L23: Backtesting - risk adjustment

# Import packages
import pandas as pd
import numpy as np
import statsmodels.api as sm
import pandas_datareader as pdr

Raw returns

Equal-weighted (EW) portfolios

# Load EW return data data
ew_ret = pd.read_pickle('../data/AG_ew_returns.zip')
ew_ret = ew_ret[:'2003-06'].copy()
ew_ret
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
mdate
1982-07 -0.003948 -0.011054 0.002117 -0.009741 -0.009275 -0.003169 -0.004816 -0.006250 -0.034018 -0.071587 0.067639
1982-08 0.043106 0.058108 0.067768 0.071293 0.070272 0.076378 0.080404 0.078438 0.076014 0.069051 -0.025945
1982-09 0.037267 0.034962 0.034323 0.037597 0.034146 0.047353 0.034485 0.030422 0.009593 -0.020953 0.058220
1982-10 0.154922 0.122708 0.106485 0.107380 0.119641 0.122316 0.112235 0.135088 0.142025 0.161400 -0.006478
1982-11 0.116354 0.115989 0.096393 0.089470 0.069214 0.090190 0.091515 0.082948 0.097205 0.050597 0.065757
... ... ... ... ... ... ... ... ... ... ... ...
2003-02 -0.019734 0.001625 -0.026126 -0.034833 -0.045058 -0.039061 -0.032196 -0.026197 -0.034444 -0.034851 0.015116
2003-03 0.055259 0.031098 0.013882 0.006659 0.009049 0.012072 0.008408 0.014281 0.005784 0.009986 0.045273
2003-04 0.148508 0.140660 0.123251 0.112512 0.095413 0.092639 0.085307 0.109632 0.085932 0.127671 0.020836
2003-05 0.315904 0.202694 0.196376 0.132280 0.140624 0.109417 0.114861 0.129849 0.140532 0.171611 0.144293
2003-06 0.135771 0.099213 0.051757 0.031807 0.047800 0.054163 0.039503 0.040638 0.038876 0.056958 0.078813

252 rows × 11 columns

# Calculate average EW returns
ew_means = ew_ret.mean()
ew_means
portf_nr
1.0       0.022429
2.0       0.019655
3.0       0.017927
4.0       0.015500
5.0       0.014881
6.0       0.014966
7.0       0.013576
8.0       0.012631
9.0       0.008677
10.0      0.001652
Spread    0.020777
dtype: float64
# Plot returns of Spread portfolio
ew_ret['Spread'].plot();

# Calculate t-statistics for each portfolio returns (H0: average portfolio return is 0)
ew_tstats = ew_means / (ew_ret.std() / ew_ret.count()**0.5)
ew_tstats
portf_nr
1.0       3.760509
2.0       4.443625
3.0       5.114522
4.0       5.082020
5.0       4.975837
6.0       4.765637
7.0       4.130154
8.0       3.400974
9.0       2.047683
10.0      0.325883
Spread    7.035854
dtype: float64
# Plot cummulative return of spread portfolio
(1 + ew_ret)['Spread'].cumprod().plot();

