Finding the Minimum-Variance Mix of Two Correlated Stocks
Summary
The document explains how to find a fully invested, long-only portfolio with minimum variance when the volatilities and correlation of two stocks are known. Let the weight in one stock vary from zero to one, express portfolio variance using both assets’ variances and their covariance, and minimize that quadratic function. The accepted answer differentiates the expression and finds that the minimum occurs at the boundary: all wealth is allocated to the lower-volatility stock. It interprets the result as a case where the higher volatility and positive correlation of the second asset prevent diversification from improving the outcome.
The explanation assumes “risk” means volatility or variance and constrains weights to a fully invested portfolio without short selling. It also gives example allocations, but their stated volatility figures do not agree with the stated inputs and variance formula. The method is useful; those examples should be checked independently. The result depends on the inputs and constraints, so it does not establish that holding only one asset is generally optimal.
Key ideas
- Portfolio variance depends on each asset’s variance, portfolio weights, and covariance between the assets.
- For two fully invested assets, vary one weight and minimize the resulting variance function.
- Under the stated long-only constraint, the calculation places the entire allocation in the lower-volatility stock.
- Positive correlation and a much higher volatility for one asset can make diversification fail to reduce risk.
- The example portfolio volatility figures in the document conflict with its stated inputs and formula.
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Full text
# How to Calculate Minimun total Risk? # How to Calculate Minimun total Risk? 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? ``` ## Answer by Richi Wa (score 2, accepted) https://quant.stackexchange.com/a/26070 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%. ## Answer by Dr_Be (score 0) https://quant.stackexchange.com/a/26069 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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