Estimates from Factor Model¶

We assume a factor model where there are \(k\) assets and \(T\) time periods (months in our case), m is the common factors (topic weights in our case). \(r_{it}\) is the return for asset \(i\) at time \(t\).

\[\begin{split} \begin{align} r_{it} = &\alpha_i + \sum_{j=1}^{m} \beta_{ij} f_jt + \epsilon_{it}, t = 1, \dots, T, i = 1, \dots, k\\ \\ R_{k \times T} = & B_{k \times m} \cdot coef_{m \times T} + E_{k \times T} \end{align} \end{split}\]

In our analysis, \(R_{k \times T}\) is the returns matrix imported, \(B_{k \times m}\) is the topic modeling matrix, and \(coef_{m \times T}\) is the coefficient matrix computed from the linear regression of returns matrix on topic modeling matrix. \(E_{k \times T}\) is the residual matrix.

In our factor model,

\[\Sigma_R = B \Sigma_{coef} B^T + D, \text{ where } D = diag(\sigma^2_1, \dots, \sigma^2_k) \text{ and Var}(\epsilon_i) = \sigma^2_i\]

With the covariance developed from the factor model, we are able to convert the covariance into correlation matrix. Then, we use this correlation matrix and sample return standard deviation to calculate the estimated covariance.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import re
import json
import string
import nltk
from nltk.corpus import wordnet
from nltk.stem import WordNetLemmatizer 
from nltk.tokenize import word_tokenize
from nltk.corpus import stopwords

r_selected = pd.read_csv("data/filtered_r.csv")
# get the mean of all 
r_selected.set_index("name", inplace = True)
mu = r_selected.mean(axis = 1)
# compute the covariance matrix 
cov = r_selected.T.cov()

df = pd.read_csv('../data/preprocessed.csv',
                 usecols = ['reportingDate', 'name', 'CIK', 'coDescription',
                           'coDescription_stopwords', 'SIC', 'SIC_desc'])
df = df.set_index(df.name)

Sent-LDA¶

We ran the coherence score benchmarking over a range of 3 to 40 topics, incrementing by 3.

First, we fit the LDA model to all business description using the number of topics selected from coherence score benchmarking.

Then, we assume each sentence only represents one topic; get the frequency of the topics revealed in the whole document (business description for one company) and calculate the probability of each topics in the whole document.

Coherence Score Plot¶

Based on the above Coherence Score, we choose up to 12 topics since it gives the highest score up to here.

data = df.loc[:,"coDescription_stopwords"].to_list()

from sklearn.feature_extraction.text import CountVectorizer
from sklearn.decomposition import LatentDirichletAllocation

# LDA can only use raw term counts for LDA because it is a probabilistic graphical model
tf_vectorizer = CountVectorizer(max_df=0.85, min_df=2, max_features=600)
tf = tf_vectorizer.fit_transform(data)
tf_feature_names = tf_vectorizer.get_feature_names_out()
lda = LatentDirichletAllocation(n_components=12, random_state=0).fit(tf)

We show the top 10 words by weights in the 12 topics LDA model generates in the below table.

std_func.get_topics(lda, tf_vectorizer, 12)

	Topic # 01	Topic # 02	Topic # 03	Topic # 04	Topic # 05	Topic # 06	Topic # 07	Topic # 08	Topic # 09	Topic # 10	Topic # 11	Topic # 12
0	could	loan	share	product	investment	gas	hotel	patient	bank	customer	million	cell
1	gas	mortgage	stock	drug	income	oil	facility	treatment	capital	service	tax	cancer
2	regulation	real	note	fda	asset	natural	tenant	trial	institution	data	asset	tumor
3	oil	estate	issued	clinical	reit	production	lease	study	federal	product	net	product
4	future	commercial	preferred	patent	real	reserve	operating	phase	act	solution	income	therapy
5	natural	bank	date	approval	tax	proved	estate	clinical	banking	software	cash	therapeutic
6	price	interest	amount	trial	share	drilling	million	disease	holding	platform	expense	clinical
7	affect	rate	september	application	interest	regulation	real	drug	regulation	technology	value	technology
8	ability	million	per	regulatory	distribution	net	center	therapy	deposit	application	note	research
9	adversely	security	director	candidate	estate	water	portfolio	data	asset	sale	statement	license

Frequency of the Topics in Each Sentence¶

n_components = 12
prob = pd.DataFrame(0, index = df.name, columns = range(n_components))

for j in range(len(df)):
    LIST_sent = pd.Series(df.coDescription[j].split('.')).apply(std_func.lemmatize_sentence).apply(std_func.remove_nums).apply(std_func.remove_stopwords)
    
    X = tf_vectorizer.transform(LIST_sent.tolist())
    sent = lda.transform(X)
    sent_df = pd.DataFrame(sent)
    # drop the values that are smaller than 1/12
    # if the maximum value is 1/12, the probability of each topic in that sentence is the same
    # we cannot determine which topic to choose
    sent_df = sent_df[sent_df.max(axis = 1) > 1/12].reset_index(drop = True)

    for i in range(n_components):
        prob.iloc[j][i] = list(sent_df.idxmax(axis = 1)).count(i)
    
