This algorithm performs a standard mean-variance optimization over a group of large cap US stocks. The algorithm constructs an efficient frontier of allocations and allows the user to choose an allocation based on risk preference.

Ryan

Clone Algorithm

457

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Backtest from
to
with
initial capital

Cumulative performance:

Algorithm
Benchmark

Custom data:

Total Returns

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Alpha

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Beta

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Sharpe

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Sortino

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Max Drawdown

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Benchmark Returns

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Volatility

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Returns | 1 Month | 3 Month | 6 Month | 12 Month |

Alpha | 1 Month | 3 Month | 6 Month | 12 Month |

Beta | 1 Month | 3 Month | 6 Month | 12 Month |

Sharpe | 1 Month | 3 Month | 6 Month | 12 Month |

Sortino | 1 Month | 3 Month | 6 Month | 12 Month |

Volatility | 1 Month | 3 Month | 6 Month | 12 Month |

Max Drawdown | 1 Month | 3 Month | 6 Month | 12 Month |

from numpy import matrix, array, zeros, empty, sqrt, ones, dot, append, mean, cov, transpose, linspace import numpy as np import scipy.optimize import math import random # This algorithm performs a Markowitz-style mean-variance optimization with a # universe of the 14 highest market capitalization stocks in the S&P 500. The # algorithm constructs an efficient frontier and then selects a weighting from # the efficient frontier. Uses ideas from the Global Minimum Variance Portfolio # algorithm posted on Quantopian. # This is the lwindow that governs the pulling of historical data and # portfolio rebalancing. window = 100 refresh_rate = 10 # Compute the expected return of the portfolio. def compute_mean(W,R): return sum(R*W) # Compute the variance of the portfolio. def compute_var(W,C): return dot(dot(W, C), W) # Combination of the two functions above - mean and variance of returns calculation. def compute_mean_var(W, R, C): return compute_mean(W, R), compute_var(W, C) def fitness(W, R, C, r): # For given level of return r, find weights which minimizes portfolio variance. mean, var = compute_mean_var(W, R, C) # Penalty for not meeting stated portfolio return effectively serves as optimization constraint # Here, r is the 'target' return penalty = 0.1*abs(mean-r) return var + penalty # Given risk-free rate, asset returns, and covariances, this function # calculates the weights of the tangency portfolio with regard to Sharpe # ratio maximization. def fitness_sharpe(W, R, C, rf): mean, var = compute_mean_var(W, R, C) utility = (mean - rf)/sqrt(var) return 1/utility # Solves for the optimal portfolio weights using the Sharpe ratio # maximization objective function. def solve_weights(R, C, rf): n = len(R) W = ones([n])/n # Start optimization with equal weights b_ = [(0.,1.) for i in range(n)] # Bounds for decision variables c_ = ({'type':'eq', 'fun': lambda W: sum(W)-1. }) # Constraints - weights must sum to 1 optimized = scipy.optimize.minimize(fitness, W, (R, C, rf), method='SLSQP', constraints=c_, bounds=b_) if not optimized.success: raise BaseException(optimized.message) return optimized.x # Solve for the efficient frontier using the variance + penalty minimization # function fitness. def solve_frontier(R, C, rf): frontier_mean, frontier_var, frontier_weights = [], [], [] n = len(R) # Number of assets in the portfolio for r in linspace(min(R), max(R), num=20): # Iterate through the range of returns on Y axis W = ones([n])/n # Set initial guess for weights b_ = [(0,1) for i in range(n)] # Set bounds on weights c_ = ({'type':'eq', 'fun': lambda W: sum(W)-1. }) # Set constraints optimized = scipy.optimize.minimize(fitness, W, (R, C, r), method='SLSQP', constraints=c_, bounds=b_) #if not optimized.success: # raise BaseException(optimized.message) # Add point to the efficient frontier frontier_mean.append(r) frontier_var.append(compute_var(optimized.x, C)) # Min-variance based on optimized weights frontier_weights.append(optimized.x) return array(frontier_mean), array(frontier_var), frontier_weights # Weights - array of asset weights (derived from market capitalizations) # Expreturns - expected returns based on historical data # Covars - covariance matrix of asset returns based on historical data def assets_meanvar(daily_returns): # Calculate expected returns expreturns = array([]) (rows, cols) = daily_returns.shape for r in range(rows): expreturns = append(expreturns, mean(daily_returns[r])) # Compute covariance matrix covars = cov(daily_returns) # Annualize expected returns and covariances # Assumes 255 trading days per year expreturns = (1+expreturns)**255-1 covars = covars * 255 return expreturns, covars def initialize(context): # Set day context.day = 0 # Set risk-free rate # 14 largest stocks by market cap in S&P in 2010 context.securities = [sid(24),sid(26578),sid(3149), sid(5061), sid(3766),sid(23112),sid(4151),sid(5938),sid(5923),sid(6653),sid(700), sid(1900), sid(8347), sid(8229)] # Set Max and Min positions in security context.max_notional = 1000000.1 context.min_notional = -1000000.0 # Set commission def handle_data(context, data): # Get 40 days of prices for each security all_prices = get_past_prices(data) # Circuit breaker in case transform returns none if all_prices is None: return # Circuit breaker, only calculate every 20 days if context.day%refresh_rate is not 0: context.day = context.day+1 return daily_returns = np.zeros((len(context.securities),window)) # Calculate daily returns into daily_returns security_index = 0; for security in context.securities: if data.has_key(security): for day in range(0,window): day_of = all_prices[security][day] day_before = all_prices[security][day-1] daily_returns[security_index][day] = (day_of-day_before)/day_before security_index = security_index + 1 expreturns, covars = assets_meanvar(daily_returns) R = expreturns C = covars rf = 0.015 frontier_mean, frontier_var, frontier_weights = solve_frontier(R, C, rf) f_w = array(frontier_weights) (row_1, col_1) = f_w.shape log.info(row_1) log.info(col_1) wts = frontier_weights[5] new_weights = wts #log.info(new_weights) #set leverage to 1 leverage = sum(abs(new_weights)) portfolio_value = (context.portfolio.positions_value + context.portfolio.cash)/leverage #reweight portfolio security_index = 0 for security in context.securities: current_position = context.portfolio.positions[security].amount new_position = (portfolio_value*new_weights[security_index])/all_prices[security][window-1] order(security,new_position-current_position) security_index = security_index+1 context.day = context.day+1 @batch_transform(refresh_period=refresh_rate, window_length=window) def get_past_prices(data): prices = data['price'] return prices

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