**Question 1**

**1(a) Selection of the Stock**

For the purpose of analysis in this study, the selected stocks are Burberry, London stock exchange group, Easyjet, Prudential and Barclay. The reason for selecting the five stocks is because of the weight of their portfolios and the efficiency of their performance (Galor 2005, p. 37-38). The study of the market prices identifies these stocks as the strongest and the most efficient considering the economic factors between 2012 and 2014 as observed by Walker (2014, p. 38). The data gathered for this analysis was taken from the Yahoo Finance library. The stocks assists in the comparison between various prices and the correlations that links the economic variables in the UK market such as the Consumer Price Index, rates of inflation, rates of economic Growth and the Volumes of imports and exports (Fister 2011, p. 40). The two stocks fall into two categories of portfolio; the restricted portfolio and the non-restricted portfolio (Hueting 2011, p. 32 – 33).

**Discussion**

**Burberry**

Burberry is a company belonging to the Fashion industry. It is large because of its portfolio with more than 9000 employees (Martenson 2011, p. 31). It has opened more than 100 branches and subsidiaries in the United Kingdom and beyond. In terms of its financial position, Burberry has a capital of more than GB₤ 6 billion as estimated by Mandelbrot & Hudson (2004, p. 34). The average dividend value during the three years of study is 5% of the shareholders equity (Vásquez 2008, p. 32). The market share for Burberry in the London Stock Exchange is about 2.8%.

**London stock exchange group**

London stock exchange group the stock exchange authority in the UK. It is large because it covers more than 250 companies in the United Kingdom. It has a capital of more than GB₤ 200 billion. According to Grier (2008, p. 45), the average dividend value during the three years of study is 10% of the shareholders equity.

**Easyjet**

Easyjet is a company in the aviation industry. It is large having more than 2000 employees and running more than 30 branches in the United Kingdom as stated by Stratton et al (2009, p. 52). Easyjet has a capital of more than GB₤ 10 billion. The average dividend value during the three years of study is 8% of the shareholders equity as stated by Tobin (1958, p. 28). The market share for Easyjet in the London Stock Exchange is about 7.6% (Shan, McGuin & Waller 2007, p. 7).

**Prudential Plc.**

Prudential is a company in the financial sector. It is large because of its portfolio with more than 26,000 employees. It has opened more than 120 branches and subsidiaries in the United Kingdom and beyond (Lintner 1965, p. 27). In terms of its financial position, Prudential has a capital of more than GB₤ 25 billion (Merton 2001, p. 49). The average dividend value during the three years of study is 12% of the shareholders equity. The market share for Burberry in the London Stock Exchange is about 3% (Jones 2002, p. 43).

**Barclays**

Barclays is a company belonging to the Banking sector. It is large because of its portfolio with more than 150,000 employees (Argyrous, Forstater & Mongiovi 2004, p. 76). It has opened more than 500 branches and subsidiaries in the United Kingdom and other regions. In terms of its financial position, Barclays has a capital of more than GB₤ 37 billion (Markowitz 1959, p. 43). The average dividend value during the three years of study is 6.7% of the shareholders equity (Bhalla 2010, p. 53). The market share for Burberry in the London Stock Exchange is about 1.8% (Skaf 1999, p. 58).

**1(b). Calculations of Mean Return and Standard Deviation**

**Calculation of Mean Return**

**London stock exchange group**

Return for year 1 (R1) = 0.577

Return for year 2 (R2) = 0.25

Mean = (R1 + R2) / 2

Mean = (0.577 + 0.25) / 2 = 0.4135

**Burberry**

Return for year 1 (R1) = 0.526

Return for year 2 (R2) = 0.599

Mean = (R1 + R2) / 2

Mean = (0.526 + 0.599) / 2 = 0.5625

**Easyjet**

Return for year 1 (R1) = 0.994

Return for year 2 (R2) = 1.164

Mean = (R1 + R2) / 2

Mean = (0.994 + 1.164) / 2 = 1.079

**Prudential**

Return for year 1 (R1) = 1.47

Return for year 2 (R2) = 1.724

Mean = (R1 + R2) / 2

Mean = (1.47 + 1.724) / 2 = 1.597

**Barclay**

Return for year 1 (R1) = 0.862

Return for year 2 (R2) = 1.109

Mean = (R1 + R2) / 2

Mean = (0.862 + 1.109) / 2 = 0.9855

**Calculations of Standard Deviation**

**London Stock Exchange Group**

Standard Deviation = (0.414 + 0.34) / 2 = 0.377

**Burberry**

Standard Deviation = (0.4577 + 0.512) / 2 = 0.48485

**Easyjet**

Standard Deviation = (0.951 + 1.098) / 2 = 1.0245

**Prudential**

Standard Deviation = (1.56 + 1.82) / 2 = 1.69

**Barclay**

Standard Deviation = (0.889 + 1.051) / 2 = 0.97

**1(c) Calculation of Covariance and Correlation**

The calculation of Covariance and Mean for the Stocks was done in Excel Spread sheet. The results are summarized in the table below. R refers to restricted Portfolio, UNR refers to unrestricted Portfolio and RD refers to the portfolio weight of the Random Distribution (Denney 2005, p. 62).

**Table 1: Covariance and Correlation Table**

A. Inputs on stocks: mean, standard deviation, and correlation matrix | |||

Standard | Expected | ||

Stock | Deviation | Return | |

RD | 0.3 | 0.23 | |

UNR | 0.23 | 0.34 | |

R | 0.4 | 0.34 | |

RD | UNR | R | |

St. Dev | 0.3 | 0.23 | 0.4 |

Mean | 0.23 | 0.34 | 0.34 |

Correlation Matrix | |||

RD | UNR | R | |

RD | 1 | ||

UNR | 1.5 | 1 | |

R | 0.3 | 1.2 | 1 |

B. Covariance Matrix | |||

RD | UNR | R | |

RD | 0.09 | 0.1035 | 0.036 |

UNR | 0.1035 | 0.0529 | 0.1104 |

R | 0.036 | 0.1104 | 0.16 |

C. Equally-Weighted Portfolio | |||

RD | UNR | R | |

Weights | 0.3333 | 0.3333 | 0.3333 |

0.3333 | 0.01 | 0.0115 | 0.004 |

0.3333 | 0.0115 | 0.005877778 | 0.012266667 |

0.3333 | 0.004 | 0.012266667 | 0.017777778 |

1.0000 | 0.0255 | 0.0296 | 0.0340 |

Variance | 0.0892 | ||

St. Dev | 0.298645089 | ||

R * weight | 0.076666667 | 0.113333333 | 0.113333333 |

Mean | 0.303333333 | ||

Minimizing Variance given Mean for portfolio | |||

Minimizing the variance in Excel Solver | |||

Portfolio | RD | UNR | R |

Weight | -0.7193 | 2.7430 | -0.7237 |

-0.719336508 | 0.046570051 | -0.204220112 | 0.018740114 |

2.743001587 | -0.204220112 | 0.398022653 | -0.219145597 |

-0.723665079 | 0.018740114 | -0.219145597 | 0.083790584 |

1.3000 | -0.1389 | -0.0253 | -0.1166 |

Variance | 0.2808679 | ||

StDev | 0.5299697 | ||

R x weight | -0.165447397 | 0.93262054 | -0.246046127 |

Mean | 0.521127016 |

**1 (d) Computation of Beta for each Portfolio of Assets**

The CAPM Method uses the formula below to calculate the expected return (r’_{a}) using the risk free rate (Feldstein, Hines & Hubbard 2007, p. 57) and (Sward 2006, p. 84).

r’a = r_{f} + B_{a} (r’_{m} – r_{f})

Where

r_{f} is the Risk Free Rate

B_{a} is the beta of the security

r’_{m} is the expected market Return

r’a = r_{f} + B_{a} (r’_{m} – r_{f})

Ba = (r’a – rf) / (r’m – r’f)

This question makes the following assumptions relating to CAPM

- The risk free rate is assumed to be 4%
- The portfolio required market return is 20%
- There is a linear relationship between stock return and the market portfolio return

**Prudential Plc**.

15.97 = 4% + Ba (20% – 4%)

0.17 = (0.04 + Ba (0.2 – 0.04))

B_{a} = (0.1597 – 0.04) / (0.2 – 0.04)

B_{a} = 0.81

Beta (Prudential Plc.) = 0.81

**London Stock Exchange**

41.35 = 4% + Ba (20% – 4%)

0.4135 = (0.04 + Ba (0.2 – 0.04))

B_{a} = (0.4135 – 0.04) / (0.2 – 0.04)

B_{a} = 2.33

Beta (London Stock Exchange) = 2.33

**Burberry**

56.25 = 4% + Ba (20% – 4%)

0.5625 = (0.04 + Ba (0.2 – 0.04))

B_{a} = (0.5625 – 0.04) / (0.2 – 0.04)

B_{a} = 3.27

Beta (Burberry) = 3.27

**Easyjet**

10.7 = 4% + Ba (20% – 4%)

0.107 = (0.04 + Ba (0.2 – 0.04))

B_{a} = (0.107 – 0.04) / (0.2 – 0.04)

B_{a} = 0.42

Beta (Easyjet) = 0.42

**Barclays**

9.85 = 4% + Ba (20% – 4%)

0.0985 = (0.04 + Ba (0.2 – 0.04))

B_{a} = (0.0985 – 0.04) / (0.2 – 0.04)

B_{a} = 0.37

Beta (Barclays) = 0.37

See the security market line in the excel sheet (Question One (d))

**1 (e) Estimation of the Mean Return of Each Stock Using CAPM**

London stock exchange group = 0.4135

Burberry = 0.5625

Easyjet = 1.079

Prudential = 1.597

Barclay = 0.9855

**London Stock Exchange Group**

Mean Return r’a = r_{f} + B_{a} (r’_{m} – r_{f})

r’a = (0.14 + 2(1.2 – 0.7) = 1.14 = 114%

The actual mean Return = 0.4135 = 41.35%

The stock was overpriced

The suggested action is to increase the number of assets in the portfolio as well as making sure that the pricing of the securities is done over a greater time period so as to reduce the instances of errors.

**Burberry**

Mean Return = r’a = r_{f} + B_{a} (r’_{m} – r_{f})

r’a = (0.08 + 2(1.08 – 0.7) = 0.84

r’a = 84%

The actual mean Return = 0.5625 = 56.25%

The stock was overpriced

The suggested action is to increase the number of assets of Burberry in the portfolio as well as making sure that the pricing of the securities is done over a greater time period in order to minimize chances of errors.

**Easyjet**

Mean Return = r’a = r_{f} + B_{a} (r’_{m} – r_{f})

r’a = 0.2 + 2(1.2 – 0.7) = 1.2

r’a = 120%

The actual mean Return = 1.079 = 107.9%

The stock was overpriced

The suggested action is to increase the number of stocks in the portfolio as well as making sure that the pricing of Easyjet stocks is carried out over a wider period of time to increase the accuracy.

**Prudential Plc.**

Mean Return = r’a = r_{f} + B_{a} (r’_{m} – r_{f})

Mean Return = 0.06 + 2(1.14 – 0.14)

r’a = 2.10

r’a = 210%

The Actual Mean return = 1.597 = 159.7%

The stock was Under-priced

The suggested action is to increase the stocks of Prudential Plc. assets in the market portfolio as well as ensuring that the pricing of the stock is carried out over a wider period of time so as to increase accuracy.

**Barclay**

Mean Return = r’a = r_{f} + B_{a} (r’_{m} – r_{f})

r’a = 0.14+2(1.17- 0.17) = 2.14

r’a = 214%

The Actual Mean return = 0.9855 = 98.55%

The stock was overpriced

The suggested action is to increase the number of stocks in the portfolio so as to increase accuracy as well as pricing the security over a wider period of time to increase accuracy.

**Question 2**

**2 (a)**

The selection of the two stocks is because they have a predictable efficiency as shown in their efficient portfolio frontiers being straight lines with a consistent growth (Kester 2011, p. 35) and (Talebi & Molaei 2010, p. 52). Secondly, the two stocks have the same properties, being overpriced. The best stock in this case is Prudential

The two selected Stock are Prudential and Burberry. The portfolios are constructed as shown below.

**2 (b)**

See the excel sheet (Question Two (b))

Prudential | |||||

Portfolio | Portfolio | ||||

x | y | St. Dev | Return | ||

1 | 0 | 0 | 0.2 | 0.37 | |

0.2 | 0.5 | 1.7 | 0.5480328 | 0.454 | |

0.4 | 0.1 | 1.5 | 0.478251 | 0.456 | |

0.6 | 1.5 | 3.1 | 1.0544477 | 0.962 | |

0.8 | 2 | 3.8 | 1.3088315 | 1.216 | |

1 | 2.5 | 4.5 | 1.5634897 | 1.47 | |

1.2 | 3 | 5.2 | 1.8183069 | 1.724 | |

MEAN | 0.9959085 | 1.597 |

Table 2: Portfolio Construction for Prudential

Figure 1: Efficient Frontier for Prudential

Burberry | |||||

Portfolio weight | Portfolio | Portfolio | |||

x | y | St. Dev | Return | ||

1 | 0 | 0 | 0.2 | 0.37 | |

0.3 | 0.2 | 0.9 | 0.3002999 | 0.307 | |

0.4 | 0.4 | 1 | 0.3516589 | 0.38 | |

0.5 | 0.6 | 1.1 | 0.4043414 | 0.453 | |

0.6 | 0.8 | 1.2 | 0.4578908 | 0.526 | |

0.7 | 1 | 1.3 | 0.5120352 | 0.599 | |

MEAN | 0.3710377 | 0.5625 |

Table 3: Portfolio Construction for Burberry

Figure 2: Efficient Frontier For Burberry

The Prudential Stock has a return of 1.597

The Burberry Stock has a return of 0.5625

The total Return for the 2 stocks = 2.1595

The total Return for all the stocks = 4.5925

Calculation of annual Return = (2.1 / 4.5925) * 100

Annual Return = 45.73%

This is greater than the 20% expected annual return

**2 (c)**

Prudential and Burberry | ||||||

Portfolio | Portfolio | Portfolio | ||||

x | y | St. Dev | Return- Prudential | Return – Burberry | ||

1 | 0 | 0 | 0.2 | 0.37 | 0.307 | |

0.3 | 0.2 | 0.9 | 0.3002999 | 0.307 | 0.37 | |

0.4 | 0.4 | 1 | 0.3516589 | 0.38 | 0.38 | |

0.5 | 0.6 | 1.1 | 0.4043414 | 0.453 | 0.453 | |

0.6 | 0.8 | 1.2 | 0.4578908 | 0.526 | 0.526 | |

0.7 | 1 | 1.3 | 0.5120352 | 0.599 | 0.599 | |

1.2 | 3 | 5.2 | 1.8183069 | 1.724 | 1.244 | |

MEAN | 0.5777904 | 1.1615 | 0.9215 |

**Table 4: Mixed Asset – ****Portfolio Frontier for Burberry and Prudential**

Figure 3: Asset Portfolio Frontier – Burberry and Prudential

**2 (d)**

Covariance Matrix | |||

RD | London Stock Exchange | Easyjet | |

0.09 | 0.1035 | 0.036 | |

London Stock Exchange Easyjet | 0.1035 | 0.0529 | 0.1104 |

0.036 | 0.1104 | 0.16 | |

Stock | Minimum Expected Return | Maximum Expected Return | RISK |

1 | 1 | 1 | |

London Stock Exchange | 0.3459 | 0.545 | 0.36 |

Easyjet | 0.36 | 0.45 | 0.34 |

**Table 5: Variance Portfolio**

The portfolio mixes that represents the minimum variance is London Stock Exchange and Easyjet. The efficient frontier is presented below.

**Figure 4: Efficient Frontier – London Stock Exchange and Easyjet**

The risk is measured as 0.34 or 34% and the maximum return of the minimum variance Portfolio is 0.45.

The portfolio weights are presented below:

| Portfolio | RD | UNR | R |

| Weight | -0.7193 | 2.7430 | -0.7237 |

Risk Asset 1 | -0.719336508 | 0.046570051 | -0.204220112 | 0.018740114 |

Risk Asset 2 | 2.743001587 | -0.204220112 | 0.398022653 | -0.219145597 |

Risk Asset 3 | -0.723665079 | 0.018740114 | -0.219145597 | 0.083790584 |

Mean Return | 1.3000 | -0.1389 | -0.253 | -0.1166 |

| Variance | 0.280867903440844 | ||

| StDev | 0.529969719 | ||

| R x weight | -0.165447397 | 0.93262054 | -0.246046127 |

| Mean | 0.521127016 |

**Table 6: Portfolio Weights**

The portfolio that suits the clients required annual return is the unrestricted portfolio (UNR). This is because the mean return is 0.25 or 25%, which is above the required 20%.

**2 (e)**

Sharpe Ratio is the ratio between the actual mean return and the expected annual return by the client (Cooper, Scott & Elko 1998, p. 46). It is used to measure the success in reaching the objective measured by the return obtained.

The Sharpe Ratio is calculated as shown below:

Sharpe Ratio = Mean return / Expected annual return

Sharpe Ratio = 0.25 / 0.20 = 1.25

The Sharpe ratio is important since it facilitates the comparison of one portfolio vis-a-vis another portfolio by making an adjustment for risk. Therefore, the ratio is critical in analyzing portfolio performance.

**Question 3**

**3 (a)**

| Equally-Weighted Portfolio | |||

RD | UNR | R | ||

| Weights | 0.3333 | 0.3333 | 0.3333 |

Risk Free Asset 1 | 0.3333 | 0.01 | 0.0115 | 0.004 |

Risk Free Asset 2 | 0.3333 | 0.0115 | 0.005877778 | 0.012266667 |

Risk Free Asset 3 | 0.3333 | 0.004 | 0.012266667 | 0.017777778 |

Mean Return | 1.0000 | 0.0255 | 0.0296 | 0.3404 |

| Variance | 0.3956 | ||

| St. Dev | 0.628958575 | ||

| R * weight | 0.076666667 | 0.113333333 | 0.113333333 |

| Mean | 0.303333333 |

**Table 7: Weights of Risk Free Assets**

Portfolio B in this case is the restricted Portfolio. The selection of this portfolio is because the weight of the mean return for the restricted portfolio with risk free assets is 0.3404 or 34.04% (Cohen 2007, p. 7). This far outweighs the expected annual return of 20% by the client.

**3 (b)**

The pink tangent line as shown below represents the capital Market Line

**Figure 5: Efficient Fronties with Capital Market line**

The right estimate of the expected return and the standard deviation is the point of meeting of the two lines (Sharpe, 1964, p. 45). At this point, the expected return is 20% and the standard deviation is 25%.

**3 (c)**

The calculations were obtained as shown in table 8 below:

Mean Return | St. Dev | RD | UNR | R | |

Portfolio Weights | Portfolio Weights | Portfolio Weights | |||

S&P500 | 0.03 | 0.270149834 | -0.781947262 | 2.474306964 | -0.692359703 |

FSTE 100 | 0.02 | 0.222987279 | -0.590263687 | 2.128465177 | -0.53820149 |

**Table 8: Mean Return of FSTE 100 and S & P 500**

The mean return for S & P 500 is 0.03 or 3% while FSTE 100 is 0.02 or 2%. The mean for UNR (Portfolio A) is 2.4 in S & P 500 and 2.12 in FSTE 100. This gives a Sharpe Ratio as:

Sharpe Ratio = 2.4 / 2.12 = 1.132

%. The mean for R (Portfolio B) is -0.69 in S & P 500 and -0.53 in FSTE 100. This gives a Sharpe Ratio as:

Sharpe Ratio = -0.69 / -0.53 = 1.301

Portfolio B is stronger than portfolio A in the market and is therefore the portfolio of Choice. This is because; portfolio B (1.301) has got a greater Sharpe Ratio compared to Portfolio A (1.132).

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