    # calculate the probability
    prob = prob.div(prob.sum(axis=1), axis=0)

	0	1	2	3	4	5	6	7	8	9	10	11
name
MONGODB, INC.	0.014652	0.007326	0.021978	0.036630	0.018315	0.010989	0.040293	0.021978	0.014652	0.739927	0.047619	0.025641
SALESFORCE COM INC	0.010811	0.010811	0.005405	0.016216	0.000000	0.021622	0.037838	0.005405	0.005405	0.821622	0.054054	0.010811
SPLUNK INC	0.010274	0.003425	0.013699	0.020548	0.013699	0.003425	0.023973	0.000000	0.003425	0.839041	0.058219	0.010274
OKTA, INC.	0.020305	0.000000	0.015228	0.050761	0.015228	0.005076	0.040609	0.000000	0.015228	0.786802	0.030457	0.020305
VEEVA SYSTEMS INC	0.093245	0.012845	0.094196	0.028544	0.035205	0.008563	0.019981	0.010466	0.017602	0.315414	0.352046	0.011893
...	...	...	...	...	...	...	...	...	...	...	...	...
AMERICAN REALTY CAPITAL NEW YORK CITY REIT, INC.	0.075472	0.084906	0.122642	0.009434	0.471698	0.009434	0.075472	0.000000	0.000000	0.066038	0.084906	0.000000
CYCLACEL PHARMACEUTICALS, INC.	0.027460	0.000000	0.029748	0.272311	0.011442	0.009153	0.018307	0.308924	0.000000	0.029748	0.013730	0.279176
ZOETIS INC.	0.036519	0.018868	0.074254	0.033475	0.034084	0.013999	0.035301	0.018868	0.020694	0.053561	0.644553	0.015825
STAG INDUSTRIAL, INC.	0.181818	0.016529	0.066116	0.016529	0.132231	0.016529	0.396694	0.008264	0.033058	0.074380	0.057851	0.000000
EQUINIX INC	0.024768	0.003096	0.030960	0.006192	0.015480	0.012384	0.061920	0.009288	0.009288	0.801858	0.018576	0.006192

675 rows × 12 columns

Factor Modelling¶

The common factors in our factor model are the 12 topics selected from LDA model. We use the calculated probability matrix of each topic for each companies as the topic modelling matrix \(B\). Then a linear regression of returns matrix on topic modelling matrix will give us the coefficient matrix for the 12 factors.

At each time \(t\), we run a linear regression of \(r_t\) on the topic modelling matrix (common factor matrix) \(B\) to generate a coefficient vector for time \(t\). At the same time, a residual vector \(\epsilon_t\) can be calculated using the diffrence of the actual \(r_t\) along with the predicted value \(\hat r_t\).

After \(T\) times (31 months in our case) of linear regression, we have a coefficient matrix \(coef_{T\times m}\) with 12 topics as columns and 31 months as rows as well as a residual matrix with 31 months as columns and the number of companies as rows. We will construct the diagonal matrix \(D\) using the diagonal values of covariance of the resiudal matrix which are the variance of residuals \(\text{Var}(\epsilon_{i1}, \epsilon_{i2}, \dots, \epsilon_{iT})\) for each company.

Demonstration in Pharmaceutical Preparations Industry¶

# get the names of the companies in the pharmaceutical preparations industry
Pharm = df[df.SIC == 2834]
Pharm_list = Pharm.index

# get the companies name that match return data and business description data
SET = (set(Pharm_list) & set(r_selected.index))
LIST = [*SET, ]

B_matrix = prob.T[LIST].T
B_matrix = B_matrix[~B_matrix.index.duplicated(keep="first")]

Topic Matrix: \({B_{k \times m}}\)¶

	0	1	2	3	4	5	6	7	8	9	10	11
name
AQUINOX PHARMACEUTICALS, INC	0.068259	0.003413	0.061433	0.607509	0.020478	0.003413	0.023891	0.071672	0.051195	0.040956	0.006826	0.040956
ASSEMBLY BIOSCIENCES, INC.	0.012658	0.000000	0.050633	0.012658	0.000000	0.012658	0.000000	0.594937	0.012658	0.025316	0.075949	0.202532
MANNKIND CORP	0.062500	0.015000	0.092500	0.370000	0.015000	0.025000	0.047500	0.125000	0.032500	0.105000	0.030000	0.080000
RIGEL PHARMACEUTICALS INC	0.018832	0.016949	0.047081	0.290019	0.011299	0.013183	0.007533	0.242938	0.013183	0.069680	0.071563	0.197740
GALECTIN THERAPEUTICS INC	0.016667	0.000000	0.050000	0.033333	0.000000	0.000000	0.008333	0.641667	0.016667	0.033333	0.000000	0.200000
...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.034591	0.011006	0.048742	0.496855	0.022013	0.004717	0.023585	0.139937	0.026730	0.064465	0.062893	0.064465
PULMATRIX, INC.	0.030303	0.006061	0.009091	0.439394	0.009091	0.006061	0.021212	0.336364	0.009091	0.042424	0.015152	0.075758
REGENERON PHARMACEUTICALS INC	0.034420	0.009058	0.054348	0.329710	0.016304	0.016304	0.028986	0.175725	0.016304	0.047101	0.083333	0.188406
CHIASMA, INC	0.021739	0.000000	0.013043	0.230435	0.004348	0.017391	0.021739	0.617391	0.000000	0.021739	0.008696	0.043478
IMPRIMIS PHARMACEUTICALS, INC.	0.000000	0.000000	0.106509	0.349112	0.011834	0.017751	0.029586	0.142012	0.017751	0.106509	0.088757	0.130178

124 rows × 12 columns

r_Pharm = r_selected.T[LIST].T

coef_mat = pd.DataFrame(0, index = r_Pharm.columns, columns = range(n_components))
res_mat = pd.DataFrame(0, index = r_Pharm.index, columns = r_Pharm.columns)

from sklearn.linear_model import LinearRegression

for i in range(len(r_Pharm.columns)):
    LR = LinearRegression()
    date = r_Pharm.columns[i]
    r_t_i = r_Pharm[date] 
    r_t_i_demean = r_t_i - r_t_i.mean()
    reg = LR.fit(B_matrix, r_t_i_demean)
    coef_mat.iloc[i] = reg.coef_
    prediction = B_matrix.dot(reg.coef_)
    residual_t_i = r_t_i_demean - prediction
    res_mat[date] = residual_t_i

