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Is it possible to calculate Minimum Total Risk mathematically for below problem.

Stock A has 25 percent risk, stock B has 50 percent risk, and their returns are 50  
percent correlated. What fully invested portfolio of A and B has minimum total  
risk?
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I assume that risk it measured here in volatility. Then a portfolio with 100*$w$ percent invested in A and 100*$(1-w)$ percent invested in B has the annual variance $$ v = w^2 0.25^2 + 2* 0.5 w(1-w) 0.25*0.5 + (1-w)^20.5^2. $$

Searching for the portfolio with the samllest variance is equivalent to searching for the smallest volatility.

To get the minimum we take the derivative of $v$ w.r.t. $w$ $$ dv/dw = 2 w 0.25^2 + (1-2w) 0.25*0.5 + (2w-2)0.5^2. $$ Searching the root of this we get that $w^* = 1$.

Thus we should invest all our wealth in stock A and the minimal risk is 25%.

If the volatolity of B were smaller it could reduce risk to invest in B. However as the volatility of B is that high and the correlation is rather large diversification does not work here.

For example the 50/50 portfolio has a volatility of approx 74%. The 80/20 has 59% and 90/10 has 56.7%.

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    $\begingroup$ That no diversification is possible (true in this case even if shorting is feasible) results from the fact that correlation equals the relation of the two volatilities, i.e. $\rho = 0.5= \frac{\sigma_A}{\sigma_B}$. $\endgroup$ – Dr_Be May 17 '16 at 9:18
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    $\begingroup$ Yes, good point. This is right for the correlation. On the other hand not betting on negative correlation - if I have an asset with twice the risk then it would be counterintuitive in any regard that I would reduce risk in buying shares of it. $\endgroup$ – Ric May 17 '16 at 11:36
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You are looking for the minimum variance portfolio of two assets, assuming "risk" translates into volatility (variance) here. So what you would do mathematically speaking is introducing a variable $w\in[0,1]$ which is the weight of stock A (say) in the portfolio, calculate the "risk" - which is the variance - of the portfolio $wA+(1-w)B$ and then solve for the $w_0$ for which then minimum variance is achieved.

So the answer to your formal question is - yes.

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