Question

Let rho₂ (X)=−E(X)+α

Var(X), where α>0 is a constant. Check the monotonicity, translation invariance, homogeneity and subadditivity of rho₂ (X). Is rho₂ a coherent risk measure?

Answer 1

The function rho₂(X) combines the expected value and variance of a random variable X to measure risk. We can check its properties to determine if it is a coherent risk measure.

Step-by-step explanation:

The function rho₂(X)=−E(X)+α Var(X) is a risk measure that combines the expected value and variance of a random variable X. Let's check its properties:

Monotonicity: To check if rho₂(X) is monotonic, we need to verify if rho₂(X) ≤ rho₂(Y) whenever X ≤ Y. This can be proven by comparing the expected values and variances of X and Y.

Translation invariance: To check if rho₂(X) is translation invariant, we need to verify if rho₂(X+c) = rho₂(X) + c for any constant c. This can be proven by substituting X+c into the formula and simplifying it.

Homogeneity: To check if rho₂(X) is homogeneous, we need to verify if rho₂(aX) = a * rho₂(X) for any constant a. This can be proven by substituting aX into the formula and simplifying it.

Subadditivity: To check if rho₂(X) is subadditive, we need to verify if rho₂(X + Y) ≤ rho₂(X) + rho₂(Y) for any two random variables X and Y. This can be proven by comparing the expected values and variances of X + Y, X, and Y.

Based on these properties, rho₂(X) is a coherent risk measure if it satisfies all of these properties.