Does the Optimal Portfolio Deliver the Best Performance in the Stock Market

UNIVERSITY OF SOUTHAMPTON SCHOOL OF MATHEMATICS

Does the Optimal Portfolio deliver the best Performance in the Stock Market?

A Project Report Submitted for the Award of B.Sc. in Mathematics with Economics

Martin Yau

21/5/2009

Supervisor: Dr. G. Kennedy

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Summary

Portfolio theory describes how investors optimise their portfolios using mean and variance from a universe of stocks. We investigate the performance of optimal portfolios against those using investment strategies. We use MATLAB to construct the portfolios from the stocks listed on the London Stock Exchange excluding investment trusts. We briefly go through the history of Modern Portfolio Theory and other descendent theories. We explain the application of the theory under various conditions. We also explain the basic concepts of behavioural finance and the various formulas for evaluating the performance of portfolios. We conclude that the optimal portfolios perform worse than those using the investment strategies although additional investigation needs to be carried out due to the methods we used for the experiment.

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Contents 1. Introduction .................................................................................................................... 9

2. Measures of Risk ........................................................................................................... 9

3. Modern Portfolio Theory............................................................................................... 10

3.1. Specifying the Opportunity Set .............................................................................. 10

3.1.1. Assumptions .................................................................................................. 10

3.1.2. Preliminaries .................................................................................................. 10

3.2. Building a portfolio ................................................................................................ 11

3.3. Examining Efficient Portfolios ................................................................................ 13

3.3.1. No short selling .............................................................................................. 13

3.3.2. With short selling ........................................................................................... 17

3.3.3. Including Riskless Assets and Riskless Borrowing and Lending .................... 18

3.4. Optimal Portfolio ................................................................................................... 19

3.5. Finding the Optimal Portfolio ................................................................................. 20

3.5.1. Short selling with Riskless Borrowing and Lending ........................................ 20

3.5.2. Short selling without Riskless Borrowing and Lending ................................... 22

3.5.3. No Short selling with Riskless Borrowing and Lending ................................... 22

3.5.4. No Short selling and Riskless Borrowing and Lending ................................... 23

3.5.5. Including additional constraints ...................................................................... 23

4. Behavioural Finance .................................................................................................... 24

5. Investment Strategies .................................................................................................. 25

5.1. Contrarian Investment Strategy ............................................................................. 25

5.2. Value Investing ..................................................................................................... 27

5.3. Zulu Principle ........................................................................................................ 28

6. Evaluating Investment Performance............................................................................. 29

6.1. Rate of Return ...................................................................................................... 29

6.2. Risk Adjusted Returns........................................................................................... 29

7. Plan of Experiment ....................................................................................................... 31

8. Results of the Experiment ............................................................................................ 34

9. Conclusion and Recommendations for Improvement ................................................... 36

Bibliography ........................................................................................................................ 39

Appendices ......................................................................................................................... 43

A.1. Capital Asset Pricing Model (CAPM) ..................................................................... 43

A.1.1. Derivation of CAPM ....................................................................................... 43

A.2. Quadratic Programming ........................................................................................ 48

A.3. An example of calculating the Optimal Portfolio .................................................... 50

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A.4. Transforming Non-Positive Semi-Definite Covariance Matrices ............................ 53

A.5. Experiment Instructions......................................................................................... 54

A.5.1. Contrarian Investment Strategy...................................................................... 54

A.5.2. Value Investing .............................................................................................. 60

A.5.3. Zulu Principle ................................................................................................. 62

A.5.4. Recording trades ........................................................................................... 63

A.5.5. Executing trades ............................................................................................ 65

A.6. Experiment Data ................................................................................................... 67

A.6.1. Data of companies matching relevant criteria for Investment Strategies ........ 67

A.6.2. Portfolio Transaction Costs ............................................................................ 71

A.6.3. Portfolio Valuation at the end of Holding Period ............................................. 77

A.6.4. Market, Bid and Offer prices .......................................................................... 79

A.6.5. Stocks Returns from Portfolios ....................................................................... 81

A.7. MATLAB codes ..................................................................................................... 83

A.8. Glossary of Terms ................................................................................................. 85

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List of Figures Figure 1: Risk-return graph for two assets when 𝜌12 = 1. .................................................... 14

Figure 2: Risk-return graph for two assets when 𝜌12 = −1. ................................................. 15

Figure 3: Risk-return graph for two assets for various values of 𝜌12. ................................... 16

Figure 4: Risk-return graph for 𝑁 assets each with various values of 𝑅, 𝜎 and 𝜌. ................ 16

Figure 5: Risk-return graph for two assets when 𝜌12 = 1 with short selling. ......................... 17

Figure 6: Risk-return graph for two assets when 𝜌12 = −1 with short selling. ...................... 17

Figure 7: Risk-return graph for 𝑁 assets each with various values of 𝑅, 𝜎 and 𝜌 between -1

and 1. .................................................................................................................................. 18

Figure 8: Risk-return graph for riskless and risky assets with short selling. ......................... 19

Figure 9: Optimal portfolio, efficient frontier and CML. ......................................................... 20

Figure 10: Optimal portfolios A, B and C for riskless rates 𝑅0(1)

, 𝑅0(2)

and 𝑅0(3)

respectively

which traces out the efficient frontier. .................................................................................. 22

Figure 11: Visual Tools webpage on Digital Look website. 'Heat Maps' link is highlighted by

the underline. ...................................................................................................................... 54

Figure 12: Heat maps webpage on the Digital Look website. .............................................. 54

Figure 13: Average P/E values of sectors defined by LSE on the Digital Look website. ..... 55

Figure 14: Screener webpage with ‘full fundamental screener’ link highlighted on the Digital

Look website. ...................................................................................................................... 55

Figure 15: Full Screener webpage. ..................................................................................... 56

Figure 16: Example result of application of steps 1 to 10 for Contrarian Investment Strategy.

........................................................................................................................................... 56

Figure 17: Heat map showing 12 companies with positive P/E values. The values are

arranged in decreasing order so the last two are in the two lowest quintiles. ....................... 57

Figure 18: Summary page of a company. ............................................................................ 57

Figure 19: Final results showing accounts for a company going back up to five years. ....... 58

Figure 20: 'UK Shares' webpage showing indices. 'FTSE 100' is highlighted by the underline.

........................................................................................................................................... 58

Figure 21: Example result of application of steps 1 to 5 for Value Investing. ....................... 60

Figure 22: Example of application of steps 1 to 5 for the Zulu Principle. .............................. 62

Figure 23: 'Portfolio Selection' window with 'Portfolio Valuation' webpage in the background.

........................................................................................................................................... 63

Figure 24: 'Trade Security' webpage to record transactions. ............................................... 63

Figure 25: 'Portfolio summary' webpage displaying types of portfolios with their values. ..... 65

Figure 26: ‘Share Trading Portfolio’ webpage showing stocks held in the portfolio with

'PLACE A TRADE' link above. ............................................................................................ 65

Figure 27: Illustration of step 4. ........................................................................................... 66

Figure 28: 'Stock Trading' webpage illustrating an example of a trade to buy £10,000 in

Royal Bank of Scotland Group plc....................................................................................... 66

Figure 29: MATLAB code for Statdata_and_find_eigenvalues.m......................................... 83

Figure 30: MATLAB code for Gettranscovmatrix.m. ............................................................ 84

Figure 31: MATLAB code for Optimal_Portfolio.m. .............................................................. 84

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List of Tables Table 1: Timetable of experiment showing each of the optimal portfolios going to be

compared with each of the portolios constructed using the investment strategies ............... 31

Table 2: Optimal Stock proportions for Optimal Portfolio 1 with relevant data...................... 34



Table 5: Investment Portfolio weightings ............................................................................. 34

Table 6: LIBOR used to calculate the optimal portfolios and evaluate the performance of

portfolios for respective holding periods .............................................................................. 35

Table 7: Overall portfolio values based on all stocks held in the portfolios over the whole

holding period ..................................................................................................................... 35

Table 8: Portfolio and market returns using various measures defined in the report ............ 35

Table 9: Companies with the relevant ratios which satisfy at least all compulsory critera for

Contrarian Investment Strategy ........................................................................................... 67

Table 10: Sector average values for the Contrarian ratios ................................................... 67

Table 11: Companies with the relevant ratios which satisfy at least all compulsory criteria for

Value Investing ................................................................................................................... 68

Table 12: Criteria satisfied by companies for respective sub-strategies for Value Investing 69

Table 13: Companies with the relevant ratios which satisfy at least all compulsory criteria for

the Zulu Principle ................................................................................................................ 69

Table 14: Information which satisfies the Zulu Principle ...................................................... 70

Table 15: Parts of the Zulu Principle satisfied by companies ............................................... 70

Table 16: Transaction costs for the Contrarian Portfolio ...................................................... 71

Table 17: Transaction costs for the Value Investing Portfolio .............................................. 72

Table 18: Transaction costs for the Zulu Principle Portfolio ................................................. 73

Table 19: Transaction costs for the Optimal Portfolio 1 ....................................................... 74



Table 22: Portfolio valuation of Contrarian Portfolio at the end of the holding period before

selling ................................................................................................................................. 77

Table 23: Portfolio valuation of Value Investing Portfolio at the end of the holding period

before selling ...................................................................................................................... 77

Table 24: Portfolio valuation of Zulu Principle Portfolio at the end of the holding period before

selling ................................................................................................................................. 77

Table 25: Portfolio valuation of Optimal Portfolio 1 at the end of the holding period before

selling ................................................................................................................................. 78


selling ................................................................................................................................. 78


selling ................................................................................................................................. 78

Table 28: Market, Bid and Offer prices at the time of sale of stocks for the Contrarian

Portfolio............................................................................................................................... 79

Table 29: Market, Bid and Offer prices at the time of sale of stocks for the Value Investing

Portfolio............................................................................................................................... 79

Table 30: Market, Bid and Offer prices at the time of sale of stocks for the Zulu Principle

Portfolio............................................................................................................................... 79

Table 31: Market, Bid and Offer prices at the time of sale of stocks for the Optimal Portfolio 1

........................................................................................................................................... 80

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........................................................................................................................................... 80


........................................................................................................................................... 80

Table 34: Stock Returns after transaction costs for the Contrarian Portfolio ........................ 81

Table 35: Stock Returns after transaction costs for the Value Investing Portfolio ................ 81

Table 36: Stock Returns after transaction costs for the Zulu Principle Portfolio ................... 81

Table 37: Stock Returns after transaction costs for Optimal Portfolio 1 ............................... 82



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List of Files on the CD Excel 1997-2003 files: LSE Data.xls LSE LISTDATE HISTORIC 2009.xls Optimal Portfolio Weights.xls Stock and Sector Ratios.xls MATLAB project files: Optimal Portfolio Project.mat Optimal Portfolio Project 2.mat Optimal Portfolio Project 3.mat MATLAB code files: Gettranscovmatrix.m Statdata_and_find_eigenvalues.m Optimal_Portfolio.m

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1. Introduction Portfolio theory is about rational investors using mean and variance to optimise their portfolios from a universe of stocks, which consist from two to a very large number of assets. This involves calculating the returns data from the time series of stock prices. Then this is in turn used to calculate the expected returns vector and covariance matrix. These are used to calculate the efficient frontier. The riskless rate of return is used calculate the optimal portfolio, which is determined at the point which the efficient frontier is tangent to the line of expected returns and standard deviation (or risk) with the intercept of the riskless rate. This report is aimed at final year mathematics students who have no knowledge of this field. The glossary of terms is in the Appendix A.8. The theory was first proposed in (Markowitz, 1952) and it was subsequently studied extensively by (Blume, 1970), (Davis & Norman, 1990), (Pardolos, Sandström, & Zopounidis, 1994), (Black & Litterman, 1992) and others under various cases. These cases include continuous time, transaction costs, portfolio optimisation from a universe of domestic and international stock and others. The theory leads to the Capital Asset Pricing Model in single and multiperiods, Behavioural Portfolio Theory, Post-Modern Portfolio Theory and Stochastic Portfolio Theory. From (Shefrin & Statman, 2000), Behavioural Portfolio Theory is about investors optimise their portfolios using expected wealth, desire for security and potential, aspiration levels, and probabilities of achieving aspiration levels. Post-Modern Portfolio Theory is just an extension of Modern Portfolio Theory since the downside semi-variance (this is explained in the next section) is used instead of variance to calculate the efficient frontier and optimal portfolio (see (Post-modern portfolio theory comes of age, 1993)). Stochastic Portfolio Theory basically takes into account the random movements of asset prices (see (Fernholz, 2002) for more details). The details of these are beyond the scope of this report. In this report we will compare the performance of the optimal portfolios against those using the investment strategies. The portfolios will be constructed from the universe of stocks listed on the London Stock Exchange (LSE) excluding investment trusts using the classical Portfolio Theory. In the next section we define and explain the different measures of risk. In section 3 we explain the concept of Modern Portfolio Theory and how it is applied to find optimal portfolios. In section 4 we explain the basic concepts of behavioural finance and the reasons why investor are irrational. This is important because one of the investment strategies was defined to help investors to be more rational in decision making. In section 5 we define the investment strategies and explain the history of each briefly. In section 6 we describe the various measures of portfolio performance evaluation. In section 7 we outline the assumptions and plans of the experiment. In section 8 we present the results and explain them in detail. In the final section we present the key findings from the results and suggest potential improvements to the experiment to future researchers.

2. Measures of Risk The variance of return is defined as

𝜇 − 𝑥 2𝑓 𝑥 𝑑𝑥+∞

−∞

where 𝜇 is the mean return at the end of the chosen period and 𝑓 𝑥 is the probability of

density function of return. The advantages of using this as a measure of risk are it is easy to use and this leads to neat optimal portfolio solutions. The disadvantage is that the distribution of returns can be asymmetric so using the variance is inappropriate in that case. The downside semi-variance of return is

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𝜇 − 𝑥 2𝑓 𝑥 𝑑𝑥𝜇

−∞

.

This is less easy to use and it takes no account of variability above the mean. This measure is useful in the cases where the distribution of returns asymmetric. Hence, this is used in Post-Modern Portfolio Theory.

3. Modern Portfolio Theory The entire section is based on (Elton, Gruber, Brown, & Goetzmann, 2007, pp. 44-246) and (Levy & Sarnat, 1984, pp. 235-266). Modern portfolio theory is about how investors can construct a portfolio that gives the maximum return for a given level of risk, or a level of return for minimum risk. This is also known as Mean-Variance theory. It is divided into two parts;

First, the specification of the opportunity set, which is the universe of portfolio choices

available to the investor. The higher the number of assets available to the investor, the larger the opportunity set.

Second, determination of the portfolio out of all feasible portfolios in the opportunity set.

3.1. Specifying the Opportunity Set

3.1.1. Assumptions

We assume the following; 1) Investors build portfolios based on the expected return and variance of that return of the

assets over a single time period. 2) Investors prefer a portfolio with a higher return to one with a lower return at a given level

of risk. 3) Investors are risk averse, where they prefer a portfolio with lower risk to one with higher

risk for a given level of return.

3.1.2. Preliminaries

An efficient portfolio is one that has the highest possible expected return with the lowest

possible risk. Therefore the investor cannot find a better one with higher expected return and lower risk than the former. A portfolio is inefficient if the investor can find another one with the same expected return with lower risk, or same level of risk with higher expected returns. Other portfolios can be ignored when the set of efficient portfolios has been identified.

We assume that asset 𝑖 has a return 𝑅𝑖 , which is known in various time periods 𝑡𝑣 where 𝑣 = 1,… , 𝑉. Then the expected return on 𝑅𝑖 is defined as

𝐸 𝑅𝑖 = 𝑅 𝑖 =1

𝑉 𝑅𝑖 𝑡𝑣 .

𝑉

𝑣=1

The length of time period can be in days, months, or years. In this report, we assume that returns have equal probabilities. The variance of 𝑅𝑖 is defined as

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𝜎𝑖2 = 𝐸 𝑅𝑖 𝑡𝑣 − 𝑅 𝑖

2 =1

𝑉 𝑅𝑖 𝑡𝑣 − 𝑅 𝑖

2.

𝑉

𝑣=1

The variance measures the distance of returns of assets from the mean return. In financial terms, variance is risk so the higher the variance means higher risk and vice versa. Standard deviation of return is also risk due to the fact that it is the square root of variance.

The covariance 𝜎𝑖𝑗 between assets 𝑖 and 𝑗 is defined as

The covariance measures the correlation between two assets. We can represent equation (1) as

From equation (2) we see that the covariance matrix is symmetrical since 𝜎𝑖𝑗 = 𝜎𝑗𝑖 and

𝜎𝑖𝑖 = 𝜎𝑖2. The values of the non-diagonal elements can be positive, zero or negative and the

maximum value is specified by

𝜎𝑖2𝜎𝑗

2 = 𝜎𝑖𝜎𝑗 .

If 𝜎𝑖𝑗 is near the above value, then the prices of the two assets move together, or they are

closely correlated with each other. If 𝜎𝑖𝑗 = 0, then the price movements of the assets are

independent and they have no correlation. If 𝜎𝑖𝑗 = − 𝜎𝑖𝜎𝑗 , then prices of the assets move in

opposite directions. However, it is often easier to use correlation 𝜌𝑖𝑗 between the two

assets and this is defined by

𝜌𝑖𝑗 =𝜎𝑖𝑗

𝜎𝑖𝜎𝑗=

𝜎𝑖𝑗

𝜎𝑖𝑖𝜎𝑗𝑗.

The values of 𝜌𝑖𝑗 lie between -1 and 1.

