# Introduction This report introduces the concept of optimal

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Last updated: August 14, 2019

IntroductionThis report introduces the concept of optimal portfoliodecision making through the usage of matrix algebra, with a given time seriesdata set of stocks of five companies (Barclays, HSBC, GSK, Tesco and BP).

Thedata consists of 72 data sets spanning across almost six years, starting from January 2006 and ending in December 2011. The reportwill detail the findings of 4 portfolio. The first consists of two companieswhere investment is equally distributed in the two companies’ shares, thesecond scenario three companies where investment is equally distributed in thethree companies’ shares, the third scenario four companies where investment isequally distributed in the four companies’ shares and the final scenario fivecompanies where investment is equally distributed among the five companies’shares. For the final portfoliowhen 20% of overall funds is invested in all 5 companies the investmentstrategy aims to maximise the Sharpe Ratio for the portfolio and the monthlyrisk-free rate of return is set at 0.2%. The report will explain concepts,techniques and results using relevant mathematical notations and formulae. Concepts ofOptimal Portfolio investmentsAverage Monthly Return in shares:This is calculated for the companies (Barclays, HSBC, GSK, Tesco and BP) in theportfolio basket by the equation                        , where n is the numberof months.

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Risk associated with the shares: The risk (variance) is                                                   Note that in calculating the variance strictly speaking thedenominator should be n-1 instead of n.The Average Portfolio Return: Is calculated using thisequation P= weight of company 1* average monthly returns company 1 + weight ofcompany 2* average monthly returns company 2 when there are 2 companies.where P is the average portfolio returnsWhen 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 monthlyreturns company 3When 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 monthlyreturns company 3 + weight of company 4* average monthly returns of company 4When there are 5 companies: P= weight of company 1* averagemonthly returns company 1 + weight of company 2* average monthly returnscompany 2 + weight of company 3* average monthly returns company 3 + weight of company 4* average monthly returns of company4+ weight of company 5* average monthly returns of company 5Portfolio Risk:Variances between the five shares reveals the risk associated with theportfolio. The portfolio risk for 2 shares portfolio isrepresented by:  Theportfolio 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 perunit of volatility, where excess return is measured as (portfoliomean return – return on a riskless asset) so the Sharpe ratio is theportfolio mean return – return on a riskless asset/standard deviation of theportfolio. Usually the larger the Sharpe ratio the more desirable therisk-adjusted return.EmpiricalStructure of Optimal Portfolio Investments:Percentage Gain/Losson each Share Over Whole PeriodThe percentage gain on each of the shares over the wholeJanuary 2006 to December 2011 can be calculated by finding the differencebetween the initial and last share price and dividing by the initial shareprice then multiplying by 100. The results show that Tesco had the greatestpercentage gain at 26.

87% and Barclays had the lowest with at -70.71%.Average MonthlyReturnThe initial phase in getting an optimal portfolio investmentis to calculate the average monthly returns of the shares to obtain theindividual monthly returns of each share in the portfolio basket. The resultsreveal that Tesco share had the greatest averagemonthly return at 0.51%, with only Tesco and GSK recording positive returnsand the remaining companies (Barclays, HSBC and BP) all recording negativereturns.

HSBC had the lowest average monthly return with a value of -0.62%.Variance/RiskAccompanied with SharesThe risk accompanying with the shares of the companies inthe portfolio basket is calculated using the variances of the monthly returnsand the volatility can be measured by computing the standard deviation of themonthly returns and dividing them by the average monthly return multiplied by100. The results determine Barclays shares to be the riskiest, with a varianceof 300.36% and second in volatility (-3984.32%) only to BP, which had thesecond highest variance. GSK had the lowest risk (24.

15%) with Tesco having thesecond lowest risk (35.10%) but the lowest volatility.AveragePortfolio ReturnThe average portfolio returns arecalculated with the assumption that all the companies share with equal weightsare included for each of the four portfolios. The first portfolio included 2companies (Barclays and HSBC) had an average monthly portfolio return of-0.

53%. The second portfolio included 3 companies (Barclays, HSBC and GSK) thisportfolio had an average monthly portfolio return of -0.30%. the thirdportfolio included 4 companies (Barclays, HSBC, GSK and Tesco) this portfoliohad an average monthly portfolio return of -0.10%.

The final portfolio included5 companies (Barclays, HSBC, GSK, Tesco and BP) this portfolio had an averagemonthly portfolio return of -0.11%. There was a trend of increasing portfolioreturns as more companies where added until the final portfolio where theaddition of BP share led to slight decline in portfolio returns.PortfolioRisk/VarianceThe risk associated with the monthly portfolio return thiswas done by using variances and the covariance based on the multiple assets ineach of the four portfolios. The first portfolio included 2 companies (Barclaysand HSBC) had a portfolio variance of 118.66%. Thesecond portfolio included 3 companies (Barclays, HSBC and GSK) this portfoliohad a portfolio variance of 57.

89%. the third portfolio included 4 companies(Barclays, HSBC, GSK and Tesco) this portfolio had a portfolio variance of41.52%. The final portfolio included 5 companies (Barclays, HSBC, GSK, Tescoand BP) this portfolio had a portfolio variance of 36.56%. There was a trend ofdecreasing portfolio risk as more companies.DiscussionSince the investment objective aims to be maximise theSharpe 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 Sharperatio. A higher Sharpe ratio can be achieved by increasing the weights of theportfolio allocation with an objective of achieving a higher Sharpe ratio. Theinitial Sharpe ratio was -0.

05. The optimal portfolio allocation maximised theSharpe ratio to 0.05 by allocating all the shares to Tesco this reduced theportfolio variance from 36.56% to 34.60%. Markowitz (1952) suggest that thetheory of optimal portfolio intends to minimise the risks associated withmultiples asset to generate superior returns and this is show in my results asthe portfolio variance decreases.

The portfolio returns also rose substantiallyfrom -0.11% to 0.51%.  The first step inthe investigation to find the optimal portfolio investments shows higher andpositive average monthly returns on Tesco shares 0.

51%. This corresponds withthe higher Sharpe ratio 0.05 when all investments are allocated to Tesco. Kanand Zhou (2007) argue that an optimal portfolio intends to minimize variancesand does not suffer from the error-in-means problem. ConclusionFrom the results it can be concluded that the purpose of anoptimal portfolio investment is to generate greater returns and minimise therisk in doing so. The results above show that the Sharpe ratio is a goodindicator in determining the optimal portfolio investment.

The weights in theportfolio can be adjusted in order to obtain larger returns. The greater returnin shares for Tesco over the risk-free rate of return is a good investment incomparison to the other shares in the portfolio. ReferencesKan, R. and Zhou, G. (2007). Optimal Portfolio Choice withParameter Uncertainty.

Journal of Financial and Quantitative Analysis, online42(03), p.621. Available at: http://www.

jstor.org.ezproxy.herts.ac.uk/stable/27647314Accessed 7 Jan. 2018.

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