# Plot cummulative return of portfolios 1 and 10
(1+ew_ret)[1].cumprod().plot();
(1+ew_ret)[10].cumprod().plot();

Value-weighted (VW) portfolios

# Load VW return data data
vw_ret = pd.read_pickle('../data/AG_vw_returns.zip')
vw_ret = vw_ret[:'2003-06'].copy()
vw_ret
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
mdate
1982-07 -0.016496 -0.032506 -0.008927 -0.020168 -0.008148 -0.014216 -0.016144 -0.026691 -0.052065 -0.063027 0.046531
1982-08 0.112879 0.126619 0.133671 0.111961 0.122147 0.108101 0.125463 0.129065 0.111937 0.149069 -0.036190
1982-09 0.035342 -0.011639 0.009310 0.021978 0.013155 0.008232 0.015515 0.021661 0.006081 -0.006955 0.042297
1982-10 0.180624 0.147793 0.118260 0.096087 0.082554 0.109715 0.099294 0.108276 0.163611 0.183572 -0.002948
1982-11 0.075535 0.093501 0.050342 0.041856 0.039193 0.049904 0.040065 0.032300 0.074103 0.070121 0.005414
... ... ... ... ... ... ... ... ... ... ... ...
2003-02 0.013984 -0.000944 -0.012868 -0.007327 -0.035330 -0.029244 -0.015190 -0.005070 -0.002535 -0.009106 0.023090
2003-03 0.039739 0.007937 -0.010142 -0.014830 0.008708 0.016905 -0.013148 0.030607 0.045923 0.003247 0.036492
2003-04 0.136213 0.078133 0.103938 0.091120 0.065227 0.060584 0.080776 0.078677 0.054610 0.089784 0.046430
2003-05 0.202974 0.166204 0.109430 0.086531 0.074955 0.047940 0.108823 0.013190 0.051810 0.101467 0.101507
2003-06 0.024044 0.023553 -0.024031 0.005133 0.010631 0.001055 0.025350 0.038687 0.004150 0.023575 0.000469

252 rows × 11 columns

# Calculate average EW returns
vw_means = vw_ret.mean()
vw_means
portf_nr
1.0       0.014354
2.0       0.013263
3.0       0.013213
4.0       0.013390
5.0       0.011904
6.0       0.012645
7.0       0.013343
8.0       0.012511
9.0       0.010194
10.0      0.006706
Spread    0.007648
dtype: float64
# Plot returns of Spread portfolio
vw_ret['Spread'].plot();

# Calculate t-statistics for each portfolio returns (H0: average portfolio return is 0)
vw_tstats = vw_means / (vw_ret.std() / vw_ret.count()**0.5)
vw_tstats
portf_nr
1.0       3.473671
2.0       3.927155
3.0       4.702159
4.0       5.177343
5.0       4.574888
6.0       4.439000
7.0       4.407662
8.0       3.497476
9.0       2.501369
10.0      1.531934
Spread    2.654032
dtype: float64
# Plot cummulative return of spread portfolio
(1 + vw_ret)['Spread'].cumprod().plot();

Risk-adjusted performance

# Load data on Fama-French factors
ff3f = pdr.DataReader(name='F-F_Research_Data_Factors', data_source='famafrench', 
                      start='1972-01-01')[0]/100
#ff3f.index.rename('mdate', inplace = True)
ff3f
Mkt-RF SMB HML RF
Date
1972-01 0.0249 0.0585 0.0224 0.0029
1972-02 0.0287 0.0130 -0.0279 0.0025
1972-03 0.0063 -0.0024 -0.0161 0.0027
1972-04 0.0029 0.0011 0.0012 0.0029
1972-05 0.0125 -0.0269 -0.0270 0.0030
... ... ... ... ...
2022-10 0.0783 0.0009 0.0805 0.0023
2022-11 0.0460 -0.0340 0.0138 0.0029
2022-12 -0.0641 -0.0068 0.0132 0.0033
2023-01 0.0665 0.0502 -0.0405 0.0035
2023-02 -0.0258 0.0110 -0.0080 0.0034

