Question

Suppose that the profit of a portfolio of assets (X) has the probability mass function P(X=10)=0.1, P(X=5)=0.3,P(X=−2)=0.4,P(X=−10)=0.17,P(X=−20)=0.03. (i) Write down the equation and plot the CDF of X. (ii) Calculate the 4% VaR of the profit X. (iii) Calculate the expected shortfall at 4% level of the profit X. (iv) Calculate the tail conditional expectation at 4% level of the profit X

Answer 1

To answer the given question, we need to understand what a probability mass function (PMF) and a cumulative distribution function (CDF) are. We can then write down the PMF and plot the CDF of the profit of a portfolio of assets. We can calculate the 4% Value at Risk (VaR), the Expected Shortfall, and the Tail Conditional Expectation at the 4% level of the profit X.

Step-by-step explanation:

In order to answer the given question, we need to understand what a probability mass function (PMF) and a cumulative distribution function (CDF) are.

(i) The PMF is a function that gives the probability that a discrete random variable takes on a specified value. In this case, the PMF for the profit of the portfolio of assets (X) is given as:

P(X=10)=0.1, P(X=5)=0.3, P(X=-2)=0.4, P(X=-10)=0.17, and P(X=-20)=0.03.

(ii) The CDF is a function that gives the probability that a random variable takes on a value less than or equal to a given value. To plot the CDF, we calculate the cumulative probabilities for each value of X:

P(X<=10)=0.1, P(X<=5)=0.4, P(X<=-2)=0.8, P(X<=-10)=0.97, and P(X<=-20)=1.0. We can then plot these cumulative probabilities on a graph.

(iii) The 4% Value at Risk (VaR) of the profit X is the largest value of X such that the probability of X being less than or equal to that value is less than or equal to 4%. To calculate the 4% VaR, we find the value of X for which P(X<=X) = 0.04. In this case, the 4% VaR is -2.

(iv) The Expected Shortfall at the 4% level of the profit X is the average value of X for which the probability of X being less than or equal to that value is less than or equal to 4%. To calculate the Expected Shortfall, we find the average value of X for which P(X<=X) = 0.04. In this case, the Expected Shortfall is -5.325.

(v) The Tail Conditional Expectation (TCE) at the 4% level of the profit X is the conditional expected value of X given that X is less than or equal to the 4% VaR. To calculate the TCE, we find the conditional expected value of X given that X<=-2. In this case, the TCE is -8.125.