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Mean-Variance Analysis - Minimum Variance

Mean-Variance Analysis - Minimum Variance

Mean-variance analysis is a mathematical framework that examplifies the trade-off between return and risk. It is used to create diversified portfolios based on investors’ expectation. There are one main approach used in this report. We have the minimum volatility portfolio that concentrates on minimizing the risk of the portfolio. Mimimum variance portfolios constructed can help us investigate if document embedding can reveal information on returns.

Mimimum variance portfolio are constructed based on three estimates of covariance:

a) the sample covariance of returns

b) the cosine similarity of the business descriptions

c) factor model based on business descriptions and return data

When we construct portfolios, to avoid high weights on certain companies, we set constraint that each company selected can only occupy the whole portfolio up to 20%. Also, we assume no short selling in the portfolios.

Monthly Returns

Since we will generate cosine similarity estimate and factor model estimate using business description of companies from 2016 to 2018, we only consider monthly returns before 2019. We group the monthly stock returns data extracted from Wharton Research Data Services’s (WRDS) CRSP/Compustat database by company name and date first. After a selection of time span, from June 2016 to December 2018, we are able to get 31-month returns for 719 companies.

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
1ST SOURCE CORP -0.044261 0.043223 0.052960 0.008618 -0.026754 0.138889 0.134654 0.010076 0.039459 0.005138 ... 0.026358 0.027262 0.027308 0.004701 0.058581 -0.005658 -0.060189 -0.114595 0.043357 -0.165840
2U, INC. 0.056394 0.189391 0.010292 0.083475 -0.089580 -0.051635 -0.088022 0.129022 0.073737 0.085089 ... 0.015100 -0.042128 0.177786 -0.118565 -0.094543 0.181073 -0.158572 -0.163320 -0.071849 -0.148484
3D SYSTEMS CORP 0.022405 -0.021914 0.082898 0.237931 -0.227298 -0.001442 -0.040433 0.240783 -0.078229 -0.015789 ... 0.220000 -0.133736 0.231076 0.117314 -0.118755 0.672145 -0.071253 -0.360847 0.024834 -0.178514
ABBOTT LABORATORIES -0.008075 0.145001 -0.061006 0.006426 -0.065973 -0.029817 0.008931 0.094376 0.079243 -0.014862 ... -0.006796 -0.025200 0.058490 -0.008776 0.079193 0.019835 0.097546 -0.056434 0.074122 -0.023228
ABBVIE INC. -0.016208 0.078986 -0.032161 -0.016069 -0.106548 0.089996 0.029934 -0.013893 0.011946 0.053687 ... -0.182854 0.030217 0.024754 -0.063574 0.005828 0.040659 -0.014586 -0.166737 0.210918 -0.022064
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
ZIOPHARM ONCOLOGY INC -0.295250 -0.114754 0.043210 0.110454 0.008881 0.140845 -0.174383 0.106542 0.072635 -0.001575 ... 0.050938 0.076531 0.161137 -0.383673 -0.142384 0.127413 0.095890 -0.381250 0.691919 -0.441791
ZOETIS INC. 0.002847 0.063422 0.012483 0.017808 -0.080946 0.055962 0.062525 0.028302 -0.029669 0.001125 ... 0.032773 0.001150 0.002635 0.017802 0.016622 0.047641 0.010596 -0.015400 0.042662 -0.088740
ZOGENIX, INC. -0.201389 0.134161 -0.044907 0.310780 -0.286964 0.546012 -0.035714 -0.337449 0.285714 0.048309 ... -0.055425 -0.018727 0.082697 0.038778 0.283937 -0.148899 0.026915 -0.158065 -0.016044 -0.112679
ZOSANO PHARMA CORP -0.294444 0.377953 -0.542800 -0.012748 -0.253070 0.372881 -0.037037 0.538462 1.233333 -0.291045 ... -0.192308 -0.150794 -0.028037 -0.019231 0.024510 -0.002392 -0.019185 -0.171149 0.103245 -0.433155
ZYNERBA PHARMACEUTICALS, INC. -0.216247 0.290511 -0.040724 0.538915 -0.117241 0.092014 0.239269 0.105837 0.328306 -0.122271 ... -0.107803 0.181818 0.038948 -0.087160 -0.317248 0.207519 0.016189 -0.355392 -0.083650 -0.383817