614 rows × 4 columns

Equal-weighted portfolios

# Merge EW monthly portfolio returns with the risk factors
alldata = ew_ret.join(ff3f)
alldata
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread Mkt-RF SMB HML RF
mdate
1982-07 -0.003948 -0.011054 0.002117 -0.009741 -0.009275 -0.003169 -0.004816 -0.006250 -0.034018 -0.071587 0.067639 -0.0319 0.0083 0.0009 0.0105
1982-08 0.043106 0.058108 0.067768 0.071293 0.070272 0.076378 0.080404 0.078438 0.076014 0.069051 -0.025945 0.1114 -0.0414 0.0095 0.0076
1982-09 0.037267 0.034962 0.034323 0.037597 0.034146 0.047353 0.034485 0.030422 0.009593 -0.020953 0.058220 0.0129 0.0295 0.0028 0.0051
1982-10 0.154922 0.122708 0.106485 0.107380 0.119641 0.122316 0.112235 0.135088 0.142025 0.161400 -0.006478 0.1130 0.0234 -0.0366 0.0059
1982-11 0.116354 0.115989 0.096393 0.089470 0.069214 0.090190 0.091515 0.082948 0.097205 0.050597 0.065757 0.0467 0.0467 -0.0187 0.0063
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
2003-02 -0.019734 0.001625 -0.026126 -0.034833 -0.045058 -0.039061 -0.032196 -0.026197 -0.034444 -0.034851 0.015116 -0.0188 -0.0045 -0.0138 0.0009
2003-03 0.055259 0.031098 0.013882 0.006659 0.009049 0.012072 0.008408 0.014281 0.005784 0.009986 0.045273 0.0109 0.0103 -0.0191 0.0010
2003-04 0.148508 0.140660 0.123251 0.112512 0.095413 0.092639 0.085307 0.109632 0.085932 0.127671 0.020836 0.0822 0.0066 0.0117 0.0010
2003-05 0.315904 0.202694 0.196376 0.132280 0.140624 0.109417 0.114861 0.129849 0.140532 0.171611 0.144293 0.0605 0.0474 0.0050 0.0009
2003-06 0.135771 0.099213 0.051757 0.031807 0.047800 0.054163 0.039503 0.040638 0.038876 0.056958 0.078813 0.0142 0.0177 0.0015 0.0010

252 rows × 15 columns

Cycle through all portfolios and regress excess returns on risk factors

# First, create empty tables to store portfolio alphas and their tstats
ew_portf_coeff = pd.DataFrame(np.nan, index = ['const', 'Mkt-RF','SMB', 'HML'], columns = ew_ret.columns)
ew_portf_coeff
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Mkt-RF NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
SMB NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
HML NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 
ew_portf_tstats = ew_portf_coeff.copy()
ew_portf_tstats
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Mkt-RF NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
SMB NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
HML NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 
# Regressions for each portfolio
for p in ew_ret.columns:
    #Set up the data
        # Dependent variable is excess return on the portfolio
    y = alldata[p] - alldata['RF']
        # Except for the spread portfolio (which is alread an excess return)
    if p == 'Spread':
        y = alldata[p] 
        
        # Independent variables are the risk factors
    X = alldata[['Mkt-RF','SMB','HML']].copy()
    X['const'] = 1
    
    # Run the regression
    res = sm.OLS(y, X[['const','Mkt-RF','SMB','HML']], missing = 'drop').fit()
    # Adjust for autocorrelation in residuals
    res_robust = res.get_robustcov_results(cov_type = 'HAC', maxlags = 6)
    
    # Store the results
    ew_portf_coeff.loc[:,p] = res_robust.params 
    ew_portf_tstats.loc[:,p] = res_robust.tvalues 

print(res_robust.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                 Spread   R-squared:                       0.197
Model:                            OLS   Adj. R-squared:                  0.187
Method:                 Least Squares   F-statistic:                     7.715
Date:                Wed, 26 Apr 2023   Prob (F-statistic):           6.01e-05
Time:                        07:21:58   Log-Likelihood:                 441.74
No. Observations:                 252   AIC:                            -875.5
Df Residuals:                     248   BIC:                            -861.4
Df Model:                           3                                         
Covariance Type:                  HAC                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.0214      0.004      6.008      0.000       0.014       0.028
Mkt-RF        -0.1747      0.062     -2.812      0.005      -0.297      -0.052
SMB            0.6059      0.174      3.474      0.001       0.262       0.949
HML            0.1271      0.154      0.825      0.410      -0.176       0.431
==============================================================================
Omnibus:                       53.275   Durbin-Watson:                   1.728
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              143.375
Skew:                           0.936   Prob(JB):                     7.35e-32
Kurtosis:                       6.185   Cond. No.                         45.2
==============================================================================

