An investor endowed with an initial wealth \(W_{0}=1\) (e.g., in euro) maximizes the expected value of the quadratic utility function \(u(x)=a x -\quad b x^{2} /2\), where \(x\) is the terminal wealth obtained by investing in \(n\) risky assets. Accordingly, the investor chooses a portfolio. Another investor, with a different initial wealth of \(W_{0}=K\), by optimizing the same utility function, chooses a different portfolio (in the sense that the asset weights are different).

- How do you explain the difference?

- If the second investor changes the coefficient \(b\) to \(b\), he finds the same portfolio as the first investor. What is the relationship between \(b\) and \(b\) ?

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fx(x) a X b FIGURE 7.5 PDF of a triangular distribution.