Let the random variable X be the payoff an investment that may involve gain (X 0) as well as loss (X < 0).

(a) Show that (X) = E[X], i.e. the expected loss, is a coherent risk measure. Note here positive risk means loss.

(b) The standard deviation of X (assuming it exists), denoted as (X), is often used as a risk measure of the investment. Is (X) = (X) a coherent risk measure? If yes, prove it. If no, give counterexamples for all the violated axioms.

(c) The risk measure (X) = E[X] + (X) combines the expected loss with the spread (standard deviation) of loss. However, show that (X) is not a coherent risk measure by giving counterexamples for all the violated axioms.1

(d) Now let X be the loss, so X > 0 means loss and X 0 means gain. A widely used risk measure is the so-called Value-at-Risk (VaR). VaR of loss X with confidence level (0,1) is defined as V aR(X) = min{z|P(X z) }, where FX(z) = Pr(X z) is the cumulative distribution function of X. In other words, V aR(X) is the lower -percentile of X. Suppose X is a random loss uniformly distributed on [-$10, $50]. Compute V aR(X) as a function of . Now suppose the loss X is an exponential random variable with density f(x) = ex for x 0. Calculate the V aR(X) as a function of , .

(e) Although VaR has its intuitive appeal, it has serious drawbacks. First, show that V aR(X) is not a coherent risk measure by giving examples of random variable X whose VaR violates axioms. Second, the VaR is usually computationally difficult to evaluate.

(f) Another very popular risk measure is called the conditional-value-at-risk (CVaR). CVaR of loss X with confidence level (0,1) is defined as CV aR(X) = E[X|X V aR(X)] i.e. it is the expected loss X conditional on that the loss is at least V aR(X). By definition, CVaR is larger than VaR. So if we can minimize CVaR, then automatically VaR is also minimized. It turns out that CVaR is a coherent risk measure. However, proving this is not trivial. Calculate CV aR(X) for the uniform and exponential distributions in (d). You need to write down the explicit forms of CVaR.