Notes:
[1] Standard Errors are heteroscedasticity and autocorrelation robust (HAC) using 6 lags and without small sample correction

Take a look at the results

print("\n Portfolio alphas and factor loadings:\n")
ew_portf_coeff

 Portfolio alphas and factor loadings:
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const 0.009559 0.006966 0.004510 0.002879 0.002097 0.002345 0.000996 -0.000359 -0.004288 -0.011835 0.021394
Mkt-RF 1.062260 0.961886 0.958992 0.891349 0.905635 0.927476 0.953022 1.034324 1.105890 1.236997 -0.174737
SMB 1.563695 1.179390 0.836426 0.688619 0.666368 0.702386 0.720628 0.828599 0.876110 0.957753 0.605943
HML 0.073249 0.223328 0.419966 0.353247 0.368794 0.289732 0.234368 0.185963 0.053055 -0.053822 0.127071 
print("\n T-statistics:\n")
ew_portf_tstats

 T-statistics:
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const 2.619291 3.055189 3.110097 2.709521 2.296160 2.428906 1.200616 -0.338180 -2.958751 -5.159747 6.007953
Mkt-RF 11.486691 15.931878 22.489307 26.761871 40.593130 39.773473 43.670124 37.855299 30.527413 20.069321 -2.812017
SMB 8.961525 10.707320 7.599961 8.404802 9.502792 12.176715 8.882976 9.840857 6.720156 5.728980 3.473969
HML 0.318961 1.472370 4.116269 5.334134 6.354676 5.332194 4.049948 3.267815 0.562896 -0.352292 0.824591

Value-weighted portfolios

# Merge VW monthly portfolio returns with the risk factors
alldata = vw_ret.join(ff3f)
alldata
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread Mkt-RF SMB HML RF
mdate
1982-07 -0.016496 -0.032506 -0.008927 -0.020168 -0.008148 -0.014216 -0.016144 -0.026691 -0.052065 -0.063027 0.046531 -0.0319 0.0083 0.0009 0.0105
1982-08 0.112879 0.126619 0.133671 0.111961 0.122147 0.108101 0.125463 0.129065 0.111937 0.149069 -0.036190 0.1114 -0.0414 0.0095 0.0076
1982-09 0.035342 -0.011639 0.009310 0.021978 0.013155 0.008232 0.015515 0.021661 0.006081 -0.006955 0.042297 0.0129 0.0295 0.0028 0.0051
1982-10 0.180624 0.147793 0.118260 0.096087 0.082554 0.109715 0.099294 0.108276 0.163611 0.183572 -0.002948 0.1130 0.0234 -0.0366 0.0059
1982-11 0.075535 0.093501 0.050342 0.041856 0.039193 0.049904 0.040065 0.032300 0.074103 0.070121 0.005414 0.0467 0.0467 -0.0187 0.0063
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
2003-02 0.013984 -0.000944 -0.012868 -0.007327 -0.035330 -0.029244 -0.015190 -0.005070 -0.002535 -0.009106 0.023090 -0.0188 -0.0045 -0.0138 0.0009
2003-03 0.039739 0.007937 -0.010142 -0.014830 0.008708 0.016905 -0.013148 0.030607 0.045923 0.003247 0.036492 0.0109 0.0103 -0.0191 0.0010
2003-04 0.136213 0.078133 0.103938 0.091120 0.065227 0.060584 0.080776 0.078677 0.054610 0.089784 0.046430 0.0822 0.0066 0.0117 0.0010
2003-05 0.202974 0.166204 0.109430 0.086531 0.074955 0.047940 0.108823 0.013190 0.051810 0.101467 0.101507 0.0605 0.0474 0.0050 0.0009
2003-06 0.024044 0.023553 -0.024031 0.005133 0.010631 0.001055 0.025350 0.038687 0.004150 0.023575 0.000469 0.0142 0.0177 0.0015 0.0010

