A portfolio manager wants to create portfolio with all three securities by minimizing its variance, while she has set target portfolio return of $0\mathrm{.}10($i$.$e$.,$ E$($rp$)=10\%).$ Which are the exact weights $($in terms of wealth$)$ she will place to each security?

$($i$).$ What$$s the standard deviation of her portfolio?

$($b$)$ Suppose a risk averse investor can choose a portfolio from among N assets with independently distributed returns, all of which have identical expected returns $[$i$.$e$.,]$ and identical variances $($i$.$e$.,.$ What will be the composition of her minimum variance portfolio if the only constraint considered is that the weights add to $100\%?$ Use calculus to mathematically prove your $($for part A you need to use the Lagrangian Method$)$