How to calculate weight of two stocks without knowing their correlation?

I have difficulty to solve attached question. The question asks me to find new weight of stocks by just changing standard deviation of the portfolio. I really do not have any idea how to solve it. Maybe the key point is the risk-free asset but I am not sure, can somebody give me a hint?

You are supposed to create a new portfolio using the tangency portfolio $P_t$ and the risk-free rate $r$.

You know that the volatility of the tangency portfolio $\sigma_{P_t}=0.20$.

You also know that the risk-free asset has:

• No risk: $\sigma_r=0$
• Is not correlated with anything $\rho_{P_t,r} = 0$

So you're asked to create a portfolio with a higher risk, which means you are going to need to borrow some money to buy more of the tangency portfolio (you leverage).

You know that, by definition of volatility, the volatility of a portfolio $P$ formed of the tangency portfolio with weight $w$ and the risk-free asset with weight $1-w$ is :

$$\sigma_P^2 = w^2 \sigma_{P_t}^2 + (1-w)^2 \sigma_r^2 + 2w(1-w)\sigma_{P_t}\sigma_r\rho_{P_t,r} = w^2 \sigma_{P_t}^2$$

You hence get

$$w = \frac{\sigma_P}{\sigma_{P_t}} = \frac{0.24}{0.20} = 1.2$$

So the overall weight of the tangency portfolio will be $w=1.2$, the weight of the risk-free asset will be $1-w = -0.2$ (short). Within tangency portfolio, nothing has change in terms of weighting so you just multiply the 1.2 by the original weights which yields $w_A = 0.60 \cdot 1.2 =0.72$ and $w_B = 0.40 \cdot 1.2 =0.48$.

• I do not know how to thank you my friend. All the best wishes for you.. – auhan Dec 21 '15 at 2:51