Coefficient Matrix: \(coef_{T \times m}\)¶

	0	1	2	3	4	5	6	7	8	9	10	11
2016-06-30	0.993361	-1.875375	0.195258	-0.337758	1.312631	2.051489	1.238693	-0.392438	-2.489539	-0.470602	-0.236897	0.011175
2016-07-31	-0.587373	2.486985	-0.051766	-0.119139	0.599033	-0.509589	0.034452	-0.130789	-1.116928	-0.347125	0.009258	-0.267018
2016-08-31	-0.088389	-2.184883	0.309753	0.127562	0.202514	-0.612011	-1.106752	0.233257	1.801470	0.803841	0.164920	0.348719
2016-09-30	-0.122107	1.875134	0.171593	0.108377	0.836037	-0.015983	-1.795211	-0.121855	-0.571359	-0.409085	0.001294	0.043164
2016-10-31	-0.236524	-2.536547	-0.143964	-0.035811	-0.187329	1.460662	-0.469362	-0.034221	1.959008	0.200099	0.077996	-0.054007
2016-11-30	-1.151701	1.936744	0.147886	-0.116144	0.417320	-1.095169	-1.533299	-0.054423	1.213919	-0.113511	0.200694	0.147685
2016-12-31	0.483848	-1.161150	-0.100632	-0.193967	-0.265334	0.825843	0.407563	-0.141296	1.436270	-0.580140	-0.186887	-0.524118
2017-01-31	-0.277906	1.977522	0.574171	0.256089	-1.584684	-3.236387	0.679985	0.346124	1.308807	-0.275143	0.050787	0.180636
2017-02-28	0.895630	3.177223	0.208617	-0.125386	-3.827412	-2.817838	0.297242	0.148517	2.594685	-0.329282	-0.044365	-0.177630
2017-03-31	-1.031747	1.838094	0.056902	-0.172604	-0.423448	-1.919721	-0.909357	-0.001361	3.564327	-0.665482	-0.290407	-0.045196
2017-04-30	-0.211334	-1.602075	-0.224406	-0.074999	1.339630	2.564717	1.782393	-0.065947	-3.099536	-0.262565	-0.009339	-0.136537
2017-05-31	0.648542	-0.663393	0.028541	0.062360	1.617295	-0.347683	0.631139	-0.103693	-2.235411	0.330108	-0.039340	0.071534
2017-06-30	0.463373	-0.674562	-0.236003	0.036023	-1.249023	0.703260	-0.299994	-0.085970	1.889070	-0.466909	-0.000894	-0.078371
2017-07-31	0.177977	2.140741	-0.235111	-0.194055	-0.116186	0.649325	-0.868795	-0.172467	-0.588006	-0.213592	-0.290586	-0.289245
2017-08-31	-0.186677	0.941224	0.053929	-0.093700	1.319535	-1.558089	-0.083888	-0.019411	0.684707	-0.952684	-0.019513	-0.085431
2017-09-30	0.316972	3.418921	0.449367	0.154292	-1.203242	-4.995160	-0.760192	0.307751	2.031288	0.106942	0.156732	0.016328
2017-10-31	1.102453	0.842594	-0.141848	-0.241794	0.370936	-0.095343	-0.453912	-0.239645	-0.450735	-0.424791	-0.172505	-0.095411
2017-11-30	0.813015	-1.930563	-0.081636	-0.256725	-0.500748	0.846789	-0.582951	-0.160954	1.791315	0.320750	-0.355372	0.097077
2017-12-31	0.917035	0.115870	-0.266767	-0.232480	0.053044	0.875878	-0.565072	-0.232319	0.126186	-0.347361	-0.248846	-0.195167
2018-01-31	0.510904	-1.424801	0.525894	0.376281	-0.103350	-2.715219	-1.475502	0.339420	3.212658	0.237646	0.242735	0.273334
2018-02-28	0.092100	-0.271018	0.392836	-0.053891	1.497478	-0.632567	-0.492316	-0.113717	0.696161	-0.951320	0.025789	-0.189535
2018-03-31	0.250350	-0.051754	-0.145994	-0.254493	1.377327	-0.671779	0.020492	-0.394691	0.523717	-0.575754	-0.282293	0.204871
2018-04-30	-0.306677	-0.355225	0.010576	0.045465	0.429732	0.335406	-1.145255	0.021852	0.837205	0.212868	-0.017595	-0.068353
2018-05-31	0.373257	-2.238666	0.102341	0.062301	-2.039520	3.767960	0.052930	0.080370	-1.161794	0.381085	0.087377	0.532358
2018-06-30	-0.562615	0.509221	-0.566240	-0.184665	0.417973	4.326845	0.844668	-0.152939	-5.142850	0.755770	-0.093239	-0.151928
2018-07-31	-0.936502	0.468251	-0.050995	0.024653	-0.288721	-0.567886	0.436411	-0.073319	1.170531	-0.030022	-0.052548	-0.099853
2018-08-31	-1.410095	-1.756273	0.335407	0.328625	1.468389	-0.571324	-0.224952	0.310353	0.193843	0.839583	0.175877	0.310568
2018-09-30	0.666432	-0.886211	0.482306	0.478674	-1.332732	-0.169247	1.223540	0.287162	-1.172659	0.241639	0.186752	-0.005655
2018-10-31	0.309981	2.743814	-0.470120	-0.223289	-1.423760	1.122517	0.128998	-0.210256	-1.681892	0.404869	-0.318367	-0.382494
2018-11-30	0.430086	-2.222911	0.248783	-0.114050	-0.226333	1.541043	-0.856399	-0.140060	1.962801	-0.443735	0.035656	-0.214880
2018-12-31	0.691217	-0.083460	-0.044176	-0.109118	0.947424	-0.069292	-0.720454	-0.163557	0.323619	-0.386821	-0.210655	-0.174728

