176k views
3 votes
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?

User Ihor Lavs
by
8.0k points

1 Answer

4 votes

Final answer:

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.

User Djv
by
7.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.