Here we are maximising the expected utility by choosing the proportion of money 0<a<1 to investin the Asset A. So, the investor invests a*w0 in Asset A and the remaining (1-a)*w0 in Asset B. Suppose Scenario A prevails. If the investor invests aw0 in Asset A and it returns -100%, then the value of that investment will be zero. If she invests (1-a)w0 in Asset B and it returns +50%, then the value of that investment will be 1.5(1-a)w0. So, if Scenario A arises she ends up with wealth of 0 + 1.5(1-a)w0, which gives her a utility of ln[0 + 1.5(1-a)w0]. Conversely if Scenario B occurs, then if she invests aw0 in Asset A and it returns +300%, then the value of that investment will 4aw0. If she invests (1-a)w0 in Asset B and it returns +50%, then the value of that investment will again be (1-a)w0. So, if Scenario B arises she ends up with wealth of 4aw0 + 1.5(1-a)w0, which gives here a utility of ln[4aw0 + 1.5(1-a)w0]. Each scenario has a probability of 1/2 of arising and so her expected utility is: EU = 1/2 * ln[0 + 1.5(1-a)w0] + 1/2 * ln[4aw0 + 1.5(1-a)w0] We then find the value of "a" that maximises her expected utility by differentiating with repsect to a, setting the resulting expression to zero and solving for a.