Residual Matrix¶

res_mat

	2016-06-30	2016-07-31	2016-08-31	2016-09-30	2016-10-31	2016-11-30	2016-12-31	2017-01-31	2017-02-28	2017-03-31	...	2018-03-31	2018-04-30	2018-05-31	2018-06-30	2018-07-31	2018-08-31	2018-09-30	2018-10-31	2018-11-30	2018-12-31
name
AQUINOX PHARMACEUTICALS, INC	0.154922	0.376646	0.055042	0.187457	-0.089496	0.370412	0.280747	-0.214152	-0.195240	-0.038969	...	0.110603	-0.155788	0.063595	-0.404264	0.138140	-0.214698	-0.394826	0.271679	-0.092694	0.131095
ASSEMBLY BIOSCIENCES, INC.	0.326680	0.107196	-0.181379	0.178080	1.111857	-0.157130	0.162014	0.216233	-0.036756	0.206368	...	0.104757	-0.113084	-0.337129	0.105990	0.225364	-0.423837	-0.281614	-0.005226	0.201520	0.207516
MANNKIND CORP	0.456006	-0.058118	-0.369472	-0.230926	-0.171823	0.191022	0.489506	-0.123330	-0.364644	-0.337926	...	-0.048994	-0.237040	-0.063584	0.086175	-0.138489	-0.531817	0.353167	0.263774	0.020654	-0.129442
RIGEL PHARMACEUTICALS INC	0.196890	0.083178	0.254775	0.006443	-0.101194	-0.065924	0.173420	-0.390079	0.100915	0.397049	...	0.120304	0.018212	-0.355162	-0.005518	0.059101	-0.110906	-0.291213	0.204859	0.105055	0.122484
GALECTIN THERAPEUTICS INC	0.380841	0.227081	-0.237329	-0.320427	-0.187439	0.091932	0.410407	-0.367103	0.701588	0.154896	...	0.387516	-0.288879	0.266132	0.498471	-0.121522	-0.061167	-0.302299	0.100248	0.252778	0.022014
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.504481	0.095034	-0.233910	0.196890	0.131270	0.241172	1.458790	-0.041756	0.372147	0.127169	...	0.422389	0.096134	-0.270544	-0.071477	-0.148716	-0.574215	-0.553092	0.306263	-0.502079	0.037460
PULMATRIX, INC.	0.170737	0.150913	-0.451489	0.032823	-0.028021	-0.489517	0.176836	2.342886	0.811992	-0.070448	...	-0.398212	-0.102640	-0.151729	0.100987	0.080103	-0.358161	-0.498207	0.326749	0.102522	-0.045197
REGENERON PHARMACEUTICALS INC	0.123785	0.289575	-0.262475	-0.028181	0.035840	0.093260	0.177957	-0.262176	-0.007960	0.124937	...	0.219150	-0.094996	-0.274320	0.285485	0.131329	-0.182917	-0.271993	0.180788	0.162024	0.315493
CHIASMA, INC	0.246003	-0.008955	-0.244587	0.208747	-0.049651	-0.069338	0.162285	-0.270802	-0.312447	0.142753	...	0.277882	-0.017701	-0.223102	0.072804	0.023282	0.365140	0.200864	0.335350	0.198473	0.095848
IMPRIMIS PHARMACEUTICALS, INC.	0.248464	0.114470	-0.170438	-0.108541	-0.162678	-0.005414	0.139754	-0.337944	0.090436	0.812665	...	0.172954	0.288501	-0.301488	0.085804	0.098330	-0.097924	-0.381156	0.884198	0.179173	0.537175

124 rows × 31 columns

Diagonal Matrix: \(D_{k \times k}\)¶

\(D = diag(\sigma^2_1, \dots, \sigma^2_k) \text{ and Var}(\epsilon_i) = \sigma^2_i\)

D_mat = pd.DataFrame(np.diag(np.diag(res_mat.T.cov()))).set_index(B_matrix.index)
D_mat.columns = B_matrix.index
D_mat

name	AQUINOX PHARMACEUTICALS, INC	ASSEMBLY BIOSCIENCES, INC.	MANNKIND CORP	RIGEL PHARMACEUTICALS INC	GALECTIN THERAPEUTICS INC	FORTRESS BIOTECH, INC.	BIOSPECIFICS TECHNOLOGIES CORP	BIOMARIN PHARMACEUTICAL INC	LEXICON PHARMACEUTICALS, INC.	WAVE LIFE SCIENCES LTD.	...	SAREPTA THERAPEUTICS, INC.	AMICUS THERAPEUTICS INC	CHEMBIO DIAGNOSTICS, INC.	NATURES SUNSHINE PRODUCTS INC	HEAT BIOLOGICS, INC.	ACHAOGEN INC	PULMATRIX, INC.	REGENERON PHARMACEUTICALS INC	CHIASMA, INC	IMPRIMIS PHARMACEUTICALS, INC.
name
AQUINOX PHARMACEUTICALS, INC	0.049207	0.000000	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.000000	0.000000
ASSEMBLY BIOSCIENCES, INC.	0.000000	0.091779	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.000000	0.000000
MANNKIND CORP	0.000000	0.000000	0.109918	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.000000	0.000000
RIGEL PHARMACEUTICALS INC	0.000000	0.000000	0.000000	0.04958	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.000000	0.000000
GALECTIN THERAPEUTICS INC	0.000000	0.000000	0.000000	0.00000	0.085131	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.000000	0.000000
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.000000	0.000000	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.14366	0.00000	0.0000	0.000000	0.000000
PULMATRIX, INC.	0.000000	0.000000	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.24829	0.0000	0.000000	0.000000
REGENERON PHARMACEUTICALS INC	0.000000	0.000000	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0365	0.000000	0.000000
CHIASMA, INC	0.000000	0.000000	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.046287	0.000000
IMPRIMIS PHARMACEUTICALS, INC.	0.000000	0.000000	0.000000	0.00000	0.000000	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.00000	0.00000	0.0000	0.000000	0.088088

