import yfinance as yf
import numpy as np
import pandas as pd
import matplotlib
import matplotlib.pyplot as pltPortfolio Index, Econometric Models
Abstract
This is an example to create an index for any portfolio of stocks. It also illustrate how to create premium returns and prepare a dataset before running a regression model
1 Download prices
I load the libraries to collect, process and visualize stock data from Yahoo Finance:
I download monthly stock prices for several firms and the S&P500 for at least 3 years
data = yf.download('NVDA MSFT GOOG AMZN ^GSPC', start='2020-12-01', end='2024-07-31', interval='1mo')
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[*********************100%%**********************] 5 of 5 completedI select only adjusted prices:
adjprices = data['Adj Close']
adjprices.head() AMZN GOOG MSFT NVDA ^GSPC
Date
2020-12-01 162.846497 87.494438 215.425507 13.020117 3756.070068
2021-01-01 160.309998 91.682671 224.665512 12.958867 3714.239990
2021-02-01 154.646500 101.727242 225.072327 13.681893 3811.149902
2021-03-01 154.703995 103.313934 228.881668 13.316516 3972.889893
2021-04-01 173.371002 120.369019 244.812225 14.978984 4181.169922The first 4 are stock prices. The 5th column is the S&P500 index. For now I will use only the stock prices to create the portfolio index.
2 Creating an index for an equally-weighted portfolio:
I calculate the holing period returns (HPR) for each price. The HPR is the cumulative return for a stock over time, asumming that we hold the investment for all periods.
HPR = adjprices / adjprices.iloc[0] - 1
# adjprices.iloc[0] refers to the FIRST ROW of the adjprices dataset, which is the FIRST MONTH OF DATAI create the equally-weighted portfolio:
# I create a dataset with the growth factors for each stock.
# I assume an equally-weighted portfolio. For 4 stocks I assign 25% each:
GROWTHFACTOR1 = 1 + (0.25* HPR.iloc[:,0] + 0.25* HPR.iloc[:,1] + 0.25* HPR.iloc[:,2] + 0.25* HPR.iloc[:,3])
# I create the Portfolio Index starting in 100:
INDEXPORT1 = 100 * GROWTHFACTOR1
# I show the first rows:
INDEXPORT1.head()Date
2020-12-01 100.000000
2021-01-01 101.762005
2021-02-01 105.198109
2021-03-01 105.400813
2021-04-01 118.180587
dtype: float64type(INDEXPORT1)<class 'pandas.core.series.Series'>INDEXPORT1 is a Series object since it has only 1 column. I convert it to a data frame
I have monthly returns from Dec 2020:
INDEXPORT1 = INDEXPORT1.to_frame()
# I name the column as INDEXPORT1:
INDEXPORT1.columns = ['INDEXPORT1']
INDEXPORT1.head() INDEXPORT1
Date
2020-12-01 100.000000
2021-01-01 101.762005
2021-02-01 105.198109
2021-03-01 105.400813
2021-04-01 118.180587I can visualize the Index over time:
plt.plot(INDEXPORT1['INDEXPORT1'])
plt.title('AI Portfolio Index')
plt.show()
3 Adding the INDEXPORT to the price dataset
Now I merge the INDEXPORT1 with the adjprices data frame before I calculate returns:
# To add the INDEXPORT1 as a column to adjprices, I join the INDEXPORT into the adjprices:
adjprices = adjprices.join(INDEXPORT1)
adjprices.head() AMZN GOOG ... ^GSPC INDEXPORT1
Date ...
