The document analyzes three portfolio strategies - an equally weighted portfolio, a minimum variance portfolio, and a maximum Sharpe ratio portfolio - using returns data from six stocks over a six-year period. The minimum variance portfolio assigns weights to minimize risk and achieves an annual return of 11.4% and annual risk of 12.5%. The maximum Sharpe ratio portfolio assigns weights to maximize risk-adjusted return and achieves an annual return of 19.4% and annual risk of 15.2%. Overall, diversifying into one of the optimized portfolios reduces risk compared to the equally weighted portfolio.

1. Portfolio Diversification
Tutor: Andrea Peric
Tutorial Number: 8
Group: 1
Max Berben, Edoardo Falchetti, Tim Schrader
1.Introduction
BlackRock is the world’s largest investment management company, with assets under
management (AUM) of over $ 5.1 T. Recently, a client of our company has come to ask for
financial advice. Specifically, he was wondering whether were it possible to make the
investment portfolio more stable.
The aim of this paper is to show the differences among the statistics of the starting equally
weighted portfolio (1/n weight on each stock), and other two more stable portfolios.
The following sections analyse the data set, with a general overview of the most relevant
summary statistics. Later, we will describe both the minimum variance portfolio and the
maximum Sharpe ratio portfolio (optimal portfolio), and then our analysis will focus on the
comparison among these three different portfolios.
At last, the paper will end with some final recommendations and a conclusion.
2.Summary statistics
The data set examined is based on a sample period from March 1996 to May 2002. The time
series show the realized daily returns for six different stocks. For expositional purposes, the
summary statistics at the portfolio level have been annualized, i.e. scaled by the approximate
number of trading days in a year (252).
At first glance, we can notice that the stocks with the highest daily returns are A, F (both
provide a daily return of 0.08%), and E (0.09%). Individually, only stock C yields a negative
average daily return (-0.02%) (Appendix, Table 1).
From a risk perspective, E represents the least risky asset, with a daily standard deviation of
0.99%. The riskiest assets in the group are A and C, with standard deviation values of 1.59%
and 1.63%, respectively.
After having calculated the values relative to the maximum and minimum for each stock, we
computed the ranges, which represent the edges around which the returns of the stocks
moved for the sample period analysed. The range values we found out are consistent with the
previous calculated values regarding the standard deviations: A and C have the highest range
values, respectively of 17.93% and 15.91%, which is consistent with the fact that these two
assets also have the highest standard deviation. Conversely, by symmetrical arguments, stock
E has the lowest range value (7.23%).

2. 3.Equally weighted portfolio
In an equally weighted portfolio, the same weight is assigned to each stock. In order to find
the return and the standard deviation of such a portfolio, at first we computed the excess
returns for each stock, which are given by the difference between a stock’s return and its
average mean. From the table of excess returns, we were able to carve out a Variance-
Covariance matrix, in which are listed the joint variabilities of one stock with another (the
joint variability of a stock with itself is equal to the variance of that stock). Furthermore, the
Variance-Covariance matrix has been adjusted for the relative weights of the assets in
question, to obtain the weighted Variance-Covariance matrix. From the assumption of equal
weights, each asset was assigned with a weight of ⅙ (since it is a portfolio comprising 6
stocks).
The matrix multiplication between the vector of the average returns of each asset and the
vector of the relative weights brought as result the portfolio return, which is equal to an
annualized value of 13.77%.
The standard deviation is given by the square root of the sum of all the values in the weighted
Variance-Covariance matrix, and is equal to 14.54% (annualized) (Appendix, Table 2).
4. Minimum Variance Portfolio
The minimum variance portfolio is the portfolio that minimizes the volatility, given an
expected return. It is a unique portfolio that lies on the efficient frontier, if we consider a
portfolio of only risky investments.
The tool used to identify the optimal combination of weights was the Excel solver, which
assigns weights of 10.13, 18.08, 9.27, 11.06, 51.45, and 0%, respectively to A, B, C, D, E,
and F (Appendix, Table 3). For the optimization, the constraint used concerned the weights in
the portfolio: Their sum must be equal to 1, which means that no short sales are allowed. The
weight allocated to asset E is 51.45%, and it is consistent with the lower standard deviation of
this stock compared to the other ones. Even though the risk and return measures for assets A
and F are remarkably close to each other, a positive weight is assigned only to asset A. The
reason for this partly lies in the extent to which each asset is correlated to the others. It is
clear from the correlation matrix that the stock F is, on average, 10% more correlated to the
other stocks if compared to stock A. A lower correlation implies a lower standard deviation,
and therefore a lower volatility.
In support of this, stock E is the one with the lowest level of Kurtosis (0.56), which indicates
that its distribution of the returns has lighter tails on the sides compared to the other stocks
and to an average normal distribution. Stock A, oppositely, has the highest value of Kurtosis
(3.37).
The return for this portfolio is 11.43%, and the standard deviation is equal to 12.49%
(Appendix, Table 4).

