python - 如何使用 CAPM 在 Python 中找到最佳投资组合的预期收益？

def show_optimal_portfolio(optimum, returns, preturns, pvariances): plt.figure(figsize=(10,6)) plt.scatter(pvolatility,preturns,c=preturns/pvolatility,marker='o') plt.grid(True) plt.xlabel('Expected Volatility') plt.ylabel('Expected Return') plt.colorbar(label='Sharpe Ratio') plt.plot(statistics(optimum['x'],returns)[1],statistics(optimum['x'],returns)[0], 'bs',markersize=10.0) plt.show() show_optimal_portfolio(optimum, returns, preturns, pvolatility) def min_func_sharpe(weights,returns): # [2] means that we want to maximize according to the Sharpe-ratio return -statistics(weights,returns)[2] # constraints remain the same in optimising based on variance constraints = ({'type':'eq','fun': lambda x: np.sum(x)-1}) #the sum of weights is 1

optimum_sr=optimization.minimize(fun=min_func_sharpe,x0=weights,args=returns,method='SLSQP',bounds = bounds,constraints=constraints) print("Optimal weights (Sharpe Ratio based):", optimum_sr['x'].round(3)) print("Expected return, volatility and Sharpe ratio (Sharpe Ratio based):", statistics(optimum_sr['x'].round(3),returns)) def show_optimal_portfolio(optimum, optimum_sr, returns, preturns, pvolatility): plt.figure(figsize=(10,6)) # defining figure size plt.scatter(pvolatility,preturns,c=preturns/pvolatility,marker='o') # creating scatter plot plt.grid(True) plt.xlabel('Expected Volatility') # labelling x axis plt.ylabel('Expected Return') # labelling y axis plt.colorbar(label='Sharpe Ratio') # colorbar that depends on the range of sharpe ratio values plt.plot(statistics(optimum['x'],returns)[1],statistics(optimum['x'],returns)[0], 'gs',markersize=10.0) # plotting a green square to show the optimum value depending on minimum variance plt.plot(statistics(optimum_sr['x'],returns)[1],statistics(optimum_sr['x'],returns)[0], 'b*',markersize=20.0) # plotting a blue star to show the optimum value depending on sharpe ratio plt.show() show_optimal_portfolio(optimum, optimum_sr, returns, preturns, pvolatility) def min_func_variance_target(weights, returns):return statistics(weights, returns)[1] **2

optimum_trg=optimization.minimize(fun=min_func_variance_target,x0=weights,args=returns,method='SLSQP' ,bound =bounds,constraints=constraints) print("Optimal weights:", optimum_trg['x'].round(3)) print("Expected return, volatility and Sharpe ratio:", statistics(optimum_trg['x'].round(3),returns)) def show_optimal_portfolio_trg(optimum_trg, returns, preturns, pvariances): plt.figure(figsize=(10,6)) plt.scatter(pvolatility,preturns,c=preturns/pvolatility,marker='o') plt.grid(True) plt.xlabel('Expected Volatility') plt.ylabel('Expected Return') plt.colorbar(label='Sharpe Ratio') plt.plot(statistics(optimum_trg['x'],returns)[1],statistics(optimum_trg['x'],returns)[0], 'gs',markersize=10.0) plt.show() show_optimal_portfolio_trg(optimum_trg, returns, preturns, pvolatility)

risk_free_rate = 0.00 start=dt.datetime(2011,1,1) end=dt.datetime(2020,12,31) #.today() tickers = ['SBUX','MSFT','REGN', 'PYPL', 'F','Market'] names = ['Starbucks', 'Microsoft','Regen', 'Paypal','Ford', 'Market'] #get the data from Yahoo Finance data = web.DataReader(tickers,data_source='yahoo',start=start,end=end)['Adj Close'] data = np.log(data/data.shift(1)) data.columns = names data = data.dropna() #covariance matrix: the diagonal items are the variances - off diagonals are the covariances #the matrix is symmetric: cov[0,1] = cov[1,0] varcov = data.cov() varcov = np.array(varcov) #calculating beta according to the formula beta_starbucks = varcov[0,5]/varcov[5,5]#= cov(x,m)/var(m) beta_microsoft = varcov[1,5]/varcov[5,5]#B_m=cov(msft, market)/var(market) beta_regen = varcov[2,5]/varcov[5,5] beta_paypal = varcov[3,5]/varcov[5,5] beta_ford = varcov[4,5]/varcov[5,5] print("Beta from formula:", beta_starbucks) print("Beta from formula:", beta_microsoft) #using linear regression to fit a line to the data [stock_returns, market_returns] - slope is the beta beta_starbucks,alpha_starbucks = np.polyfit(data['Market'], data['Starbucks'], deg=1) print("Beta from regression:", beta_starbucks) print("Alpha from regression:", alpha_starbucks) #plot plt.figure(figsize=(10,6)) plt.scatter(data['Market'], data['Starbucks'], label="Data points") plt.plot(data['Market'], alpha_starbucks + beta_starbucks*data['Market'], color='green', label="CAPM Line") plt.title('Capital Asset Pricing Model', fontsize=18) plt.xlabel('Market return $r_m$', fontsize=18) plt.ylabel('Stock return(Starbucks) $r_i$', fontsize=18) plt.text(0.05, 0.05, r'$r_i = \alpha + \beta r_m$', color='red', fontsize=18) plt.legend() plt.grid(True, axis='both') plt.show() #calculate the expected return according to the CAPM formula expected_return = risk_free_rate + beta_starbucks*(data['Market'].mean()*365-risk_free_rate) print("Expected return:", expected_return)