May you can help me undertanding the following conclusion: Suppose we have an agent who has preferences over contingent claims, represented by a concave function $U$. This simply means that $\mathbb{E}U(X)\le\mathbb{E}U(Y)$ where $X,Y$ represent two claims. Now suppose that if two contingent claims have the same mean and the same variance they are indifferent for the agent.

If we consider distributions concentrated on three points than the function $U$ must be quadratic.

I do not understand the last part, may you could show me that mathematically?



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