# Applied CAPM¶

Let’s study an example of calculating a CAPM $$\beta$$ using real market data. Begin by loading the pandas_datareader module for importing data. We’ll also import the datetime module so that we can select stock prices over a given date range.

import pandas_datareader.data as web
from datetime import datetime

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
<ipython-input-1-3972f8269bfb> in <module>
----> 1 import pandas_datareader.data as web
2 from datetime import datetime



We’ll use AlphaVantage for stock prices. Load up the API key that we stored earlier.

import pickle
with open('../pickle_jar/av_key.p', 'rb') as f:
print(api_key[0:3])

VAS


The stock we’ll use for our study is AAPL. Let’s collect return data over 2015-2019.

start = datetime(2015, 1, 1)
end = datetime(2019, 12, 31)



Print the head to see a snapshot of the data.

aapl.head()

open high low close adjusted close volume dividend amount split coefficient
2015-01-02 111.39 111.44 107.350 109.33 24.779987 53204626 0.0 1.0
2015-01-05 108.29 108.65 105.410 106.25 24.081896 64285491 0.0 1.0
2015-01-06 106.54 107.43 104.630 106.26 24.084162 65797116 0.0 1.0
2015-01-07 107.20 108.20 106.695 107.75 24.421875 40105934 0.0 1.0
2015-01-08 109.23 112.15 108.700 111.89 25.360219 59364547 0.0 1.0

Create a column of stock returns using the adjusted close data.

aapl['ret'] = aapl['adjusted close'].pct_change()
aapl['ret'].plot()

<AxesSubplot:>


Now we need some market return data. The Fama-French data library has that available.

Note that Fama-French API calls via pandas_datareader return a dictionary of values, rather than a DataFrame.

ff = web.DataReader('F-F_Research_Data_Factors_daily', 'famafrench', start, end)
ff.keys()

dict_keys([0, 'DESCR'])


The 'DESCR' gives some details about the data while the 0 key stores the actual DataFrame.

print(ff['DESCR'])

F-F Research Data Factors daily
-------------------------------

This file was created by CMPT_ME_BEME_RETS_DAILY using the 202107 CRSP database. The Tbill return is the simple daily rate that, over the number of trading days in the month, compounds to 1-month TBill rate from Ibbotson and Associates Inc. Copyright 2021 Kenneth R. French

0 : (1258 rows x 4 cols)

ffdf = ff[0]

Mkt-RF SMB HML RF
Date
2015-01-02 -0.12 -0.62 0.08 0.0
2015-01-05 -1.84 0.33 -0.68 0.0
2015-01-06 -1.04 -0.78 -0.31 0.0
2015-01-07 1.19 0.20 -0.65 0.0
2015-01-08 1.81 -0.11 -0.28 0.0

One tricky issue is that, while the head printouts for the two DataFrames make the index items appear similar, they are not. Data from Fama-French returns an datetime variable as an index (which is more useful) whereas data from the AlphaVantage API call returns the index as a string.

print(aapl.index.dtype)
print(ffdf.index.dtype)

object
datetime64[ns]


We’ll want the index of the AlphaVantage data to be a datetime (to match the Fama-French data). To do this, we simply us the .to_datetime() function on the .index variable of the AlphaVantage DataFrame.

import pandas as pd
aapl.index = pd.to_datetime(aapl.index, format='%Y-%m-%d')
print(aapl.index.dtype)

datetime64[ns]


With both DataFrames using the same index, we can merge the two using the index value as the merge key. Thus, rather than using left_on and right_on to specify the merge variable, we specify left_index=True and right_index=True to tell Python that the merging variable will be the index.

df = aapl.merge(ffdf, left_index=True, right_index=True)

open high low close adjusted close volume dividend amount split coefficient ret Mkt-RF SMB HML RF
2015-01-02 111.39 111.44 107.350 109.33 24.779987 53204626 0.0 1.0 NaN -0.12 -0.62 0.08 0.0
2015-01-05 108.29 108.65 105.410 106.25 24.081896 64285491 0.0 1.0 -0.028172 -1.84 0.33 -0.68 0.0
2015-01-06 106.54 107.43 104.630 106.26 24.084162 65797116 0.0 1.0 0.000094 -1.04 -0.78 -0.31 0.0
2015-01-07 107.20 108.20 106.695 107.75 24.421875 40105934 0.0 1.0 0.014022 1.19 0.20 -0.65 0.0
2015-01-08 109.23 112.15 108.700 111.89 25.360219 59364547 0.0 1.0 0.038422 1.81 -0.11 -0.28 0.0

Remember that the $$y$$ variable in a CAPM equation is the excess return of the stock. So, subtract the company’s return from the risk free rate.

df['eret'] = df['ret']*100 - df['RF']


We can now estimate a CAPM $$\beta$$.

import statsmodels.formula.api as smf

mod = smf.ols('eret ~ Q("Mkt-RF")', data=df).fit()
print(mod.summary())
print(mod.params)

                            OLS Regression Results
==============================================================================
Dep. Variable:                   eret   R-squared:                       0.442
Method:                 Least Squares   F-statistic:                     995.8
Date:                Sun, 26 Sep 2021   Prob (F-statistic):          2.17e-161
Time:                        22:26:10   Log-Likelihood:                -1978.4
No. Observations:                1257   AIC:                             3961.
Df Residuals:                    1255   BIC:                             3971.
Df Model:                           1
Covariance Type:            nonrobust
===============================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------
Intercept       0.0417      0.033      1.264      0.206      -0.023       0.106
Q("Mkt-RF")     1.2009      0.038     31.556      0.000       1.126       1.276
==============================================================================
Omnibus:                      190.235   Durbin-Watson:                   1.884
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2298.267
Skew:                           0.241   Prob(JB):                         0.00
Kurtosis:                       9.607   Cond. No.                         1.16
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Intercept      0.041726
Q("Mkt-RF")    1.200855
dtype: float64


Putting this all together, we get the code below.

def get_beta(start, end, ticker):
with open('../../pickle_jar/av_key.p', 'rb') as fout:
ff = web.DataReader('F-F_Research_Data_Factors_daily', 'famafrench', start, end)[0]
stock.index = pd.to_datetime(stock.index, format='%Y-%m-%d')
df = stock.merge(ff, left_index=True, right_index=True)
df['eret'] = df['ret'] - df['RF']
mod = smf.ols('eret ~ Q("Mkt-RF")', data=df)
res = mod.fit()
print(res.summary())
alpha, beta = res.params
return beta

get_beta(start = datetime(2018,1,1), end = datetime(2019,12,31), ticker='GOOG')

                            OLS Regression Results
==============================================================================
Dep. Variable:                   eret   R-squared:                       0.587
Method:                 Least Squares   F-statistic:                     711.9
Date:                Sun, 26 Sep 2021   Prob (F-statistic):           3.49e-98
Time:                        22:31:28   Log-Likelihood:                -741.01
No. Observations:                 502   AIC:                             1486.
Df Residuals:                     500   BIC:                             1494.
Df Model:                           1
Covariance Type:            nonrobust
===============================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------
Intercept       0.0016      0.047      0.035      0.972      -0.091       0.095
Q("Mkt-RF")     1.3093      0.049     26.681      0.000       1.213       1.406
==============================================================================
Omnibus:                      147.463   Durbin-Watson:                   1.922
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             7110.333
Skew:                           0.390   Prob(JB):                         0.00
Kurtosis:                      21.421   Cond. No.                         1.05
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

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