9.3. Expanding returns

9.3.1. The problem

You know the charts that show cumulative returns if you’d bought and held a stock since some long ago date? Let’s make one!

This is called “expanding returns” because you get the total returns from day 0 to day N, then from day 0 to day N+1, and so on; the window is expanding instead of having a fixed number of units or containing a specific increment of time.

9.3.2. Download the returns

We need a dataset with firm, date, and the faily return. Let’s build it:

#!pip install pandas_datareader # uncomment and run this ONE TIME ONLY to install pandas data reader
import pandas as pd
import numpy as np
import pandas_datareader as pdr # you might need to install this (see above)
from datetime import datetime

# choose your firms and dates 
stocks = ['SBUX','AAPL','MSFT']
start  = datetime(1980, 1, 1)
end    = datetime(2022, 7, 31)

Tip

The code in the next block is explained more thoroughly in handouts/factor_loading_simple.ipynb in the textbook repo, because that file prints the status of the data throughout. Looking at this might help.

# download stock prices 
# here, from yahoo: not my fav source, but quick. 
# we need to do some data manipulation to get the data ready 
stock_prices = pdr.get_data_yahoo(stocks, start=start, end=end)
stock_prices = stock_prices.filter(like='Adj Close') # reduce to just columns with this in the name
stock_prices.columns = stocks # put their tickers as column names

# refmt from wide to long
stock_prices = stock_prices.stack().swaplevel().sort_index().reset_index()
stock_prices.columns = ['Firm','Date','Adj Close']

# add return var = pct_change() function compares to prior row
# EXCEPT: don't compare for first row of one firm with last row of prior firm!
# MAKE SURE YOU CREATE THE VARIABLES WITHIN EACH FIRM - use groupby
stock_prices['ret'] = stock_prices.groupby('Firm')['Adj Close'].pct_change()
stock_prices['ret'] = stock_prices['ret'] 
stock_prices.head(15)
Firm Date Adj Close ret
0 AAPL 1980-12-12 0.100040 NaN
1 AAPL 1980-12-15 0.094820 -0.052171
2 AAPL 1980-12-16 0.087861 -0.073398
3 AAPL 1980-12-17 0.090035 0.024751
4 AAPL 1980-12-18 0.092646 0.028992
5 AAPL 1980-12-19 0.098300 0.061029
6 AAPL 1980-12-22 0.103084 0.048670
7 AAPL 1980-12-23 0.107434 0.042199
8 AAPL 1980-12-24 0.113088 0.052628
9 AAPL 1980-12-26 0.123527 0.092310
10 AAPL 1980-12-29 0.125267 0.014083
11 AAPL 1980-12-30 0.122222 -0.024304
12 AAPL 1980-12-31 0.118743 -0.028468
13 AAPL 1981-01-02 0.120048 0.010988
14 AAPL 1981-01-05 0.117438 -0.021738

9.3.3. Getting the expanding returns

Notice that this dataset has the simple return for a period, not the gross returns (defined here).

To compute \(R_i[0,T]\) for all firms \(i\) and each time \(T\) in the dataset, you’re going to need to use groupby. You have two equivalent options from there:

  1. For each firm, get the cumprod() of the gross return over its time series.

    df.assign(R=1+df['r']).groupby('firm')['R'].cumprod()
    
  2. For each firm, take the product of \(1+r\) for all prior periods using the expanding window functionality.

    df.groupby('firm')['r'].expanding().apply(lambda x: np.prod(1+x))
    

Which you choose is up to you, but in my testing the cumprod approach is 2.5x faster.

stock_prices['cumret'] = \
(
    stock_prices
    .assign(ret=1+stock_prices['ret'])
    .groupby('Firm')
    ['ret']
    .cumprod()
)

9.3.4. Plotting the total returns

If only we could turn back time.

(stock_prices.set_index('Date').groupby('Firm')['cumret']
 .plot(title="If you bought $1 back when, you'd have this now",
       figsize=(6,5))
);
../../_images/05a_expanding_8_0.png