In the case of 𝑁 assets, the arguments are very similar. Now we can assume that the portfolio has

𝑁 risky assets 𝑆𝑖 , where 𝑖 = 1,…𝑁.

An expected return for each risky asset.

A variance of return regarding each asset.

A covariance between each pair of risky assets.

In this report we assume that investors hold no riskless assets.

3.2. Building a portfolio

We will explain how a portfolio Π is constructed by considering two assets 𝑆1 and 𝑆2. We will assume that 𝜆1 + 𝜆2 = 1 where 𝜆1 and 𝜆2 are proportions of each asset in a portfolio. We note that the portfolio return is the weighted average of return of individual assets. The weight applied to each individual return is the fraction of the portfolio invested in that asset.

𝜎𝑖𝑗 = 𝐸 𝑅𝑖 𝑡𝑣 − 𝑅 𝑖 𝑅𝑗 𝑡𝑣 − 𝑅 𝑗 =

1

𝑉 𝑅𝑖 𝑡𝑣 − 𝑅 𝑖 𝑅𝑗 𝑡𝑣 − 𝑅 𝑗 .

𝑉

𝑣=1

(1)

𝜎 =

𝜎11 𝜎12 𝜎13

𝜎21 𝜎22 𝜎23

𝜎31 𝜎32 𝜎33

. (2)

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Therefore, the expected return of a portfolio 𝑅 Π is a weighted average of the expected returns of individual assets so we note that

Using the same weighted average argument, the variance of a portfolio 𝜎Π2 is also the

weighted average of the variance of the return on the portfolio from the mean return on the portfolio. We have 𝜎Π

2 = 𝐸 𝑅Π − 𝑅 Π 2

= 𝐸 𝜆1𝑅1 + 𝜆2𝑅2 − 𝜆1𝑅 1 − 𝜆2𝑅 2 2

= 𝐸 𝜆1 𝑅1 − 𝑅 1 + 𝜆2 𝑅2 − 𝑅 2 2

= 𝐸 𝜆12 𝑅1 − 𝑅 1

2 + 2𝜆1𝜆2 𝑅1 − 𝑅 1 𝑅2 − 𝑅 2 + 𝜆22 𝑅2 − 𝑅 2

2

= 𝜆12𝜎1

2 + 2𝜆1𝜆2𝜎12 + 𝜆22𝜎2

2

= 𝜆12𝜎1

2 + 2𝜆1𝜆2𝜌12𝜎1𝜎2 + 𝜆22𝜎2

2. (4)

We notice from equation (4) that if 𝜌12 > 0 then the portfolio variance is larger than the

weighted sum of the individual variances therefore risk is higher. Conversely, if 𝜌12 < 0 then the risk is lower.

In the case of 𝑁 assets, the above argument can be applied so suppose that a portfolio is given by

The expected return is

𝑅 Π = 𝜆𝑖E 𝑅𝑖

𝑁

𝑖=1

= 𝜆𝑖𝑅 𝑖

𝑁

𝑖=1

and the variance is

Regarding equation (6), if the covariances of all assets are zero, that is the prices of the

assets move independently, then 𝜌𝑖𝑗 = 0 and we end up with

𝜎Π2 = 𝜆𝑖

2𝜎𝑖2

𝑁

𝑖=1

.

Suppose that each asset is held in equal proportions, that is 𝜆𝑖 = 1 𝑁 for all 𝑖, then

𝜎Π2 =

1

𝑁

1

𝑁 𝜎𝑖

2

𝑁

𝑖=1

.

As 𝑁 tends to infinity 𝜎Π2 tends to zero, implying that in theory holding more uncorrelated

assets reduces risk and this disappears as the number of holdings increases to a very large number. However, assets are almost never uncorrelated in reality. Diversification may

𝑅 Π = E 𝜆1𝑅1 + 𝜆2𝑅2 = 𝜆1𝑅 1 + 𝜆2𝑅 2. (3)

Π = 𝜆𝑖𝑆𝑖 ,

𝑁

𝑖=1

where 𝜆𝑖

𝑁

𝑖=1

= 1.

𝜎Π

2 = 𝜆𝑖2𝜎𝑖

2

𝑁

𝑖=1

+ 𝜆𝑖𝜆𝑗𝜌𝑖𝑗 𝜎𝑖𝜎𝑗

𝑁

𝑗=1 𝑗≠𝑖

𝑁

𝑖=1

. (5)

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reduce the portfolio risk to a level which is less than the risk of an individual asset. We again

assume that 𝜆𝑖 = 1 𝑁 for all 𝑖 with 𝜌𝑖𝑗 ≠ 0, then the portfolio variance will be the same as

equation (4). Again assume 𝜆𝑖 = 1 𝑁 , so we have

𝜎Π2 =

1

𝑁

1

𝑁 𝜎𝑖

2

𝑁

𝑖=1

+ 1

𝑁2𝜌𝑖𝑗 𝜎𝑖𝜎𝑗

𝑁

𝑗=1 𝑗≠𝑖

𝑁

𝑖=1

and factoring out 𝑁 − 1 𝑁 from the second summation we arrive at

𝜎Π2 =

1

𝑁

1

𝑁 𝜎𝑖

2

𝑁

𝑖=1

+ 𝑁 − 1

𝑁

1

𝑁 𝑁 − 1 𝜌𝑖𝑗 𝜎𝑖𝜎𝑗

𝑁

𝑗=1 𝑗≠𝑖

𝑁

𝑖=1

=1

𝑁𝜎 𝑖

2 +𝑁 − 1

𝑁𝜎 𝑖𝑗

The first and second terms in brackets are the average variance and average covariance respectively. As the number of assets tends to infinity then the risk tends to the average covariance therefore the individual risk of assets can be diversified away but the risk caused by covariances cannot be diversified away. An example is shown in (Elton, Gruber, Brown, & Goetzmann, 2007, p. 59) in Table 4.8 Effect of Diversification.

3.3. Examining Efficient Portfolios

In this section we analyse the combinations of two assets with and without short selling.

3.3.1. No short selling

Case 1: 𝜌12 = 1.

Since the assets are perfectly correlated and from equations (3) and (4), we have

and

or

In order to examine the effect of varying the proportions of the assets, we let 𝜆1 = 𝐴 and

𝜆2 = 1 − 𝐴, where 𝐴 ∈ 0,1 . Then equations (7) and (8) becomes

and

Starting off with equation (10), we solve for 𝐴 to get

𝐴 =𝜎Π − 𝜎2

𝜎1 − 𝜎2

𝑅 Π = 𝜆1𝑅 1 + 𝜆2𝑅 2 (7)

𝜎Π2 = 𝜆1

2𝜎12 + 2𝜆1𝜆2𝜎1𝜎2 + 𝜆2

2𝜎22,

𝜎Π = 𝜆1𝜎1 + 𝜆2𝜎2. (8)

𝑅 Π = 𝐴𝑅 1 + 1 − 𝐴 𝑅 2 (9)

𝜎Π = 𝐴𝜎1 + 1 − 𝐴 𝜎2. (10)

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and substituting this into equation (9) yields 𝑅 Π =

𝜎Π − 𝜎2

𝜎1 − 𝜎2𝑅 1 + 1 −

𝜎Π − 𝜎2

𝜎1 − 𝜎2 𝑅 2

= 𝑅 2 −

𝑅 1 − 𝑅 2

𝜎1 − 𝜎2𝜎2 +

𝑅 1 − 𝑅 2

𝜎1 − 𝜎2 𝜎Π

which is the equation of a straight line with gradient 𝑅 1 − 𝑅 2 𝜎1 − 𝜎2 in the risk-return graph shown in Figure 1. From the graph, diversification cannot reduce risk of the portfolio and that proportionally higher risk leads to proportionally higher return.

Figure 1: Risk-return graph for two assets when 𝝆𝟏𝟐 = 𝟏.

Case 2: 𝜌12 = −1.

Since the assets are perfectly negatively correlated and from equations (3) and (4), we have

and

or

Again, in order to examine the effect of varying the proportions of the assets, we again let

𝜆1 = 𝐴 and 𝜆2 = 1 − 𝐴, where 𝐴 ∈ 0,1 . Then we have

and

We notice that when

𝜆1 =𝜎2

𝜎1 + 𝜎2,

𝑅 Π = 𝜆1𝑅 1 + 𝜆2𝑅 2

𝜎Π2 = 𝜆1

2𝜎12 − 2𝜆1𝜆2𝜎1𝜎2 + 𝜆2

2𝜎22,

𝜎Π = 𝜆1𝜎1 − 𝜆2𝜎2 . (11)

𝑅 Π = 𝐴𝑅 1 + 1 − 𝐴 𝑅 2 (12)

𝜎Π = 𝐴𝜎1 − 1 − 𝐴 𝜎2. (13)

𝑅 Π

𝑅 2

𝑅 1

𝜎Π 𝜎2 𝜎1

𝐴 = 0, 𝜆1 = 0, 𝜆2 = 1

𝐴 = 1, 𝜆1 = 1, 𝜆2 = 0

15

risk is non-existent so 𝜎Π = 0 therefore a risk free portfolio can be constructed by diversifying

and choosing 𝜆1. From the above the return is 𝜎1𝑅 2 + 𝜎2𝑅 1 𝜎1 + 𝜎2 . Solving for 𝐴 in equation (13) to get

𝐴 =𝜎Π − 𝜎2

𝜎1 + 𝜎2

and substituting this into equation (12) and manipulating the subsequent equation we arrive at

𝑅 Π =

𝜎1𝑅 2 + 𝜎2𝑅 1

𝜎1 + 𝜎2 +

𝑅 2 − 𝑅 1

𝜎1 + 𝜎2 𝜎Π if 𝐴 <

𝜎2

𝜎1 + 𝜎2

𝜎1𝑅 2 + 𝜎2𝑅 1

𝜎1 + 𝜎2 +

𝑅 1 − 𝑅 2

𝜎1 + 𝜎2 𝜎Π if 𝐴 >

𝜎2

𝜎1 + 𝜎2

,

which the risk-return curve is two straight lines (see Figure 2).

Figure 2: Risk-return graph for two assets when 𝜌12 = −1.

Case 3: 𝜌12 = 0

Since the assets are uncorrelated and their prices move independently, we have

and

As before, we have

and

Differentiating equation (14) yields

𝜕𝜎Π2

𝜕𝐴= 2𝐴𝜎1

2 − 2 1 − 𝐴 𝜎22 = 0

𝑅 Π = 𝜆1𝑅 1 + 𝜆2𝑅 2

𝜎Π2 = 𝜆1

2𝜎12 + 𝜆2

2𝜎22.

𝑅 Π = 𝐴𝑅 1 + 1 − 𝐴 𝑅 2

𝜎Π2 = 𝐴2𝜎1

2 + 1 − 𝐴 2𝜎22. (14)

𝑅 Π

𝑅 2

𝑅 1

𝜎Π 𝜎2 𝜎1

𝐴 = 0, 𝜆1 = 0, 𝜆2 = 1

𝐴 = 1, 𝜆1 = 1, 𝜆2 = 0

Efficient frontier

16

and solving for 𝐴 yields the minimum value for 𝜎Π2 when

𝐴 =𝜎2

𝜎1 + 𝜎2,

which 𝐴 is between 0 and 1 for all values of 𝜎1 and 𝜎2 . This implies that there is a combination of assets that has a lower risk than the risk of each asset on its own therefore this is preferred by investors who are risk averse. The risk-return curve is curved (see Figure 3).

Figure 3: Risk-return graph for two assets for various values of 𝝆𝟏𝟐.

General case

The existence of minimum 𝜎Π value depends on the values of 𝜎1, 𝜎2 and 𝜌12. The minimum

value of 𝜎Π always exist when 𝜌12 ≤ 0 and only exits for 𝜌12 ≥ 0 when 𝜌12 < 𝜎2 𝜎1 . The risk-

return curve always lie between 𝜌12 = −1 and 𝜌12 = 1. Also the curve will always be convex and that concave curves cannot be possible. This is because the risk from a combination of assets cannot be equal to or greater than the risk of individual assets. The only case that occurs is perfect correlation. In reality, perfect correlation does not exist. 𝑵 assets

In the case of 𝑁 assets, the arguments in all of the above cases are the same than the case of two assets. Since lower correlation between assets will result in higher payoff from diversification, we will get a region of possible portfolios as well as the risk-return curve on the risk-return graph (see Figure 4).

Figure 4: Risk-return graph for 𝑵 assets each with various values of 𝑹 , 𝝈 and 𝝆.

𝑅 Π

𝑅 N

𝑅 1

𝜎Π 𝜎𝑁

σ1

σ1

𝐴 = 0, 𝜆1 = 0, 𝜆2 = 1

𝐴 = 1, 𝜆1 = 1, 𝜆2 = 0

𝑅 Π

𝑅 2

𝑅 1

𝜎Π 𝜎2 𝜎1

𝐴 = 0, 𝜆1 = 0, 𝜆2 = 1

𝐴 = 1, 𝜆1 = 1, 𝜆2 = 0

Efficient frontier

𝜌12 = 0

𝜌12 =1

2

𝜌12 = 1

𝜌12 = −1

𝜌12 = −1

2

17

3.3.2. With short selling

Case 1: 𝜌12 = 1.

The argument is the same as in the case without short selling but the risk-return curve extends from both sides indefinitely (see Figure 5). These extensions correspond to 𝐴 < 0,

where the investor short sells asset 1 and invests the proceeds into asset 2, and 𝐴 > 1, where the investor short sells asset 2 and invests the proceeds into asset 1.

Figure 5: Risk-return graph for two assets when 𝝆𝟏𝟐 = 𝟏 with short selling.

Case 2: 𝜌12 = −1. Again, the argument is identical to the one without short selling but with two straight lines extended (see Figure 6). Risk-free portfolio can still be achieved through perfect hedging of assets, though it is difficult to do so in reality.

Figure 6: Risk-return graph for two assets when 𝝆𝟏𝟐 = −𝟏 with short selling.

Case 3: −1 ≤ 𝜌12 ≤ 1

Again, the case is identical to the one without short selling, except that curves are extended in both directions.

𝑅 Π

𝑅 2

𝑅 1

𝜎Π 𝜎2 𝜎1

𝐴 = 0

𝐴 = 1

Efficient frontier

𝜆1 < 0,𝜆2 > 1

𝜆1 > 0,𝜆2 < 1

𝑅 Π

𝑅 2

𝑅 1

𝜎Π 𝜎2 𝜎1

𝐴 = 0

𝐴 = 1

𝜆1 < 0,𝜆2 > 1

𝜆1 > 0,𝜆2 < 1

18

Figure 7: Risk-return graph for 𝑵 assets each with various values of 𝑹 , 𝝈 and 𝝆 between -1 and 1.

𝑵 assets with short selling

The differences between this case and the one without short selling is that the region of possible portfolios is in a convex shape, as well as the risk-return curve, and there are portfolios with returns are arbitrarily large, due to the fact that selling of assets with low returns and purchasing of assets with high returns with the proceeds takes place. Therefore, the efficient frontier has no upper bound.

3.3.3. Including Riskless Assets and Riskless Borrowing and Lending

Suppose that a portfolio has one riskless asset 𝑆0 with return 𝑅0 and one risky asset 𝑆1. This

asset has no variance and correlation with any of the risky assets so that 𝑅 0 = 𝑅0 and

𝜎0 = 𝜌0𝑖 = 0. Suppose that a portfolio has 𝜆0 invested in the riskless asset and the rest

𝜆1 = 1 − 𝜆0 in the risky asset. Then the portfolio is

Π = 𝜆0𝑆0 + 𝜆1𝑆1 and we have

and

or

Solving equation (16) in terms of 𝜆1 yields

𝜆1 =𝜎Π𝜎1

and substituting this into equation (15) yields

𝑅 Π = 𝜆0𝑅0 + 𝜆1𝑅 1 = 1 − 𝜆1 𝑅0 + 𝜆1𝑅 1 (15)

𝜎Π2 = 𝜆1

2𝜎12,

𝜎Π = 𝜆1 𝜎1. (16)

𝑅 Π

𝑅 N

𝑅 1

𝜎Π 𝜎𝑁

σ1

σ1

𝐴 = 0,

𝐴 = 1

𝜆1 < 0, 𝜆2 > 1

𝜆1 > 0, 𝜆2 < 1

19

𝑅 Π =

1 −

𝜎Π𝜎1

𝑅0 +𝜎Π𝜎1

𝑅 1 = 𝑅0 + 𝑅 1 − 𝑅0

𝜎1 𝜎Π if 𝜆1 > 0

1 +𝜎Π𝜎1

𝑅0 −𝜎Π𝜎1

𝑅 1 = 𝑅0 − 𝑅 1 − 𝑅0

𝜎1 𝜎Π if 𝜆1 < 0

,

which the risk-return curve is a pair of straight lines (see Figure 8). When 𝜆0 < 0 and 𝜆1 > 1,

the investor is borrowing from the bank at 𝑅0 and investing the loan in more risky asset. Conversely, when 𝜆0 > 1 and 𝜆1 < 0 , short selling of risky asset takes place and the

proceeds are deposited in the bank account to earn interest at 𝑅0.

Figure 8: Risk-return graph for riskless and risky assets with short selling.

