Portfolio Expected Return and Utility
Consider a portfolio consisting of two assets, \(A\) and \(B\), with returns \(r_A\) and \(r_B\). The portfolio return is
\[ r_P = w_A r_A + w_B r_B \]
where \(w_A + w_B = 1\). Assume \(r_A\) and \(r_B\) are independent and identically distributed with mean \(\mu\) and variance \(\sigma^2\). Let \(U(\cdot)\) be a concave utility function representing a risk-averse investor.
Part 1
Show that the expected portfolio return is equal to the weighted average of the expected returns of the individual assets:
\[ \mathbb{E}[r_P] = w_A \mathbb{E}[r_A] + w_B \mathbb{E}[r_B]. \]
Part 2 (Extension)
Compare the expected utility of the portfolio, \(\mathbb{E}[U(r_P)]\), with the utility of the expected return, \(U(\mathbb{E}[r_P])\). Explain why
\[ \mathbb{E}[U(r_P)] \leq U(\mathbb{E}[r_P]) \]
and how this motivates diversification for a risk-averse investor.