HW 2 Part 1¶

By: Chengyi (Jeff) Chen

%load_ext nb_black
%matplotlib inline

import matplotlib.pyplot as plt

plt.rcParams["figure.dpi"] = 300
plt.rcParams["figure.figsize"] = (16, 12)

import pandas as pd
import numpy as np
import cvxpy as cp
import scipy as sp
from collections import namedtuple
from statsmodels.graphics.gofplots import qqplot
from datetime import datetime, timedelta
from tqdm import tqdm

/opt/anaconda3/envs/ml/lib/python3.7/site-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.
  import pandas.util.testing as tm

SPY and US Dollar Index Monthly Returns¶

spy_monthly_returns = pd.read_excel(
    "./data/HW2.xlsx", sheet_name=2, usecols=[0, 1], header=[0], index_col=[0]
)
spy_monthly_returns.index = pd.to_datetime(spy_monthly_returns.index)
spy_monthly_returns.head()

	S&P - 500 Index
2001-07-31	-0.984416
2001-08-31	-6.260163
2001-09-30	-8.075231
2001-10-31	1.906875
2001-11-30	7.670629

us_dollar_monthly_returns = pd.read_excel(
    "./data/HW2.xlsx", sheet_name=2, usecols=[0, 2], header=[0], index_col=[0]
)
us_dollar_monthly_returns.index = pd.to_datetime(us_dollar_monthly_returns.index)
us_dollar_monthly_returns.head()

	US Dollar Index
2001-07-31	-1.916799
2001-08-31	-3.208739
2001-09-30	-0.008817
2001-10-31	1.278547
2001-11-30	1.105694

S&P 500 Equities Monthly Returns¶

equity_monthly_returns = (
    pd.read_excel("./data/HW2.xlsx", sheet_name=1, header=[1]).iloc[:, 4:].T
)
equity_monthly_returns.columns = pd.MultiIndex.from_frame(
    pd.read_excel("./data/HW2.xlsx", sheet_name=1, header=[1], usecols=[0, 1, 2, 3])
)
equity_monthly_returns.index = pd.to_datetime(equity_monthly_returns.index)
equity_monthly_returns.head()

NAME	3m Co	Abbott Labs	Abbvie Inc	Abiomed Inc	Accenture Plc Ireland	Activision Blizzard Inc	Adobe Sys Inc	Advance Auto Parts	Advanced Micro Devic	Aes Corp	...	Willis Towers Watson Pu	Wynn Resorts Ltd	Xcel Energy Inc	Xerox Corp	Xilinx Inc	Xylem Inc	Yum Brands Inc	Zimmer Hldgs Inc	Zions Bancorp	Zoetis Inc
TICKER	MMM	ABT	ABBV	ABMD	ACN	ATVI	ADBE	AAP	AMD	AES	...	WLTW	WYNN	XEL	XRX	XLNX	XYL	YUM	ZBH	ZION	ZTS
SECTOR	INDU	HLTH	HLTH	HLTH	INFT	TCOM	INFT	DSCR	INFT	UTIL	...	FINA	DSCR	UTIL	INFT	INFT	INDU	DSCR	HLTH	FINA	HLTH
Market Cap ($ Mil)	110949	127036	138674	14640	89887	35535	110435	11478	18449	9577	...	19733	10755	25326	4708	21552	11991	28707	21156	7830	41098
2001-07-31	-1.945100	12.112976	NaN	-19.423240	NaN	-13.577718	-20.234087	NaN	-36.816609	-11.033682	...	-2.816904	NaN	-4.077114	-16.614480	-3.006790	NaN	4.214608	NaN	-0.932129	NaN
2001-08-31	-6.446533	-7.258951	NaN	-1.315789	-0.401070	9.226306	-10.349165	NaN	-25.794085	-13.524804	...	9.043619	NaN	1.707327	15.288163	-2.400000	NaN	-6.842003	-4.895105	-1.698511	NaN
2001-09-30	-5.475902	4.326061	NaN	-6.826667	-14.429530	-26.532794	-28.616782	NaN	-39.852399	-61.292271	...	24.348706	NaN	4.106134	-15.760939	-39.728484	NaN	-7.977550	2.022059	-6.287271	NaN
2001-10-31	6.077282	2.599966	NaN	24.441900	37.803922	32.808711	10.091781	NaN	20.736196	8.034321	...	-0.427533	NaN	0.461629	-9.677419	29.281768	NaN	28.990256	11.387387	-10.696854	NaN
2001-11-30	10.339559	3.812730	NaN	-12.741490	28.628344	3.113693	21.515301	NaN	37.804878	19.277978	...	1.674522	NaN	-3.429815	20.000215	18.704799	NaN	-6.206153	4.367519	1.164570	NaN