3. 5. Optimal Portfolio
The optimal portfolio is commonly defined as the portfolio which maximizes the Sharpe ratio
(excess return of a portfolio over its volatility). The risk-free rate employed was the 1-month
T-Bill rate, which is considered to be the one with the lowest volatility due to its shortest
maturity, and therefore, basically, risk-free.
In order to find the portfolio with the highest Sharpe ratio, the steps are similar to finding the
portfolio with the lowest variance. Relying on the same assumption as before - no short sales
are allowed - the following weights are assigned to the optimal portfolio by the Excel solver:
24.63, 41.11, 22.62, and 11.63%, respectively to stocks A, D, E, and F, with no weights
assigned to stocks B and C (Appendix, Table 5). The optimal portfolio has a Sharpe ratio of
1.21.
The weights of the assets can be partly explained by the returns for each stock: Asset D yields
the highest return compared to the other ones, and therefore the weight assigned to this stock
is higher. Conversely, B and C provide the lower returns (C even has a negative return), and
for this reason the weights are lower (in this case 0). Keeping in mind the benefits of
diversification, at first glance it might look strange to exclude some of the stocks from the
optimal combination. But since the Sharpe ratio considers both the excess return and the
standard deviation, it is assumable that, considering B and C’s returns, the benefits provided
by the lower risk values of stocks B and C are not sufficient to justify an investment in those
stocks.
The return for this portfolio is 19.42%, and the standard deviation is equal to 15.24%
(Appendix, Table 6).
6.Conclusion
In the previous sections, three different approaches have been analysed. After calculating
different portfolios and analysing the risk and the return, we can conclude that a portfolio of 6
stocks provides several allocation options between high and low (even negative) returns, and
possibilities to reduce risk by diversification.
The risk and return values for the equally weighted portfolio are between the same values for
the other two portfolios.
In order to make the portfolio more stable, it is advisable to focus on either minimizing the
variance or maximizing the Sharpe ratio.
In case of only risky assets, by choosing to minimize the variance, our client would have a
unique choice, given by the risk-return combination which lies at the extreme left on the
efficient frontier. Likewise, if he opted for a maximization of the Sharpe ratio he would
choose the portfolio given by the tangency point between the efficient frontier including a
risk-free investment and the efficient frontier of only risky investments.

4. Appendix:
Table 1: Summary statistics
A B C D E F
Mean
0.0
8% Mean
0.0
5% Mean
-
0.02
% Mean
0.0
9% Mean
0.0
4% Mean
0.0
8%
Standar
d
Deviati
on
1.5
9%
Standar
d
Deviati
on
1.2
3%
Standar
d
Deviati
on
1.62
%
Standar
d
Deviati
on
1.4
1%
Standar
d
Deviati
on
0.9
9%
Standar
d
Deviati
on
1.5
3%
Sample
Varianc
e
0.0
3%
Sample
Varianc
e
0.0
2%
Sample
Varianc
e
0.03
%
Sample
Varianc
e
0.0
2%
Sample
Varianc
e
0.0
1%
Sample
Varianc
e
0.0
2%
Kurtosis
3.3
7 Kurtosis
2.0
9 Kurtosis 2.62 Kurtosis
1.7
2 Kurtosis
0.5
6 Kurtosis
1.3
5
Range
17.
93
% Range
12.
94
% Range
15.9
1% Range
13.
78
% Range
7.2
3% Range
13.
73
%
Minimu
m
-
7.9
1%
Minimu
m
-
6.5
4%
Minimu
m
-
8.22
%
Minimu
m
-
7.8
8%
Minimu
m
-
3.2
6%
Minimu
m
-
6.1
1%
Maxim
um
10.
02
%
Maxim
um
6.4
0%
Maxim
um
7.69
%
Maxim
um
5.9
0%
Maxim
um
3.9
6%
Maxim
um
7.6
2%
Table 2: Statistics for the Equally Weighted Portfolio
Monthly Annual
Portfolioreturn 0.0546% 13.7695%
Portfoliovariance 0.0084% 2.1152%
PortfolioSD 0.9162% 14.5437%
Sharpe ratio 0.88

5. Table 3: Weights for the Minimum Variance Portfolio
Weights
A 10.13%
B 18.08%
C 9.27%
D 11.06%
E 51.45%
F 0.00%
Table 4: Statistics for the Minimum Variance Portfolio
Monthly Annual
Portfolioreturn 0.0453% 11.4258%
Portfoliovariance 0.0062% 1.5589%
PortfolioSD 0.7865% 12.4855%
Sharpe ratio 0.83
Table 5: Weights for the Optimal Portfolio
Weights
A 24.63%
B 0.00%
C 0.00%
D 41.11%
E 22.62%
F 11.63%
Table 6: Statistics for the Optimal Portfolio
Monthly Annual
Portfolioreturn 0.0771% 19.4235%
Portfoliovariance 0.0092% 2.3211%
PortfolioSD 0.9597% 15.2350%
Sharpe ratio 1.208303