Hey all - These things have appeared on several exams, and I never gave them much thought. Now that I'm looking a little closer, I don't understand how monotonicity and positive homogeneity can both be true.
Definitions:
Monotonicity: If W1 <= W2, then p(W1) >= p(W2)
Homogeneity: For a>0, p(a*W) = a*p(W)
I'm thinking that if W2 is always greater than or equal to W1, then there is some constant (let's call it 'a'), greater than or equal to 1, such that a*W1 = W2.
If W2 = a*W1, it follows that p(W2) = p(a*W1)
And p(a*W1) = a*p(W1), by homogeneity
But a*p(W1) >= p(W1), because a >=1
So p(W2) = a*p(W1) >= p(W1), which contradicts monotonicity.
I must be missing something, but this is bothering me. Any thoughts?
Definitions:
Monotonicity: If W1 <= W2, then p(W1) >= p(W2)
Homogeneity: For a>0, p(a*W) = a*p(W)
I'm thinking that if W2 is always greater than or equal to W1, then there is some constant (let's call it 'a'), greater than or equal to 1, such that a*W1 = W2.
If W2 = a*W1, it follows that p(W2) = p(a*W1)
And p(a*W1) = a*p(W1), by homogeneity
But a*p(W1) >= p(W1), because a >=1
So p(W2) = a*p(W1) >= p(W1), which contradicts monotonicity.
I must be missing something, but this is bothering me. Any thoughts?
Coherent Risk Measures
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