5 rows × 505 columns

A - Calculate historical factor monthly returns for the following factors based on Arbitrage Pricing Theory¶

Equity market factor. Using SP500 index return as equity market return proxy. (Data see SPX_DXY_MonthlyReturns)
US Dollar factor. Using DXY Index returns as proxy. (Data see SPX_DXY_MonthlyReturns)
Sector factors. (Note: you should leave one sector out to make sure proper running of Fama Macbeth regression. Let’s leave out the sector of UTIL.)
Size factor. At each month-end reconstitution date, we use the same company market cap data (column D on SP500Stocks) for simplicity. (Note: size exposure for a company should be sector neutral, i.e., the z-score of the log(mkt_cap) should be w.r.t. peer companies within the same sector.)

OLS Closed Form Solution:

(61)¶\[\begin{align} \hat\beta &= (X^TX)^{-1}X^Ty \end{align}\]

intersect = lambda a, b: sorted(set(a) & set(b))


def β_hat(X, y, dropna=True):
    if dropna:
        X, y = X.dropna(), y.dropna()
        common_index = intersect(X.index, y.index)
        X, y = X.loc[common_index], y.loc[common_index]
    else:
        X, y = X.fillna(0.0), y.fillna(0.0)
    return (
        pd.DataFrame(np.linalg.inv(X.T @ X), columns=X.columns, index=X.columns)
        @ X.T
        @ y.values
    )

Fama-Macbeth Two-step Regression ¶

Step 1 - Regress the Time Series of Returns of the Stock against Time Series of Factors to get Factor Exposures $\beta_{\text{factor}}$¶

For $m$ factors:

(62)¶\[\begin{align} R_{\text{Stock A}} &= \alpha + \beta_{F_1}F_1 + \beta_{F_2}F_2 + \cdots + \beta_{F_m}F_m + \epsilon \end{align}\]

Factor Exposures of each stock to S&P 500 and US Dollar¶

factor_exposures = equity_monthly_returns.apply(
    lambda equity: β_hat(
        X=pd.concat(
            [
                spy_monthly_returns,
                us_dollar_monthly_returns,
                pd.Series(1, name="α", index=spy_monthly_returns.index),
            ],
            axis=1,
        ),
        y=equity,
        dropna=True,
    ),
    axis=0,
).T
factor_exposures.reset_index(level=[1, 2, 3], drop=True, inplace=True)
factor_exposures.rename(
    {
        "S&P - 500 Index": "β (equity market factor exposure)",
        "US Dollar Index": "β (us dollar factor exposure)",
    },
    axis=1,
    inplace=True,
)
factor_exposures.head()

	β (equity market factor exposure)	β (us dollar factor exposure)	α
NAME
3m Co	0.897279	0.137059	0.411085
Abbott Labs	0.501031	-0.096830	0.635259
Abbvie Inc	1.450096	-0.556901	0.662926
Abiomed Inc	1.263956	0.059849	1.770226
Accenture Plc Ireland	1.152855	0.193686	0.772685

Sector factor one-hot encoding, drop UTIL sector column¶

sector_factor = pd.concat(
    [
        pd.Series(equity_monthly_returns.columns.get_level_values("NAME")),
        pd.Series(equity_monthly_returns.columns.get_level_values("SECTOR")),
    ],
    axis=1,
)
sector_factor.set_index("NAME", inplace=True)
sector_factor = pd.get_dummies(sector_factor).drop(["SECTOR_UTIL"], axis=1)
sector_factor.head()

