Setup
Let variables $X$ and $Y$ to be the returns on two assets that following some unknown distributions and $Z = w_x * X + w_y * Y$ to be the return of a portfolio, with proportion $w_x$ and $w_y$ invested in $X$ and $Y$ respectively.
Now by basic property of variance we have (1) $$Var(Z) = w_x^2 Var(X) + w_y^2Var(Y) + 2w_xw_yCov(X, Y)$$
Problem
If we have two vectors $\textbf{x}$ and $\textbf{y}$ representing the historical returns of the two assets, then it follows that $\textbf{z} = w_x\textbf{x} + w_y\textbf{y}$ is the historical return of the portfolio.
If we want to estimate $Var(Z)$ based on the historical information, there are two possible ways.
Using formula (1) by estimating $Var(X)$, $Var(Y)$, $Cov(X, Y)$ with sample statistics $var(\textbf{x})$, $var(\textbf{y})$, and $cov(\textbf{x}, \textbf{y})$.
Calculating the sample variance directly from $\textbf{z}$.
According to this blog post, when estimating the variance of a portfolio, one should always use the first method. While I can understand why the first method is correct, I fail to see why the second is incorrect.
Afterall, $\textbf{z}$ is a sample of $Z$. Why cannot we just estimate the population variance directly from sample variance?