719 rows × 31 columns

Sample Mean

name
1ST SOURCE CORP                  0.009076
2U, INC.                         0.024622
3D SYSTEMS CORP                  0.009901
ABBOTT LABORATORIES              0.022606
ABBVIE INC.                      0.018973
                                   ...   
ZIOPHARM ONCOLOGY INC           -0.022408
ZOETIS INC.                      0.020733
ZOGENIX, INC.                    0.084996
ZOSANO PHARMA CORP              -0.045029
ZYNERBA PHARMACEUTICALS, INC.   -0.003030
Length: 719, dtype: float64

Sample Covariance

name 1ST SOURCE CORP 2U, INC. 3D SYSTEMS CORP ABBOTT LABORATORIES ABBVIE INC. ABEONA THERAPEUTICS INC. ABRAXAS PETROLEUM CORP ACADIA PHARMACEUTICALS INC ACADIA REALTY TRUST ACELRX PHARMACEUTICALS INC ... XBIOTECH INC. XENCOR INC XENON PHARMACEUTICALS INC. XOMA CORP ZEDGE, INC. ZIOPHARM ONCOLOGY INC ZOETIS INC. ZOGENIX, INC. ZOSANO PHARMA CORP ZYNERBA PHARMACEUTICALS, INC.
name
1ST SOURCE CORP 0.004037 0.002210 0.002809 0.000770 0.002418 0.003306 0.005177 0.002316 0.001162 0.003289 ... -0.003115 0.003075 0.002632 0.004194 0.004987 0.005336 0.001121 0.009492 0.006947 0.007938
2U, INC. 0.002210 0.011917 0.010599 0.001859 0.003255 0.008266 0.000459 0.002350 0.001803 0.007566 ... -0.002568 0.006610 0.002465 0.005714 0.002574 0.005000 0.001211 0.006308 0.006805 0.011199
3D SYSTEMS CORP 0.002809 0.010599 0.041611 0.002045 0.001621 0.003528 0.002866 0.000601 0.002762 0.008999 ... -0.004730 0.007320 0.016933 -0.010924 0.002617 0.010725 0.002968 0.003055 0.006912 0.016312
ABBOTT LABORATORIES 0.000770 0.001859 0.002045 0.002716 0.002223 0.000001 -0.001260 0.004953 0.000644 0.003408 ... -0.001995 0.000060 0.001042 0.005094 0.000346 0.002999 0.000776 0.003065 0.008432 0.003382
ABBVIE INC. 0.002418 0.003255 0.001621 0.002223 0.007276 0.003242 0.000717 0.003293 -0.000128 0.000062 ... -0.002213 0.004581 -0.002855 0.013811 0.008295 0.008498 0.001434 0.014807 0.003906 0.005539
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
ZIOPHARM ONCOLOGY INC 0.005336 0.005000 0.010725 0.002999 0.008498 0.016045 0.002711 0.009927 0.001973 0.008192 ... -0.002800 0.013566 -0.003391 0.011584 0.024891 0.045590 0.001967 0.004574 0.017793 0.012347
ZOETIS INC. 0.001121 0.001211 0.002968 0.000776 0.001434 -0.000931 0.001524 0.000304 0.000496 0.002061 ... -0.002579 0.001551 0.001696 0.001430 0.002570 0.001967 0.002037 0.001968 0.001314 0.004093
ZOGENIX, INC. 0.009492 0.006308 0.003055 0.003065 0.014807 0.020618 0.013220 0.009577 0.001537 0.028441 ... -0.015704 0.007531 -0.003133 0.059678 0.009011 0.004574 0.001968 0.153881 0.024156 0.033034
ZOSANO PHARMA CORP 0.006947 0.006805 0.006912 0.008432 0.003906 -0.002526 -0.001789 0.021359 0.005113 0.024977 ... -0.000696 0.003778 0.010329 0.010033 0.013489 0.017793 0.001314 0.024156 0.106290 0.037329
ZYNERBA PHARMACEUTICALS, INC. 0.007938 0.011199 0.016312 0.003382 0.005539 0.010700 0.015823 0.004340 0.003028 0.020225 ... -0.013969 0.006585 0.008309 0.006049 0.011897 0.012347 0.004093 0.033034 0.037329 0.060301

719 rows × 719 columns

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Estimates from Sample Covariance