## Introduction This report introduces the concept of optimal

Introduction

This report introduces the concept of optimal portfolio
decision making through the usage of matrix algebra, with a given time series
data set of stocks of five companies (Barclays, HSBC, GSK, Tesco and BP). The
data consists of 72 data sets spanning across almost six years, starting from January 2006 and ending in December 2011. The report
will detail the findings of 4 portfolio. The first consists of two companies
where investment is equally distributed in the two companies’ shares, the
second scenario three companies where investment is equally distributed in the
three companies’ shares, the third scenario four companies where investment is
equally distributed in the four companies’ shares and the final scenario five
companies where investment is equally distributed among the five companies’
shares.

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For the final portfolio
when 20% of overall funds is invested in all 5 companies the investment
strategy aims to maximise the Sharpe Ratio for the portfolio and the monthly
risk-free rate of return is set at 0.2%. The report will explain concepts,
techniques and results using relevant mathematical notations and formulae.

Concepts of
Optimal Portfolio investments

Average Monthly Return in shares:
This is calculated for the companies (Barclays, HSBC, GSK, Tesco and BP) in the
portfolio basket by the equation                        , where n is the number
of months.

Risk associated with the shares: The risk (variance) is

Note that in calculating the variance strictly speaking the
denominator should be n-1 instead of n.

The Average Portfolio Return: Is calculated using this
equation P= weight of company 1* average monthly returns company 1 + weight of
company 2* average monthly returns company 2 when there are 2 companies.
where P is the average portfolio returns

When there are 3 companies: P=
weight of company 1* average monthly returns company 1 + weight of company 2*
average monthly returns company 2 + weight of company 3* average monthly
returns company 3

When there are 4 companies: P=
weight of company 1* average monthly returns company 1 + weight of company 2*
average monthly returns company 2 + weight of company 3* average monthly
returns company 3 + weight of company 4* average monthly returns of company 4

When there are 5 companies: P= weight of company 1* average
monthly returns company 1 + weight of company 2* average monthly returns
company 2 + weight of company 3* average monthly returns company 3 + weight of company 4* average monthly returns of company
4+ weight of company 5* average monthly returns of company 5

Portfolio Risk:
Variances between the five shares reveals the risk associated with the
portfolio.

The portfolio risk for 2 shares portfolio is
represented by:

The
portfolio risk for 3 shares portfolio is represented by:

The portfolio risk for 4 shares portfolio is represented by:

?²(port) = w1²?1² + w2²?2² + w3²?3² + w3²?3² + w5²?5² +
2w1w2?1?2?(1,2) + 2w1w3?1?3?(1,3) + 2w1w4?1?4?(1,4) + 2w2w3?2?3?(2,3) +
2w2w4?2?4?(2,4) + 2w3w4?3?4?(3,4)

The portfolio risk for five shares portfolio is represented by:
?²(port) = w1²?1² + w2²?2² + w3²?3² + w3²?3² + w5²?5² + 2w1w2?1?2?(1,2) +
2w1w3?1?3?(1,3) + 2w1w4?1?4?(1,4) + 2w1w5?1?5?(1,5) + 2w2w3?2?3?(2,3) +
2w2w4?2?4?(2,4) + 2w2w5?2?5?(2,5) + 2w3w4?3?4?(3,4) + 2w3w5?3?5?(3,5) +
2w4w5?4?5?(4,5)

Where 1= Barclays, 2= HSBC, 3= GSK, 4= Tesco, 5= BP and w1=
20%,w2= 20%,w3= 20%, w4= 20%, w5= 20%

Sharpe Ratio:
Sharpe ratio is a measure of the risk-return trade-off: the excess return per
unit of volatility, where excess return is measured as (portfolio
mean return – return on a riskless asset) so the Sharpe ratio is the
portfolio mean return – return on a riskless asset/standard deviation of the
portfolio. Usually the larger the Sharpe ratio the more desirable the

Empirical
Structure of Optimal Portfolio Investments:

Percentage Gain/Loss
on each Share Over Whole Period

The percentage gain on each of the shares over the whole
January 2006 to December 2011 can be calculated by finding the difference
between the initial and last share price and dividing by the initial share
price then multiplying by 100. The results show that Tesco had the greatest
percentage gain at 26.87% and Barclays had the lowest with at -70.71%.

