import numpy as np import pandas as pd import matplotlib.pyplot as plt from numpy.polynomial.polynomial import Polynomial # Read the S&P 500 data df = pd.read_csv('/path/to/your/csv/file.csv') # Filter the data to start from March 4, 1957 df_1957 = df[df['Date'] >= '1957-03-04'].reset_index(drop=True) # Convert the dates to numerical values (number of days since March 4, 1957) x_data = np.arange(len(df_1957)) # Normalize the S&P 500 data to start from 1 normalized_s_1957 = df_1957['Close'] / df_1957['Close'].iloc[0] # Fit a first-degree polynomial (a line) to the logarithmic data p = Polynomial.fit(x_data, np.log(normalized_s_1957), 1) # Generate y-values based on the fit y_fit_log = p(x_data) # Plotting the logarithmic graph and the fitted line plt.figure(figsize=(15, 6)) plt.semilogy(x_data, normalized_s_1957, label='Normalized Actual S&P 500 (Log scale)', color='b') plt.semilogy(x_data, np.exp(y_fit_log), label=f'Fitted Line', color='r') plt.title('Logarithmic Plot of the Normalized Actual S&P 500 and Fitted Line (Starting from March 4, 1957)') plt.xlabel('Number of Days') plt.ylabel('Close Price (Log Scale, Normalized)') plt.legend() plt.grid(True) plt.show() # Show the coefficients of the fitted line print("Coefficients of the fitted line:", p.convert().coef)

import numpy as np import pandas as pd import matplotlib.pyplot as plt # Read the S&P 500 data df = pd.read_csv('/path/to/your/csv/file.csv') # Filtering the dataset to only include data from January 1, 1990, onwards df_1990 = df[df['Date'] >= '1990-01-01'].reset_index(drop=True) # Number of data points in the dataset starting from 1990 n_1990 = len(df_1990) # Calculate the daily returns for the actual S&P 500 data starting from 1990 s_actual_1990 = df_1990['Close'].values daily_returns_1990 = (s_actual_1990[1:] / s_actual_1990[:-1]) - 1 # Define the annual fee rate for 2x, 3x, and 4x leveraged products (1% annual fee) annual_fee_rate = 0.01 daily_fee_multiplier = 1 - (annual_fee_rate / 252) # Assuming 252 trading days in a year # Initialize arrays for 2x, 3x, and 4x leveraged products with daily rebalancing and fees starting from 1990 s_2x_leverage_daily_rebalance_fee_1990 = np.zeros(n_1990) s_3x_leverage_daily_rebalance_fee_1990 = np.zeros(n_1990) s_4x_leverage_daily_rebalance_fee_1990 = np.zeros(n_1990) s_2x_leverage_daily_rebalance_fee_1990[0] = s_actual_1990[0] s_3x_leverage_daily_rebalance_fee_1990[0] = s_actual_1990[0] s_4x_leverage_daily_rebalance_fee_1990[0] = s_actual_1990[0] # Perform the simulation with daily rebalancing and fees for t in range(1, n_1990): daily_return = (s_actual_1990[t] / s_actual_1990[t-1]) - 1 s_2x_leverage_daily_rebalance_fee_1990[t] = s_2x_leverage_daily_rebalance_fee_1990[t-1] * (1 + 2 * daily_return) * daily_fee_multiplier s_3x_leverage_daily_rebalance_fee_1990[t] = s_3x_leverage_daily_rebalance_fee_1990[t-1] * (1 + 3 * daily_return) * daily_fee_multiplier s_4x_leverage_daily_rebalance_fee_1990[t] = s_4x_leverage_daily_rebalance_fee_1990[t-1] * (1 + 4 * daily_return) * daily_fee_multiplier # Normalize the leveraged products and actual S&P 500 to start from 100 normalized_s_actual_1990 = s_actual_1990 / s_actual_1990[0] * 100 normalized_s_2x_leverage_daily_rebalance_fee_1990 = s_2x_leverage_daily_rebalance_fee_1990 / s_2x_leverage_daily_rebalance_fee_1990[0] * 100 normalized_s_3x_leverage_daily_rebalance_fee_1990 = s_3x_leverage_daily_rebalance_fee_1990 / s_3x_leverage_daily_rebalance_fee_1990[0] * 