124 rows × 124 columns

Covariance from Factor Model: \(\Sigma_{R\{k \times k\}}\)¶

\[\Sigma_R = B \Sigma_{coef} B^T + D, \text{ where } D = diag(\sigma^2_1, \dots, \sigma^2_k)\]

cov_Factor_Model = pd.DataFrame(np.array(B_matrix.dot(coef_mat.cov()).dot(B_matrix.T)) + 
                                np.diag(np.diag(res_mat.T.cov()))).set_index(B_matrix.index)
cov_Factor_Model.columns = B_matrix.index
cov_Factor_Model

name	AQUINOX PHARMACEUTICALS, INC	ASSEMBLY BIOSCIENCES, INC.	MANNKIND CORP	RIGEL PHARMACEUTICALS INC	GALECTIN THERAPEUTICS INC	FORTRESS BIOTECH, INC.	BIOSPECIFICS TECHNOLOGIES CORP	BIOMARIN PHARMACEUTICAL INC	LEXICON PHARMACEUTICALS, INC.	WAVE LIFE SCIENCES LTD.	...	SAREPTA THERAPEUTICS, INC.	AMICUS THERAPEUTICS INC	CHEMBIO DIAGNOSTICS, INC.	NATURES SUNSHINE PRODUCTS INC	HEAT BIOLOGICS, INC.	ACHAOGEN INC	PULMATRIX, INC.	REGENERON PHARMACEUTICALS INC	CHIASMA, INC	IMPRIMIS PHARMACEUTICALS, INC.
name
AQUINOX PHARMACEUTICALS, INC	0.082664	0.027580	0.021975	0.023906	0.031289	0.022151	0.017689	0.026934	0.015996	0.022890	...	0.024623	0.026814	0.017966	0.017976	0.025234	0.027905	0.026684	0.022341	0.024222	0.025094
ASSEMBLY BIOSCIENCES, INC.	0.027580	0.121400	0.021564	0.025520	0.032224	0.021936	0.020872	0.025931	0.021005	0.024203	...	0.024283	0.026607	0.020009	0.015124	0.030050	0.025998	0.026621	0.022971	0.026178	0.026942
MANNKIND CORP	0.021975	0.021564	0.128645	0.019494	0.023330	0.016550	0.016915	0.020663	0.016230	0.019876	...	0.019500	0.021384	0.018122	0.014170	0.021800	0.020677	0.021561	0.018664	0.021170	0.021865
RIGEL PHARMACEUTICALS INC	0.023906	0.025520	0.019494	0.072990	0.027582	0.019454	0.019165	0.023705	0.019305	0.022260	...	0.022541	0.024455	0.020153	0.014569	0.027107	0.023675	0.024321	0.021053	0.023424	0.024748
GALECTIN THERAPEUTICS INC	0.031289	0.032224	0.023330	0.027582	0.120964	0.023394	0.021352	0.028389	0.021646	0.025895	...	0.026476	0.028824	0.019961	0.016570	0.032010	0.028752	0.029106	0.024567	0.028157	0.028425
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.027905	0.025998	0.020677	0.023675	0.028752	0.021043	0.018753	0.025612	0.017820	0.022553	...	0.023734	0.025881	0.020043	0.016172	0.025490	0.169572	0.025857	0.021777	0.024290	0.025221
PULMATRIX, INC.	0.026684	0.026621	0.021561	0.024321	0.029106	0.020715	0.020334	0.026036	0.019779	0.023535	...	0.024154	0.026431	0.020827	0.015695	0.026326	0.025857	0.275105	0.022459	0.026065	0.026166
REGENERON PHARMACEUTICALS INC	0.022341	0.022971	0.018664	0.021053	0.024567	0.018554	0.018188	0.022061	0.017901	0.021015	...	0.020891	0.022790	0.019760	0.013850	0.024470	0.021777	0.022459	0.056436	0.021823	0.023276
CHIASMA, INC	0.024222	0.026178	0.021170	0.023424	0.028157	0.019913	0.020919	0.024883	0.020438	0.022984	...	0.022901	0.025401	0.020539	0.014173	0.025562	0.024290	0.026065	0.021823	0.072737	0.025849
IMPRIMIS PHARMACEUTICALS, INC.	0.025094	0.026942	0.021865	0.024748	0.028425	0.021703	0.021905	0.025278	0.021344	0.024408	...	0.024026	0.026551	0.025019	0.016308	0.028788	0.025221	0.026166	0.023276	0.025849	0.116258

124 rows × 124 columns

Perform Mean-Variance Analysis¶

For demonstration, we only use the Pharmaceutical Preparations industry data to generate portfolio based on Mean-Variance Analysis. We estimate the covariance matrix based on the factor model constructed above.

from pypfopt import EfficientFrontier
from pypfopt import risk_models
from pypfopt import expected_returns
from pypfopt import objective_functions
from pypfopt import plotting

Sample Mean for the Pharmaceutical Preparations Industry¶

mu_Pharm = mu[LIST]
mu_Pharm

name
AQUINOX PHARMACEUTICALS, INC     -0.004622
ASSEMBLY BIOSCIENCES, INC.        0.072839
MANNKIND CORP                    -0.002810
RIGEL PHARMACEUTICALS INC         0.011020
GALECTIN THERAPEUTICS INC         0.064165
                                    ...   
ACHAOGEN INC                      0.007742
PULMATRIX, INC.                   0.009480
REGENERON PHARMACEUTICALS INC     0.002351
CHIASMA, INC                      0.018143
IMPRIMIS PHARMACEUTICALS, INC.    0.031240
Length: 124, dtype: float64