Examining Figure 8 carefully, if the investor can borrow but cannot lend at the riskless rate,

then one would invest all of the funds in the risky asset to earn 𝑅 1. Additionally, borrowing

may take place to invest in more of the risky asset to earn higher return than 𝑅 1. If one had some funds invested in the riskless asset then one would sell that asset and invest the proceeds in the risky asset. If one cannot borrow but can lend at the riskless rate, then one would invest all funds in the riskless asset to earn interest at that rate. Short selling of the risky asset does not take place because the rational investor would not take on higher risk to earn lower return than the riskless rate. If the lending and borrowing rates are different then the efficient frontier would

change. If the lending rate is below the borrowing rate 𝑅 0′ then the efficient frontier would

start from 𝑅 0 going through 𝑆1 and 𝑆1′ onto the line with intercept at the borrowing rate. In the

case of one riskless asset and 𝑁 risky assets the above arguments are essentially the same.

3.4. Optimal Portfolio

In this section we explain the basic ideas concerning optimal portfolios. If all investors have the same expectations, then they face the same efficient frontier of risky assets. Also if they borrow and lend at the sane risk-free interest rate, then the efficient frontier is just a straight

line with intercept 𝑅0 in 𝑅 0,𝜎 space. Therefore there is a unique point that lies within the

opportunity set of 𝑁 risky assets and the efficient frontier comprising a portfolio of 𝑁 risky assets and one riskless asset. This point also maximises the gradient of the line of possible risky and riskless portfolios. Hence that a rational investor chooses to invest in a combination of riskless asset and portfolio of risky assets at a point on the efficient frontier tangent to the line of maximum gradient, which that line is called the Capital Market Line (CML). This is because other mixtures of riskless and portfolio of risky assets lie on the efficient frontier which would result in a lower returns than the point tangent to the CML. Also there are no combinations of

𝜆0 > 1,𝜆1 < 0

𝑅 Π

𝑅 1

𝑅0

𝜎Π 𝜎1

𝑆1

𝜆0 < 0,𝜆1 > 1

𝜆0 = 0,𝜆1 = 1 𝜆0 = 1,𝜆1 = 0 𝑅0′

𝑆1′

20

riskless and risky assets with higher returns proportional to risk is possible since the CML would lie above the efficient frontier. From an assumption of CAPM (see Appendix A.1), all investors are rational, if one invests at a point of the CML, others should do the same so that they will achieve the same rate of return proportional to risk. If all investors invest on the CML, then all of them should have the same portfolio comprising the same proportions of risky assets. The risky part of the portfolio is called the Optimum Portfolio, which is point 𝑀, and gradient of the CML is called the Market Price of Risk because of the above reason. The Market Price of Risk

measures the amount of profit over the riskless rate for accepting a certain risk level (see Figure 9). The point the investor chooses on the CML depends on the amount of risk they would tolerate. Looking at the following cases, if the investor wants to;

Invest only in risky assets, then no proportion of their wealth is invested in the riskless

asset, that is 𝜆0 = 0.

Increase their return and risk they will increase their holdings in the risky asset by borrowing from the bank.

Reduce risk and return by depositing some of their wealth in the bank, that is 𝜆0 > 0.

Figure 9: Optimal portfolio, efficient frontier and CML.

Adapting the expected return and variance expressions from previous section, the proportions of the portfolio add up to one, that is 𝜆0 + 𝜆𝑖 = 1, and the CML is defined by

𝑅 Π = 𝑅0 +𝑅 𝑃 − 𝑅0

𝜎𝑃𝜎Π ,

the risk-return curve is two straight lines. Hence, the portfolio can be optimised by maximising the gradient of the CML, in other words finding the portfolio with the greatest ratio of excess return to risk, which we will show in the next section.

3.5. Finding the Optimal Portfolio

3.5.1. Short selling with Riskless Borrowing and Lending

In this case we maximise the objective function

𝜃 =𝑅 𝑃 − 𝑅0

𝜎𝑃

subject to the constraint

𝑀

𝑅 Π

𝑅 P

𝑅0

𝜎Π 𝜎P

Optimum Portfolio Capital Market Line

Efficient Frontier

Inefficient portfolios

21

𝜆𝑖

𝑁

𝑖=1

= 1.

There are numerous methods for solving the above problem such as Lagrangian multipliers and substitution of constraint into the objective function and maximise it without constraints.

In this case we will use the latter method. First we write 𝑅0 as

Then we substitute equation (17) and stating the expected return and standard deviation of return in the general form to get

Using the chain and product rules, we differentiate equation (18) with respect to each

variable 𝜆𝑖 and set it to zero to get a system of simultaneous equations:

𝜕𝜃

𝑑𝜆1= 0

𝜕𝜃

𝑑𝜆2= 0

⋮ 𝜕𝜃

𝑑𝜆𝑁= 0,

where

𝜕𝜃

𝑑𝜆𝑖= − 𝛾𝜆1𝜎1𝑖 + 𝛾𝜆2𝜎2𝑖 + ⋯𝛾𝜆𝑖𝜎𝑖

2 + ⋯+ 𝛾𝜆𝑁−1𝜎𝑁−1𝑖 + 𝛾𝜆𝑁𝜎𝑁𝑖 + 𝑅 𝑖 − 𝑅0 = 0,

where 𝛾 is a constant. Then we use a trick which involves modifying the derivative. Let 𝑍𝑖 = 𝛾𝜆𝑖. Since we know that 𝜆𝑖 is a proportion of each asset then the 𝑍𝑖 is proportional to

this proportion. In order to solve for 𝜆𝑖 after solving for 𝑍𝑖 , we divide each 𝑍𝑖 by the sum of 𝑍𝑖 . We substitute 𝑍𝑖 for 𝛽𝛾 and moving the covariance terms to the right-hand side to obtain

𝑅 𝑖 + 𝑅0 = 𝑍1𝜎1𝑖 + 𝑍2𝜎2𝑖 + ⋯𝑍𝑖𝜎𝑖2 + ⋯+ 𝑍𝑁−1𝜎𝑁−1𝑖 + 𝑍𝑁𝜎𝑁𝑖 .

There is one equation similar to the above for each 𝑖 so we solve the following system of simultaneous equations:

We can determine optimum investment amount in each asset by solving equation (19). The solution is the optimum proportions of investment in asset 𝑖 is 𝜆𝑖, where

𝑅0 = 1𝑅0 = 𝜆𝑖

𝑁

𝑖=1

𝑅0 = 𝜆𝑖𝑅0

𝑁

𝑖=1

. (17)

𝜃 =

𝜆𝑖 𝑅 𝑖 − 𝑅0 𝑁𝑖=1

𝜆𝑖

2𝜎𝑖2 +𝑁

𝑖=1 𝜆𝑖𝜆𝑗𝜎𝑖𝑗𝑁𝑗=1 𝑗≠𝑖

𝑁𝑖=1

12

. (18)

𝑅 1 + 𝑅0 = 𝑍1𝜎11 + 𝑍2𝜎12 …+ 𝑍𝑁𝜎1𝑁

𝑅 2 + 𝑅0 = 𝑍1𝜎12 + 𝑍2𝜎22 …+ 𝑍𝑁𝜎2𝑁

⋮ 𝑅 𝑁 + 𝑅0 = 𝑍1𝜎1𝑁 + 𝑍2𝜎2𝑁 …+ 𝑍𝑁𝜎𝑁𝑁 .

(19)

22

𝜆𝑖 =𝑍𝑖

𝑍𝑖𝑁𝑖=1

.

An example is shown in Appendix A.3.

3.5.2. Short selling without Riskless Borrowing and Lending

In this case, we assume that there are different lending and borrowing rates. The method presented in the previous section can be adapted for this case. We first arbitrarily choose a

riskless rate 𝑅0 1

and then find the optimal portfolio which corresponds to that rate. Then we

choose another riskless rate 𝑅0 2

and again find the optimal portfolio which corresponds to

the second rate. We repeat this process again and again until the full efficient frontier is

determined (see Figure 10). Portfolio A would be selected if the riskless rate is 𝑅0 1

. Portfolio

B would be chosen if riskless rate is 𝑅0 2

and C is selected if the rate is 𝑅0 3

. Note that the

efficient frontier can be determined if more riskless rates are taken into account.

Figure 10: Optimal portfolios A, B and C for riskless rates 𝑹𝟎

𝟏 , 𝑹𝟎

𝟐 and 𝑹𝟎

𝟑 respectively which traces out the

efficient frontier.

3.5.3. No Short selling with Riskless Borrowing and Lending

This problem is very similar to the case of riskless lending and borrowing with short selling. Again, at the point on the CML tangent to the efficient frontier we maximise the gradient of the CML. However, assets cannot be held in negative quantities. Therefore we maximise the objective function

𝜃 =𝑅 𝑃 − 𝑅0

𝜎𝑃

subject to the constraints

𝜆𝑖

𝑁

𝑖=1

= 1

and

𝜆𝑖 ≥ 0 ∀𝑖.

𝑅 Π

𝑅0 1

𝜎Π

𝑅0 2

𝑅0 3

𝐴

𝐵

𝐶

23

Unfortunately, we cannot solve the problem using the method described in section 3.5.1

because it is a mathematical programming problem, due to the inequality constraint on 𝜆𝑖. This problem is called a quadratic programming problem because the objective function

contains 𝜆𝑖2 and 𝜆𝑖𝜆𝑗 terms, hence it is non-linear. Therefore quadratic programming is used

to solve the problems of this type, which will explain in Appendix A.2.

3.5.4. No Short selling and Riskless Borrowing and Lending

This problem is similar to the one in the previous section. In this case, we minimise the portfolio risk subject to the constraints from the previous section and the return being some level. Mathematically, we minimise

𝜎Π2 = 𝜆𝑖

2𝜎𝑖2

𝑁

𝑖=1

+ 𝜆𝑖𝜆𝑗𝜎𝑖𝑗

𝑁

𝑗=1 𝑗≠𝑖

𝑁

𝑖=1

subject to the constraints

𝜆𝑖

𝑁

𝑖=1

= 1

𝜆𝑖𝑅 𝑖

𝑁

𝑖=1

= 𝑅 Π

and

𝜆𝑖 ≥ 0 ∀𝑖. From section 3.5.3, we assume that the riskless rate exists and varying this determines the efficient frontier. Since the above problem is quadratic programming problem, we cannot solve the problem using the method described in section 3.5.1.

3.5.5. Including additional constraints

Any requirement that can be formulated as a linear function of investment weight can be imposed on the solution. For example, choosing optimal portfolios given that the dividend

yield on these is at some minimum, for example 4%. Let 𝑑𝑖 be dividend yield on asset 𝑖 and

𝐷 be target dividend yield. Then this requirement can be imposed as a constraint

𝜆𝑖𝑑𝑖

𝑁

𝑖=1

= 𝐷

on any of the problems described in sections 3.5.1 to 3.5.4. In general, if inequality constraints are imposed on the problem then it has to be solved through the use of quadratic programming. Otherwise, the method of a system of simultaneous equations is used.

24

4. Behavioural Finance This section is based on (Montier, 2002, pp. 1-15), (Elton, Gruber, Brown, & Goetzmann, 2007, pp. 485-492) and (Thaler, 1985). We included this section in this report because the Contrarian investment strategy is designed to minimise the investor’s tendency to make choices based on facts instead of their perceptions and preferences. In other words, the strategy helps the investors behave more rationally. Behavioural Finance is the study of psychology the investor and their subsequent

effects on financial markets. This field explores the reasons private and professional investors, traders and other participants make predictable and systematic errors. In other words, the following alters investors’ perception regarding reality, decision making more bias towards certain choices and preferences:

Over-optimism is where investors tend to exaggerate their own abilities, for example, regarding selecting the right stocks to earn high returns on their portfolio. Investors are often deceived by randomness. Illusion of control is where investors believe that they are in control of a situation more often than they are. Self-attribution bias is reference to situations where good outcomes are attributed to good skill whilst bad outcomes are attributed to lack of luck. Hence, illusion of control and self-attribution bias tends to lead investors to be over-optimistic. There are other psychological biases which also lead to over-optimism.

Over-confidence is where investors tend to believe that their judgements of a situation

are correct more often than they are. This and over-optimism lead to investors to overestimate their knowledge, underestimate risk and exaggerate their ability to control the situation.

Cognitive dissonance is the mental conflict that investors experience when they are

presented with evidence which confirms that their beliefs or assumptions are incorrect. This leads to self-denial and they jump through mental hurdles to reduce or avoid mental inconsistencies.

Confirmation bias is the investor’s need to discover information that agrees with their

view. Information which conflict with their view is ignored whilst ones that support their view is over weighted.

Conservatism bias is a tendency to cling tenaciously to a view or forecast. When this

happens, most investors find it very difficult to change their view. When change occurs it takes place very slowly, which causes under-reaction to events.

Anchoring is a tendency to rely heavily on a single piece of information when making decisions.

Representativeness heuristic is a rule of thumb (heuristic) that refers a tendency to

evaluate the likelihood of something is with reference to how closely it resembles to rather than using probabilities. For example, an investor notices that a company has many periods of good earnings and, from the law of small numbers, leads to their belief that the company is one with high earnings growth, and hence likely to continue to deliver high earnings growth in the future. If the company has some average earnings in some periods then its earnings would appear average in their view even if there are many periods of high earnings.

Availability bias is a heuristic which the decision makers assess the frequency of class

or probability of an event based on the ease of examples can be brought to mind. Ceteris paribus, common events are easier to come up with than rare events. For example, an investor is faced with a choice of thousands of stocks. Their choice is affected by the press coverage of stocks, such as record earnings of a company and tipping. Therefore, one or more stocks featured will be selected. Another example is that events which occurred recently are more easily remembered than the events which occurred in the distant past.

25

Ambiguity aversion is where the investor has preference for the known over unknown,

for example, they choose a stock they have knowledge of instead of another they have no knowledge of.

Narrow framing (mental accounting) is the failure of the investor to consider all parts

of the portfolio, where each part is already allocated, separate and non-transferrable. For example, an investor allocates their money into two separate halves for different investment purposes. The first and second parts are for investing in stocks in FTSE 100 and FTSE AIM indices. The investor already has invested one stock in each index and

decides to diversify to reduce the portfolio risk but only considers FTSE AIM stocks. Therefore, their portfolio is not sufficiently diversified and the portfolio risk is higher than it should be.

Herd behaviour where investors follow market trends. For example, investors bought

Internet stocks in the late 1990s, due to the forecasted high earnings growth. The stock prices collapsed in the early 2000s when that did not materialise and most made huge losses. One or more of the above can cause herd behaviour.

There are other factors which affect the investor’s psychology other than those mentioned. However, these are beyond the scope and relevance to this report. The above concepts will provide enough information in order to build and run the portfolio using the Contrarian investment strategy successfully without making the above errors.

5. Investment Strategies

5.1. Contrarian Investment Strategy

This section is based on (Dreman, 1982) and (Dreman, 1998). From (Investopedia ULC, 2009), Contrarian Investment Strategy is one that involves investing in an approach which goes against prevailing market trends by buying stocks that are perform poorly, that most investors do not want to hold and sell, and sell them when they perform well, when other investors buy in large quantities. The first strategy was outlined in (Dreman, 1980) in an attempt to prevent investors from making the same predictable mistakes as outlined in the previous section. This strategy involved purchasing stocks with low price to earnings (P/E) ratios and going against the market trend. The strategy was refined in (Dreman, 1982). In (Dreman, 1998), the strategy was expanded considerably to purchasing stock with low price to book value (P/BV), low price to cash flow (P/CF) and high dividend yield ratios, as

well as P/E below their market values (or averages). There are other variants which are devised by others. The strategies have been researched by many, such as (Chan K. C., 1988) and (Gregory, Harris, & Michou, 2001). They generally indicate that stocks with low P/E, P/BV, P/CF and dividend yield ratios consistently on average earn above-market returns, capital and income, than their high value counterparts. In this report, we will use the strategy based on (Dreman, 1998, pp. 160-173 and 406-410). Mandatory: 1. Lowest possible P/E, P/BV and P/CF for a stock, while using the P/E as a core ratio.

These ratios have to be below market value for the industry or index. The companies have to lie within the two lowest quintiles (lowest 40% of companies based on the ratios mentioned).

2. Dividend yield should be highest possible, which has to be above market yield. Companies have to lie within two highest quintiles.

3. Dividend yield which is above average which the company can sustain and increase. Dividend cover of 2 or more is desirable. The value must be more than 1.5.

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4. Current assets should be twice the amount of current liabilities is desirable, that is current ratio of 2:1. This value varies from industry to industry. Minimum ratio is 1:1.

Optional:

1. Many favourable financial ratios as possible; high return on capital employed (ROCE)

and return on assets (ROA), low price to sales (P/S) and PEG less than1. For ROCE,

values over 20% are attractive in general. For the latter two, values greater than 15 are unsuitable.

2. Earnings per share (EPS) should be higher than FTSE 100 in the previous year.

3. Earnings should always lean on the conservative side.

Selling: Sell a stock when it’s P/E, P/BV, P/CF or dividend yield approaches that of the overall market, regardless of how favourable prospects may appear. Replace it with another contrarian stock. We will abide by the following points in order to avoid the main pitfalls of behavioural finance: 1. Favoured stocks underperform the market, while out-of-favour companies outperform the

market, but the reappraisal often happens slowly, even glacially. 2. Buy only contrarian stocks because of their superior performance characteristics. 3. Buy the least expensive stocks within an industry, as determined by the, four contrarian

strategies, regardless of how high or low the general price of the industry group. 4. Look beyond obvious similarities between a current investment situation and one that

appears equivalent in the past. Consider other important factors that may result in a markedly different outcome.

5. Don't be influenced by the short-term record of a money manager, broker, analyst, or advisor, no matter how impressive; do not accept cursory economic or investment news without significant substantiation.