252 rows × 15 columns

Cycle through all portfolios and regress excess returns on risk factors

# First, create empty tables to store portfolio alphas and their tstats
vw_portf_coeff = pd.DataFrame(np.nan, index = ['const', 'Mkt-RF','SMB', 'HML'], columns = vw_ret.columns)
vw_portf_coeff
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Mkt-RF NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
SMB NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
HML NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 
vw_portf_tstats = vw_portf_coeff.copy()
vw_portf_tstats
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Mkt-RF NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
SMB NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
HML NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 
# Regressions for each portfolio
for p in vw_ret.columns:
    #Set up the data
        # Dependent variable is excess return on the portfolio
    y = alldata[p] - alldata['RF']
        # Except for the spread portfolio (which is alread an excess return)
    if p == 'Spread':
        y = alldata[p] 
        
        # Independent variables are the risk factors
    X = alldata[['Mkt-RF','SMB','HML']].copy()
    X['const'] = 1
    
    # Run the regression
    res = sm.OLS(y, X[['const','Mkt-RF','SMB','HML']], missing='drop').fit()
    res_robust = res.get_robustcov_results(cov_type = 'HAC', maxlags = 6)
    
    # Store the results
    vw_portf_coeff.loc[:,p] = res_robust.params 
    vw_portf_tstats.loc[:,p] = res_robust.tvalues 

print(res_robust.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                 Spread   R-squared:                       0.227
Model:                            OLS   Adj. R-squared:                  0.217
Method:                 Least Squares   F-statistic:                     15.12
Date:                Wed, 26 Apr 2023   Prob (F-statistic):           4.53e-09
Time:                        07:23:08   Log-Likelihood:                 452.67
No. Observations:                 252   AIC:                            -897.3
Df Residuals:                     248   BIC:                            -883.2
Df Model:                           3                                         
Covariance Type:                  HAC                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.0054      0.003      1.817      0.070      -0.000       0.011
Mkt-RF        -0.0781      0.080     -0.974      0.331      -0.236       0.080
SMB            0.4752      0.149      3.181      0.002       0.181       0.769
HML            0.6611      0.122      5.439      0.000       0.422       0.901
==============================================================================
Omnibus:                       40.358   Durbin-Watson:                   1.891
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               95.899
Skew:                           0.749   Prob(JB):                     1.50e-21
Kurtosis:                       5.625   Cond. No.                         45.2
==============================================================================

Notes:
[1] Standard Errors are heteroscedasticity and autocorrelation robust (HAC) using 6 lags and without small sample correction

Take a look at the results

print("\n Portfolio alphas and factor loadings:\n")
vw_portf_coeff

 Portfolio alphas and factor loadings:
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const 0.000290 -0.000428 0.000185 0.001408 0.000363 0.000519 0.001905 0.002063 0.000197 -0.005083 0.005374
Mkt-RF 1.137942 1.079517 0.961564 0.883103 0.880913 0.950440 0.976145 1.006447 1.066098 1.216012 -0.078070
SMB 0.737353 0.169837 0.042593 -0.075015 -0.049715 -0.136981 -0.102068 0.076246 0.065937 0.262185 0.475168
HML 0.268259 0.307791 0.359662 0.250077 0.146542 0.170523 -0.041658 -0.341530 -0.553634 -0.392884 0.661143 
print("\n T-statistics:\n")
vw_portf_tstats

 T-statistics:
portf_nr 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Spread
const 0.138785 -0.247346 0.181093 1.489228 0.424545 0.551499 2.117366 1.610279 0.185540 -2.927644 1.816907
Mkt-RF 18.169942 24.861527 26.683369 26.409845 30.260815 28.649934 36.383952 31.229472 22.818687 26.260559 -0.974375
SMB 7.450553 1.837578 0.807182 -2.127936 -1.136439 -4.092208 -2.521685 1.509005 1.186412 3.443714 3.180851
HML 3.423134 2.726291 5.445035 4.055598 3.168255 1.787967 -0.805742 -7.028478 -7.056546 -5.343526 5.439341