There are 3 assets $S_1$, $S_2$ and $S_3$. $S_1$ has a mean return of $0.17$ and standard deviation of $0.2$, $S_2$ has a mean return of $0.13$ and standard deviation of $0.4$ and $S_3$ has a mean return of $0.09$ and standard deviation of $0.15$. The correlation between $S_1$ and $S_2$ is $0.5$, the correlation between $S_1$ and $S_3$ is $0$, and the correlation between $S_2$ and $S_3$ is $0.3$. Given that the risk free rate if $5$%,
(a) Determine the Minimum Variance and Tangency portfolios of these assets, and calculate each portfolio's mean returns and standard deviations.
(b) Write the equation for the efficient frontier of these three assets.
(c) Find the portfolio of $S_1$, $S_2$, $S_3$ that gives the lowest possible variance for a return of $0.13$. Also, find the portfolio that gives the highest possible return for a standard deviation of $0.15$. Calculate the Sharpe ratios of these two portfolios.
(d) Suppose an investor also has the option of investing in a risk-free asset. Repeat the steps in part (c) above and compare the results. What is happening graphically?
Here is what I tried
I derived the correlation matrix of $S_i$ for $i = 1, 2, 3$, which gave
$$C = \begin{pmatrix} 0.04 & 0.04 & 0\\ 0.04 & 0.16 & 0.018\\ 0 & 0.018 & 0.0225\\ \end{pmatrix}$$
The diagonals were the square of the given standard deviations and the remaining entries were calculated with the formula for covariance using correlation.
The weights of the minimum variance portfolio are given by
$$W^* = \frac{C^{-1} O}{O' C^{-1} O}$$
for a covariance matrix $C$ and matrix of ones $O$ (This is the formula I found and not sure if it's actually the right one...)
The result was $0.431, -0.1, 0.669$ as the weights.
For the tangency I first subtracted the risk-free rate from a vector (or $3\times 1$ matrix) of the mean returns, giving:
$$E = \begin{pmatrix} 0.12 \\ 0.08 \\ 0.04 \end{pmatrix}$$
The formula for tangency weights is:
$$W^{**} = \frac{C^{-1} E}{O' C^{-1} E}$$
(This is again the formula I found and not sure if it's actually the right one...)
This gave me $0.692, -0.128, 0.436$.
The means of a portfolio are $M = WR$ where $M$ is the matrix of means and $R$ is the vector of returns. The standard deviations are $S = \sqrt{W^tCW}$.
The results I got were $0.121$ and $0.115$ for the mean and standard deviation of the minimum variance and $0.140$ and $0.130$ for the tangency (Would really appreciate if someone can verify).
I'm unsure how to proceed after this. I got the equation of efficient frontier from the internet to be $R_p = R_f + \frac{E[R_t-R_f]}{\sigma_t}\cdot \sigma_p$ where $R_p$ are returns from portfolio, $R_t$ and $\sigma_t$ are tangency mean returns and standard deviation, $R_f$ is the risk free rate and $\sigma_p$ is the portfolio standard deviations. This gives the equation $R_p = 0.05 + 0.692 \sigma_p$. Is this correct?
Not sure how to proceed with (c) and (d). If anyone can verify (a) and (b) above and help out with (c) and (d) this would be appreciated.