LaTeX macros (hidden cell) $ \newcommand{\Q}{\mathcal{Q}} \newcommand{\ECov}{\boldsymbol{\Sigma}} \newcommand{\EMean}{\boldsymbol{\mu}} \newcommand{\EAlpha}{\boldsymbol{\alpha}} \newcommand{\EBeta}{\boldsymbol{\beta}} $

Imports and configuration

In [11]:
import sys
import os
import re
import datetime as dt

import numpy as np
import pandas as pd
%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt
from matplotlib.colors import LinearSegmentedColormap

from mosek.fusion import *

from import ConfigManager

from portfolio_tools import data_download, DataReader
In [12]:
# Version checks
print('matplotlib: {}'.format(matplotlib.__version__))

# Jupyter configuration
c = ConfigManager()
c.update('notebook', {"CodeCell": {"cm_config": {"autoCloseBrackets": False}}})  

# Numpy options
np.set_printoptions(precision=5, linewidth=120, suppress=True)

# Pandas options
pd.set_option('display.max_rows', None)

# Matplotlib options
plt.rcParams['figure.figsize'] = [12, 8]
plt.rcParams['figure.dpi'] = 200
3.6.9 (default, Jan 26 2021, 15:33:00) 
[GCC 8.4.0]
matplotlib: 3.3.4

Prepare input data

Here we compute the optimization input variables, the vector $\EMean$ of expected returns, and the covariance matrix $\ECov$, from raw data. The data consists of daily stock prices of $8$ stocks from the US market.

Download data

In [13]:
# Data downloading:
# If the user has an API key for, then this code part will download the data. 
# The code can be modified to download from other sources. To be able to run the examples, 
# and reproduce results in the cookbook, the files have to have the following format and content:
# - File name pattern: "daily_adjusted_[TICKER].csv", where TICKER is the symbol of a stock. 
# - The file contains at least columns "timestamp", "adjusted_close", and "volume".
# - The data is daily price/volume, covering at least the period from 2016-03-18 until 2021-03-18, 
# - Files are for the stocks PM, LMT, MCD, MMM, AAPL, MSFT, TXN, CSCO.
list_stocks = ["PM", "LMT", "MCD", "MMM", "AAPL", "MSFT", "TXN", "CSCO"]
list_factors = []
alphaToken = None
list_tickers = list_stocks + list_factors
if alphaToken is not None:
    data_download(list_tickers, alphaToken)   

Read data

We load the daily stock price data from the downloaded CSV files. The data is adjusted for splits and dividends. Then a selected time period is taken from the data.

In [14]:
investment_start = "2016-03-18"
investment_end = "2021-03-18"
In [15]:
# The files are in "stock_data" folder, named as "daily_adjusted_[TICKER].csv"
dr = DataReader(folder_path="stock_data", symbol_list=list_tickers)
df_prices, _ = dr.get_period(start_date=investment_start, end_date=investment_end)
Found data files: 

Using data files: 

Compute yearly return statistics

Here we use the loaded daily price data to compute the corresponding yearly mean return and covariance matrix. The below logic is implemented also in the portfolio_tools module as a function. It is used in the other notebooks.

1. Compute weekly logarithmic return

First we convert the daily prices to weekly prices.

In [16]:
df_weekly_prices = df_prices.resample('W').last()

Convert the weekly prices to weekly logarithmic return.

In [17]:
df_weekly_log_returns = np.log(df_weekly_prices) - np.log(df_weekly_prices.shift(1))
df_weekly_log_returns = df_weekly_log_returns.dropna(how='all')
df_weekly_log_returns = df_weekly_log_returns.fillna(0)

We can see based on the histograms that the distribution of weekly logarithmic return is approximately normal.

In [18]:
array([[<AxesSubplot:title={'center':'PM'}>, <AxesSubplot:title={'center':'LMT'}>,
       [<AxesSubplot:title={'center':'MMM'}>, <AxesSubplot:title={'center':'AAPL'}>,
       [<AxesSubplot:title={'center':'TXN'}>, <AxesSubplot:title={'center':'CSCO'}>, <AxesSubplot:>]], dtype=object)

2. Compute the distribution of weekly logarithmic return

Assuming that the distribution is normal, we estimate the mean and covariance of the weekly logarithmic return.

In [19]:
return_array = df_weekly_log_returns.to_numpy()
T = return_array.shape[0]
m_weekly_log = np.mean(return_array, axis=0)
S_weekly_log = np.cov(return_array.transpose())

3. Project the distribution to the investment horizon

Next we project the logarithmic return statistics to the investment horizon of 1 year.

In [20]:
m_log = 52 * m_weekly_log
S_log = 52 * S_weekly_log

4. Compute the distribution of yearly linear return

We recover the distribution of prices from the distribution of logarithmic returns.

In [21]:
p_0 = df_weekly_prices.iloc[0].to_numpy()
m_P = p_0 * np.exp(m_log + 1/2*np.diag(S_log))
S_P = np.outer(m_P, m_P) * (np.exp(S_log) - 1)

Finally we convert the distribution of prices to the distribution of yearly linear returns.

In [22]:
m = 1 / p_0 * m_P - 1
S = 1 / np.outer(p_0, p_0) * S_P