Consider an investor with initial wealth $w$ and has to decide how to invest it. There is a riskless asset with rate of return $r$. The risky asset has return $x_i$ with probability $\pi_i$ for $i=1,2,3,...,n$. Denote by $\alpha$ the fraction of wealth that investor puts into the risky asset, so that $1- \alpha$ is the fraction that he puts in the riskless asset. Write down the investor's optimization problem.
This is a question from my homework test. I want to specifically confirm my answer of part a), I wrote $$max \sum_{i=1}^{n} \pi _{i}[U((1-\alpha )w(1+r)-\alpha w+\sum_{i=1}^{n}x_i]$$
This, we have to maximise w.r.t. $\alpha$ Can anyone confirm?