Sample Covariance for the Pharmaceutical Preparations Industry¶

tmp = cov[LIST].T
cov_Pharm = tmp[LIST]
cov_Pharm

name	AQUINOX PHARMACEUTICALS, INC	ASSEMBLY BIOSCIENCES, INC.	MANNKIND CORP	RIGEL PHARMACEUTICALS INC	GALECTIN THERAPEUTICS INC	FORTRESS BIOTECH, INC.	BIOSPECIFICS TECHNOLOGIES CORP	BIOMARIN PHARMACEUTICAL INC	LEXICON PHARMACEUTICALS, INC.	WAVE LIFE SCIENCES LTD.	...	SAREPTA THERAPEUTICS, INC.	AMICUS THERAPEUTICS INC	CHEMBIO DIAGNOSTICS, INC.	NATURES SUNSHINE PRODUCTS INC	HEAT BIOLOGICS, INC.	ACHAOGEN INC	PULMATRIX, INC.	REGENERON PHARMACEUTICALS INC	CHIASMA, INC	IMPRIMIS PHARMACEUTICALS, INC.
name
AQUINOX PHARMACEUTICALS, INC	0.044662	-0.000043	-0.001594	0.009369	0.002725	0.009361	0.002458	0.004787	0.008030	0.010651	...	0.006712	0.007105	-0.001749	0.006418	0.020250	0.022941	0.008470	0.001158	0.004271	-0.000662
ASSEMBLY BIOSCIENCES, INC.	-0.000043	0.071030	-0.008567	-0.006169	-0.014516	0.005300	-0.006172	-0.001668	-0.002814	0.018593	...	-0.005024	0.002169	0.001177	-0.004359	0.004144	-0.000960	0.045995	-0.006473	-0.006780	-0.017609
MANNKIND CORP	-0.001594	-0.008567	0.099741	-0.004992	-0.008867	-0.002842	-0.002372	-0.004764	-0.005258	0.001322	...	-0.011109	-0.006724	-0.000207	0.006063	-0.004294	0.004404	0.007041	-0.000822	0.020805	-0.031491
RIGEL PHARMACEUTICALS INC	0.009369	-0.006169	-0.004992	0.033500	0.015401	0.008834	0.000461	0.000356	0.000150	0.012268	...	0.010023	0.003802	-0.000883	0.004908	0.021613	0.000028	-0.006501	-0.000615	0.005843	0.014332
GALECTIN THERAPEUTICS INC	0.002725	-0.014516	-0.008867	0.015401	0.083509	0.021772	0.002357	0.009969	0.016729	-0.008727	...	0.002776	0.011818	0.012040	0.001235	0.001660	0.014341	0.031317	0.006681	-0.001683	0.006423
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.022941	-0.000960	0.004404	0.000028	0.014341	0.012751	0.007767	0.000741	0.004936	-0.008309	...	-0.001360	-0.001465	-0.000546	-0.001501	-0.018732	0.097615	0.031299	-0.003298	-0.011172	-0.004221
PULMATRIX, INC.	0.008470	0.045995	0.007041	-0.006501	0.031317	0.003358	-0.009037	0.011645	0.016356	0.006861	...	0.016898	0.021201	-0.012933	-0.015419	-0.008802	0.031299	0.306222	-0.001204	-0.000617	-0.007801
REGENERON PHARMACEUTICALS INC	0.001158	-0.006473	-0.000822	-0.000615	0.006681	0.006683	0.001614	0.004185	0.005760	-0.005950	...	0.006381	0.004394	0.000644	0.004890	0.000815	-0.003298	-0.001204	0.009307	0.003194	-0.001211
CHIASMA, INC	0.004271	-0.006780	0.020805	0.005843	-0.001683	0.000563	0.006284	0.001363	0.000866	0.006205	...	0.018008	0.000039	-0.003681	0.000256	0.000472	-0.011172	-0.000617	0.003194	0.049106	-0.000313
IMPRIMIS PHARMACEUTICALS, INC.	-0.000662	-0.017609	-0.031491	0.014332	0.006423	-0.002731	0.002582	0.000078	-0.004131	0.001506	...	-0.006600	-0.003028	-0.006985	0.000745	0.002129	-0.004221	-0.007801	-0.001211	-0.000313	0.045175

124 rows × 124 columns

Correlation Matric Converted from Covariance Matrix of Factor Model¶

def correlation_from_covariance(covariance):
    v = np.sqrt(np.diag(covariance))
    outer_v = np.outer(v, v)
    correlation = covariance / outer_v
    correlation[covariance == 0] = 0
    return correlation

cor_Factor_Model = correlation_from_covariance(cov_Factor_Model)
cor_Factor_Model

name	AQUINOX PHARMACEUTICALS, INC	ASSEMBLY BIOSCIENCES, INC.	MANNKIND CORP	RIGEL PHARMACEUTICALS INC	GALECTIN THERAPEUTICS INC	FORTRESS BIOTECH, INC.	BIOSPECIFICS TECHNOLOGIES CORP	BIOMARIN PHARMACEUTICAL INC	LEXICON PHARMACEUTICALS, INC.	WAVE LIFE SCIENCES LTD.	...	SAREPTA THERAPEUTICS, INC.	AMICUS THERAPEUTICS INC	CHEMBIO DIAGNOSTICS, INC.	NATURES SUNSHINE PRODUCTS INC	HEAT BIOLOGICS, INC.	ACHAOGEN INC	PULMATRIX, INC.	REGENERON PHARMACEUTICALS INC	CHIASMA, INC	IMPRIMIS PHARMACEUTICALS, INC.
name
AQUINOX PHARMACEUTICALS, INC	1.000000	0.275315	0.213099	0.307769	0.312901	0.287328	0.269061	0.372618	0.263411	0.295830	...	0.264436	0.358893	0.245833	0.306985	0.249898	0.235693	0.176946	0.327089	0.312367	0.255979
ASSEMBLY BIOSCIENCES, INC.	0.275315	1.000000	0.172552	0.271105	0.265919	0.234792	0.261976	0.296029	0.285437	0.258116	...	0.215190	0.293873	0.225926	0.213127	0.245561	0.181195	0.145670	0.277525	0.278575	0.226780
MANNKIND CORP	0.213099	0.172552	1.000000	0.201177	0.187022	0.172079	0.206249	0.229151	0.214251	0.205922	...	0.167875	0.229431	0.198774	0.193983	0.173055	0.139994	0.114610	0.219045	0.218847	0.178793
RIGEL PHARMACEUTICALS INC	0.307769	0.271105	0.201177	1.000000	0.293535	0.268544	0.310234	0.349001	0.338327	0.306166	...	0.257617	0.348344	0.293460	0.264772	0.285683	0.212805	0.171630	0.328031	0.321473	0.268653
GALECTIN THERAPEUTICS INC	0.312901	0.265919	0.187022	0.293535	1.000000	0.250846	0.268485	0.324680	0.294671	0.276666	...	0.235048	0.318933	0.225793	0.233916	0.262056	0.200754	0.159552	0.297337	0.300183	0.239696
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.235693	0.181195	0.139994	0.212805	0.200754	0.190580	0.199167	0.247398	0.204895	0.203510	...	0.177966	0.241863	0.191483	0.192821	0.176249	1.000000	0.119718	0.222611	0.218711	0.179628
PULMATRIX, INC.	0.176946	0.145670	0.114610	0.171630	0.159552	0.147289	0.169547	0.197445	0.178543	0.166734	...	0.142191	0.193925	0.156215	0.146926	0.142911	0.119718	1.000000	0.180248	0.184258	0.146311
REGENERON PHARMACEUTICALS INC	0.327089	0.277525	0.219045	0.328031	0.297337	0.291276	0.334824	0.369376	0.356767	0.328710	...	0.271535	0.369177	0.327227	0.286254	0.293280	0.222611	0.180248	1.000000	0.340613	0.287350
CHIASMA, INC	0.312367	0.278575	0.218847	0.321473	0.300183	0.275359	0.339220	0.366987	0.358804	0.316674	...	0.262186	0.362446	0.299608	0.258031	0.269868	0.218711	0.184258	0.340613	1.000000	0.281093
IMPRIMIS PHARMACEUTICALS, INC.	0.255979	0.226780	0.178793	0.268653	0.239696	0.237385	0.280960	0.294891	0.296377	0.265997	...	0.217571	0.299665	0.288679	0.234838	0.240394	0.179628	0.146311	0.287350	0.281093	1.000000

