Portfolio Variance of a Portfolio of N Assets in Python

Portfolio variance is calculated as:

port_var = W'_p * S * W_p

for a portfolio with N assest where

W'_p = transpose of vector of weights of stocks in portfolios
S = sample covariance matrix
W_p = vector of weights o开发者_如何学Cf stocks in portfolios

I have the following numpy matrixes.

Array (vector) of weights of stocks in the portfolio (there are 10 stocks):

weights = np.array(
     [[ 0.09],
      [ 0.05],
      [ 0.15],
      [ 0.10],
      [ 0.15],
      [ 0.15],
      [ 0.08],
      [ 0.08],
      [ 0.1 ],
      [ 0.05]])

Covariance matrix of stock returns:

covar = np.array([[ 0.00154474  0.00079555  0.00099691  0.00052596  0.0005363   0.00062005
0.00064031  0.00037494  0.00018826  0.00132809],
[ 0.00079555  0.00287429  0.00058536  0.00091774  0.00046885  0.00110434
0.00137141  0.00046724  0.00030414  0.0016615 ],
[ 0.00099691  0.00058536  0.00155757  0.00056336  0.00052395  0.00060104
0.00057223  0.00021365  0.00017057  0.00130247],
[ 0.00052596  0.00091774  0.00056336  0.00126312  0.00031941  0.00088137
0.00024493  0.00025136  0.00011519  0.00135475],
[ 0.0005363   0.00046885  0.00052395  0.00031941  0.00054093  0.00045649
0.00042927  0.00021928  0.00016835  0.00093471],
[ 0.00062005  0.00110434  0.00060104  0.00088137  0.00045649  0.00133081
0.00060353  0.0003967   0.00024983  0.00168281],
[ 0.00064031  0.00137141  0.00057223  0.00024493  0.00042927  0.00060353
0.00468731  0.00059557  0.00020384  0.00078669],
[ 0.00037494  0.00046724  0.00021365  0.00025136  0.00021928  0.0003967
0.00059557  0.00082333  0.00017191  0.00066816],
[ 0.00018826  0.00030414  0.00017057  0.00011519  0.00016835  0.00024983
0.00020384  0.00017191  0.00036348  0.0004505 ],
[ 0.00132809  0.0016615   0.00130247  0.00135475  0.00093471  0.00168281
0.00078669  0.00066816  0.0004505   0.00530036]])

When I compute

weights.T * covar * weights

The result is an array the same size as covar. I am new to portfolio theory but I would imagine that the variance of the portfolio should be a scalar (single value).

Does anyone have experience with this that might help?

---

np.dot(weights.T,np.dot(covar,weights))
# array([[ 0.00064654]])

For 2D numpy arrays, np.dot is equivalent to matrix multiplication.

For a 2D array np.dotted with a 1D array, np.dot is equivalent to matrix-vector multiplication.

For 1D arrays, np.dot is equivalent to the inner product.

For numpy arrays, the * performs element-wise multiplication (with broadcasting if necessary).

---

weights.T*np.matrix(covar)*weights
#matrix([[ 0.00064654]])

Alternatively, if you convert covar to a np.matrix, then * is equivalent to matrix multiplication.