2020-12-01 162.846497 87.494438 ... 3756.070068 100.000000
2021-01-01 160.309998 91.682671 ... 3714.239990 101.762005
2021-02-01 154.646500 101.727242 ... 3811.149902 105.198109
2021-03-01 154.703995 103.313934 ... 3972.889893 105.400813
2021-04-01 173.371002 120.369019 ... 4181.169922 118.180587
[5 rows x 6 columns]4 Calculate returns for all prices and the Index
# I create returns as the first difference of log prices
returns = np.log(adjprices) - np.log(adjprices.shift(1))
# I drop the first row that has null values:
returns = returns.dropna()
returns.head() AMZN GOOG MSFT NVDA ^GSPC INDEXPORT1
Date
2021-01-01 -0.015699 0.046758 0.041998 -0.004715 -0.011199 0.017467
2021-02-01 -0.035968 0.103962 0.001809 0.054293 0.025757 0.033209
2021-03-01 0.000372 0.015477 0.016783 -0.027068 0.041563 0.001925
2021-04-01 0.113920 0.152790 0.067286 0.117643 0.051097 0.114443
2021-05-01 -0.073076 0.000597 -0.009963 0.079071 0.005471 0.0019435 Download the risk-free rate
I download the Treasury Bills rate of the US:
import pandas_datareader.data as pdr
import datetime
# I define start as the month Jan 2020
start = datetime.datetime(2021,1,1)
# I define the end month as July 2024
end = datetime.datetime(2024,7,1)
Tbills = pdr.DataReader('TB3MS','fred',start,end)We see the content of Tbills:
Tbills.head() TB3MS
DATE
2021-01-01 0.08
2021-02-01 0.04
2021-03-01 0.03
2021-04-01 0.02
2021-05-01 0.02The TB3MS serie is given in percentage and in annual rate. I divide it by 100 and 12 to get a monthly simple rate since I am using monthly rates for the stocks:
rfrate = Tbills / 100 / 12Now I get the continuously compounded return from the simple return:
rfrate = np.log(1+rfrate)7 Visualize relationship between premium returns
I do a scatter plot putting the S&P500 premium returns as the independent variable (X) and the Index1 premium return as the dependent variable (Y). I also add the regression line to the plot:
import seaborn as sb
plt.clf()
x = returns['SP500_Premr']
y = returns['INDEX1_Premr']
# I plot the (x,y) values along with the regression line that fits the data:
sb.regplot(x=x,y=y)
plt.xlabel('Market Premium returns')
plt.ylabel('INDEX Premium returns')
plt.xticks(np.arange(-0.3,0.4,0.1))([<matplotlib.axis.XTick object at 0x000002016E569AE0>, <matplotlib.axis.XTick object at 0x000002016E569F90>, <matplotlib.axis.XTick object at 0x000002016E54F8E0>, <matplotlib.axis.XTick object at 0x000002016E5D40A0>, <matplotlib.axis.XTick object at 0x000002016E593880>, <matplotlib.axis.XTick object at 0x000002016E593430>, <matplotlib.axis.XTick object at 0x000002016E5937F0>], [Text(0, 0, ''), Text(0, 0, ''), Text(0, 0, ''), Text(0, 0, ''), Text(0, 0, ''), Text(0, 0, ''), Text(0, 0, '')])plt.show()
8 RUNNNG THE CAPM FOR THE INDEX1 PORTFOLIO
import statsmodels.formula.api as smf
# I estimate the OLS regression model:
mkmodel = smf.ols('INDEX1_Premr ~ SP500_Premr',data=returns).fit()
# I display the summary of the regression:
print(mkmodel.summary()) OLS Regression Results
==============================================================================
Dep. Variable: INDEX1_Premr R-squared: 0.620
Model: OLS Adj. R-squared: 0.611
Method: Least Squares F-statistic: 67.02
Date: jue., 29 ago. 2024 Prob (F-statistic): 3.68e-10
Time: 20:08:29 Log-Likelihood: 60.751
No. Observations: 43 AIC: -117.5
Df Residuals: 41 BIC: -114.0
Df Model: 1
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
Intercept 0.0164 0.009 1.761 0.086 -0.002 0.035
SP500_Premr 1.5916 0.194 8.186 0.000 1.199 1.984
==============================================================================
Omnibus: 1.837 Durbin-Watson: 1.991
Prob(Omnibus): 0.399 Jarque-Bera (JB): 0.942
Skew: 0.076 Prob(JB): 0.624
Kurtosis: 3.709 Cond. No. 21.1
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.