6. Don't rely solely on the short term. Take into account the long term the prior probabilities of profit or loss.

7. It is far safer to project a continuation of the psychological reactions of investors than it is to project the visibility of the companies themselves. (Very essential)

8. Political and financial crises lead investors to sell stocks. This is precisely the wrong reaction. Buy during a panic, don't sell. (Very essential)

9. In a crisis, carefully analyse the reasons put forward to support lower stock prices – more often than not they will disintegrate under scrutiny.

10. a. Diversify extensively. No matter how cheap a group of stocks looks, you never

know for sure that you aren't getting a clinker. b. In a crisis, these criteria get dramatically better as prices plummet (that is P/E,

P/BV and P/CF falls and dividend yields rises), markedly improving your chances of a big score.

11. A given in markets is that perceptions change rapidly.

Optional: Small-cap investing: 1. Buy companies that are strong financially (normally no more than 60% debt in the capital

structure for a manufacturing firm). 2. Buy companies with increasing and well-protected dividends that also provide an above-

market yield. 3. Pick companies with above-average earnings growth rates.

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4. Diversify widely, particularly in small companies because these issues have far less liquidity. A good portfolio should contain about twice as many stocks as an equivalent large-cap one.

5.2. Value Investing

This section is based on (Graham, Zweig, & Buffett, The Intelligent Investor: The Definitive Book on Value Investing, 2003). Value investing involves investing in companies that are worth less than their intrinsic value or underpriced, which is usually measured by P/BV. The strategy was first published in (Graham & Dodd, 1934) and was refined in subsequent editions ending with the final form of the strategy in (Graham, Dodd, & Cottle, 1963). The strategy was simplified in (Graham, 1973). There has been studied in various sources, such as (Capaul, Rowley, & Sharpe, 1993) and (Chan & Lakonishok, 2004). These usually indicate that P/E and P/BV ratios have higher returns than their high value counterparts, hence, support that the strategy on average produces above market returns. We will use the strategy based on (Graham, Zweig, & Buffett, 2003). In general: 1. The size of the company must be adequate, that is, in terms in turnover which must be at

least £30m. 2. Sufficiently strong financial condition. Current ratio must be at least equal to 2. 3. Continued dividends for the previous 5 years. 4. No earnings loss for the previous 5 years. 5. 4 year EPS growth must be at least 1/3. 6. P/BV must be at most 1.5. 7. Average P/E must be at most 15 for the previous 3 years. For defensive strategy: 1. As in general, the size of the company must be adequate. However, turnover which must

be at least £70m for manufacturers and at least £30m for utilities. 2. Net current assets must be at least twice the amount of current liabilities (Current ratio of

at least equal to 2). For manufacturers, long term debt (those due in more than 1 year) should be less than net current assets. For utilities, debt should less than twice net asset value.

3. Continued dividends for the previous 5 years. 4. No earnings loss for the previous 5 years. 5. 4 year EPS growth must be at least 1/3. 6. P/BV must be at most 1.5. 7. Average P/E must be at most 15 for the previous 3 years. For enterprising strategy, as in general but with the following: 1. Current assets at least 1.5 times current liabilities (Current ratio of 1.5). For

manufacturers, debt must be at most 110% of net current assets. Replaces criteria 2. 2. No earnings loss for the previous 5 years. 3. Some dividends paid. Replaces criteria 3. 4. Market capitalisation less than 120% of net tangible assets. Selling: Only sell when the situation changes to the extent that the company no longer satisfies the above criteria.

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Regarding requirement 1 for both general and defensive strategies, (Graham, Zweig, & Buffett, 2003) does not define this requirement clearly in terms of pounds sterling (pound or £). Therefore, a reasonable value is used based on the exchange rate between the US dollar and the pound in the year the book was published.

5.3. Zulu Principle

This section is based heavily on (Slater, 1992) and (Slater, 1998). The Zulu Principle is about dedicating a disproportionate amount of effort to become an expert within a narrow subject though learning. From (Slater, 1992), this name is applied to the investment strategy because the Jim Slater’s wife read an Reader’s Digest article on the Zulu people and

subsequently she had more knowledge than him within a few minutes. He thought that if she carried on studying on the same subject and lived in a Zulu village from a fixed period of time she would become a leading expert. The investor can be very successful by applying the same method to the stock market. One can become an expert in blue chips, growth companies, turnarounds and/or cyclical companies. There are sources, such as (Bary, 1997) and (O'Shaughnessy, 2005), prove that low P/E, P/BV, P/S, price to earnings growth (PEG) and price to research (P/R) ratios, high values of relative strength and EPS

produce higher returns than their opposite counterparts. Therefore, the strategy produces above market returns on average. The strategy we will be using is extracted from (Slater, 1992) and (Slater, 1998): Mandatory: 1. PEG less than 1 for large companies and less than 0.75 for smaller companies.

Preferably PEG below 0.66. PEG works best if companies have EPS growth rate between 15%-30 and P/E between 12 and 20.

2. P/E less than 20. 3. Strong cash flow and in particular cash flow per share greater than EPS for last reported

year and average for the last five years. 4. Low gearing, preferably below 50%. Positive cash balances are better. 5. High relative strength (RS) in the last 12 months with high relative strength in previous

one month or three months. Alternatively, relative strength index (RSI) of at least 83

1/3%. 6. Strong competitive advantage, for example, strong brand names, patents, copyrights,

legal monopolies, dominance in an industry and established position in a niche market. 7. No active selling by a group of directors. Highly desirable (optional): 1. Increasing EPS, preferably a company can clone its activities, for example Ikea opening

a new store using a successful business formula. 2. A number of directors buying new shares. 3. Market capitalisation between £30m-£250m. 4. Dividend yield. Bonus (optional): 1. Low P/S. 2. Something new, for example, new events in the industry, new acquisitions, new

product/service, new management or director. 3. Low P/R. This value must be at most 10. 4. Reasonable asset position. That is low P/BV at most 1/3.

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We note that very few companies meet all of the above mandatory and highly desirable requirements. A company which lacks one of these requirements then compromise a little. Regarding mandatory requirement 5, no number for relative strength is stated in (Slater, 1998) and relative strength is not available on the Digital Look website. Therefore we use the relative strength index (RSI) which is defined as

𝑅𝑆𝐼 = 100 −100

1 + 𝑅𝑆.

The range for the above is 0 ≤ 𝑅𝑆𝐼 ≤ 100. Higher values indicate that better performance of

the stock. An arbitrary value of 𝑅𝑆 = 5 is used instead to calculate the value for RSI.

6. Evaluating Investment Performance This section is based on (Elton, Gruber, Brown, & Goetzmann, 2007, pp. 632-658), (Kevin, 2007, pp. 197-211) and (Maginn, Tuttle, McLeavey, & Pinto, 2007, pp. 720-723). In this section we will use various measures to measure investment performance of the portfolios, constructed using Portfolio Theory and investment strategies outlined in this report.

6.1. Rate of Return

The rate of return 𝑟Π in percent over one period is the difference of the net asset value

(NAV), dividend payments and capital gains 𝐶𝑡 . Then the return of the portfolio is

𝑟Π = 𝑁𝐴𝑉𝑡 −𝑁𝐴𝑉𝑡−1 + 𝛿𝑡 + 𝐶𝑡

𝑁𝐴𝑉𝑡−1

where 𝑁𝐴𝑉𝑡−1 and 𝑁𝐴𝑉𝑡 are NAVs at the start and end of the holding period respectively, 𝛿𝑡 and 𝐶𝑡 are dividend payments and capital gains during the holding period. Portfolios may have different returns because they have different compositions and proportions of assets. Hence, the returns have to be adjusted before comparisons can be made. In this report the NAV at the end of the holding period will be the sale value of the shares held at the time after transaction costs.

6.2. Risk Adjusted Returns

Sharpe Ratio

The Shape ratio (SR) was developed in (Sharpe, Mutual Fund Performance, 1966). This is the ratio of the difference of portfolio return and riskless return to risk. The expression is defined as

𝑆𝑅 =𝑅Π − 𝑅0

𝜎Π

where 𝑅Π is the realised return on the portfolio after transaction costs. Higher values indicate that the portfolio is performing better than the ones with lower values.

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Treynor Ratio

The Treynor ratio (TR) was devised in (Treynor, 1965). This is the ratio of the difference of

portfolio return and riskless return to volatility (or systematic risk) measured by beta. The equation is defined as

𝑇𝑅 =𝑅Π − 𝑅0

𝛽Π

where 𝛽Π is the portfolio beta, which is covariance of asset 𝑖 with the market portfolio divided by the variance of the market portfolio. This measure the correlation compared to the whole market. Similarly for the Treynor ratio, higher values indicate that the portfolio is performing better than the ones with lower values. The values calculated will be different for both ratios because the Sharpe ratio uses standard deviation, while the Treynor ratio uses the systematic risk measured. If the portfolio is well diversified then the latter would be appropriate to use. Otherwise, the former is appropriate to use. Jensen Ratio The Jensen ratio (JR) was developed in (Jensen, 1968). This measures the difference

between the actual return on the portfolio and its expected return at a given level of risk. This indicates the portfolio manager’s skill level. The Security Market Line (SML) is

𝑅 Π = 𝑅0 + 𝑅 𝑀 − 𝑅0 𝜎𝑖𝑀

𝜎𝑀2 = 𝑅0 + 𝑅 𝑀 − 𝑅0 𝛽Π

where 𝑅 𝑀 is the market index expected return. The formula for JR is

𝛼Π = 𝑅Π − 𝑅 Π

where 𝛼Π and 𝑅Π are the differential and actual returns respectively. If 𝛼Π > 0 , then it indicates that the portfolio performance is better than the market and the excess return is contributed to good management skills. If 𝛼Π = 0, then it indicates neutral performance since the portfolio performance is identical to an unmanaged portfolio with a buy and hold strategy

or the market. If 𝛼Π < 0, then the portfolio performance is worse than that of the market or a randomly selected portfolio with the same level of risk. Fama's Net Selectivity

The Fama's net selectivity (FNS) was first defined in (Fama, 1972). This measure provides

a breakdown of the portfolio's performance into different components. The total return on a portfolio is defined as

Total return = Risk free return + Excess return. Excess return is defined as

Excess return = Risk premium + Return from stock selection where risk premium is reward for taking on risk. Since no portfolio would be fully diversified in reality, it would have systematic risk and small amount of diversifiable risk. Therefore we have

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Risk premium = Return from systematic risk + Return from diversifiable risk. Thus the portfolio's total return can be decomposed into four components. We have

Total return = Risk free return + Return from systematic risk + Return from diversifiable risk + Return from pure selectivity.

We can represent the above mathematically as 𝑅Π = 𝑅0 + 𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐 (20)

where 𝑅𝑎 is the return from market risk, which is defined as

𝑅𝑎 = 𝑅𝑀 − 𝑅0 𝛽Π ,

𝑅𝑏 is the return from diversifiable risk, which is

𝑅𝑏 = 𝑅𝑀 − 𝑅0 𝜎Π𝜎𝑀

− 𝛽Π ,

and 𝑅𝑐 is the return from pure selectivity, which is

𝑅𝑐 = 𝑅Π − 𝑅0 + 𝑅𝑎 + 𝑅𝑏 . We can simplify equation (20) to get

𝐹𝑁𝑆 = 𝑅Π − 𝑅0 +𝜎Π𝜎𝑀

𝑅𝑀 − 𝑅0 .

This basically measures the actual return of the portfolio and the return mandated by its total

risk. If 𝐹𝑁𝑆 > 0 then the portfolio has achieved the return equal or above its expected return

justified by its total risk due to good stock selection. If 𝐹𝑁𝑆 < 0 then the portfolio has not achieved its expected return justified by its total risk due to poor stock selection.

7. Plan of Experiment We will go through the details of the experiment. We will construct three optimal portfolios and a portfolio using each of the strategies outlined in this report, starting with the first optimal portfolio and portfolio using the Contrarian Investment strategy. Then we run them for two weeks and compare the performance of both to find out which produces the highest return during that period. We repeat the process with the second optimal portfolio and Value Investing portfolio. Then we repeat again with the third optimal portfolio and the Zulu Principle portfolio. The timetable of the experiment is shown below.

Table 1: Timetable of experiment showing each of the optimal portfolios going to be compared with each of the portolios constructed using the investment strategies

Period Start Date End Date Optimal Portfolio Type of strategy used for portfolio

1 9/3/2009 20/3/2009 1 Contrarian Investment

2 23/3/2009 3/4/2009 2 Value Investing

3 6/4/2009 17/4/2009 3 Zulu Principle

We will use the following assumptions in order to make the experiment realistic as possible:

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1) There are transaction costs when buying and selling stocks.

a. When buying stocks on the LSE, the following costs will be incurred; i. Broker commission of £10. ii. Stamp duty of 0.5% on the value of the stock excluding commission iii. Panel of Takeovers and Mergers (PTM) levy of £1 per transaction if the

value of stock is over £10,000. The levy is £1.50 per transaction if the value is greater than £15,000. These values are from (The Panel on Takeovers and Mergers, 2009).

b. When selling stocks, only ai and aiii apply. 2) There are capital gains tax and income tax on dividends, though the latter is very unlikely

during each two week period. 3) Bid/offer spread. This is the difference between the buying (offer) and selling (bid) price. 4) Only integer quantities of stock can be bought or sold. 5) No short selling because private investors cannot short sell stocks whereas institutional

investors can. 6) Borrowing and lending rates are the same. The interest rate we will use is the two week

London Interbank Offered Rate (LIBOR). 7) The FTSE All-Share index is the market for performance comparison purposes because

it represents approximately 98% of the total Main Market capitalisation, which also represents approximately 98% of the LSE capitalisation. Therefore assume that it represents the value of the Main Market. We also assume that the Main Market represents LSE.

Regarding the following assumptions:

Number 4, in real life any positive quantity of stocks can be bought and sold through some share dealing accounts, such as Halifax Sharebuilder (Halifax Share Dealing Ltd, 2009). For example, an investor buys 100.5 RBS shares. The stockbroker buys 101 shares on the LSE. The former receives the required number of shares while the latter keeps the remaining fraction.

Number 5, if private investors are allowed to short sell stocks the market would be more volatile.

Number 6, investors in real life do not borrow and lend using the 2-week LIBOR rate. Banks do not allow investors to borrow funds for investment purposes. Investors can lend at the annual interest rate the bank offers on the savings account. Rates differ across different savings accounts and banks.

Number 7, we will use LSE capitalisations values as of 31/03/2009. We will use the following in order to conduct the experiment:

Digital Look® website (which the web address is www.digitallook.com) to: o Look up:

Portfolio and FTSE All-Share values at the end of the holding period in order to evaluate their performance.

Relevant ratios and information at the end of each trading day during the holding period to determine which stock no longer matches the criteria and sell them.

o Keep track of the portfolios. o Select suitable stocks, defined by the criteria from the strategies, though the

stock screener. o Execute transactions. o Construct and run portfolios.

Microsoft Excel® to: o Record transactions.

http://www.digitallook.com/

33

o Calculate: Portfolio betas, standard deviation and expected returns. Overall holding period portfolio betas, standard deviation and expected

returns (Details are shown below). Value of portfolio after selling all stocks at the end of the period.

o Record: Portfolio values at the end of the holding period in order to evaluate their

performance. Relevant ratios and information at the end of each trading day during the

holding period to determine which stock no longer matches the criteria and sell them.

o Evaluate portfolio performance using the ratios outlined in the previous section.

Thomson Datastream to download: o Past weekly stock prices for the past year of all stocks listed on the LSE in order

to construct optimal portfolios. o Past weekly share prices for the past year of stocks held in the portfolios to

evaluate their performance. o Stock betas.

BullBearings website (which the web address is www.bullbearings.co.uk) to: o Execute stock transactions. o Run the portfolios.

MATLAB® to o Calculate the efficient frontier of 20 portfolios and optimal portfolio from the

universe of 1914 stocks, which is all of the companies, excluding investment

trusts, listed on the LSE, using portopt and portalloc commands

respectively. o If necessary, transform non-positive semi-definite covariance matrix into positive

semi-definite one using the eigenvalue method of (Rousseeuw & Molenberghs, 1993, pp. 971-973) (see Appendix A.4).

The instructions regarding the use of the Digital Look and BullBearings websites are in Appendix A.5. The MATLAB codes for transforming the non-positive semi-definite covariance matrix into positive semi-definite one and calculating the efficient frontier and optimal portfolio are in Appendix A.7 We can calculate portfolio expected returns, standard deviation and betas for the

whole holding period 𝐻 by using the following:

𝐻𝑅 Π =1

𝑕1𝑅 Π1 +

1

𝑕2𝑅 Π2 ⋯+

1

𝑕𝑛𝑅 Π𝑛

𝐻𝜎Π =1

𝑕1𝜎Π1 +

1

𝑕2𝜎Π2 ⋯+

1

𝑕𝑛𝜎Π𝑛

and

𝐻𝛽Π =1

𝑕1𝛽Π1 +

1

𝑕2𝛽Π2 ⋯+

1

𝑕𝑛𝛽Π𝑛

where 𝑕𝑖 is the number of days in a sub-period, where 𝑖 = 1,… ,𝑛. Note that 𝑕𝑖 are not necessarily equal to each other. We will use these to evaluate the performance of the investment portfolios because these take into account of all the stocks held during the holding period and this give an accurate figure for the portfolio. Regarding the universe of LSE stocks, we exclude investment trusts because some have assets consisting of assets in one or more sectors or indices on the exchange, some have assets which are unlisted and other have assets that are listed solely in other stock

http://www.bullbearings.co.uk/

34

exchanges. Including investment trusts in the universe will result in the optimal portfolio not having optimal allocations of companies listed on the LSE.

