Market portfolio and portfolio with three risky assets

Question

I'm trying to solve a problem with portfolios, but I cannot get to the solution. There are three risky assets A, B and C whose betas are respectively 0.6 (A), 1.5 (B) and 1.1 (C). Besides, we know there's a risk-free asset with a 1% return. The market portfolio has a 5% return and a volatility of 13%.

I am asked to find the $\beta$ and the correlation between the market portfolio and the portfolio given by the combination of assets A, B and C, whose weights are respectively 30% (A), 50% (B) and 20% (C). This portfolio has a volatility of 16%.

I tried to use the data on the betas and the volatility of the market portfolio to find the covariance between the risky assets and the market, but this doesn't help me. In fact, I always miss one value to apply the formulas such as:

$\sigma_p^2=w_a^2\sigma_a^2+w_b^2\sigma_b^2+w_c^2\sigma_c^2+2w_aw_bCov(a,b)+2w_aw_cCov(a,c)+2w_bw_cCov(b,c)$

$Cov(a,m)=\rho_{a,m}\sigma_a\sigma_m$

So far, I computed the values of the Covariance of A, B and C with the market using the definition of beta:

$\beta_a=Cov(a,m)/\sigma_m^2 , \\Cov(a,m)=0.010, \\Cov(b,m)=0.025, \\Cov(c,m)=0.019$

What do I miss to find the $\beta$ between the market and the portfolio and their correlation?

Thanks in advance for your help!

Answer 1

$\beta_{portfolio} = 0.6 \times 0.3 + 1.5 \times 0.5 + 1.1 \times 0.2 = 1.15$

$\rho_{portfolio,market} = 1.15 \times \frac{0.13}{0.16} = 0.93$

Edit: To reply to the comment: $\sigma^2_{port} = \beta_{port}^2 \sigma_m^2 + Idiosyncratic risk$

So just do $\frac{\beta_{port}^2 \sigma_m^2}{\sigma^2_{port}}$