	SECTOR_DSCR	SECTOR_ENER	SECTOR_FINA	SECTOR_HLTH	SECTOR_INDU	SECTOR_INFT	SECTOR_MATS	SECTOR_REAL	SECTOR_STPL	SECTOR_TCOM
NAME
3m Co	0	0	0	0	1	0	0	0	0	0
Abbott Labs	0	0	0	1	0	0	0	0	0	0
Abbvie Inc	0	0	0	1	0	0	0	0	0	0
Abiomed Inc	0	0	0	1	0	0	0	0	0	0
Accenture Plc Ireland	0	0	0	0	0	1	0	0	0	0

Size factor - Standardize log(Market Cap) by sector¶

size_factor = (
    pd.concat(
        [
            pd.Series(equity_monthly_returns.columns.get_level_values("NAME")),
            pd.Series(equity_monthly_returns.columns.get_level_values("SECTOR")),
            pd.Series(
                equity_monthly_returns.columns.get_level_values("Market Cap ($ Mil)")
            ),
        ],
        axis=1,
    )
    .groupby("SECTOR")["Market Cap ($ Mil)"]
    .transform(lambda x: (np.log(x) - np.log(x).mean()) / np.log(x).std())
    .to_frame()
)
size_factor.index = equity_monthly_returns.columns.get_level_values("NAME")
size_factor.rename(
    {"Market Cap ($ Mil)": "β (size factor exposure)"}, axis=1, inplace=True
)
size_factor.head()

	β (size factor exposure)
NAME
3m Co	1.977538
Abbott Labs	1.350510
Abbvie Inc	1.436935
Abiomed Inc	-0.779869
Accenture Plc Ireland	1.038986

Step 2 - Regress Returns of all the Stock in the Portfolio @ time $t, t \in T$ against Factor Exposures $\beta_{\text{factor}}$ of each Stock¶

For $t \in T$ timesteps:

(63)¶\[\begin{align} R_{\text{Stock }i, t} &= \gamma_{t, F_0} + \gamma_{t, F_1}\hat{\beta_{F_1}} + \gamma_{t, F_2}\hat{\beta_{F_2}} + \cdots + \gamma_{t, F_m}\hat{\beta_{F_m}} + \epsilon \end{align}\]

γ_pure_factor_monthly_returns = equity_monthly_returns.apply(
    lambda returns: β_hat(
        X=pd.concat(
            [
                factor_exposures.drop(["α"], axis=1),
                sector_factor,
                size_factor,
                pd.Series(1, index=size_factor.index, name="γ_residual"),
            ],
            axis=1,
        ),
        y=returns,
        dropna=False,
    ),
    axis=1,
)
γ_pure_factor_monthly_returns.head()

	β (equity market factor exposure)	β (us dollar factor exposure)	SECTOR_DSCR	SECTOR_ENER	SECTOR_FINA	SECTOR_HLTH	SECTOR_INDU	SECTOR_INFT	SECTOR_MATS	SECTOR_REAL	SECTOR_STPL	SECTOR_TCOM	β (size factor exposure)	γ_residual
2001-07-31	-5.081906	0.546521	9.689164	4.497985	6.003830	3.868261	7.997561	5.971431	7.795002	3.298414	5.834931	4.522445	-0.666501	-0.172946
2001-08-31	-5.351213	-2.530621	-0.865683	-1.974351	-0.172676	0.152704	0.283117	-0.786473	2.069103	2.237107	0.399686	-0.210663	-0.887846	3.563947
2001-09-30	-16.448408	-4.242000	6.115425	5.129572	15.719172	7.981975	9.199334	8.180248	9.807025	11.983056	5.616929	8.494224	-0.404314	0.634008
2001-10-31	7.730491	4.661245	-0.237472	5.161065	-8.600545	2.401637	-3.667805	5.580222	-2.020707	-6.257782	-0.408720	0.168053	-0.612423	-2.110206
2001-11-30	9.140003	4.696506	3.084856	-7.999442	-0.934668	0.446165	1.300240	2.209795	3.643307	-0.218926	2.664973	-1.297612	1.092705	-3.667960

B - Check for each factor their historical returns are significant or not (based on T-stat)¶