100 normalized_s_4x_leverage_daily_rebalance_fee_1990 = s_4x_leverage_daily_rebalance_fee_1990 / s_4x_leverage_daily_rebalance_fee_1990[0] * 100 # Plotting the graph plt.figure(figsize=(15, 6)) plt.plot(df_1990['Date'], normalized_s_actual_1990, label='Normalized Actual S&P 500', color='r') plt.plot(df_1990['Date'], normalized_s_2x_leverage_daily_rebalance_fee_1990, label='Normalized 2x Leveraged with Fee (Daily Rebalance)', color='g') plt.plot(df_1990['Date'], normalized_s_3x_leverage_daily_rebalance_fee_1990, label='Normalized 3x Leveraged with Fee (Daily Rebalance)', color='b') plt.plot(df_1990['Date'], normalized_s_4x_leverage_daily_rebalance_fee_1990, label='Normalized 4x Leveraged with Fee (Daily Rebalance)', color='m') plt.title('Normalized Actual S&P 500, 2x, 3x, and 4x Leveraged with Daily Rebalancing and Fees (Base Value = 100, Starting from 1990)') plt.xlabel('Date') plt.ylabel('Normalized Value') plt.legend() plt.grid(True) plt.show()

import pandas as pd import numpy as np import matplotlib.pyplot as plt # Load the data df = pd.read_csv('/path/to/data.csv') df['Date'] = pd.to_datetime(df['Date']) # Filter data starting from 1997 df_1997 = df[df['Date'] >= '1997-01-01'].reset_index(drop=True) # Initialize arrays and variables n_1997 = len(df_1997) s_actual_1997 = df_1997['Close'].values # Normalize the actual S&P 500 normalized_s_actual_1997 = s_actual_1997 / s_actual_1997[0] * 100 # Initialize arrays for 2x, 3x, and 4x leveraged products s_2x_leverage_daily_rebalance_1997 = np.zeros(n_1997) s_3x_leverage_daily_rebalance_1997 = np.zeros(n_1997) s_4x_leverage_daily_rebalance_1997 = np.zeros(n_1997) s_2x_leverage_daily_rebalance_1997[0] = s_actual_1997[0] s_3x_leverage_daily_rebalance_1997[0] = s_actual_1997[0] s_4x_leverage_daily_rebalance_1997[0] = s_actual_1997[0] # Perform daily rebalancing for 2x, 3x, and 4x leveraged products for t in range(1, n_1997): daily_return = (s_actual_1997[t] / s_actual_1997[t-1]) - 1 s_2x_leverage_daily_rebalance_1997[t] = s_2x_leverage_daily_rebalance_1997[t-1] * (1 + 2 * daily_return) s_3x_leverage_daily_rebalance_1997[t] = s_3x_leverage_daily_rebalance_1997[t-1] * (1 + 3 * daily_return) s_4x_leverage_daily_rebalance_1997[t] = s_4x_leverage_daily_rebalance_1997[t-1] * (1 + 4 * daily_return) # Normalize the leveraged products normalized_s_2x_leverage_daily_rebalance_1997 = s_2x_leverage_daily_rebalance_1997 / s_2x_leverage_daily_rebalance_1997[0] * 100 normalized_s_3x_leverage_daily_rebalance_1997 = s_3x_leverage_daily_rebalance_1997 / s_3x_leverage_daily_rebalance_1997[0] * 100 normalized_s_4x_leverage_daily_rebalance_1997 = s_4x_leverage_daily_rebalance_1997 / s_4x_leverage_daily_rebalance_1997[0] * 100 # Plotting plt.figure(figsize=(15, 6)) plt.plot(df_1997['Date'], normalized_s_actual_1997, label='Normalized Actual S&P 500', color='r') plt.plot(df_1997['Date'], normalized_s_2x_leverage_daily_rebalance_1997, label='Normalized 2x Leveraged (Daily Rebalance)', color='g') plt.plot(df_1997['Date'], normalized_s_3x_leverage_daily_rebalance_1997, label='Normalized 3x Leveraged (Daily Rebalance)', color='b') plt.plot(df_1997['Date'], normalized_s_4x_leverage_daily_rebalance_1997, label='Normalized 4x Leveraged (Daily Rebalance)', color='m') plt.title('Normalized Actual S&P 500, 2x, 3x, and 4x Leveraged with Daily Rebalancing (Base Value = 100, Starting from 1997)') plt.xlabel('Date') plt.ylabel('Normalized Value') plt.legend() plt.grid(True) plt.show()