124 rows × 124 columns

Estimated Covariance¶

sd = pd.DataFrame(np.sqrt(np.diag(np.diagonal(cov_Pharm))))
sd = sd.set_index(cov_Pharm.index)
sd.columns = cov_Pharm.index
Factor_Model_cov = pd.DataFrame((np.dot(np.dot(sd, cor_Factor_Model),sd))).set_index(cor_Factor_Model.index)
Factor_Model_cov.columns = cor_Factor_Model.index
Factor_Model_cov

name	AQUINOX PHARMACEUTICALS, INC	ASSEMBLY BIOSCIENCES, INC.	MANNKIND CORP	RIGEL PHARMACEUTICALS INC	GALECTIN THERAPEUTICS INC	FORTRESS BIOTECH, INC.	BIOSPECIFICS TECHNOLOGIES CORP	BIOMARIN PHARMACEUTICAL INC	LEXICON PHARMACEUTICALS, INC.	WAVE LIFE SCIENCES LTD.	...	SAREPTA THERAPEUTICS, INC.	AMICUS THERAPEUTICS INC	CHEMBIO DIAGNOSTICS, INC.	NATURES SUNSHINE PRODUCTS INC	HEAT BIOLOGICS, INC.	ACHAOGEN INC	PULMATRIX, INC.	REGENERON PHARMACEUTICALS INC	CHIASMA, INC	IMPRIMIS PHARMACEUTICALS, INC.
name
AQUINOX PHARMACEUTICALS, INC	0.044662	0.015507	0.014223	0.011905	0.019109	0.011316	0.004450	0.006625	0.008248	0.012029	...	0.016209	0.010009	0.005952	0.008725	0.017869	0.015562	0.020693	0.006669	0.014628	0.011498
ASSEMBLY BIOSCIENCES, INC.	0.015507	0.071030	0.014524	0.013225	0.020480	0.011662	0.005464	0.006638	0.011272	0.013236	...	0.016635	0.010335	0.006899	0.007639	0.022144	0.015088	0.021484	0.007136	0.016452	0.012846
MANNKIND CORP	0.014223	0.014524	0.099741	0.011629	0.017069	0.010128	0.005097	0.006089	0.010026	0.012512	...	0.015378	0.009562	0.007192	0.008239	0.018492	0.013814	0.020030	0.006674	0.015316	0.012002
RIGEL PHARMACEUTICALS INC	0.011905	0.013225	0.011629	0.033500	0.015526	0.009160	0.004444	0.005374	0.009175	0.010782	...	0.013677	0.008413	0.006154	0.006518	0.017692	0.012169	0.017384	0.005792	0.013039	0.010451
GALECTIN THERAPEUTICS INC	0.019109	0.020480	0.017069	0.015526	0.083509	0.013509	0.006072	0.007894	0.012617	0.015383	...	0.019702	0.012162	0.007476	0.009091	0.025623	0.018126	0.025515	0.008289	0.019223	0.014722
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
ACHAOGEN INC	0.015562	0.015088	0.013814	0.012169	0.018126	0.011097	0.004870	0.006503	0.009485	0.012234	...	0.016128	0.009972	0.006854	0.008102	0.018632	0.097615	0.020698	0.006710	0.015142	0.011928
PULMATRIX, INC.	0.020693	0.021484	0.020030	0.017384	0.025515	0.015189	0.007342	0.009192	0.014639	0.017752	...	0.022823	0.014161	0.009904	0.010935	0.026758	0.020698	0.306222	0.009623	0.022595	0.017209
REGENERON PHARMACEUTICALS INC	0.006669	0.007136	0.006674	0.005792	0.008289	0.005237	0.002528	0.002998	0.005100	0.006101	...	0.007598	0.004700	0.003617	0.003714	0.009573	0.006710	0.009623	0.009307	0.007282	0.005892
CHIASMA, INC	0.014628	0.016452	0.015316	0.013039	0.019223	0.011371	0.005883	0.006842	0.011781	0.013502	...	0.016852	0.010599	0.007607	0.007690	0.020234	0.015142	0.022595	0.007282	0.049106	0.013239
IMPRIMIS PHARMACEUTICALS, INC.	0.011498	0.012846	0.012002	0.010451	0.014722	0.009403	0.004673	0.005273	0.009334	0.010878	...	0.013413	0.008405	0.007030	0.006713	0.017288	0.011928	0.017209	0.005892	0.013239	0.045175

124 rows × 124 columns

Efficient Frontier - Pharmaceutical Preparations¶

ef1 = EfficientFrontier(mu_Pharm, Factor_Model_cov, weight_bounds=(0, 0.2))

fig, ax = plt.subplots()
plotting.plot_efficient_frontier(ef1, ax=ax, show_assets=True)

# Find and plot the tangency portfolio
ef2 = EfficientFrontier(mu_Pharm, Factor_Model_cov, weight_bounds=(0, 0.2))
# min volatility
ef2.min_volatility()
ret_tangent, std_tangent, _ = ef2.portfolio_performance()
ax.scatter(std_tangent, ret_tangent, marker="*", s=100, c="r", label="Min Volatility")