T = γ_pure_factor_monthly_returns.shape[0]

γ_pure_factor_risk_premium = γ_pure_factor_monthly_returns.mean(axis=0)

γ_pure_factor_standard_error = γ_pure_factor_monthly_returns.std(axis=0) / np.sqrt(T)

t_statistic = γ_pure_factor_risk_premium / γ_pure_factor_standard_error

# One-sample t-test, df = n - 1
p = 1 - sp.stats.t.cdf(t_statistic, df=T - 1)

α = 0.95  # Confidence level
for pure_factor, p_val in zip(γ_pure_factor_risk_premium.index[:-1], p[:-1]):
    if p_val < 1 - α:
        print(f"{pure_factor} monthly risk premium is statistically significant.")
    else:
        print(f"{pure_factor} monthly risk premium is not statistically significant.")

β (equity market factor exposure) monthly risk premium is not statistically significant.
β (us dollar factor exposure) monthly risk premium is not statistically significant.
SECTOR_DSCR monthly risk premium is not statistically significant.
SECTOR_ENER monthly risk premium is not statistically significant.
SECTOR_FINA monthly risk premium is not statistically significant.
SECTOR_HLTH monthly risk premium is not statistically significant.
SECTOR_INDU monthly risk premium is not statistically significant.
SECTOR_INFT monthly risk premium is not statistically significant.
SECTOR_MATS monthly risk premium is not statistically significant.
SECTOR_REAL monthly risk premium is not statistically significant.
SECTOR_STPL monthly risk premium is not statistically significant.
SECTOR_TCOM monthly risk premium is not statistically significant.
β (size factor exposure) monthly risk premium is statistically significant.

C - Using the last month in the back-test, i.e., 12/31/2018, show¶

1. All factor portfolios are long-short neutral portfolio, i.e., the total weights sum to 0¶

X = pd.concat(
    [
        factor_exposures.drop(["α"], axis=1),
        sector_factor,
        size_factor,
        pd.Series(1, index=size_factor.index, name="γ_residual"),
    ],
    axis=1,
)
y = equity_monthly_returns.loc["2018-12-31"]
pure_factor_portfolio_weights = np.linalg.inv(X.T @ X) @ X.T
pure_factor_portfolio_weights.index = X.columns

Check that the allocation of portfolio weights for each stock sums to 0 for all pure factor portfolios

# Exclude γ_residual
np.allclose(pure_factor_portfolio_weights.sum(axis=1)[:-1], 0)

True

2. For any factor portfolio, it has unit exposure to its own factor, but zero exposure to all other factors in the model¶

for pure_factor in X.columns[:-1]:
    stock_exposures_to_pure_factors = X[pure_factor]
    title = "|| " + pure_factor + " ||"
    print("=" * len(title))
    print(title)
    print("=" * len(title))
    for (
        name,
        pure_factor_portfolio_weights_for_one_factor,
    ) in pure_factor_portfolio_weights.iterrows():
        if (
            np.allclose(
                stock_exposures_to_pure_factors.T
                @ pure_factor_portfolio_weights_for_one_factor,
                1,
            )
            and name == pure_factor
        ):
            print(
                f"{pure_factor} has unit exposure to its own factor {name} - factor portfolio weights * stock exposure to factor == 1"
            )
        elif (
            np.allclose(
                stock_exposures_to_pure_factors.T
                @ pure_factor_portfolio_weights_for_one_factor,
                0,
            )
            and name != pure_factor
        ):
            print(
                f"{pure_factor} has 0 exposure to {name} factor - factor portfolio weights * stock exposure to factor == 1"
            )
        else:
            print("Something's wrong")
    print("\n")

=======================================
|| β (equity market factor exposure) ||
=======================================
β (equity market factor exposure) has unit exposure to its own factor β (equity market factor exposure) - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
β (equity market factor exposure) has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