8. Results of the Experiment

Table 2: Optimal Stock proportions for Optimal Portfolio 1 with relevant data

Stock Name Ticker Optimal Weighting (%) 𝑹 𝚷 (%) 𝝈𝚷𝟐 (%) 𝝈𝚷 (%)

Abcam ABC 0.26 0.87 0.16 4.06

ADL AD. 75.63 0.96 3.49 18.69

Advent Capital (Holdings) ADV 0.03 14.27 123.50 111.13

AEC Education AEC 0.15 0.93 0.24 4.92

Regenesis Group RGN 23.94 1.80 12.75 35.71



Abcam ABC 0.52 1.09 0.18 4.24

Access Intelligence ACC 1.13 0.49 2.23 14.92

ADL AD. 91.36 0.96 3.49 18.69

Advanced Computer Software ASW 1.61 0.58 0.46 6.76

Advanced Medical Solutions Group AMS 3.52 0.36 0.21 4.63




ADL AD. 88.74 1.15 3.47 18.63


Albemarle and Bond Holdings ABM 2.51 0.16 0.30 5.47

Advanced Computer Software ASW 2.23 0.60 0.46 6.76

Alexander David Securities GRP ADS 1.71 0.53 2.85 16.88

AEC Education AEC 1.63 0.93 0.24 4.92

Admiral Group ADM 0.41 0.35 0.39 6.25

Altona Energy ANR 0.22 1.26 2.78 16.68

Table 5: Investment Portfolio weightings

Portfolio Number of Stocks invested Initial Weighting (%)

Contrarian 10 10

Value Investing 8 12.5

Zulu Principle 6 16.66

We used MATLAB to calculate efficient frontier of 20 portfolios and the optimal portfolio for each of the three holding periods using the three sets of returns data and LIBORs (see Table 6). Regarding these sets of returns data, the history of returns goes back one year from the last trading day before the start of each holding period. The universes of stocks used to calculate the optimal portfolios are 294, 352 and 430 for the first, second and third portfolios respectively because the computer used struggles to calculate the solutions from the original universe of 1914 stocks. We explain this in more detail in the next section. Past index

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values, stock prices and returns, additional details of optimal portfolio allocations and MATLAB files are on the CD provided. Table 2, Table 3 and Table 4 show the optimal proportions for optimal portfolios 1, 2 and 3 respectively. For each optimal portfolio, the data of expected returns, variances and betas are shown for each stock. We calculated the efficient frontiers for each universe and they were unsatisfactory so we decided not to include them in the report. We constructed the portfolios for each investment strategy using the criteria defined in section 4 (see Appendix A.6.1). We calculated overall portfolio expected return, standard deviation and beta for each portfolio (see Table 7). We also calculated the expected return and standard deviation for FTSE All-Share, going back one year from the last trading day before the start of each holding period, because no dividends were paid during each of the holding periods. We will use these to evaluate the performance of these portfolios. Table 6: LIBOR used to calculate the optimal portfolios and evaluate the performance of portfolios for respective

holding periods

Period 2-week LIBOR (%)

1 0.82125

2 0.77

3 0.7275

Source: British Bankers Association

Table 7: Overall portfolio values based on all stocks held in the portfolios over the whole holding period

Portfolio 𝑯𝑹 𝚷 (%) 𝑯𝝈𝚷 (%) 𝑯𝜷𝚷

Contrarian -0.18 1.24 1.15

Value -0.73 10.12 0.88

Zulu Principle -0.19 4.89 0.48

Optimal 1 1.17 22.73 0.34

Optimal 2 1.18 19.60 0.40

Optimal 3 1.44% 20.09% 0.42

Table 8: Portfolio and market returns using various measures defined in the report

Period Portfolio Return measure

𝑹𝚷 (%) 𝒓𝚷 (%) SR TR JR FNS (%)

1

Contrarian 6.58 13.15 4.64 5.01 7.66 3.89

Optimal 1 -32.10 -64.20 -1.44 -96.83 -35.51 -67.14

FTSE All-Share 8.44 0.17 1.51 7.61

2

Value Investing -8.92 -17.84 -0.96 -11.01 -8.48 -13.59

Optimal 2 -20.85 -41.71 -1.1 -54.05 -21.07 -29.18

FTSE All-Share 2.76 0.06 0.39 1.99

3

Zulu Principle -10.55 -21.09 -2.31 -23.49 -10.68 -13.49

Optimal 3 -25.07 -50.15 -1.28 -61.42 -25.27 -34.88

FTSE All-Share 3.06 0.06 0.45 2.33

Note that the values are calculated using the portfolio values after incurring transaction costs during the sale of all stocks for each portfolio.

Table 8 illustrates that overall the Contrarian Portfolio performed better than Optimal Portfolio 1 and the market. This is because, partly on an economic level, there were hopes that at the London G20 (group of 20 countries with developed and emerging economies) summit leaders would agree on a global plan to prevent the global economy from going into a depression and revive it. This made the investors regain some of the confidence lost

36

during the Credit Crunch to invest in the stock market. For the Contrarian Portfolio, Inchape returned a profit of 77.11% due to the fact that, from (London Stock Exchange Plc, 2009), the company announced that the rights issue of 9 shares for every one held at a price of 6 pence would take place. This resulted in a massive demand for the stock so the price increased dramatically since investors wanted to gain massive profit from the sale of the newly issued shares (an example of arbitrage). Kazakhmys returned a profit of 38.35% probably due to the fact the copper, zinc and gold prices were rising during the holding period. A few other companies made small profits and other made small losses. Regarding the latter, this was due to transaction costs and spread between the buying and selling prices (see Table 16 and Table 28). For the Optimal 1, the loss is due to the transaction costs and large spread between buying and selling prices (see Table 20 and Table 31). This is because no significant events took place during the holding period therefore prices of the stocks hardly changed during the holding period therefore Overall, the Value Investing Portfolio performed better than Optimal Portfolio 2 but worse than the market. This was due to transaction costs and some stocks have large bid/offer spreads (see Table 17 and Table 29). The overall portfolio loss is reduced by the fact that the share price of Jersey Electricity (A shares) increased during the holding period probably due to the company performed to its expectations (see (London Stock Exchange Plc, 2009)). Also the share price of Delta increased due to fact that the company performed better than the market expected. From (Delta Plc, 2009), turnover increased by 24%, operating profit increased by 109%, cash flow increased by 82%, EPS raised by 22% and dividend has gone up by 30%. This resulted in increased demand for Delta shares. Thus, it delivered 6.25% profit. Regarding Optimal Portfolio 2, the loss was due to transaction costs and large bid/offer spreads since the shares hardly changed during the holding period (see Table 20 and Table 32). The reasons are the same as in the case of Optimal 1. Overall, the Zulu Principle Portfolio performed better than Optimal Portfolio 3 but worse than the market. This was also due to the transaction costs and most stocks have large bid/offer spreads (see Table 18 and Table 30) since the shares prices generally have not changed very much during the holding period. Even the strong first half performance from Cash Converters (see (London Stock Exchange Plc, 2009)), strong performance from Visonic (see (London Stock Exchange Plc, 2009)) and a director buying shares in Zytronic (see (London Stock Exchange Plc, 2009)) failed to raise the confidence of investors to invest in these stocks. For Optimal Portfolio 3, the loss was due to transaction costs and large bid/offer spreads since the shares hardly changed during the holding period (see Table 21 and Table 32). The reasons are the same as in the case of first two optimal portfolios. Even quite strong performances of Advanced Medical Solutions Group (see (London Stock Exchange Plc, 2009)) failed to increase the demand for the stock. Regarding the above results, the returns would have been higher if the stocks have not been sold (see Appendix A.6.3). Additional details regarding the actual returns of stock after transaction costs in each portfolio are shown in Appendix A.6.5. During the holding period, we had to sell some stocks because they no longer match the investment criteria defined in the report. Details in the Stock and Sector Ratios.xls file.

9. Conclusion and Recommendations for Improvement We used MATLAB to calculate efficient frontier and the optimal portfolio for each of the three holding periods using relevant data. For each holding period we used the past year's stock prices, going back one year from the last trading day before the start of the holding period, to calculate the data of returns. The resulting data had missing observations due to the fact that some stocks were not listed during these periods. Therefore we use nanmean and nancov

commands to calculate the expected return vector and covariance matrix from observed

data. If we used the mean and cov commands, the portopt command will not calculate the

efficient frontier since not all elements in the covariance matrix are numbers. We had to

37

transform a non-positive semi-definite covariance matrix into a positive semi-definite one using the eigenvalue method of (Rousseeuw & Molenberghs, 1993). Then we use the expected returns vector, transformed covariance matrix and 2-week LIBOR to calculate the

efficient frontier and optimal portfolio, using the portopt and portalloc commands.

MATLAB did produce quite strange looking efficient frontiers because we did not use longer stock price history, that is, too few observations for each stock. We note that when we used the transformed covariance matrix to calculate the efficient frontier and optimal portfolio, MATLAB displays the messages stating that the covariance matrix is non-symmetric and non-positive definite. This is because computer have finite amount of memory to store numbers and MATLAB only stores number up to 15 digits. Also the algorithms in MATLAB tend to round the solutions to the nearest number resulting in inaccuracies. When we investigated the matrix manually, we noted that they were symmetric and positive semi-definite. We argue the following so that we can calculate the efficient frontiers and optimal portfolios in a reasonable amount of time:

Investors will only choose stocks with positive expected returns because MATLAB takes extremely long time to calculate efficient frontier and the optimal portfolio from a universe of 1914 stocks. This reduced the universe of stocks to considerably.

Using simple regression analysis of expected returns and variance for each returns data we excluded stocks with variances greater than 300% because including these would adversely affect the solutions, that is, this would lead to optimal portfolios consisting of just one stock which is unsatisfactory. Note that we did not include the regression results in this report because we only used this to decide the number of stocks to exclude.

These substantially reduced the universe of stocks. Hence, the number of stocks used to calculate the optimal portfolios are 294, 352 and 430 for the first, second and third portfolios respectively. Using the new respective universe of stocks, MATLAB only takes about 5 minutes to calculate the efficient frontier and an optimal portfolio. Based on the evidence in the previous section, we conclude that the portfolios using the investment strategies performed better than the optimal portfolios. This is because, in general, there was good economic news throughout the holding periods, especially regarding the London G20 summit. . Regarding individual companies, significant events took place which, in general, indicated that they were performing better than the market expected. There was a case of arbitrage regarding Inchape which contributed to the massive profit for the Contrarian Portfolio during the holding period. This lead to increased investor confidence in general which resulted in increased share prices generally. When Inchape announced that the 9 to 1 rights issue at the price of 6 pence would take place, large numbers of investors bought the shares in the hope of earning a large amount of risk free profit, indicating herd behaviour. Regarding Kazakhmys, the share price increased probably due to the fact the media noted that copper, zinc and gold prices has risen during the holding period leading to increased demand for the stock. Regarding Royal Dutch Shell, the media also mentioned that oil prices started to rise from their lowest point of $40 per barrel, leading increased demand for the stock. When Delta reported strong financial results demand increased for the stock. These suggest there is some evidence of anchoring and may be over-optimism. For other stocks of companies held in investment portfolios, the share prices have not changed very much, even though the ratios of these indicate that they are in good financial health, because there was no information which indicated that these would perform better than market expectations. This suggests that the investors were over-pessimistic. Overall, these suggest that there is some qualitative evidence that some aspects of behavioural finance had an effect on share prices of stocks. The optimal portfolios performed very badly for the following reasons; firstly, we only used past share prices going back one year to calculate the returns, expected returns and variances of stocks. This resulted in biased optimal allocations of stock for each optimal

38

portfolio. These in turn probably made the portfolio evaluation results biased since these suggest that the portfolios using investment strategies in general will perform better than optimal portfolios. Secondly, the stocks held in the optimal portfolios had large bid/offer spread. Therefore, the losses are due to this and the transaction costs. Overall, two week holding period is too short so that the results in this report are inconclusive. This illustrates that the evidence presented in this report is insufficient to prove that the portfolios using investment strategies consistently perform better than the optimal portfolios. Investors do not borrow or lend using 2 week LIBOR. Banks have policies that investors cannot borrow money for investment in the stock market. However, investors can lend at the rate of a savings account they deposit their money in. Under this case and if riskless lending rate is the same, we would end up with the same optimal portfolios. The rates differ across different savings and investment accounts and different banks and building societies. In some cases it is possible for investors to borrow at 0% interest rate (if they are being dishonest at the time application of an overdraft). These are university students who have student overdrafts on their student bank accounts and people who have bank accounts which offer interest free overdraft up to a certain amount (which is usually very small amount, say £200). The amount they can borrow is determined by their credit history and usage behaviour of their accounts. Similarly to the previous case, we would have the same optimal portfolios. We suggest improvements to the experiment to potential researchers: 1) Use 5 year history of stock prices to calculate the returns data. Ideally, the history of

stock prices should be weekly because more observations usually mean more accurate results.

2) Use longer holding period so that the results are more conclusive.

3) If dividends are paid during holding periods, use the total return index to calculate 𝑅 Π , 𝑅Π

and 𝜎Π of that index because this takes into account of price movements of stock and dividends paid. Also, the calculated values are more appropriate for portfolio performance evaluation.

4) Use weighted total return values, consisting of FTSE All-Share and FTSE AIM according to the market capitalisations of each index. Based on the market capitalisation values on the LSE on 31/03/2009, the Main Market represents approximately 98% of the value on the stock exchange. FTSE AIM represents approximately 2% of the value. Using these values, we can calculate the weighted total return values as follows:

Weighted Value = (0.98 × FTSE All-Share index value) + (0.02 × FTSE AIM index value)

Then we can use the above to calculate history of weighted total index values, which in

turn we use to calculate the history of weighted returns. Then we use this to calculate 𝑅 Π, 𝑅Π and 𝜎Π of that index.

5) Use average instant access savings rate. Regarding all the above points except point 4, the actual returns of the optimal portfolios would probably be closer to their expected returns. Regarding point 4, we believe this would help future researchers to evaluate the performance of portfolios more accurately. Hence, these recommendations should make the results more reliable and conclusive.

39

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43

Appendices

A.1. Capital Asset Pricing Model (CAPM)

This section is based on (Elton, Gruber, Brown, & Goetzmann, 2007, pp. 284-286).The CAPM was suggested independently by (Sharpe, 1964), (Lintner, 1965) and (Mossin, 1966). The theory introduces a number of additional assumptions regarding the market and the behaviour of other investors in order to construct a general equilibrium of asset prices in the entire market. The assumptions of CAPM are: 1) No transaction costs and taxes, such as stockbroker commissions, stamp duty, capital

gains tax and so on. 2) Investors prefer a portfolio with a higher return to one with a lower return at a given level

of risk. 3) Investors are risk averse, where they prefer a portfolio with lower risk to one with higher

risk for a given level of return. 4) Assets are infinitely divisible. This means that investors can own any quantity of an

asset. 5) Investors are price takers, meaning that they cannot affect the asset prices through

buying and selling. 6) Investors make decisions based on Mean-Variance theory. 7) Short selling at any amount is allowed. 8) Investors can borrow and lend any amount at a given interest rate. 9) Investors are concerned with the expected return and variance in a single period, and

define the relevant period in the same way. 10) Investors have identical expectations with respect to expected return, variance and

correlation matrix. 11) All assets can be bought or sold on the market. In real life, all assumptions, expect assumption 7, do not hold. Under these assumptions, all investors face the same efficient frontier and the same risk free rate. Hence the efficient frontier is just a straight line. This line is called the CML. The optimal portfolio is at a point where the CML is tangent to the efficient frontier and that all investors have the same optimal allocations in their portfolios. Also all investors will hold a combination of risky and riskless assets.

A.1.1. Derivation of CAPM

In this section we will rigorously derive the CAPM. There are various ways of deriving the theory (see (Elton, Gruber, Brown, & Goetzmann, 2007, pp. 293-294), (Sharpe, 1964), (Lintner, 1965) and (Mossin, 1966)). From (Howls & Fitt, 2009, pp. 239-240), let there be

𝑕 = 1,… ,𝐻 investors in the market of 𝑁 risky assets and one riskless asset. The usual notations of expected returns and risk free rate of return will be used. Suppose that 𝜔𝑖𝑕 is

the proportion of the total market in asset 𝑖 that investor 𝑕 has in the portfolio. Then

𝜔𝑖𝑕 = 𝜙𝑖𝑕

𝑁

𝑖=0

where 𝜙𝑖𝑕 is the total in the economy that the investor has. From the above, there is a

relationship between the investor’s proportion of the total market of asset 𝑖, 𝜔𝑖𝑕 , and the proportion of their wealth invested in the asset, 𝜆𝑖𝑕 . Thus

44

𝜔𝑖𝑕 = 𝜆𝑖𝑕𝜙𝑖𝑕 with

𝜆𝑖𝑕 = 1

𝑁

𝑖=0

.

Then the portfolio expected return, comprising of risky and riskless assets, is

and the variance is

𝜎Π𝑕2 = 𝜆𝑖𝑕𝜆𝑗𝑕𝜎𝑖𝑗

𝑁

𝑗=1

𝑁

𝑖=1

=1

𝜙𝑕2 𝜔𝑖𝑕𝜔𝑗𝑕𝜎𝑖𝑗

𝑁

𝑗=1

𝑁

𝑖=1

.

