VaR (value-at-risk) and derive ES (expected shortfall).

What is a coherent risk measure?

Suppose you own two assets, \(A\) and \(B\), with payoffs that are independent of each other, with each asset returning either 0 with probability 0.9 or -100. Is VaR(5%) sub–additive?

Consider the assets in the last question. Is VaR(15%) sub–additive?

Give one example of a traded asset that could lead to a sub–additivity violation of VaR, carefully explaining why the payoff structure of this asset would lead to that conclusion.

Suppose you own 1$ million worth of both stocks A and B. Stock A is a small cap stock, with a market capitalization of 3 million, while stock B is a large cap stock, with a market capitalization of 3 billion. As a consequence, one of the axioms of a coherent risk measure is likely to be violated for VaR on one of the stocks but not the other. Which stock is it and why would that be the case?

State the three steps in VaR calculations.