===================================
|| β (us dollar factor exposure) ||
===================================
β (us dollar factor exposure) has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has unit exposure to its own factor β (us dollar factor exposure) - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
β (us dollar factor exposure) has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_DSCR ||
=================
SECTOR_DSCR has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has unit exposure to its own factor SECTOR_DSCR - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_DSCR has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_ENER ||
=================
SECTOR_ENER has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has unit exposure to its own factor SECTOR_ENER - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_ENER has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_FINA ||
=================
SECTOR_FINA has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has unit exposure to its own factor SECTOR_FINA - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_FINA has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_HLTH ||
=================
SECTOR_HLTH has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has unit exposure to its own factor SECTOR_HLTH - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_HLTH has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_INDU ||
=================
SECTOR_INDU has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has unit exposure to its own factor SECTOR_INDU - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INDU has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_INFT ||
=================
SECTOR_INFT has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has unit exposure to its own factor SECTOR_INFT - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_INFT has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_MATS ||
=================
SECTOR_MATS has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has unit exposure to its own factor SECTOR_MATS - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_MATS has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_REAL ||
=================
SECTOR_REAL has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has unit exposure to its own factor SECTOR_REAL - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_REAL has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_STPL ||
=================
SECTOR_STPL has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has unit exposure to its own factor SECTOR_STPL - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_STPL has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


=================
|| SECTOR_TCOM ||
=================
SECTOR_TCOM has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has unit exposure to its own factor SECTOR_TCOM - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to β (size factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
SECTOR_TCOM has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1


==============================
|| β (size factor exposure) ||
==============================
β (size factor exposure) has 0 exposure to β (equity market factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to β (us dollar factor exposure) factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_DSCR factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_ENER factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_FINA factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_HLTH factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_INDU factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_INFT factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_MATS factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_REAL factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_STPL factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to SECTOR_TCOM factor - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has unit exposure to its own factor β (size factor exposure) - factor portfolio weights * stock exposure to factor == 1
β (size factor exposure) has 0 exposure to γ_residual factor - factor portfolio weights * stock exposure to factor == 1

ΨΦ

HW 2 Part 1¶

SPY and US Dollar Index Monthly Returns¶

S&P 500 Equities Monthly Returns¶

A - Calculate historical factor monthly returns for the following factors based on Arbitrage Pricing Theory¶

Fama-Macbeth Two-step Regression ¶

Step 1 - Regress the Time Series of Returns of the Stock against Time Series of Factors to get Factor Exposures \(\beta_{\text{factor}}\)¶

Factor Exposures of each stock to S&P 500 and US Dollar¶

Sector factor one-hot encoding, drop UTIL sector column¶

Size factor - Standardize log(Market Cap) by sector¶

Step 2 - Regress Returns of all the Stock in the Portfolio @ time \(t, t \in T\) against Factor Exposures \(\beta_{\text{factor}}\) of each Stock¶

B - Check for each factor their historical returns are significant or not (based on T-stat)¶

C - Using the last month in the back-test, i.e., 12/31/2018, show¶

1. All factor portfolios are long-short neutral portfolio, i.e., the total weights sum to 0¶

2. For any factor portfolio, it has unit exposure to its own factor, but zero exposure to all other factors in the model¶

ΨΦ

HW 2 Part 1¶

SPY and US Dollar Index Monthly Returns¶

S&P 500 Equities Monthly Returns¶

A - Calculate historical factor monthly returns for the following factors based on Arbitrage Pricing Theory¶

Fama-Macbeth Two-step Regression¶

Step 1 - Regress the Time Series of Returns of the Stock against Time Series of Factors to get Factor Exposures \(\beta_{\text{factor}}\)¶

Factor Exposures of each stock to S&P 500 and US Dollar¶

Sector factor one-hot encoding, drop UTIL sector column¶

Size factor - Standardize log(Market Cap) by sector¶

Step 2 - Regress Returns of all the Stock in the Portfolio @ time \(t, t \in T\) against Factor Exposures \(\beta_{\text{factor}}\) of each Stock¶

B - Check for each factor their historical returns are significant or not (based on T-stat)¶

C - Using the last month in the back-test, i.e., 12/31/2018, show¶

1. All factor portfolios are long-short neutral portfolio, i.e., the total weights sum to 0¶

2. For any factor portfolio, it has unit exposure to its own factor, but zero exposure to all other factors in the model¶

Fama-Macbeth Two-step Regression ¶