From assumptions 2 and 3, each investor maximises their expected utility, where utility is the measurement of happiness,

𝑢 Π𝑕 = 𝑢 𝑕 𝑅 Π𝑕 ,𝜎Π𝑕2

subject to

𝜔𝑖𝑕 = 𝜙𝑖𝑕

𝑁

𝑖=0

.

Using Lagrange multipliers, the expected utility can be rewritten as

max 𝑢 𝑕 𝑅 Π𝑕 ,𝜎Π𝑕2 + 𝜇𝑕 𝜔𝑖𝑕 −𝜙𝑖𝑕

𝑁

𝑖=0

,

where 𝜇𝑕 is the Lagrange multiplier for each investor which maximises the constraint which, if satisfied, leaves the above function unchanged. The conditions for a critical point for the

above function is vanishing of all the first partial derivatives with respect to 𝜔𝑖𝑕 . Using the chain rule, this results in

Using the portfolio expected return and variance expressions on the above to become

and

𝑅 Π𝑕 = 𝜆𝑖𝑕𝑅 𝑖

𝑁

𝑖=0

=1

𝜙𝑖𝑕 𝜔𝑖𝑕𝑅 𝑖

𝑁

𝑖=0

(21)

𝜕𝑢 𝑕

𝜕𝑅 Π𝑕.𝜕𝑅 Π𝑕

𝜕𝜔𝑖𝑕+

𝜕𝑢 𝑕

𝜕𝜎Π𝑕2 .

𝜕𝜎Π𝑕2

𝜕𝜔𝑖𝑕+ 𝜇𝑕 = 0 for ∀𝑖.

𝜕𝑢 𝑕

𝜕𝑅 Π𝑕.𝑅 𝑖𝜙𝑖𝑕

+𝜕𝑢 𝑕

𝜕𝜎Π𝑕2 .

𝜔𝑗𝑕𝜎𝑖𝑗

𝜙𝑕2

𝑁

𝑗=1

+ 𝜇𝑕 = 0 for 𝑖 = 1,… ,𝑁

45

which simplifies to

and

Rearranging equation (23) in terms of 𝜇𝑕 and substituting into equation (22) to get

Consider two of these equations for two different assets 𝑖 and 𝑘, and divide one equation by the other to obtain

𝜕𝑢 𝑕𝜕𝑅 Π𝑕

. 𝑅 𝑖 − 𝑅0

𝜕𝑢 𝑕𝜕𝑅 Π𝑕

. 𝑅 𝑘 − 𝑅0 =

𝜕𝑢 𝑕𝜕𝜎Π𝑕

2 . 𝜔𝑗𝑕𝜎𝑖𝑗𝜙𝑕

𝑁𝑗=1

𝜕𝑢 𝑕𝜕𝜎Π𝑕

2 . 𝜔𝑗𝑕𝜎𝑘𝑗𝜙𝑕

𝑁𝑗=1

.

The partial derivatives and 𝜙𝑕 cancel each other out resulting in

𝑅 𝑖 − 𝑅0

𝑅 𝑘 − 𝑅0 =

𝜔𝑗𝑕𝜎𝑖𝑗𝑁𝑗=1

𝜔𝑗𝑕𝜎𝑘𝑗𝑁𝑗=1

.

This relationship holds for all investors and, hence, we sum all investors to arrive at

𝑅 𝑖 − 𝑅0

𝑅 𝑘 − 𝑅0 =

𝜔𝑗𝑕𝜎𝑖𝑗𝑁𝑗=1

𝑁𝑗=1

𝜔𝑗𝑕𝜎𝑘𝑗𝑁𝑗=1

𝑁𝑗=1

= 𝜔𝑗𝑕

𝑁𝑗=1 𝜎𝑖𝑗

𝑁𝑗=1

𝜔𝑗𝑕𝑁𝑗=1 𝜎𝑘𝑗

𝑁𝑗=1

= 𝜂𝑗𝜎𝑖𝑗

𝑁𝑗=1

𝜂𝑗𝜎𝑘𝑗𝑁𝑗=1

.

Rewriting the above as

𝑅 𝑖 − 𝑅0

𝜂𝑗𝜎𝑖𝑗𝑁𝑗=1

= 𝑅 𝑘 − 𝑅0

𝜂𝑗𝜎𝑘𝑗𝑁𝑗=1

and since this must hold for all assets 𝑖 and 𝑘, the above expression must be equal to some

constant 𝛾. That is

Multiplying the numerator and denominator by 𝜂𝑖 and sum all 𝑖 assets to obtain

𝜕𝑢 𝑕

𝜕𝑅 Π𝑕.𝑅0

𝜙𝑖𝑕+

𝜕𝑢 𝑕

𝜕𝜎Π𝑕2 . 0 + 𝜇𝑕 = 0 for 𝑖 = 0,

𝜕𝑢 𝑕

𝜕𝑅 Π𝑕.𝑅 1 +

𝜕𝑢 𝑕

𝜕𝜎Π𝑕2 .


𝜙𝑕

𝑁

𝑗 =1

+ 𝜇𝑕 = 0 for 𝑖 = 1,… ,𝑁 (22)

𝜕𝑢 𝑕

𝜕𝑅 Π𝑕.𝑅0

𝜙𝑕+ 𝜇𝑕 = 0 for 𝑖 = 0. (23)

𝜕𝑢 𝑕

𝜕𝑅 Π𝑕. 𝑅 𝑖 − 𝑅0 +

𝜕𝑢 𝑕

𝜕𝜎Π𝑕2 .


𝜙𝑕

𝑁

𝑗=1

= 0 for 𝑖 = 1,… ,𝑁.

𝑅 𝑖 − 𝑅0

𝜂𝑗𝜎𝑖𝑗𝑁𝑗=1

= 𝛾. (24)

46

Using the definitions of expected return and variance of the optimal market portfolio 𝑀,

𝑅 𝑀 = 𝑅 𝑖

𝑁

𝑖=1

and

𝜎𝑀2 = 𝜂𝑗𝜂𝑘𝜎𝑖𝑗

𝑁

𝑗=1

𝑁

𝑖=1

and the fact that

𝜂𝑖

𝑁

𝑖=1

= 1,

simplifying equation (25) yields

Since covariances are additive, the covariance of asset 𝑖 with the optimal market portfolio 𝑀 is

𝜎𝑖𝑀 = 𝜂𝑗𝜎𝑖𝑗

𝑁

𝑗=1

.

Therefore rearranging equation (26) in terms of 𝛾 and substituting this into equation (24) to get

𝑅 𝑖 − 𝑅0

𝜎𝑖𝑀=

𝑅 𝑀 − 𝑅0

𝜎𝑀2

and rearranging to arrive at

𝑅 𝑖 = 𝑅0 + 𝑅 𝑀 − 𝑅0 𝜎𝑖𝑀

𝜎𝑀2 = 𝑅0 + 𝑅 𝑀 − 𝑅0 𝛽𝑖

where 𝛽𝑖 is the beta of asset 𝑖 , which is covariance of asset 𝑖 with the market portfolio divided by the variance of the market portfolio. The above expression is the Security Market Line (SML) and it represents the expected return for all assets and portfolios of assets in the

whole economy. The linear relationship between the expected return and on any two assets is determined by their betas. The higher the 𝛽𝑖 , the equilibrium return is higher, and vice

versa. The above result can be applied to a portfolio Π of riskless and risky assets by summing over all risky assets. The CML can be derived from equation (20). First note that since 𝛽𝑖 are covariances

of asset 𝑖 with the market 𝑀 then 𝛽Π can be obtained from the weighted sum of 𝛽𝑖s. Thus, the expression is

𝜂𝑖 𝑅 𝑖 − 𝑅0

𝑁

𝑖=1

= 𝛾 𝜂𝑗𝜂𝑘𝜎𝑖𝑗

𝑁

𝑗=1

𝑁

𝑖=1

. (25)

𝑅 𝑀 − 𝑅0 = 𝛾𝜎𝑀2 . (26)

47

𝛽Π = 𝜂𝑖𝛽𝑖

𝑁

𝑖=1

.

Therefore, the expected return on portfolio Π for investor 𝑕 is

𝑅 Π𝑕 = 𝜔𝑖𝑕

𝜙𝑖𝑕 𝑅 𝑖

𝑁

𝑖=0

=

𝜔𝑖𝑕

𝜙𝑖𝑕 𝑅0

𝑁

𝑖=0

+ 𝑅0 + 𝑅 𝑀 − 𝑅0

𝜎𝑀

𝜎𝑖𝑀

𝜎𝑀2

=

𝜔𝑖𝑕

𝜙𝑖𝑕 𝑅0

𝑁

𝑖=0

+ 𝑅 𝑀 − 𝑅0

𝜎𝑀

𝜔𝑖𝑕

𝜙𝑖𝑕

𝜎𝑖𝑀

𝜎𝑀2

𝑁

𝑖=0

= 𝑅0 +

𝑅 𝑀 − 𝑅0

𝜎𝑀

𝜔𝑖𝑕

𝜙𝑖𝑕

𝜎𝑖𝑀

𝜎𝑀2

𝑁

𝑖=0

= 𝑅0 +

𝑅 𝑀 − 𝑅0

𝜎𝑀 𝜎Π𝑕 .

The above equation is the CML. Recall from section 3.4, the gradient 𝑅 𝑀 − 𝑅0 𝜎𝑀 is the

market price of risk and point 𝑀 on the CML which is tangent to the efficient frontier of the opportunity set of risky assets is the optimal portfolio at a given rate of interest (see Figure 9 in section 3.4). Thus, this maximises the market price of risk. All investors should select this part of portfolio, although the choice of proportion in relation to the risky asset depends on

their preferences. We know from section 3.3.3, if 𝜆0𝑕 < 0 and 𝜆𝑖𝑕 > 1 , the investor is

borrowing from the bank at 𝑅0 and investing the loan in more risky assets. Conversely, if 𝜆0𝑕 > 1 and 𝜆𝑖𝑕 < 0, short selling of risky assets takes place and the proceeds are deposited

in the bank account to earn interest at 𝑅0. If 𝜆0𝑕 = 𝜆𝑖𝑕 = 0, then no investment in the risky assets takes place. This completes the derivation of CAPM.

48

A.2. Quadratic Programming

This appendix is based heavily on (Elton, Gruber, Brown, & Goetzmann, 2007, pp. 122-125) and (Brandimarte, 2002, pp. 154-159). Also based on (Jucker & de Faro, 1975), (Pardolos, Sandström, & Zopounidis, 1994, pp. 240-241) and (Jacobs, Levy, & Markowitz, 2005). Recall from sections 3.5.3 to 3.5.5 that the maximisation and minimisation of the objective functions subject to inequality constraints can only be solved through the use of quadratic programming. This programming method is called Kuhn-Tucker conditions (also known as

Karush–Kuhn–Tucker conditions). This basically restricts 𝜆𝑖 to positive values. If the maximum of 𝜆𝑖 is negative, then the feasible maximum of 𝜃 is at 𝜆𝑖 = 0 where 𝜕𝜃 𝜕𝜆𝑖 < 0.

Whereas if the maximum of 𝜆𝑖 is positive then the maximum of 𝜃 is at some 𝜆𝑖 > 0 where

𝜕𝜃 𝜕𝜆𝑖 = 0. Before we proceed further, we will define this formally. Consider the following problem: min 𝑓 𝐱

s. t. 𝑕𝑖 𝐱 = 0 𝑖 ∈ 𝐸 𝑔𝑖 𝐱 ≤ 0 𝑖 ∈ 𝐼,

where 𝑓, 𝑔𝑖 and 𝑕𝑖 are functions and 𝐱 is a vector. The stationary, which means variables are

constant over time, of 𝑓 has no role in finding the optimal solution but the Lagrangian

function ℒ does have one. Consider the function

ℒ 𝐱,𝝀,𝝁 = 𝑓 𝐱 + 𝜆𝑖𝑕𝑖 𝐱

𝑖∈𝐸

+ 𝜇𝑖𝑔𝑖 𝐱

𝑖∈𝐼

,

where 𝜆 and 𝜇 are Lagrangian multipliers. The stationary of the above does play a role in the Kuhn-Tucker conditions. THEOREM 1 (Kuhn-Tucker conditions) Assume that 𝑓 , 𝑕𝑖 and 𝑔𝑖 are continuously

differentiable, and that 𝒙∗ is feasible and satisfies a constraint qualification condition. Then a necessary condition for the local optimality of 𝒙∗ is that there exist numbers 𝜆𝑖

∗ for 𝑖 ∈ 𝐸 and

𝜇𝑖∗ ≥ 0 for 𝑖 ∈ 𝐼 such that

∇𝑓 𝐱∗ 𝜆𝑖

∗∇𝑕𝑖 𝐱∗

𝑖∈𝐸

+ 𝜇𝑖∗∇𝑔𝑖 𝐱

∗

𝑖∈𝐼

= 𝟎

and

𝜇𝑖

∗𝑔𝑖 𝐱∗ = 0 ∀𝑖 ∈ 𝐼.

All of these conditions has to be satisfied to ensure that the optimal solution is feasible. Back to where we were, we write

𝜕𝜃

𝜕𝜆𝑖≤ 0.

We make the above into an equality by writing

𝜕𝜃

𝜕𝜆𝑖+ 𝑈𝑖 = 0

where 𝑈𝑖 is a dummy variable. This is the first Kuhn-Tucker condition for a maximum. If the optimum value of 𝜃 exists when 𝜆𝑖 > 0 , then 𝜕𝜃 𝜕𝜆𝑖 = 0 and 𝑈𝑖 = 0 . In addition, if the

49

optimum exists at 𝜆𝑖 = 0, then 𝜕𝜃 𝜕𝜆𝑖 < 0 and 𝑈𝑖 > 0. The second, third and forth conditions are

𝜆𝑖𝑈𝑖 = 0

𝜆𝑖 ≥ 0 and

𝑈𝑖 ≥ 0. If the proportions of 𝜆𝑖 satisfy all Kuhn-Tucker conditions, then these give the optimum portfolio.

50

A.3. An example of calculating the Optimal Portfolio

This appendix is based heavily on (Howls & Fitt, 2009, pp. 36-37). Suppose that we have

three risky assets 𝑆1, 𝑆2 and 𝑆3 and a riskless asset 𝑆0 . The respective expected returns, standard deviations and correlations are

𝑅 0 = 0.05, 𝑅 1 = 0.14, 𝑅 2 = 0.08, 𝑅 3 = 0.2,

𝜎1 = 0.06, 𝜎2 = 0.03, 𝜎3 = 0.15 and

𝜌12 = 𝜌21 = 0.5, 𝜌13 = 𝜌31 = 0.2, 𝜌23 = 𝜌32 = 0.4. Short selling is allowed. In this case we

𝜃 =𝑅 𝑃 − 𝑅0

𝜎𝑃

subject to

𝜆𝑖

3

𝑖=1

= 1.

First, we calculate the 𝜎𝑖𝑗 to get

𝜎𝑖𝑗 =1

104

36 9 189 9 18

18 18 225 .

The portfolio variance 𝜎Π2 is

𝜎Π2 = 𝜆𝑖

2𝜎𝑖2

3

𝑖=1

+ 𝜆𝑖𝜆𝑗𝜎𝑖𝑗

3

𝑗=1𝑗≠𝑖

3

𝑖=1

=1

104 36𝜆1

2 + 18𝜆1𝜆2 + 36𝜆1𝜆3 + 9𝜆22 + 36𝜆2𝜆3 + 225𝜆3

2

so the portfolio standard deviation 𝜎Π is

𝜎Π =1

100 36𝜆1

2 + 18𝜆1𝜆2 + 36𝜆1𝜆3 + 9𝜆22 + 36𝜆2𝜆3 + 225𝜆3

2.

We also have

𝑅 𝑃 − 𝑅0 = 𝜆𝑖 𝑅 𝑖 − 𝑅0

𝑁

𝑖=1

=1

100 9𝜆1 + 3𝜆2 + 15𝜆3 .

and so

51

𝜃 = 9𝜆1 + 3𝜆2 + 15𝜆3

36𝜆12 + 18𝜆1𝜆2 + 36𝜆1𝜆3 + 9𝜆2

2 + 36𝜆2𝜆3 + 225𝜆32

.

We differentiate with respect to each 𝜆𝑖 to get 𝜕𝜃

𝑑𝜆1= −𝛾 36𝜆1 + 9𝜆2 + 18𝜆3 + 14 − 5 = 0

𝜕𝜃

𝑑𝜆2= −𝛾 9𝜆1 + 9𝜆2 + 18𝜆3 + 8 − 5 = 0

𝜕𝜃

𝑑𝜆3= −𝛾 18𝜆1 + 18𝜆2 + 225𝜆3 + 20 − 5 = 0

and rearranging in terms if 𝑅 𝑖 − 𝑅0 yields

𝛾 36𝜆1 + 9𝜆2 + 18𝜆3 = 9

𝛾 9𝜆1 + 9𝜆2 + 18𝜆3 = 3

𝛾 18𝜆1 + 18𝜆2 + 225𝜆3 = 15. Using the trick described in section 3.5.1 on the above, we obtain

36𝑍1 + 9𝑍2 + 18𝑍3 = 9

9𝑍1 + 9𝜆2 + 18𝜆3 = 3

18𝑍1 + 18𝑍2 + 225𝑍3 = 15. We simplify the system of equations to get

Then we solve for each 𝑍𝑖 . First, we multiply equation (26) by 3, subtract this from equation

(27) and solve for 𝑍1 yields

9𝑍1 = 2 ⇒ 𝑍1 =2

9.

We substitute 𝑍1 into, say, equation (26) to acquire

4 2

9 + 𝑍2 + 2𝑍3 = 1,

multiply this by 6 and subtract from equation (27) to obtain

18 2

9 + 63𝑍3 = −1.

Then we solve for 𝑍3 yields

𝑍3 =1

21.

Substitute 𝑍1 and 𝑍3 into equation (26) and rearrange in terms of 𝑍2 results in

4𝑍1 + 𝑍2 + 2𝑍3 = 1 (26)

3𝑍1 + 3𝑍2 + 6𝑍3 = 1 (27) 6𝑍1 + 6𝑍2 + 75𝑍3 = 5. (28)

52

4 2

9 + 𝑍2 + 2

1

21 = 1 ⇒ 𝑍2 =

1

63.

To summarise, we have

𝑍1 =2

9, 𝑍2 =

1

63, 𝑍3 =

1

21.

Note that 𝜆𝑖3𝑖=1 = 1 so 𝑍𝑖

3𝑖=1 = 2 7 . Using 𝜆𝑖 = 𝑍𝑖 𝑍𝑖

3𝑖=1 , we find that the allocations are

𝜆1 =14

18, 𝜆2 =

1

18, 𝜆3 =

3

18

which give the optimal portfolio. The portfolio expected return is 𝑅 Π = 𝜆1𝑅 1 + 𝜆2𝑅 2 + 𝜆3𝑅 3

= 14

14

18 + 8

1

18 + 20

3

18

=

11

75≈ 0.1466

and risk is

𝜎Π =1

100 𝜆1

2𝜎12 + 𝜆1𝜆2𝜎1𝜎2 + 𝜆1𝜆3𝜎1𝜎3 + 𝜆2

2𝜎22 + 𝜆2𝜆3𝜎2𝜎3 + 𝜆3

2𝜎32

=

1

100 36

14

18

2+ 18

14

18

1

18 + 36

14

18

3

18 + 9

1

18

2

+ 36 1

18

3

18 + 225

3

18

2

and this leads to

𝜎Π = 1218

600≈ 0.05816.

The market price of risk is 𝜃 = 29 2 609 ≈ 1.66 and the CML is given by

𝑅 Π =1

20+

29 2

609𝜎 ≈ 0.05 + 1.66𝜎.

53

A.4. Transforming Non-Positive Semi-Definite Covariance Matrices

This appendix is based on (Rousseeuw & Molenberghs, 1993, pp. 971-973). Before we explain the method, we will go through the concept of transforming a non-positive semi-definite correlation matrix into positive semi-definite one. Note that non-positive semi-definite correlation matrices occur due to the method the researcher collected the data.

Since the correlation matrix 𝚸 is symmetric and positive semi-definite we can write

𝚸 = 𝐎𝐃𝐎𝑇

where 𝐃 is the diagonal matrix containing the eigenvalues of 𝚸 and 𝐎 is the orthogonal matrix of matching eigenvectors. Suppose that the eigenvalues appear in decreasing order.

Let 𝑌 𝑖 be standardised versions of initial variables. Then an orthonormal set of eigenvectors

𝑍𝑗 corresponds to the linear combinations of 𝑌 𝑖 . These linear combinations are uncorrelated

and have decreasing variance. In the case of 𝚸 is positive definite, the eigenvalues are strictly positive.

When 𝐏 non-positive semi-definite, due to some eigenvalues are negative, but symmetric, the above expression still holds. Usually, these negative values will not have large absolute values. We can simply replace the negative eigenvalues with their absolute

values. Now we let 𝐃′ be diagonal matrix containing the absolute eigenvalues. Then we calculate the positive definite correlation matrix

𝚸′ = 𝐎𝐃′𝐎𝑇 . If the diagonal elements of 𝚸′ are not equal to 1, we transform 𝚸′ to

𝚸 = 𝐃1𝐑′𝐃1,

where 𝐃1 is the diagonal matrix with diagonal elements 1 𝑝𝑖𝑖′ .

In our case, we need to transform a non-positive semi-definite covariance matrix into positive semi-definite one by transforming the former into a correlation matrix by using the

cov2corr command. Then we use the above to calculate the positive definite correlation

matrix. Then we the corr2cov command to convert the matrix into positive semi-definite

covariance matrix. Then we check the covariance matrix by using the eig command to

ensure that it is positive definite. Surprisingly, we find that the matrix is positive semi-definite since there were no negative eigenvalues.

54

A.5. Experiment Instructions

A.5.1. Contrarian Investment Strategy

1) On the Digital Look website, click on ‘Visual Tools’ link under ‘Investor Toolbox’ menu.

Then click on ‘Heat Maps’ link.

Figure 11: Visual Tools webpage on Digital Look website. 'Heat Maps' link is highlighted by the underline.

Figure 12: Heat maps webpage on the Digital Look website.

55

2) Click on ‘Sector Averages’ in the ‘Sector’ drop-down menu. 3) Find the average values of sectors by selecting the following ratios; P/E, P/BV, P/CF and

Dividend Yield. The first three are found by clicking on ‘Valuation Ratios’. The last is under ‘Income Ratios’. Both under ‘Data Plot Category’ drop-down menu. Click on the relevant ratios in the ‘Data Plot’ drop-down menu. An example is shown in Figure 13.

Figure 13: Average P/E values of sectors defined by LSE on the Digital Look website.

4) Click on ‘Screening Tools’ link under ‘Investor Toolbox’ menu. Then click on ‘Screener’.

Figure 14: Screener webpage with ‘full fundamental screener’ link highlighted on the Digital Look website.

56

5) Click on ‘full fundamental screener’ link under ‘Getting Started’ (see Figure 14). Ensure that UK flag is displayed on the next screen.

6) Select the required sector under ‘Sector’ menu.

Figure 15: Full Screener webpage.

7) Enter the market values from the required sector into the ‘Max’ fields of the following ratios: P/E, P/BV and P/CF (This is labelled as ‘Share Price / Operating Cashflow per Share’). These are under Valuation Ratios section.

Figure 16: Example result of application of steps 1 to 10 for Contrarian Investment Strategy.

57

8) Enter the market value for dividend yield into the ‘Min’ field. This is under ‘Income Ratios’ section.

9) Enter 0.1 in the ‘Min’ fields of the following ratios to weed out zero and negative values: P/CF and P/S. Enter 1.5 in the ‘Min’ field of dividend cover.

10) Click on ‘Run Screen’ at the bottom of the webpage. An example is shown in Figure 16.

Figure 17: Heat map showing 12 companies with positive P/E values. The values are arranged in decreasing

order so the last two are in the two lowest quintiles.

Figure 18: Summary page of a company.

58

11) Use ‘Heat Maps’ to check if companies lie in the two lowest quintiles by value for P/E, P/BV and P/CF and the two highest quintiles for dividend yield. This is done by selecting the relevant sector the companies are listed under in LSE. Arrange the companies in value and tick the box to hide ‘n/a’ values. Each row has up to four companies. Count the rows with four companies and add the number of companies appearing on the last row with less than four appearing. For example, there are ten rows of four companies and one row of two. Therefore, there are 42. In this case there are 12 as shown in Figure 17.

Figure 19: Final results showing accounts for a company going back up to five years.

Figure 20: 'UK Shares' webpage showing indices. 'FTSE 100' is highlighted by the underline.

59

12) To check if the companies have current ratio of at least 1: a) Click on link of a company. This loads the summary page (see Figure 18). b) Click on ‘Financials’ tab on the company webpage. c) Click on ‘Fundamentals’. This loads the final results accounts for the company. This

is indicated by the highlight of the ‘Final Results’ tab in light blue (see Figure 19). d) Look for values for the current assets and current liabilities in the latest year and

calculate the current ratio. 13) To check if the companies’ forecast 1 year EPS is greater than that of the FTSE100:

a) Click on ‘UK Market’ link under ‘Select a Market’ menu. b) Click on ‘FTSE 100’ link (see Figure 20).

14) If all mandatory requirements are satisfied, execute the trade by using the portfolio features on the website. Otherwise repeat steps 4 to 14 until all mandatory requirements are satisfied.

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A.5.2. Value Investing

1) On the Digital Look website, click on ‘Screening Tools’ link under ‘Investor Toolbox’

menu. Then click on Screener. 2) Click on ‘full fundamental screener’ link under ‘Getting Started’. 3) Enter the relevant values in the ‘Min’ field for the following: 30 for revenue (in £m). 33 for

EPS growth (average) (in percent) and select 4 years in the drop down menu on the right hand side. The former is under ‘Profit/Loss Account’ and the latter is under ‘Growth Ratios’.

4) Enter the values in the ‘Max’ fields for the following: 1.5 for P/BV and 15 for P/E. 5) Click on ‘Run Screen’ at the bottom of the webpage. 6) Click on any company.

Figure 21: Example result of application of steps 1 to 5 for Value Investing.

7) To check if the company satisfies other criteria: a) Click on link of a company. This loads the summary page. b) Click on ‘Financials’ tab on the company webpage. c) Click on ‘Fundamentals’. This loads the final results accounts for the company. This

is indicated by the highlight of the ‘Final Results’ tab in light blue. d) For general and defensive strategies:

i) Check if there are dividends paid and no earnings loss for the past five years under.

ii) Current ratio is equal to at least 2. e) For defensive strategy:

i) For manufacturers, check if liabilities due in more than one year are less than net current assets.

ii) For utility companies, ensure that liabilities are less than double the net asset value.

f) For enterprising strategy check if: i) There are at least some dividends paid for the past five years under. ii) Current ratio is equal to at least 1.5. iii) For manufacturers, debt must be at most 110% of net current assets.

61

iv) Market capitalisation is less than 120% of net tangible assets. 8) If all mandatory requirements are satisfied, execute the trade by using the portfolio

features on the website. Otherwise repeat steps 6 and 7 until all mandatory requirements are satisfied.

62

A.5.3. Zulu Principle

1) On the Digital Look website, click on ‘Screening Tools’ link under ‘Investor Toolbox’

menu. Then click on Screener. 2) Click on ‘full fundamental screener’ link under ‘Getting Started’. 3) Enter values in the ‘Max’ fields for the following: 1 for PEG, 20 for P/E and 50% for net

gearing. 4) Enter values in the ‘Min’ fields for the following: 83.33 for RSI, 0 for P/S and P/BV. This is

because to easily check if the bonus criteria are satisfied of the latter two. 5) Click on ‘Run Screen’ at the bottom of the webpage. 6) Select any company.

Figure 22: Example of application of steps 1 to 5 for the Zulu Principle.

7) To check if the company satisfies other criteria: a) Click on link of a company. This loads the summary page. Market capitalisation can

be found on the page. b) Click on ‘Financials’ tab on the company webpage. c) Click on ‘Fundamentals’. This loads the final results accounts for the company. This

is indicated by the highlight of the ‘Final Results’ tab in light blue. d) Check if the cash flow per share greater than EPS for last reported year and average

for the last five years. e) Check if EPS increases every year for the past five years. f) On the screener results webpage, click on ‘Director Deals’ link to find out if a group of

directors have bought and sold shares in the past month. g) Look at the latest annual reports. These contain information regarding brand names,

industries the company operates in and other important information. 8) If all mandatory requirements are satisfied, execute the trade by using the portfolio

features on the website. Otherwise repeat steps 6 and 7 until all mandatory requirements are satisfied.

63

A.5.4. Recording trades

1) On the Digital Look website, click on ‘Portfolios & Watchlists’ under ‘Investor Toolbox’.

This loads the Portfolio Valuation screen. 2) Click on ‘Add to portfolio’. 3) Enter the name or ticker of the company. Select LSE in the ‘Stock Exchange’ drop-down

menu (see Figure 23).

Figure 23: 'Portfolio Selection' window with 'Portfolio Valuation' webpage in the background.

Figure 24: 'Trade Security' webpage to record transactions.

64

4) Click on ‘Search’. 5) Select the required company. 6) Click on ‘add shares’. 7) On the ‘Trade Security’ webpage (see Figure 24), enter the required number of shares

taking into account of the bid, or offer price, commission of £10, stamp duty and, if the value of shares alone is over £10,000, PTM levy of £1. If over £15,000, £1.50. Also enter the date of trade.

8) Select ‘Buy’ or ‘Sell’ in the drop-down menu under ‘Action’, depending on the type of transaction.

65

A.5.5. Executing trades

1) On the BullBearings website, click on ‘Portfolio summary’. 2) Click on ‘Fantasy Stocks’ (see figure 25).

Figure 25: 'Portfolio summary' webpage displaying types of portfolios with their values.

3) Click on ‘PLACE A TRADE’ link on the ‘Share Trading Portfolio’ webpage.

Figure 26: ‘Share Trading Portfolio’ webpage showing stocks held in the portfolio with 'PLACE A TRADE' link

above.

66

4) One the ‘Place a trade’ webpage, enter the company name or ticker in the relevant field and press enter (see Figure 27).

5)

Figure 27: Illustration of step 4.

6) Since the £100,000 is split into equal proportions, enter the cash amount equal to that in ‘Cash Amount’ field. For example, an investor has £100,000 to invest in 10 stocks in equal proportions so £10,000 would be spent on each stock. Select ‘At Best’ and click on ‘Next’ (see Figure 28).

Figure 28: 'Stock Trading' webpage illustrating an example of a trade to buy £10,000 in Royal Bank of Scotland

Group plc.

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A.6. Experiment Data

A.6.1. Data of companies matching relevant criteria for Investment Strategies

Table 9: Companies with the relevant ratios which satisfy at least all compulsory critera for Contrarian Investment

Strategy

Company Ticker

Company Ratios

P/E P/BV P/CF Dividend

Yield ROCE P/S ROA

Current

Ratio

Dividend

cover PEG

Beazley Group BEZ 4.9 0.78 0.6 7.22% 15.17% 0.37 2.15% 1.2 2.85

Elementis ELM 2.6 0.4 2.21 12.08% 18.34% 0.3 3.53% 1.94 3.14 0.1

GETECH Group GTC 7.8 7.65 4.21 7.65% 17.50% 1.2 9.17% 2 1.7 0.4

Inchcape INCH 1 0.2 43.15 0.84% 31.20% 0.03 6.72% 1.2 2.35 0.3

Kazakhmys KAZ 1.2 0.31 10.84 1.71% 35.82% 0.4 19.25% 6.05 7.41

KCOM Group KCOM 3.7 0.48 1.76 16.59% 10.87% 0.2 3.39% 1 1.62

Raymarine RAY 0.8 0.49 0.37 25.92% 40.80% 0.1 5.29% 1.93 4.9

Royal Dutch Shell

'A' RDSA 4.8 0.57 1.67 7.73% 29.70% 0.2 9.30% 1.1 2.7

SIG SHI 1.4 0.25 1.37 24.84% 30.76% 0.1 5.55% 2.72 2.8 0.1

Titan Europe TSW 0.7 0.03 0.3 50.00% 11.15% 0.03 3.11% 1.23 2.8 0.02

Vitec Group VTC 2.8 0.51 1.99 11.49% 20.89% 0.2 6.54% 1.41 3.05 0.1

Yell Group YELL 0.4 0.07 0.36 77.42% 71.28% 0.05 3.04% 1.31 3.14 0.08

All companies have P/E, P/BV, P/CF and dividend yield values which are below the sector counterparts (see Table 3) at the time of purchase. They are also in the two lowest quintiles for the first three ratios and the top two for dividend yield. Note that the 1 year forecase EPS growth rate of 39.7% for Titan Europe is greater than that of FTSE 100, which the rate is 19.7%.

Table 10: Sector average values for the Contrarian ratios

Ticker Sector in which the Company is listed in Sector Ratios

P/E P/BV P/CF Dividend Yield

BEZ Non-life Insurance 11.78 1.54 6.04 6.90%

ELM Chemicals 10.17 1.47 5.59 9.70%

GTC Oil Equipment & Services 15 1.57 7.95 5.10%

INCH General Retailers 11.12 1.62 3.5 15.70%

KAZ Mining 25.59 1.71 18.73 7.80%

KCOM Fixed Line Telecommunications 9.3 1.45 23.42 12.00%

RAY Electronic & Electrical Equipment 11.68 1.58 7 9.90%

RDSA Oil & Gas Producers 234.8 1.08 10.1 3.60%

SHI Support Services 8.49 2.23 2.6 8.20%

TSW General Industrials 11.79 0.62 29.75 14%

VTC Industrial Engineering 1083.89 1.16 3.54 10.30%

YELL Media 8.68 1.61 14.91 15.50%

68

Table 11: Companies with the relevant ratios which satisfy at least all compulsory criteria for Value Investing

Company Ticker P/E Average P/E

(4 Years) P/BV P/BV x 2

Current Ratio

Market Capitalisation

Accident Exchange Group

ACE 0.9 0.86 0.15 0.3 2.59 £0.1500

Aero Inventory

AI. 2.2 3.26 0.42 0.84 4.91 £1.5600

Blacks Leisure

BSLA 9.1 146 0.2 0.4 1.50 £0.3950

Bloomsbury

Publishing BMY 8.4 9.78 0.99 2.34 2.34 £1.3475

Delta DTLA 7.6 13.7 0.62 1.24 4.82 £1.0500

Jersey

Electricity 'A' JEL 10.5 10 0.55 1.1 2.80 £67.0000

Lonmin LMI 6.3 13.23 1.35 2.7 2.25 £1.6300

Man Group EMG 3.3 11.2 1.12 2.24 3.55 £2.4300

Ocean Wilsons

OCN 1.1 4.07 0.46 0.92 5.72 £5.4750

Market

Capitalisation x 1.2

Net Tangible Assets

Net Tangible

Assets x 1.1 Revenue (m)

Net Assets (m)

Total Liabilities (m)

Accident Exchange Group

ACE £0.1800 £0.6621 £0.7283 £114.58

Aero Inventory

AI. £1.8720 £3.1999 £3.5199 £305.56

Blacks Leisure

BSLA £0.4740 £1.0930 £1.2023 £294.41

Bloomsbury Publishing

BMY £1.6170 £1.1170 £1.2287 £150.00

Delta DTLA £1.2600 £1.6434 £1.8077 £266.40 £258.80 £63.20

Jersey

Electricity 'A' JEL £80.4000 £121.0818 £133.1900 82.00 £133.26 £39.24

Lonmin LMI £1.9560 £6.8063 £7.4869 £15.49

Man Group EMG £2.9160 £1.2905 £1.4196 £310.56

Ocean Wilsons

OCN £6.5700 £11.7410 £12.9151 £280.56

Note that the net assets and total liabilities only apply to Delta and Jersey Electricity, which the former is a manufacturer and the latter is a utility

69

Table 12: Criteria satisfied by companies for respective sub-strategies for Value Investing

Criteria numbers satisfied

Ticker General Defensive Enterprising

ACE All All All

AI. All All All

BSLA 1, 3, 4, 5, 6 1, 3, 4, 5, 6 All

BMY All All 1,2,3

DTLA All All All

JEL All All All

LMI All All All

EMG All All 1,2,3

OCN All All All

Table 13: Companies with the relevant ratios which satisfy at least all compulsory criteria for the Zulu Principle

1 2 year average EPS was used.

