Suppose you can invest in only the market index fund and 1-month Treasury bill because of some regulation. Assume E[r]=10%, SD[r]=20%, and rf=2%.
You become a new portfolio manager at Fidelity. You heard your predecessor chose w=0.5 (=the weight on the market index fund) to please the CEO.
You also need to construct the portfolio that will please the CEO, but now E[r]=13%, SD[r]=10%, and rf=3%. So, w=0.5 will not be the optimal portfolio for the CEO any more.
What value of w should you pick, given the portfolio theory you learned if the CEO's risk preference (= price of risk) does not change?