Average Monthly
Return

The initial phase in getting an optimal portfolio investment
is to calculate the average monthly returns of the shares to obtain the
individual monthly returns of each share in the portfolio basket. The results
reveal that Tesco share had the greatest average
monthly return at 0.51%, with only Tesco and GSK recording positive returns
and the remaining companies (Barclays, HSBC and BP) all recording negative
returns. HSBC had the lowest average monthly return with a value of -0.62%.

Variance/Risk
Accompanied with Shares

The risk accompanying with the shares of the companies in
the portfolio basket is calculated using the variances of the monthly returns
and the volatility can be measured by computing the standard deviation of the
monthly returns and dividing them by the average monthly return multiplied by
100. The results determine Barclays shares to be the riskiest, with a variance
of 300.36% and second in volatility (-3984.32%) only to BP, which had the
second highest variance. GSK had the lowest risk (24.15%) with Tesco having the
second lowest risk (35.10%) but the lowest volatility.

Average
Portfolio Return

The average portfolio returns are
calculated with the assumption that all the companies share with equal weights
are included for each of the four portfolios. The first portfolio included 2
companies (Barclays and HSBC) had an average monthly portfolio return of
-0.53%. The second portfolio included 3 companies (Barclays, HSBC and GSK) this
portfolio had an average monthly portfolio return of -0.30%. the third
portfolio included 4 companies (Barclays, HSBC, GSK and Tesco) this portfolio
had an average monthly portfolio return of -0.10%. The final portfolio included
5 companies (Barclays, HSBC, GSK, Tesco and BP) this portfolio had an average
monthly portfolio return of -0.11%. There was a trend of increasing portfolio
returns as more companies where added until the final portfolio where the
addition of BP share led to slight decline in portfolio returns.

Portfolio
Risk/Variance

The risk associated with the monthly portfolio return this
was done by using variances and the covariance based on the multiple assets in
each of the four portfolios. The first portfolio included 2 companies (Barclays
and HSBC) had a portfolio variance of 118.66%. The
second portfolio included 3 companies (Barclays, HSBC and GSK) this portfolio
had a portfolio variance of 57.89%. the third portfolio included 4 companies
(Barclays, HSBC, GSK and Tesco) this portfolio had a portfolio variance of
41.52%. The final portfolio included 5 companies (Barclays, HSBC, GSK, Tesco
and BP) this portfolio had a portfolio variance of 36.56%. There was a trend of
decreasing portfolio risk as more companies.

Discussion

Since the investment objective aims to be maximise the
Sharpe ratio for the portfolio when the the risk-free rate of return is 0.2%.
The optimal portfolio will allocate to give the highest value of the Sharpe
ratio. A higher Sharpe ratio can be achieved by increasing the weights of the
portfolio allocation with an objective of achieving a higher Sharpe ratio. The
initial Sharpe ratio was -0.05. The optimal portfolio allocation maximised the
Sharpe ratio to 0.05 by allocating all the shares to Tesco this reduced the
portfolio variance from 36.56% to 34.60%. Markowitz (1952) suggest that the
theory of optimal portfolio intends to minimise the risks associated with
multiples asset to generate superior returns and this is show in my results as
the portfolio variance decreases. The portfolio returns also rose substantially
from -0.11% to 0.51%.  The first step in
the investigation to find the optimal portfolio investments shows higher and
positive average monthly returns on Tesco shares 0.51%. This corresponds with
the higher Sharpe ratio 0.05 when all investments are allocated to Tesco. Kan
and Zhou (2007) argue that an optimal portfolio intends to minimize variances
and does not suffer from the error-in-means problem.

Conclusion

From the results it can be concluded that the purpose of an
optimal portfolio investment is to generate greater returns and minimise the
risk in doing so. The results above show that the Sharpe ratio is a good
indicator in determining the optimal portfolio investment. The weights in the
portfolio can be adjusted in order to obtain larger returns. The greater return
in shares for Tesco over the risk-free rate of return is a good investment in
comparison to the other shares in the portfolio.

References

Kan, R. and Zhou, G. (2007). Optimal Portfolio Choice with
Parameter Uncertainty. Journal of Financial and Quantitative Analysis, online
42(03), p.621. Available at: http://www.jstor.org.ezproxy.herts.ac.uk/stable/27647314
Accessed 7 Jan. 2018.

Markowitz, H., 1952. Portfolio selection. The journal of
finance, 7(1), pp.77-91.

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