Table of content Executive Summary Part I: Introduction I. Background and motivation Different investors have different expected return and different confidence levels, demanding them to make different investment decisions. Meanwhile, portfolio managers want the lowest risk level for a given level of return objective. An optimal portfolio, as we know, is one that has the minimum risk with the given level of returns. The ultimate objective in portfolio optimization is to balance the expected return and risk via diversification and obtain the efficient frontier under various practical constraints. This study aims to help investors and portfolio managers to make the “best” choice of portfolios via using Efficient Frontier. The remainder of the paper is structured as follows. Part II provides the comparison of Markowitz and Bayes-Stein as well as briefing through the methods used to test the in-sample and out-of-sample performance of the optimal portfolios. In Part III some further studies on improving the portfolio performance are derived. And our conclusions are included in Part IV. II. Data & Methodology Data The data used in this study comprises of 8 US industry portfolios from January 1981 to December 2014. Industry portfolios used in this study are from Food, Oil, Clothes, Chemicals, Steel, Cars, Utilities and Finance sectors. For further analysis we have selected one individual stock from each industry sector above. Those stocks are Pepsico, Exxon Oil,
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Table of content
Executive Summary
Part I: Introduction
I. Background and motivation
Different investors have different expected return and different confidence levels, demanding them
to make different investment decisions. Meanwhile, portfolio managers want the lowest risk level
for a given level of return objective. An optimal portfolio, as we know, is one that has the minimum
risk with the given level of returns. The ultimate objective in portfolio optimization is to balance the
expected return and risk via diversification and obtain the efficient frontier under various practical
constraints. This study aims to help investors and portfolio managers to make the “best” choice of
portfolios via using Efficient Frontier. The remainder of the paper is structured as follows. Part II
provides the comparison of Markowitz and Bayes-Stein as well as briefing through the methods
used to test the in-sample and out-of-sample performance of the optimal portfolios. In Part III some
further studies on improving the portfolio performance are derived. And our conclusions are
included in Part IV.
II. Data & Methodology
Data
The data used in this study comprises of 8 US industry portfolios from January 1981 to December
2014. Industry portfolios used in this study are from Food, Oil, Clothes, Chemicals, Steel, Cars,
Utilities and Finance sectors.
For further analysis we have selected one individual stock from each industry sector above. Those
stocks are Pepsico, Exxon Oil, Dow Chemicals, PVH, Nucor Steel, Ford, Duke Energy Corporation
and JPMorgan Chase. The selecting criteria are: (1) all companies are enlisted in NYSE, (2)
companies have historical data from January, 1981 and (3) these companies are among top 10 in
their industry sectors. Regarding the history of the chosen companies, we tried to select companies
with longer histories. However, 1981 is the furthest point we can reach.
In the scope of our study, we also construct the efficient frontier using Fama-French formed
portfolios meant to mimic the underlying risk factors in returns related to size and book-to-market
equity. This efficient frontier built from these portfolios is somewhat similar to a benchmark for
other efficient frontiers in this study.
Methodology
Markowitz efficient frontier is the widely known method to be used in choosing the optimal
portfolios. However, this method is criticised against its poor performance on out-of-sample basis
due to its estimation error. In short, the Markowitz frontier use the mean and covariance matrix
estimated from the data sample, therefore it naturally leads to lead to: (1) Extreme weights for
“lucky” asset. (2) Under-diversification due the over-weighting of those “lucky” asset. (3) Unstable
optimal portfolio, slight changes in the inputted mean parameter may lead to substantial changes in
the estimated allocations especially when asset correlations are low.
In the other hand, Bayes-Stein is considered a more accurate estimator by using a coefficient to
shrink the means of the assets toward a global mean according to Jorion( 1986) and
Stevenson(2001). This effectively reduces the difference between extreme observations. Thus this
method can reduce the degree of estimation error. Particularly, the general form for the estimators
can be defined as:
Where is adjusted mean , is the global mean, is the original asset mean and w is the shrinkage
factor which can be estimated from a suitable prior :
Where S is the sample covariance metric and T is the sample size.
For testing the performance of our constructed portfolios, we use Jobson-Korkie test (Jobson and
Korkie 1981) to compare the performance of the first and second sub-periods.
Suppose Sharpe measures for n portfolios, the null hypothesis for the transformed differences is
Ho: Sh=0
Sh: An (n -1) × 1 vector of transformed differences Shi, i=1, 2, 3,..., n-1 respectively, and assumed
to be asymptotically multivariate normal
The first employs the Z-sum statistic, ZS= ∑j=1
j=n −1
ZSjwhich is asymptotically normal. Thus the test
statistic is zs
√( e' θ e )where e is the unit vector and θ is the estimated covariance matrix. An element of
this covariance matrix is
Sn2 ∙ S i ∙ S j − S jn ∙ S j ∙ Si − S❑ ∙ Sn ∙ S j+Sn
2 ∙ S ij+12
Si ∙ S j ∙ Sn2 −
rn ∙ r j
4 Sn ∙ Si
(S❑2 +Si
2 ∙ Sn2 )− rn ∙ ri
4 Sn ∙ S j
(S jn2 +S i
2 ∙ Sn2 )− rn
2
4 S i ∙ S i
(S ij2 +S i
2 ∙ S j2 )
θij=1T
where i,j=1,2,...n-1.
