Formula in Markowitz Optimization Problem (without riskless asset)

Question

(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

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.