Execute the following steps to find the Efficient Frontier using convex optimization.

1. Import the library:

import cvxpy as cp

2. Convert the annualized average returns and the covariance matrix to numpy arrays:

avg_returns = avg_returns.values
cov_mat = cov_mat.values

3. Set up the optimization problem:

weights = cp.Variable(n_assets)
gamma = cp.Parameter(nonneg=True)
portf_rtn_cvx = avg_returns * weights 
portf_vol_cvx = cp.quad_form(weights, cov_mat)
objective_function = cp.Maximize(portf_rtn_cvx - gamma * portf_vol_cvx)
problem = cp.Problem(objective_function, 
                     [cp.sum(weights) == 1, weights >= 0])

4. Calculate the Efficient Frontier:

N_POINTS = 25
portf_rtn_cvx_ef = np.zeros(N_POINTS)
portf_vol_cvx_ef = np.zeros(N_POINTS)
weights_ef = []
gamma_range = np.logspace(-3, 3, num=N_POINTS)

for i in range(N_POINTS):
    gamma.value = gamma_range[i]
    problem.solve()
    portf_vol_cvx_ef[i] = cp.sqrt(portf_vol_cvx).value
    portf_rtn_cvx_ef[i] = portf_rtn_cvx.value
    weights_ef.append(weights.value)

5. Plot the allocation for different values of the risk-aversion parameter:

weights_df = pd.DataFrame(weights_ef, 
                          columns=RISKY_ASSETS, 
                          index=np.round(gamma_range, 3))
ax = weights_df.plot(kind='bar', stacked=True) 
ax.set(title='Weights allocation per risk-aversion level',
       xlabel=r'$gamma$', 
       ylabel='weight')
ax.legend(bbox_to_anchor=(1,1))

In the following plot, we see the asset allocation for the considered range of risk-aversion parameters:

6. Plot the Efficient Frontier, together with the individual assets:

fig, ax = plt.subplots()
ax.plot(portf_vol_cvx_ef, portf_rtn_cvx_ef, 'g-')
for asset_index in range(n_assets):
     plt.scatter(x=np.sqrt(cov_mat[asset_index, asset_index]), 
                 y=avg_returns[asset_index], 
                 marker=MARKS[asset_index], 
                 label=RISKY_ASSETS[asset_index],
                 s=150)
ax.set(title='Efficient Frontier',
       xlabel='Volatility', 
       ylabel='Expected Returns', )
ax.legend()

The following image shows a plot of the Efficient Frontier, generated by solving the convex optimization problem:

In the There's more... section, we also compare the frontier to the one obtained in the previous recipe, Finding the Efficient Frontier using optimization with scipy.