# Format
ax.set_title("Efficient Frontier - Pharmaceutical Preparations \n Factor Model Estimates")
ax.legend()
plt.tight_layout()
plt.savefig('images/Efficient_Frontier_Returns_Pharmaceutical_Preparations.png', dpi=200, bbox_inches='tight')
plt.show()

Efficient_Frontier_Returns_Pharmaceutical_Preparations.png

Min Volatility Portfolio¶

Performance¶

ef2.portfolio_performance(verbose=True);

Expected annual return: 1.4%
Annual volatility: 3.3%
Sharpe Ratio: -0.18

Weights¶

companies = []
weights = []
for company, weight in ef2.clean_weights().items():
    if weight != 0:
        companies.append(company)
        weights.append(weight)
        
dic = {'Company_Name':companies,'Weight':weights}
min_vol = pd.DataFrame(dic)
min_vol.to_csv("data/min_vol_factor_model_Pharmaceutical_Preparations.csv", index = False)

	Company_Name	Weight
0	BIOSPECIFICS TECHNOLOGIES CORP	0.0175
1	JOHNSON & JOHNSON	0.2000
2	PFIZER INC	0.2000
3	ZOETIS INC.	0.2000
4	LILLY ELI & CO	0.1825
5	MERCK & CO., INC.	0.2000

Results for the Other 4 Industries¶

Prepackaged Software (mass reproduction of software)¶

Efficient_Frontier_Factor_Model_Estimates_Prepackaged_Software.png

Min Volatility Portfolio¶

Performance¶

Expected annual return: 1.1%
Annual volatility: 4.1%
Sharpe Ratio: -0.21

Weights¶

	Company_Name	Weight
0	AWARE INC /MA/	0.06064
1	ULTIMATE SOFTWARE GROUP INC	0.05857
2	ORACLE CORP	0.20000
3	NATIONAL INSTRUMENTS CORP	0.11657
4	ACI WORLDWIDE, INC.	0.20000
5	REALPAGE INC	0.02255
6	BLACK KNIGHT, INC.	0.20000
7	ANSYS INC	0.03257
8	SALESFORCE COM INC	0.09549
9	POLARITYTE, INC.	0.01095
10	MICROSTRATEGY INC	0.00228
11	Q2 HOLDINGS, INC.	0.00038

Crude Petroleum and Natural Gas¶

When we conduct the same analysis, there is no weight shown. Efficient frontier cannot be found.

Real Estate Investment Trusts¶

Efficient_Frontier_Factor_Model_Estimates_Real_Estate_Investment_Trusts.png

Min Volatility Portfolio¶

Performance¶

Expected annual return: 0.6%
Annual volatility: 2.4%
Sharpe Ratio: -0.57

Weights¶

	Company_Name	Weight
0	ARES COMMERCIAL REAL ESTATE CORP	0.09357
1	TWO HARBORS INVESTMENT CORP.	0.05921
2	GREAT AJAX CORP.	0.20000
3	GAMING & LEISURE PROPERTIES, INC.	0.04173
4	MFA FINANCIAL, INC.	0.00089
5	EQUITY COMMONWEALTH	0.20000
6	PUBLIC STORAGE	0.01551
7	ALEXANDRIA REAL ESTATE EQUITIES INC	0.07207
8	STARWOOD PROPERTY TRUST, INC.	0.20000
9	ESSEX PROPERTY TRUST INC	0.03164
10	SUN COMMUNITIES INC	0.05974
11	UDR, INC.	0.02051
12	RAYONIER INC	0.00513

State Commercial Banks (commercial banking)¶

Efficient_Frontier_Factor_Model_Estimates_State_Commercial_Banks.png

Min Volatility Portfolio¶

Performance¶

Expected annual return: 1.0%
Annual volatility: 3.6%
Sharpe Ratio: -0.28

Weights¶

	Company_Name	Weight
0	INVESTAR HOLDING CORP	0.20000
1	GUARANTY FEDERAL BANCSHARES INC	0.08886
2	CITIZENS & NORTHERN CORP	0.03483
3	BANK OF NEW YORK MELLON CORP	0.02348
4	HOPFED BANCORP INC	0.09023
5	MACKINAC FINANCIAL CORP /MI/	0.17768
6	BANK OF THE JAMES FINANCIAL GROUP INC	0.12467
7	VILLAGE BANK & TRUST FINANCIAL CORP.	0.03140
8	COMMERCE BANCSHARES INC /MO/	0.04634
9	CB FINANCIAL SERVICES, INC.	0.06201
10	BANK OF HAWAII CORP	0.08409
11	OLD LINE BANCSHARES INC	0.03641

Analysis of Textual data from 10K Financial Reports

Estimates from Factor Model

Contents

Estimates from Factor Model¶

Sent-LDA¶

Coherence Score Plot¶

Frequency of the Topics in Each Sentence¶

Factor Modelling¶

Demonstration in Pharmaceutical Preparations Industry¶

Topic Matrix: \({B_{k \times m}}\)¶

Coefficient Matrix: \(coef_{T \times m}\)¶

Residual Matrix¶

Diagonal Matrix: \(D_{k \times k}\)¶

Covariance from Factor Model: \(\Sigma_{R\{k \times k\}}\)¶

Perform Mean-Variance Analysis¶

Sample Mean for the Pharmaceutical Preparations Industry¶

Sample Covariance for the Pharmaceutical Preparations Industry¶

Correlation Matric Converted from Covariance Matrix of Factor Model¶

Estimated Covariance¶

Efficient Frontier - Pharmaceutical Preparations¶

Min Volatility Portfolio¶

Performance¶

Weights¶

Results for the Other 4 Industries¶

Prepackaged Software (mass reproduction of software)¶

Min Volatility Portfolio¶

Performance¶

Weights¶

Crude Petroleum and Natural Gas¶

Real Estate Investment Trusts¶

Min Volatility Portfolio¶

Performance¶

Weights¶

State Commercial Banks (commercial banking)¶

Min Volatility Portfolio¶

Performance¶

Weights¶