In [2]:

    

import numpy as np


    

In [3]:

    

muvec = np.matrix([[.06],[.02],[.04]])
Onevec = np.matrix([[1],[1],[1]])
V = np.matrix([[8, -2, 4], [-2, 2, -2], [4, -2, 8]])/1000
rf = .01


    

In [4]:

    

Xeq = np.matrix([[1/3], [1/3], [1/3]])


    

In [6]:

    

Xeq_return = muvec.T * Xeq
Xeq_return * 100


    

    
Out[6]:





matrix([[ 4.]])

In [7]:

    

Xeq_vol = np.sqrt(Xeq.T * V * Xeq)
Xeq_vol * 100


    

    
Out[7]:





matrix([[ 4.47213595]])

In [22]:

    

Xmin = 1/np.sum(V.I * Onevec) * V.I * Onevec
Xmin


    

    
Out[22]:





matrix([[ 0.16666667],
        [ 0.66666667],
        [ 0.16666667]])

In [24]:

    

Xmin_return = 1/np.sum(V.I * Onevec) * muvec.T * V.I * Onevec
Xmin_return * 100


    

    
Out[24]:





matrix([[ 3.]])

In [26]:

    

muhat = muvec - rf
Xsh = 1/np.sum(V.I * muhat) * V.I * muhat
Xsh


    

    
Out[26]:





matrix([[ 0.29166667],
        [ 0.58333333],
        [ 0.125     ]])

In [28]:

    

Xsh_return = 1/np.sum(V.I * muhat) * muvec.T * V.I * muhat
Xsh_return * 100


    

    
Out[28]:





matrix([[ 3.41666667]])

In [34]:

    

Xsh_var = 1/np.sum(V.I * muhat)**2 * muhat.T * V.I * V * V.I * muhat
Xsh_vol = np.sqrt(Xsh_var)
Xsh_vol *100


    

    
Out[34]:





matrix([[ 2.83823106]])

In [35]:

    

Sharpe_ratio = (Xsh_return - rf)/Xsh_vol
Sharpe_ratio


    

    
Out[35]:





matrix([[ 0.85146932]])

In [38]:

    

opportunity_vol = .05
opportunity_return = rf + Sharpe_ratio * opportunity_vol
opportunity_return * 100


    

    
Out[38]:





matrix([[ 5.25734659]])

In [ ]: