**Project 01. Optimal Portfolio Decision**

**01.Introduction:** This report presents the concept of optimal portfolio decision making through the application of linear programming and matrix algebra based on a given time series data set of stocks of five companies (Barclays, HSBC, GSK, Tesco and BP). The time series data includes 84 data points spanning over 7 years. It is assumed that the financial portfolio consists of all the companies’ shares with equal weights wherein 20% of the total fund is invested in each share and the investment strategy aims to maximise the Sharpe Ratio for the portfolio and the monthly risk free rate of return stands at 0.2%. The report is systematically presented in a technical language through mathematical notations and the subsequent findings are discussed from the perspective of optimal portfolio investments. The first part (Part A) of the report represents the conceptual framework of Optimal portfolio investments and second part (Part B) represents the empirical framework and the findings from the application of linear programming and matrix algebra.

**Part A: Conceptual Framework of Optimal Portfolio investments:**

It is generally argued that an optimal portfolio intends to minimize variances and does not suffer from the error-in-means problem. Hence, it is imperative to derive a good estimate of variances and covariance’s of the stocks included in to the portfolio basket

(Chopra and Ziemba, 2011). Based on this theory the data set of stocks (Barclays, HSBC, GSK, Tesco and BP) are analysed with the perspective of means and variances. The percentage change in prices represent the returns and the variance of the returns represent the risk associated. Hence, it makes it imperative to minimise the variance of the returns in order to derive an optimal portfolio.

**Average Monthly returns of the shares**: The average monthly returns on the shares of the companies (Barclays, HSBC, GSK, Tesco and BP) in the portfolio basket are calculated based on the below formula.

n= number of months

- i)
**Risk associated with the shares:**Variance is taken as a measure to calculate the risk associated with the returns of the shares as per the formula

n = number of months

The monthly returns are calculated based on the percentage change in the prices of shares between first month (P1) and second month (P2) and the same is multiplied with 100 to obtain the percentage change. The formula to calculate the percentage change in prices is given as Percentage Change (Prices of Shares) = P2-P1/P1*100

- ii)
**The average monthly portfolio return**: Since the financial portfolio in the scenario consists of all the company’s shares with equal weights ( 20 % of the total fund is invested in each share ). The average monthly portfolio return represents w1 ( 20%) for X and w2 (20%) for Y.

The equal weights of 20% are taken for multiples assets w3= 20% w4= 20% w5= 20% to compute the average monthly portfolio returns.

- ii
**) Portfolio risk:**Variances between the five shares indicates the risk associated with the portfolio. The portfolio risk for five shares portfolio is represented by

σ²(port) = w1²σ1² + w2²σ2² + w3²σ3² + w3²σ3² + w5²σ5²

+ 2w1w2σ1σ2ρ(1,2) + 2w1w3σ1σ3ρ(1,3) + 2w1w4σ1σ4ρ(1,4) + 2w1w5σ1σ5ρ(1,5)

+ 2w2w3σ2σ3ρ(2,3) + 2w2w4σ2σ4ρ(2,4) + 2w2w5σ2σ5ρ(2,5)

+ 2w3w4σ3σ4ρ(3,4) + 2w3w5σ3σ5ρ(3,5)+ 2w4w5σ4σ5ρ(4,5)

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

The investment strategy intends to maximise the Sharpe ratio for the portfolio with an assumption that the monthly risk free rate of returns stands at 0.2%. The average excess return, standard deviation of excess return and Sharpe ratio are key parameters to consider.

iii) **Average Monthly Excess Return:** The excess return is the difference between the monthly return and the risk free rate of return. *Excess Return (Er) =Monthly Return-Risk Free Rate.*

iii) **Standard Deviation of the Excess Return**: The standard deviation of the excess return indicates the risks associated with the portfolio and it is given by the standard deviation formula

*x=excess return,* *μ=average excess return, N=number of months*

iii) **Sharpe Ratio:** The effectiveness of the portfolio related the risk free rate of return is represented by the Sharpe Ratio. The Sharpe ration is taken as a benchmark to determine the effectiveness of the portfolio in generating returns against the risks associated. This is given a

This can be also represented as Sharpe Ratio= Average Monthly Excess Return/Standard Deviation of the Excess Return.

**Part B: Empirical Framework of Optimal Portfolio investments:**

**01.Data presentation:** The raw data provided in the scenario is a time series data set of share prices of five companies (Barclays, HSBC, GSK, Tesco and BP). The time series data includes 84 data points spanning over 7 years (2005-2011). It is assumed that the financial portfolio consists of all the companies’ shares with equal weights wherein 20% of the total fund is invested in each share. The share prices in the data set are provided in pence. The shares of the companies provided in the data set are from different sectors such as Banking (Barclays, HSBC), Pharma (GSK), Retail (Tesco) and Oil and Gas (BP).

**02.Empirical Results:** The concept of optimal portfolio investments decision making is investigated by the application of linear programming and matrix algebra. The first step in deriving optimal portfolio investments is to compute the average monthly returns of the shares to derive the individual monthly returns of each share in the portfolio basket. The empirical results indicate a higher and a positive average monthly return on Tesco shares. HSBC in the banking for a period of 2005-2011 reported a negative average monthly return on its shares.

Task (i) | Barclays | HSBC | GSK | Tesco | BP |

Average Monthly Return of shares |
-0.32% | -45.00% | 38.00% | 48.00% | 20.00% |

The risk associated with the shares of the companies in the portfolio basket is computed based on the variances of the monthly returns and the results indicate Barclays shares to be more risky and are highly volatile when compared to others and GSK from the Pharma sector are the shares with low volatility.

Task (i) | Barclays | HSBC | GSK | Tesco | BP |

Risk associated with shares |
258.17% | 49.23% | 22.96% | 32.05% | 67.79% |

The average monthly portfolio returns are computed based on the assumption that all the company’s shares with equal weights are included ( 20 % of the total fund is invested in each share ). The empirical results conclude an average monthly portfolio return of 6%.

Task (ii) Average Monthly Portfolio Return |
0.06 or 6% |

The risk associated with the monthly portfolio return which includes five companies’ shares is investigated by using variances and the covariance based on the multiple assets.

Task (ii) Portfolio risk ( as measured by variances) |
87.02% |

Excess return of the portfolio is computed against the risk free rate of the return to assess the risk associated with the portfolio. The benchmark risk free rate of return is assumed as 0.2% and hence the computed average monthly excess rate of return indicates an excess return on HSBC in banking and a zero excess return on BP.

Task (iii) |
Barclays | HSBC | GSK | Tesco | BP |

Average Excess Return |
-52.00% | -65.00% | 18.00% | 28.00% | 0.00% |

In terms of risk associated returns Tesco has been very successful in generating the risk adjusted returns and this is evident with the higher Sharpe ratio of Tesco against all the shares of the companies’ in the portfolio basket.

Task (iii) |
Barclays | HSBC | GSK | Tesco | BP |

Sharpe Ratio |
-0.033 | -0.093 | 0.038 | 0.05 | 0 |

Since the investment objective aims to be maximise the Sharpe ratio for the portfolio based on the assumption that the risk free rate of return stands at 0.2%. The optimal portfolio allocation can be made based on the higher values of Sharpe ratio. The higher Sharpe ratio can be achieved by increasing the weights of the portfolio allocation with an objective of achieving a higher Sharpe ratio.

Task (iii)Average Excess Return |
-52.00% | -65.00% | -18.00% | 28.00% | 0.00% |

Task (iii) STDEV of Excess Return |
16.07 | 7.02 | 4.79 | 5.66 | 8.23 |

Task (iii) Sharpe Ratio |
-0.03 | -0.093 | 0.038 | 0.05 | 0 |

Portfolio Allocation by Weight |
0.20% | 0.20% | 0.20% | 0.20% | 0.20% |

**03.Discussion:** Based on the above data presentation and the empirical results thus obtained it can be interpreted that the sectorial distribution of portfolio is highly skewed. According to Markowitz (1952) the theory of optimal portfolio intends to minimise the risks associated with multiples asset to generate superior returns. The optimal portfolio allocation intends to maximise the Sharpe ratio (0.05) and it is evident from the results that Tesco shares outperformed in generating higher risk adjusted returns than others present in the basket of portfolio. The first step in the investigation to derive optimal portfolio investments reveals higher and positive averages monthly return on Tesco shares (48%). This coincides with the higher Sharpe ratio (0.05). It is argued that an optimal portfolio intends to minimize variances and does not suffer from the error-in-means problem. Hence, it is imperative to derive a good estimate of variances and covariance’s of the stocks included in the portfolio (Chopra and Ziemba, 2011).