Due to Sh is assumed to be asymptotically multivariate normal, it may well produce a Z-sum
statistic close to zero under some circumstances. From their Monte Carlo experiments, Jobson and
Korkie find that the best procedure for the Sharpe index is to compute returns for the n portfolio
using the Chi-square statistic.
However, the standard tests are not valid when returns have tails heavier than the normal
distribution or for time series and panel datasets (see Ledoit and Wolf, 2008). In other words, the
Jobson-Korkie test is no longer reliable when it comes to testing non-iid time series data. In order to
solve this problem, Ledoit and Wolf (2008) suggested two procedures. Put it simple, if Sh = 0 is not
contained in the confidence interval, we can reject the null hypothesis that the performance of the
portfolio in the first sub-period is equal to that in the second sub-period.
Part II: Portfolio Optimisation – Markowitz vs Bayes-Stein
I. Optimisation for individual stocks
Markowitz Bayes-Stein
We observe some strange patterns of Efficient Frontier for both Markowitz and Bayes-Stein
method. The coefficient of the frontier should be upward, however in this case all of them are
downward. Often, the situation of portfolio returns smaller than risk-free rates is the explanation for
this. We will observe more “natural” versions of Efficient Frontier in the Optimisation for Industry
Portfolios section.
Plotting Markowitz and Bayes-Stein mean-variance portfolios in the same figure gives us a more
thorough understanding of the Bayes-Stein method. Apparently, the Bayes-Stein line lies “inside”
the Markowitz line due to the shrinkage factor (1-w) of Bayes-Stein method. This also means the
returns associate with Bayes-Stein efficient frontier is closer to the true value than Markowitz's.
Observing this figure leads to an interesting implication. One can logically thinks that because
Bayes-Stein EF is closer to true value than Markowitz'z, Markowitz's would probably lie inside
Bayes-Stein's in out-of-sample tests. However, in the scope of this study, we shall not try to prove
that implication.
Although Bayes-Stein is considered a better method than Markowitz, the robustness test still
presents similar patterns for both methods. We split the timeframe into 2 equal sub-periods. For
both methods, the mean-variance frontiers of the second sub-samples completely lies inside the
frontiers of the first sub-samples, which probably means the Bayes-Stein and Markowitz EF did a
poor job in predicting the future returns of portfolios.
II. Optimisation for Fama-French factor-mimicking portfolios
For all three Fama-French datasets, we observe similar patterns as in the previous section. These
patterns strongly consolidate our belief that though being superior to Markowitz Efficient Frontier,
Bayes-Stein frontier still contains estimation errors and overestimates the future returns of the
portfolios.
III. Optimisation for industry portfolios
Markowitz Bayes-Stein
As mentioned in section one, here we observe more “natural” Efficient Frontier for both methods,
which is upward rather than downward. This means we can earn higher returns than risk-free rates
from the portfolios constructed.
From two figures, it is safe to conclude that the sign of EF coefficient should not be taken lightly in
portfolio construction. Particularly, when the coefficient is negative (downward EF), it is better if
we consider replacing one or more items in the portfolio to see if we can build up an upward
efficient frontier (positive coefficient), which means we can earn excess returns over risk-free rates.
We get similar results to previous sections of Individual Stocks and Fama-French sections. It is now
safe to conclude that Bayes-Stein's Efficient Frontier still needs to be modified to give better
performance. This creates an open for further studies of more advanced techniques to lessen the
estimation errors in Efficient Frontier construction.
Above is the plot of all previous Efficient Frontier for both methods. In general, individual stock
portfolios frontier lie at the bottom of the figure. This can be interpreted that portfolios in the black
line yield the least returns for a given level of standard deviation or in vice versa, highest standard
deviation for a given level of expected returns. Meanwhile, the industry portfolios frontier lie at the
very top of the figure. This strongly illustrates the benefits of diversification which we will talk in
details in section 3 & 4 of Part III.
IV. Performance Test
We use Jobson and Korkie test to test whether the sharp ratio in the first sample is equal to that in
the second sample.
Null hypothesis : s1 = s2
Alternatives : s1≠ s2
s1: sharp ratio in the 1st period.
s2: sharp ration in the 2nd period.
Here, we perform the test at 5% significance level. The JKtest for Markowitz Tangency Portfolio
details on number of tested portfolios
Tangency portfolio Test stats Low-critical value Up-critical value Result
Industry portfolio 2.5820 -1.9600 1.9600 s1 # s2
Stock portfolio -4.0317 -1.9600 1.9600 s1 # s2
FF 2x2 -3.6861 -1.9600 1.9600 s1 # s2
FF 2x3 -1.9807 -1.9600 1.9600 s1 # s2
FF 2x2x2x2 -1.4539 -1.9600 1.9600 s1 = s2
We only fail to reject the null hypothesis for FF 2x2x2x2 portfolios. It seems like the MV EF did a
poor job here,
JKtest for Bayes-Stein tangency portfolio
Tangency portfolio
Bayes-Stein
Test
Stat
Low-critical
value
Up-critical
value
Right-up-
critical value
Result
Industry portfolio 2.1961 -1.9600 1.9600 1.6449 s1 # s2