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.

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

risk-adjusted return.

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.