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Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=begin{bmatrix} w_1 \ w_2 \ w_3 end{bmatrix} $, expected returns $R=begin{bmatrix} mu_1 \ mu_2 \ mu_3 end{bmatrix}$, and a covariance matrix $V$.

The expected return of my portfolio is $mu_p=W^TR$ and the variance of my portfolio is $sigma^2_p=W^TVW$.

I would like to run Monte Carlo simulations on my portfolio using a normal distribution.

I can do this either by:

1. Sampling from the distribution of portfolio returns $N(mu_p,sigma^2_p)$.


2. Sample from the three individual asset returns and use those three returns to compute my overall portfolio return.

First, how would I accomplish the second approach (would I be sampling from a multivariate normal distribution)?

Second, are these two approaches equivalent as long as I assume that the weights $W$ of my portfolio remain the same?

For the first case, you would directly sample $n$ random normals $x$ and compute:
$$R^p_i = mu_p + sigma_p x_i, i in [1,n]$$

For the second case, you can sample $n$ x $3$ independent normals, compute the Cholesky decomposition matrix $C$ of $V$, which is the matrix $C$ such that $V=C^t C$, and get $n$ samples of vectors $X$ of size 3.

The return $R_i$ for random draw $i$ is given by:
$$R_i = mu_p + C . X_i, i in [1,n]$$
You can check for high values of $n$ the convergence towards the limit values:
$$E(R_i) = R$$
$$Cov(R_i) = V$$
The portfolio return is then computed as:
$$R^p_i = W.T R_i$$
and you can check it converges towards the same mean and variance $mu_p$, $sigma_p^2$ for a large enough $n$.

The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals.

Basically what @sebapi said.
"The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals."

Q: How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

A: You might use scipy.stats.multivariate_normal (rv = multivariate_normal(mean=None, cov=1, allow_singular=False))