Capital Market Line

The capital market line (CML) represents portfolios that optimally combine risk and return.

def min_func_sharpe(weights, rf=0.015):
    return -statistics(weights, rf)[2]

Function min_func_sharpe export sharpe ratio.

cons = ({'type': 'eq', 'fun': lambda x:  np.sum(x) - 1})
bnds = tuple([(0, 1) for x in range(noa)])
noa * [1. / noa,]

opts = opt.minimize(min_func_sharpe, noa * [1. / noa,], method='SLSQP',
                       bounds=bnds, constraints=cons)

where opts is the optimized point contacting with optimal sharpe ratio level.

when the return of risk free is 0.015, we can make CML with the statement as follow :

rf = 0.015
slope = (statistics(opts['x'])[0] - rf ) / statistics(opts['x'])[1]
var_list = [x*slope + rf for x in np.linspace(0,0.3,5000)]

x = np.linspace(0,0.3,5000)
y = var_list

size of x (=5000) is random value. 

Visualization

plt.figure(figsize=(12,8))
plt.scatter(port_std, port_df,
            c=np.array(port_df) / np.array(port_std), marker='o', cmap='PuBuGn')
plt.plot(tvols, trets, label = 'mean-variance frontier')
plt.plot(x,y, label = 'CML with riskfree asset')
plt.plot(statistics(opts['x'])[1], statistics(opts['x'])[0],
         'r*', markersize=15.0, label = 'Portfolio with highest Sharpe Ratio')
plt.xlim(0.09,0.33)
plt.ylim(0.05,0.2)
plt.legend()
plt.grid(True)
plt.ylabel('expected return$(μ)$')
plt.xlabel('expected std $(σ)$');
plt.colorbar(label='Sharpe ratio')

the optimal CML and mean-variance frontier of economic expansion and recession is following images