Executive Summary
Group Member
Salvy Fung
Amol Kumar
Zakariya Bhikoo
Faheem
Sidharth Varma
Date: 11/05/23
To: Audrey Koski
From: Andrew Analyst
Subject: RE: Portfolio
UPI
Efun489
Akmu592
Zbhi686
Mibr036
Svar751
Thank you for the chance to develop a portfolio using these 20 stocks as my first project. Using
the Markowitz and Tikhonov methods, I have created portfolios with high alphas (Markowitz
alpha 61 bp/m, Tikohonov alpha 126 bp/m) relative to the Fama-French Three-Factor model
(FF3). The Tikhonov portfolio outperformed the Markowitz portfolio, offering higher alpha and
expected returns, hence my recommendation. We have selected six stocks: MU, TXN, CAT, MO,
BBBY, and CMA. This memo provides a comprehensive analysis, results, and the Python code
used.
Stock Selection
The six stocks I propose to construct a portfolio for the new investment strategy are as follows:
MU
TXN
CAT
MO
BBBY
CMA
Micron
Technology Inc
Texas
Instruments
Caterpillar Inc
Altria Group
Bed, Bath and
Beyond
Comerica
Incorporated
These stocks were chosen due to their characteristics that achieve a high alpha relative to the
Fama-French Three-Factor model.
The two companies with the smallest market capitalisations were selected. The SMB factor of
the FF3 model accounts for the fact that small market cap companies have historically
outperformed the market. Bed, Bath & Beyond Inc (BBBY) and Comerica Incorporated (CMA)
have a market cap of 64.81 Million and 4.80 Billion.
The two companies with the highest beta were selected. High beta companies are generally more
volatile but have higher returns under the FF3 model. Micron Technology Inc (MU) has a beta of
1.9786. Texas Instruments Inc (TXN) has a beta of 1.8847.
The two companies with the lowest P/E ratios were selected. The HML factor of the FF3 model
accounts for the fact that companies that are undervalued (low P/E ratio) tend to outperform
overvalued stocks. Caterpillar Inc (CAT) has a low P/E of 15.90 and Altria Group Inc (MO) has
a low P/E of 15.00.
Preliminary Analysis
Please see below a table that summarises the historical characteristics of the stocks chosen.
TXN
MU
CMA
BBY
CAT
MO
count
120.000000
120.000000
120.000000
120.000000
120.000000
120.000000
mean
0.020813
0.017087
0.005138
0.047106
0.015531
0.015189
std
0.141524
0.186377
0.026941
0.182194
0.087599
0.089634
min
-0.325342
-0.499335
-0.067800
-0.486924
-0.173785
-0.263254
25%
-0.065501
-0.092942
-0.011475
-0.069001
-0.041976
-0.031853
50%
0.016877
0.014624
0.002550
0.033890
0.010449
0.028437
75%
0.115445
0.111933
0.019300
0.132455
0.067646
0.055905
max
0.541763
0.559799
0.090600
0.684567
0.407891
0.342653
Markowitz Optimal Portfolio
The following is a table showing the correlation matrix for the 6 stocks in the proposed portfolio.
TXN
MU
CMA
BBY
CAT
MO
TXN
1
0.5687
-0.2603
0.4135
0.2655
0.0477
MU
0.5687
1
-0.3408
0.3538
0.1472
0.0481
CMA
-0.2603
-0.3408
1
-0.3975
0.0157
0.1405
BBY
0.4135
0.3538
-0.3975
1
0.1471
-0.0835
CAT
0.2655
0.1472
0.0157
0.1471
1
0.0814
MO
0.0477
0.0481
0.1404
-0.0835
0.0814
1
The Markowitz optimal portfolio maximises expected returns while minimising portfolio risk.
Here are the associated weights of each stock to achieve the Markowitz optimal portfolio.
Stock
Weight
TXN
0.0124
MU
0.0013
CMA
0.6285
BBY
0.1503
CAT
0.0805
MO
0.1270
The following is a table that compares the Markowitz portfolio expected return against the
average market return both monthly and annually.