The strategies implemented to achieve an optimal portfolio depend on the risk appetite and the objective of investment. The results indicate that a higher allocation of portfolio in Tesco shares can maximise the generation of risk adjusted returns. For instance, the portfolio risk adjusted returns can be maximised by revising the portfolio allocation based on weighted average. The primary purpose of constructing an optimal portfolio is to address the investment needs of the customers and the risk appetite. Optimal portfolio management with an objective of maximisation of Sharpe ratio aims to generate superior risk adjusted returns. There are many benchmarks to assess the superior risk adjusted returns. An optimal portfolio investments surpasses the bench mark index over a period. The below results forms a strong basis to conduct an optimal portfolio decision making based on the Sharpe Ratio. Tesco shares have generated an average excess return and risk adjusted returns over other shares and the weights of the portfolio can be allocated to prepare an optimal portfolio.

Task ( iii ) |
Barclays |
HSBC |
GSK |
Tesco |
BP |

Average Excess Return |
-52.00% | -65.00% | -18.00% | 28.00% | 0.00% |

STDEV of Excess Return |
16.07 | 7.02 | 4.79 | 5.66 | 8.23 |

Sharpe Ratio |
-0.03 | -0.093 | 0.038 | 0.05 | 0 |

Portfolio Allocation by Weight |
0.20% | 0.20% | 0.20% | 0.20% | 0.20% |

**04.Conclusion:** Based on the above empirical results and discussion, it can be concluded that the purpose of decision making in optimal portfolio investments aims to generate superior returns with minimum risk. Optimal portfolio strategy is dependent on the portfolio objective and the risk appetite of investors. The data provided and the results subsequently obtained indicates that Sharpe ratio serves as a significant variable to determine the optimal portfolio investments. The weights of the individual stocks in the portfolio can be revised based on the Sharpe ratio and the same can be applied to obtain excess returns. The optimal portfolio allocation intends to maximise the Sharpe ratio (0.05) and it is evident from the results that Tesco shares outperformed in generating higher risk adjusted returns than others present in the basket of portfolio. The first step in the investigation to derive optimal portfolio investments reveals higher and positive averages monthly return on Tesco shares (48%). This coincides with the higher Sharpe ratio (0.05). It is argued that an optimal portfolio intends to minimize variances and does not suffer from the error-in-means problem. Hence it is imperative to derive a good estimate of the variances and covariance prior to the construction of the optimal portfolio. Excess returns over the risk free rate of returns can be taken as a yardstick to evaluate the risk adjusted returns. The excess returns of Tesco shares (28%) over the risk free rate of return (0.2%) serves as a good investment option over other shares in the basket of portfolio. The standard deviation of excess return which is highest for Barclays (16.07) also indicates the volatility of bank stocks and their peers and this can form a sound basis for the allocation of sectorial stocks while making decisions on optimal portfolio investments.

**05.References: **

- Chopra, V. K., & Ziemba, W. T. (2011). The effect of errors in means, variances, and covariances on optimal portfolio choice.
*The Kelly Capital Growth Investment Criterion: Theory and Practice*,*3*, 249. - Markowitz, H., 1952. Portfolio selection.
*The journal of finance*,*7*(1), pp.77-91. - Niu, C., Wong, W.K. and Zhu, L., 2016. First Stochastic Dominance and Risk Measurement
- Poornima, S. and Remesh, A.P., 2016. Construction of Optimal Portfolio using Sharpe’s Single Index Model: A Study with Reference to Automobiles and Pharmaceutical Sector.
*International Journal*,*4*(3). - Niu, C., Wong, W.K. and Zhu, L., 2016. First Stochastic Dominance and Risk Measurement.

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