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

Stock portfolio -3.7602 -1.9600 1.9600 1.6449 s1 # s2

Add results for FF portfolio as well.? Necessary?

However, the results are still dubious because of the unrealistic assumption previously mentioned.

Therefore, we did a more general test which is Ledoit and Wolf test for non-i.i.d data in the next

section.

Ledoit-Wolf test

The Ledoit-Wolf test is proved to be efficient, yet the performance of block bootstrap critically

depends on the choice of block length (Nordman and Lahiri 2014) . Ledoit and Wolf (2008)

suggests that we could use the calibration method (Loh 1987) to choose an optimal block size .

However, we will instead use the method of Politics and White (2003) since it is much more

intuitive. We also modified the MATLAB code follows this study, which provided by Andrew

Patton from Duke University. The original code is made for choosing one optimal block size for

one time series data at a time, we modified it in order to take out one common optimal block size

for more than one time series data.

Null hypothesis : s1 – s2 = 0 (delta = 0)

Alternatives : delta # 0

The confidence interval of delta of Markowitz EF are as below.

Tangency portfolio Lower value Upper value Results JK test results

Industry portfolio 0.4187 0.4670 s1 # s2 s1 # s2

Stock portfolio -0.5259 -0.1759 s1 # s2 s1 # s2

FF 2x2 -0.4742 -0.4563 s1 # s2 s1 # s2

FF 2x3 -0.3210 -0.3127 s1 # s2 s1 # s2

FF 2x2x2x2 -0.1247 -0.1079 s1 # s2 s1 = s2

This result is quite similar to JK test except the result for FF 2x2x2x2 portfolios. However, the

result is consistent with the conclusion we drew in the section 3 of Part II. In general, there is no

evidence that the performance of the second sub-period can match with that of the first sub-period.

Part III: Further studies

I. Optimisation with constraints

Since Markowitz mean–variance portfolio optimization is known to be overly sensitive to

estimation error in risk-return estimates and have poor out-of-sample performance characteristics.

Adding constraints and Resampled Efficiency techniques are useful methods to address the

problems of estimation errors and improve out-of-sample performance.

Here we add one constraint—no short sales and also use resampling method to see if there is any

improvement in Markowitz EF's performance.

Compare efficient frontier with short sales and without short sales Industry Portfolios

Stock Portfolios - Short sales and no short sales

FF 2x2 - Short sales and no short sales

FF 2x3 - short sales and no short sales

FF 2x2x2x2 - short sales and no short sales

Put everything in 1 figure? Or remove some?

From the results we can see that the patterns for industry portfolio, stock portfolio, Fama-French 3

factors, Fama-French 5 factors (2x3) and Fama/French (2x2x2x2) are similar. The figures illustrate

that no short sale efficient frontier lies “inside” the efficient frontier that allows short sales. Because

the proceeds of the short sale can be used to purchase securities with a higher expected return.

Theoretically, short sales can increase possible returns infinitely (Merton, 1972), since any number

of stocks can be sold short to purchase higher-yielding securities. However, the risk would also

greatly increase as the expected returns increase. Thus, risk-averse investors may prefer the

portfolio without shortsales.

Additionally, this extension is based on the assumption of perfect market, which is unrealistic, since

margin must be posted, but it does allow at least a theoretical extension of the efficient frontier.

% One resample

Multi resample

Average resample

We cannot see clear patterns when for one resampling or ten seperated sampling. But the pattern

becomes quite obvious when we average all ten resampling. The average resampling EF is inside

Markowitz's, yet not so far from the original one. However, this still consolidates the theory of

Markowitz EF overestimates the performance of portfolios and can be taken into consideration as

an efficient way to improve the performance of Markowitz Efficient Frontier or Bayes-Stein

Efficient Frontier.

II. International diversification effects

III. Inter-asset class diversification

Investors are always advised to diversify their portfolios, the key to diversify is not the number of

different investments, but the type of investment. This is an important consideration for anyone who

considering using bonds to diversify a portfolio that is heavily invested in stocks.

Because bonds generally may not move in tandem with stock investments, they help provide

diversification in an investor's portfolio. They also seek to provide investors with a steady income.

Plot the whole sample for stock & corporate bond in the same figure

To test the above concepts, we compared industry portfolio’s performance with and without

government state bonds. From the volatility value of 0.045 to 0.06, for the same level of risk,

portfolio with bonds earns more returns than the portfolio without bonds. Hence it is vivid that

diversified portfolio performs better than industry portfolios which don’t have bonds included in

them.

The above analysis is repeated by including corporate bonds instead of state bonds,

As corporate bonds have higher interest rates and also high volatility when compared to state bonds,

for the first sub sample the returns were higher than portfolios, which include those bonds. At the

same time for the second sub sample industry portfolio without corporate bonds performs better, but

this may be because of the unrealistic assumptions of Markowitz efficient frontier and its estimation

errors.??????

Add JK test

IV. Further analysis

Part IV: Conclusion