Monthly
Annually
Sharpe Ratio
0.3502
1.2131
Expected Optimal Portfolio Returns
0.0108
0.1378
Average Market Returns
0.0080
0.0954
The portfolio created has an annual Sharpe Ratio of 1.2 which is considered good as it is greater
than 1. For every 1% of additional risk taken, our portfolio's excess return is expected to increase
by 1.2%. The proposed Markowitz optimal portfolio has historically outperformed market annual
returns by 4.24%.
Tikhonov Regularisation
The Markowitz weights are reasonable but the weighting of MU is quite small. This is fine for a
large portfolio value but may be difficult to buy with low AUM and without buying partial
shares. Tikhonov regularisation may be useful due to high correlation between some of the
stocks.
The following table indicates weights used to create the Tikhonov optimal portfolio.
Stock
Weight
TXN
0.047
MU
-0.0271
CMA
0.2153
BBY
0.3058
CAT
0.1858
MO
0.273
Performance Analysis
The following table shows the monthly mean return for the Markowitz and Tikhonov portfolios.
Portfolio
Mean Monthly Return
Markowitz Mean
0.0138
Tikhonov Mean
0.0231
The following table indicates monthly alphas for portfolios under 3 different factor models.
Alpha
Markowitz
Thikinov
CAPM
0.0098
0.0164
FF3
0.0061
0.0126
FF5
0.0048
0.0126
Here is a breakdown of factors attributes for the FF5 model for the Markowitz portfolio.
Markowitz Optimal Portfolio
Factor
Coefficient
MktRF
0.4292
SMB
0.0687
Sign
Economic Significance
Positive Used as a benchmark for economic significance.
Contribution = 0.21%.
Positive SMB contributes 0.03%. This is more than 5% of
P-value
0.000
Statistical
Significance
Significant
0.355
Not significant
HML
0.2041
CMA
0.4595
RMW
0.0242
the MktRF contribution and is economically
significant.
Positive HML contributes 0.12%. This is more than 5% of
the MktRF contribution and is economically
significant.
Positive CMA contributes 0.24%. This is more than 5%
of the MktRF contribution and is economically
significant.
Positive RMW contributes 0.01%. This is less than 5% of
the MktRF contribution and is economically
insignificant.
0.103
Not significant
0.001
Significant
0.825
Not significant
All 5 factors have a positive contribution to expected return. Only MktRF and CMA are
statistically significant and have a P-value of less than 0.05. All statistically significant factors
were also economically significant. MktRF was used as a benchmark to test economic
significance. CMA is economically significant as its contribution of 0.24% was more than 5% of
the MktRF factor’s contribution to expected return.
Here is a breakdown of factors attributes for the FF5 model for the Tikhonov portfolio.
Tikhonov Regularised Portfolio
Factor Coefficient
Sign
Economic Significance
P-value
Used as a benchmark for economic significance.
Contribution = 0.46%
SMB contributes 0.05%. This is more than 5% of
the MRP contribution and is economically
significant.
HML contributes 0.26%. This is more than 5% of
the MRP contribution and is economically
significant.
CMA contributes -0.04%. This is more than 5% of
the MRP contribution and is economically
significant.
SMB contributes 0.02%. This is less than 5% of
the MRP contribution and is economically
insignificant.
0.000
MktRF
0.9119
Positive
SMB
0.1370
Positive
HML
0.4412
Positive
CMA
-0.0735
Negative
RMW
0.0634
Positive
Statistical
Significance
Significant
0.370 Not significant
0.087 Not significant
0.795 Not significant
0.778 Not significant
All factors in the Tikhonov Regularised Portfolio had a positive contribution to expected return,
except for CMA which is negative. Only MktRF is statistically significant and has a P-value less
than 0.05. MktRF was deemed to be economically significant as it was used as a benchmark to
test the economic significance of the other factors. All factors except for RMW were
economically significant but are irrelevant as they are not statically significant.
