I understand the concept of the efficient frontier and am able to calculate it in Python. But even when generating 50'000 random 10 asset portfolios, the single portfolios are not even close to the efficient frontier.
I see that, for example, the maximum sharpe ratio portfolio has very pronounced allocation (most of the 10 asset get 0 allocation).
Since this work is very critical for myself I just wanted to ask the community if you experienced similar behaviour? Is it normal that when generating random portfolios not even one lies near the efficient frontier?
[![Portfolios vs Efficient frontier][1]][1]
Please find the code below:
def portfolio_annualised_performance(weights, mean_returns, cov_matrix):
returns = np.sum(mean_returns*weights )
std = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
return std, returns
def random_portfolios(num_portfolios, mean_returns, cov_matrix, risk_free_rate):
results = np.zeros((3,num_portfolios))
weights_record = []
for i in range(num_portfolios):
weights = abs(np.random.randn(len(mean_returns)))
weights /= np.sum(weights)
weights_record.append(weights)
portfolio_std_dev, portfolio_return = portfolio_annualised_performance(weights, mean_returns, cov_matrix)
results[0,i] = portfolio_std_dev
results[1,i] = portfolio_return
results[2,i] = (portfolio_return - risk_free_rate) / portfolio_std_dev
return results, weights_record
def neg_sharpe_ratio(weights, mean_returns, cov_matrix, risk_free_rate):
p_var, p_ret = portfolio_annualised_performance(weights, mean_returns, cov_matrix)
return -(p_ret - risk_free_rate) / p_var
def max_sharpe_ratio(mean_returns, cov_matrix, risk_free_rate):
num_assets = len(mean_returns)
args = (mean_returns, cov_matrix, risk_free_rate)
constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
bound = (0.0,1.0)
bounds = tuple(bound for asset in range(num_assets))
result = sco.minimize(neg_sharpe_ratio, num_assets*[1./num_assets,], args=args,
method='SLSQP', bounds=bounds, constraints=constraints)
return result
def portfolio_volatility(weights, mean_returns, cov_matrix):
return portfolio_annualised_performance(weights, mean_returns, cov_matrix)[0]
def min_variance(mean_returns, cov_matrix):
num_assets = len(mean_returns)
args = (mean_returns, cov_matrix)
constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
bound = (0.0,1.0)
bounds = tuple(bound for asset in range(num_assets))
result = sco.minimize(portfolio_volatility, num_assets*[1./num_assets,], args=args,
method='SLSQP', bounds=bounds, constraints=constraints)
return result
def efficient_return(mean_returns, cov_matrix, target):
num_assets = len(mean_returns)
args = (mean_returns, cov_matrix)
def portfolio_return(weights):
return portfolio_annualised_performance(weights, mean_returns, cov_matrix)[1]
constraints = ({'type': 'eq', 'fun': lambda x: portfolio_return(x) - target},
{'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
bounds = tuple((0.0,1) for asset in range(num_assets))
result = sco.minimize(portfolio_volatility, num_assets*[1./num_assets,], args=args, method='SLSQP', bounds=bounds, constraints=constraints)
return result
def efficient_frontier(mean_returns, cov_matrix, returns_range):
efficients = []
for ret in returns_range:
efficients.append(efficient_return(mean_returns, cov_matrix, ret))
return efficients
def display_calculated_ef_with_random(mean_returns, cov_matrix, num_portfolios, risk_free_rate):
results, _ = random_portfolios(num_portfolios,mean_returns, cov_matrix, risk_free_rate)
max_sharpe = max_sharpe_ratio(mean_returns, cov_matrix, risk_free_rate)
sdp, rp = portfolio_annualised_performance(max_sharpe['x'], mean_returns, cov_matrix)
max_sharpe_allocation = pd.DataFrame(max_sharpe.x,index=curr_w_terms,columns=['allocation'])
max_sharpe_allocation.allocation = [round(i*100,4)for i in max_sharpe_allocation.allocation]
max_sharpe_allocation = max_sharpe_allocation.T
min_vol = min_variance(mean_returns, cov_matrix)
sdp_min, rp_min = portfolio_annualised_performance(min_vol['x'], mean_returns, cov_matrix)
min_vol_allocation = pd.DataFrame(min_vol.x,index=curr_w_terms,columns=['allocation'])
min_vol_allocation.allocation = [round(i*100,4)for i in min_vol_allocation.allocation]
min_vol_allocation = min_vol_allocation.T
print("-"*80)
print("Maximum Sharpe Ratio Portfolio Allocation\n")
print("Annualised Return:", round(rp,4))
print("Annualised Volatility:", round(sdp,4))
print("\n")
print(max_sharpe_allocation)
print("-"*80)
print("Minimum Volatility Portfolio Allocation\n")
print("Annualised Return:", round(rp_min,4))
print("Annualised Volatility:", round(sdp_min,4))
print("\n")
print(min_vol_allocation)
plt.figure(figsize=(10, 7))
plt.scatter(results[0,:],results[1,:],c=results[2,:],cmap='YlGnBu', marker='o', s=10, alpha=0.3)
plt.colorbar()
plt.scatter(sdp,rp,marker='*',color='r',s=500, label='Maximum Sharpe ratio')
plt.scatter(sdp_min,rp_min,marker='*',color='g',s=500, label='Minimum volatility')
target = np.linspace(rp_min, 0.05, 20)
efficient_portfolios = efficient_frontier(mean_returns, cov_matrix, target)
plt.plot([p['fun'] for p in efficient_portfolios], target, linestyle='-.', color='black', label='efficient frontier')
plt.title('Calculated Portfolio Optimization based on Efficient Frontier')
plt.xlabel('annualised volatility')
plt.ylabel('annualised returns')
plt.legend(labelspacing=0.8)
plt.ylim([-0.005,0.03])
plt.xlim([0.0,0.05])
display_calculated_ef_with_random(log_ret, new_cov, 50000, 0)
I haven't annualised the Covar-Matrix since I already have annual return estimates as well as covar estimates.
My very question is: is this plausible or not?
**EDIT**
Since the weight generation process of my random portfolios seems to preffer too similar portfolio I changed the following function:
def random_portfolios(num_portfolios, mean_returns, cov_matrix, risk_free_rate):
results = np.zeros((3,num_portfolios))
weights_record = []
for i in range(num_portfolios):
weights = abs(np.random.randn(len(mean_returns)))
weights[weights<1] = 0
if sum(weights)==0:
print("sum=0")
indexes = np.unique(np.random.randint(0,10,3)).tolist()
weights[indexes] = abs(np.random.randn(len(indexes)))
weights /= np.sum(weights)
weights_record.append(weights)
portfolio_std_dev, portfolio_return = portfolio_annualised_performance(weights, mean_returns, cov_matrix)
results[0,i] = portfolio_std_dev
results[1,i] = portfolio_return
results[2,i] = (portfolio_return - risk_free_rate) / portfolio_std_dev
return results, weights_record
After doing so, the Portfolios are way better distributed:
[![Portfolios after new weighting scheme][2]][2]
**So, can we then agree that the above code does what it should and I can continue from here?**
[1]: https://i.sstatic.net/RZq6J.png
[2]: https://i.sstatic.net/XZKaU.png