(hope this is not too basic, I'm new to this forum) Im struggling to understand the optimization problem (global minimum variance portfolio) formula in Markowitz Theory:

$$\arg\ \min\ Var(Return\ x) = [\max_x (-\frac{1}{2} x^{\mathrm{T}}Vx)]$$

The only thing I dont understand is where the -1/2 is coming from, in all the sources I could find it wasn't explained and just taken as given...

Thanks in advance


1 Answer 1


Note that the solution to the problem is the same with or without the $\frac{1}{2}$, since multiplying it only changes the value of the objective function, but not where its extrema are located. The "$-$" comes from the fact that you switched it from a minimization problem to a maximization problem.

The reason behind the choice of $\frac{1}{2}$ simply makes the derivative nicer. Take the 2-dimensional case, where $x$ is squared. Then the $\frac{1}{2}$ cancels out the $2$ from taking the derivative with respect to $x$, and the first order conditions look neater.


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