The Tikhonov regularised portfolio appears to prioritise MktRF and HML factors more heavily
than the Markowitz portfolio. The Markowitz portfolio shows a preference for more conservative
firms, whereas the Tikhonov regularised portfolio exhibits a negative coefficient for the CMA
factor.
From a statistical significance perspective (p<0.05), the Markowitz portfolio has two factors
(MktRF and CMA) that are statistically significant, while the Tikhonov portfolio presents only
one such factor (MktRF).
All factors except for RMW are economically significant in both the Markowitz and Tikhonov
portfolios.
The Tikhonov portfolio had a higher monthly mean return of 2.31% compared to the Markowitz
portfolio return of 1.38%. The Tikhonov Portfolio is on average expected to outperform the
Markowitz portfolio by 0.93% monthly.
The Tikhonov portfolio has a monthly alpha of 164 bp/m, 126 bp/m and 126 bp/m for the
CAPM, FF3 & FF5 models respectively. The Markowitz optimal portfolio has a monthly alpha
of 98 bp/m, 61 bp/m and 48 bp/m for the CAPM, FF3 & FF5 models respectively. The Tikhonov
regularised portfolio consistently has higher values of alpha than the Markowitz optimal
portfolio.
Recommendation
After a thorough analysis of our portfolio using both the Markowitz optimal portfolio and
Tikhonov regularisation methods, I recommend the Tikhonov regularised portfolio to the
investment committee. It has demonstrated a superior performance, generating a higher alpha of
126 bp/m compared to the 61 bp/m alpha of the Markowitz optimal portfolio when regressed
using the Fama-French 3-factor model, and similarly, higher alpha with both the CAPM and FF5
models.
Appendix
The following indicates annualised alphas for portfolios under different factor models.
Monthly alphas were annualised using the formula: (1 + Monthly Alpha)^ 12 - 1.
Alpha (annualised)
CAPM
FF3
FF5
Markowitz
Thikinov
0.1242
0.2156
0.0757
0.1621
0.0591
0.1621
The following indicates monthly and annual mean returns for each portfolio.
Monthly mean return was calculated using an OLS regression. Annual mean was calculated
using the monthly mean and the formula: (1 + Monthly Mean )^ 12 - 1.
Portfolio Return
Markowitz Optimal
Tikhonov Regularised
Monthly
0.0138
0.0231
Annual
0.1788
0.3153
The following table contains the monthly and annualised factor means.
Monthly means were calculated by averaging the factor data over the total months in the dataset.
The monthly factor mean was annualised using the formula: (1 + Monthly Factor Mean )^ 12 - 1.
Means
Monthly
Annualised
MktRF
SMB
0.004994 0.003728
0.05993
0.04474
HML
RMW
CMA
RF
MKT
0.005933
0.003488
0.005138
0.002953
0.007947
0.0712
0.04186
0.06165
0.03543
0.09536
Below are the expected FF5 Factor contributions for Markowitz towards the expected return.
Beta was calculated using an OLS regression. Monthly factor mean was calculated by averaging
the factor over the number of months in the dataset. Expected contribution of each factor was
calculated using the formula: Beta * Monthly Factor Mean = Contribution.
Factor
Beta
Monthly Factor Mean
Contribution to expected return
MktRF
0.4292
0.004994
0.2143%
SMB
0.0687
0.003728
0.0256%
HML
0.2041
0.005933
0.1211%
CMA
0.4595
0.003488
0.1603%
RMW
0.0242
0.005138
0.0124%
Below are the expected FF5 Factor contributions for Tikonov towards the expected return.
Beta was calculated using an OLS regression. Monthly factor mean was calculated by averaging
the factor over the number of months in the dataset. Expected contribution of each factor was
calculated using the formula: Beta * Monthly Factor Mean = Contribution.
Factor
Beta
Monthly Factor Mean
Contribution to expected return
MktRF
0.9119
0.004994
0.4554%
SMB
0.137
0.003728
0.0511%
HML
0.4412
0.005933
0.2618%
CMA
-0.0735
0.003488
-0.0256%
RMW
0.0634
0.005138
0.0326%