Markovitz Portfolio Optimization- Why do we associate risk with $\Sigma$? I'm trying to interpret Markovitz portfolio optimization problems. The specific problem that I'm looking at is as follows: minimize $x^T\Sigma x$ subject to $p^Tx\geq p_{\min}$ $x^T1=1$ $x\geq0$ I know that I can interpret this as minimizing the risk, $x^T\Sigma x$, of the portfolio, subject to the constraint that the return, $p^T x$, is greater than a minimum amount, $p_{\min}$. Additionally, no shorting is allowed, which corresponds to the constraint that $x\geq0$. I don't understand the purpose of the remaining constraint; namely, that $x^T1=1$. I also don't understand why we write $p^T x$ rather than simply writing $p$. Furthermore, why do we associated the risk of the portfolio with the covariance matrix $\Sigma$, and why do we write it as the quadratic form $x^T\Sigma x$, rather than simply writing $\Sigma$?

The $x$ are a column vector of _weights_ on each asset in the portfolio. The constrant $1^Tx=1$ means the elements of $x$ all add up to one, i.e. the weights add up to one like weights should.

$p$ is a column vector of returns on the assets. To get the porfolio return, we take the average return weighted by the portfolio weights. This is precisely what $p^T x$ is.

$\Sigma$ is the covariance matrix of the asset returns, i.e. it is $E(pp^T)- E(p)E(p^T).$ The variance of portfolio return is $$E((p^Tx)^2)-E(p^Tx)^2 = x^T(E(pp^T)-E(p)E(p^T))x = x^T\Sigma x.$$