Company Ticker PEG P/E EPS Average EPS

(5 year) CF

per share RSI

Net Gearing

Cash Converters CCVU 0.3 5.8 6.28AU¢ 3.92AU¢ 13.12AU¢ 88.24% 12.25%

Dewhurst DWHT 0.1 4.8 38.92p 27.78p 92.56p 100% 22.85%

Jersey Electricity 'A' JEL 0.4 10.9 641p 373.92p 1359.34p 100% 25.53%

Prologic PGC 0.1 2.7 13.24p 8.8p1 24.6p 100% 34.74%

Visonic VSC 0.01 7 8.9US¢ 9.38US¢ 8.12US¢ 100% 17.99%

Zytronic ZYT 0.1 14.8 8.4p 5.02p 13.55p 90% 47.69%

Current Ratio

Quick Ratio

P/S Dividend

Yield P/BV P/R

Cash Converters CCVU 2.4 2.2 0.7 8.2% 0.7

Dewhurst DWHT 2.98 2.23 0.8 3.1% 0.3

Jersey Electricity 'A' JEL 2.8 2.31 0.94 3.3% 0.58

Prologic PGC 1.17 1.16 0.31 5.6% 0.32 3.17

Visonic VSC 2.4 1.7 0.3 3.4% 0.57 4.76

Zytronic ZYT 1.92 1.22 1.24 3.2% 2.05

70

Table 14: Information which satisfies the Zulu Principle

Table 15: Parts of the Zulu Principle satisfied by companies

Ticker Criteria numbers satisfied

Mandatory Highly Desirable Bonus

CCVU All 1, 4 1, 2

DWHT All 1, 4 1, 2, 4

JEL All 1, 2, 4 1

PGC All 1, 2, 4 All

VSC All 1, 4 1

ZYT All 1, 2, 4 2

Ticker Other Information

CCVU

Strong brand

Well established in pawn broking

Opening a new branch in Norwich

DWHT Specialist in input devices

New product launches

JEL Legal monopoly of electricity market in Jersey

PGC

Niche market – provides computer systems and consultancy to business in fashion and lifestyle sector

Directors bought shares on 31/3/2009

Stock taking time drops by 50% at FatFace after new system goes live

VSC

Strong brand in the security market

In possession of 75 unique patents

45 patents pending

ZYT

Niche market – specialists in touch screens, laminated products and processes.

In possession of patents for the above

Directors bought shares on 20/2/2009 and 30/3/2009.

71

A.6.2. Portfolio Transaction Costs

Table 16: Transaction costs for the Contrarian Portfolio

Buy

Date Ticker Quantity Price

Value excluding

Stamp

Duty

Stamp Duty

Value PTM Levy

Commission Total

20/03/2009 BEZ 15,341 £0.8825 £13,538.43 £67.69 £13,606.12 £1.00 £10.00 £13,617.12

09/03/2009 ELM 40,990 £0.2425 £9,940.08 £49.70 £9,989.78 £0.00 £10.00 £9,999.78

09/03/2009 GTC 53,731 £0.1850 £9,940.24 £49.70 £9,989.94 £0.00 £10.00 £9,999.94

09/03/2009 INCH 26,158 £0.3800 £9,940.04 £49.70 £9,989.74 £0.00 £10.00 £9,999.74

09/03/2009 KCOM 56,801 £0.1750 £9,940.18 £49.70 £9,989.88 £0.00 £10.00 £9,999.88

09/03/2009 KAZ 3,768 £2.6375 £9,938.10 £49.69 £9,987.79 £0.00 £10.00 £9,997.79

09/03/2009 RAY 77,963 £0.1275 £9,940.28 £49.70 £9,989.98 £0.00 £10.00 £9,999.98

09/03/2009 RDSA 657 £15.1100 £9,927.27 £49.64 £9,976.91 £0.00 £10.00 £9,986.91

09/03/2009 SHI 9,161 £1.0375 £9,504.54 £47.52 £9,552.06 £0.00 £10.00 £9,562.06

20/03/2009 TSW 104,147 £0.1300 £13,539.11 £67.70 £13,606.81 £1.00 £10.00 £13,617.81

09/03/2009 VTC 6,098 £1.6300 £9,939.74 £49.70 £9,989.44 £0.00 £10.00 £9,999.44

09/03/2009 YELL 65,182 £0.1525 £9,940.26 £49.70 £9,989.96 £0.00 £10.00 £9,999.96

Sell


Value

excluding Stamp Duty

Stamp Duty

Value PTM Levy

Commission Total

20/03/2009 BEZ 15,341 £0.8800 £13,500.08 £0.00 £13,500.08 £1.00 £10.00 £13,489.08

20/03/2009 ELM 40,990 £0.2500 £10,247.50 £0.00 £10,247.50 £1.00 £10.00 £10,236.50

20/03/2009 GTC 53,731 £0.1500 £8,059.65 £0.00 £8,059.65 £0.00 £10.00 £8,049.65

20/03/2009 INCH 26,158 £0.6775 £17,722.05 £0.00 £17,722.05 £1.00 £10.00 £17,710.55

20/03/2009 KCOM 56,801 £0.1700 £9,656.17 £0.00 £9,656.17 £0.00 £10.00 £9,646.17

20/03/2009 KAZ 3,768 £3.6800 £13,866.24 £0.00 £13,866.24 £1.00 £10.00 £13,855.24

20/03/2009 RAY 77,963 £0.1100 £8,575.93 £0.00 £8,575.93 £0.00 £10.00 £8,565.93

20/03/2009 RDSA 657 £15.6800 £10,301.76 £0.00 £10,301.76 £1.00 £10.00 £10,290.76

20/03/2009 SHI 9,161 £1.0375 £9,504.54 £0.00 £9,504.54 £0.00 £10.00 £9,494.54

20/03/2009 TSW 104,147 £0.14 £14,059.85 £0.00 £14,059.85 £1.00 £10.00 £14,048.85

20/03/2009 VTC 6,098 £1.5500 £9,451.90 £0.00 £9,451.90 £0.00 £10.00 £9,441.90

20/03/2009 YELL 65,182 £0.1375 £8,962.53 £0.00 £8,962.53 £0.00 £10.00 £8,952.53

Total Spent £126,780.39 Total Spent at start of holding period £99,545.46

Total Sold £133,781.68 Total Sold at end of holding period £106,576.60

72

Table 17: Transaction costs for the Value Investing Portfolio

Buy


Value


Stamp Duty

Value PTM Levy

Commission Total

23/03/2009 ACE 77,667 £0.16 £12,426.72 £62.13 £12,488.85 £1.00 £10.00 £12,499.85

23/03/2009 AI. 7,815 £1.59 £12,425.85 £62.13 £12,487.98 £1.00 £10.00 £12,498.98

23/03/2009 BSLA 31,460 £0.40 £12,426.70 £62.13 £12,488.83 £1.00 £10.00 £12,499.83

23/03/2009 BMY 9,205 £1.35 £12,426.75 £62.13 £12,488.88 £1.00 £10.00 £12,499.88

23/03/2009 JEL 172 £72.00 £12,384.00 £61.92 £12,445.92 £1.00 £10.00 £12,456.92

23/03/2009 LMI 813 £15.28 £12,422.64 £62.11 £12,484.75 £1.00 £10.00 £12,495.75

23/03/2009 EMG 5,773 £2.15 £12,426.38 £62.13 £12,488.51 £1.00 £10.00 £12,499.51

23/03/2009 OCN 2,239 £5.55 £12,426.45 £62.13 £12,488.58 £1.00 £10.00 £12,499.58

26/03/2009 DTLA 10,756 £1.08 £11,562.70 £57.81 £11,620.51 £1.00 £10.00 £11,631.51

Sell


Value excluding

Stamp

Duty

Stamp Duty

Value PTM Levy

Commission Total

26/03/2009 OCN 2239 £5.20 £11,642.80 £0.00 £11,642.80 £1.00 £10.00 £11,631.80

03/04/2009 ACE 77,667 £0.11 £8,155.04 £0.00 £8,155.04 £0.00 £10.00 £8,145.04

03/04/2009 AI. 7,815 £1.50 £11,722.50 £0.00 £11,722.50 £1.00 £10.00 £11,711.50

03/04/2009 BSLA 31,460 £0.33 £10,381.80 £0.00 £10,381.80 £1.00 £10.00 £10,370.80

03/04/2009 BMY 9,205 £1.20 £11,046.00 £0.00 £11,046.00 £1.00 £10.00 £11,035.00

03/04/2009 JEL 172 £65.00 £11,180.00 £0.00 £11,180.00 £1.00 £10.00 £11,169.00

03/04/2009 LMI 813 £14.86 £12,081.18 £0.00 £12,081.18 £1.00 £10.00 £12,070.18

03/04/2009 EMG 5,773 £2.47 £14,230.45 £0.00 £14,230.45 £1.00 £10.00 £14,219.45

03/04/2009 DTLA 10,756 £1.15 £12,369.40 £0.00 £12,369.40 £1.00 £10.00 £12,358.40



73

Table 18: Transaction costs for the Zulu Principle Portfolio

Buy


Value


Stamp Duty

Value PTM Levy

Commission Total

06/04/2009 CCVU 87,222 £0.19 £16,572.18 £82.86 £16,655.04 £1.50 £10.00 £16,666.54

06/04/2009 DWHT 8,722 £1.90 £16,571.80 £82.86 £16,654.66 £1.50 £10.00 £16,666.16

06/04/2009 JEL 220 £75.00 £16,500.00 £82.50 £16,582.50 £1.50 £10.00 £16,594.00

06/04/2009 PGC 41,430 £0.40 £16,572.00 £82.86 £16,654.86 £1.50 £10.00 £16,666.36

06/04/2009 VSC 36,827 £0.45 £16,572.15 £82.86 £16,655.01 £1.50 £10.00 £16,666.51

06/04/2009 ZYT 12,947 £1.28 £16,572.16 £82.86 £16,655.02 £1.50 £10.00 £16,666.52

Sell


Value


Stamp Duty

Value PTM Levy

Commission Total

17/04/2009 CCVU 87,222 £0.17 £14,827.74 £0.00 £14,827.74 £1.50 £10.00 £14,816.24

17/04/2009 DWHT 8,722 £1.82 £15,874.04 £0.00 £15,874.04 £1.50 £10.00 £15,862.54

17/04/2009 JEL 220 £65.00 £14,300.00 £0.00 £14,300.00 £1.50 £10.00 £14,288.50

17/04/2009 PGC 41,430 £0.32 £13,257.60 £0.00 £13,257.60 £1.50 £10.00 £13,246.10

17/04/2009 VSC 36,827 £0.42 £15,467.34 £0.00 £15,467.34 £1.50 £10.00 £15,455.84

17/04/2009 ZYT 12,947 £1.22 £15,795.34 £0.00 £15,795.34 £1.50 £10.00 £15,783.84



74

Table 19: Transaction costs for the Optimal Portfolio 1

Buy


Value

excluding Stamp

Duty

Stamp Duty

Value PTM Levy

Commission Total

09/03/2009 ABC 44 £5.6750 £249.70 £1.25 £250.95 £0.00 £10.00 £260.95

09/03/2009 AD. 327,127 £0.2300 £75,239.21 £376.20 £75,615.41 £1.50 £10.00 £75,626.91

09/03/2009 ADV 11 £1.4000 £15.40 £0.08 £15.48 £0.00 £10.00 £25.48

09/03/2009 AEC 671 £0.2000 £134.20 £0.67 £134.87 £0.00 £10.00 £144.87

09/03/2009 RGN 3,779,278 £0.0063 £23,809.45 £119.05 £23,928.50 £1.50 £10.00 £23,940.00

Sell


Value excluding

Stamp Duty

Stamp Duty

Value PTM Levy

Commission Total

20/03/2009 ABC 44 £5.7750 £254.10 £0.00 £254.10 £0.00 £10.00 £244.10

20/03/2009 AD. 327,127 £0.1800 £58,882.86 £0.00 £58,882.86 £1.50 £10.00 £58,871.36

20/03/2009 ADV 11 £1.1000 £12.10 £0.00 £12.10 £0.00 £10.00 £2.10

20/03/2009 AEC 671 £0.1700 £114.07 £0.00 £114.07 £0.00 £10.00 £104.07

20/03/2009 RGN 3,779,278 £0.0023 £8,692.34 £0.00 £8,692.34 £1.50 £10.00 £8,680.84



75


Buy


Value excluding

Stamp Duty

Stamp

Duty Value

PTM

Levy Commission Total

23/03/2009 ABC 79 £6.4000 £505.60 £2.53 £508.13 £0.00 £10.00 £518.13

23/03/2009 ACC 42,454 £0.0263 £1,114.42 £5.57 £1,120.00 £0.00 £10.00 £1,129.99

23/03/2009 AD. 395,178 £0.2300 £90,890.94 £454.45 £91,345.39 £1.50 £10.00 £91,356.89

23/03/2009 ASW 7,342 £0.2175 £1,596.89 £7.98 £1,604.87 £0.00 £10.00 £1,614.87

23/03/2009 AMS 10,652 £0.3275 £3,488.53 £17.44 £3,505.97 £0.00 £10.00 £3,515.97

23/03/2009 ADV 1,316 £1.4000 £1,842.40 £9.21 £1,851.61 £0.00 £10.00 £1,861.61

Sell


Value excluding

Stamp Duty

Stamp

Duty Value

PTM

Levy Commission Total

03/04/2009 ABC 79 £5.5900 £441.61 £0.00 £441.61 £0.00 £10.00 £431.61

03/04/2009 ACC 42,454 £0.0238 £1,008.28 £0.00 £1,008.28 £0.00 £10.00 £998.28

03/04/2009 AD. 395,178 £0.1800 £71,132.04 £0.00 £71,132.04 £1.50 £10.00 £71,120.54

03/04/2009 ASW 7,342 £0.2025 £1,486.76 £0.00 £1,486.76 £0.00 £10.00 £1,476.76

03/04/2009 AMS 10,652 £0.3250 £3,461.90 £1.00 £3,462.90 £0.00 £10.00 £3,452.90

03/04/2009 ADV 1,316 £1.2750 £1,677.90 £0.00 £1,677.90 £0.00 £10.00 £1,667.90



76


Buy


Value


Stamp Duty

Value PTM Levy

Commission Total

06/04/2009 AD. 383,848 £0.2300 £88,285.04 £441.43 £88,726.47 £1.50 £10.00 £88,737.97

06/04/2009 ADM 44 £8.8750 £390.50 £1.95 £392.45 £0.00 £10.00 £402.45

06/04/2009 ASW 9,518 £0.2325 £2,212.94 £11.06 £2,224.00 £0.00 £10.00 £2,234.00

06/04/2009 ADV 1,608 £1.5750 £2,532.60 £12.66 £2,545.26 £0.00 £10.00 £2,555.26

06/04/2009 AEC 8,079 £0.2000 £1,615.80 £8.08 £1,623.88 £0.00 £10.00 £1,633.88

06/04/2009 ABM 1,217 £2.0400 £2,482.68 £12.41 £2,495.09 £0.00 £10.00 £2,505.09

06/04/2009 ADS 212,146 £0.0080 £1,686.56 £8.43 £1,694.99 £0.00 £10.00 £1,704.99

06/04/2009 ANR 5,510 £0.0372 £204.97 £1.02 £206.00 £0.00 £10.00 £216.00

Sell


Value excluding

Stamp

Duty

Stamp Duty

Value PTM Levy

Commission Total

17/04/2009 AD. 383,848 £0.1700 £65,254.16 £326.27 £65,580.43 £1.50 £10.00 £65,568.93

17/04/2009 ADM 44 £9.1650 £403.26 £2.02 £405.28 £0.00 £10.00 £395.28

17/04/2009 ASW 9,518 £0.2350 £2,236.73 £11.18 £2,247.91 £0.00 £10.00 £2,237.91

17/04/2009 ADV 1,608 £1.2500 £2,010.00 £10.05 £2,020.05 £0.00 £10.00 £2,010.05

17/04/2009 AEC 8,079 £0.1600 £1,292.64 £6.46 £1,299.10 £0.00 £10.00 £1,289.10

17/04/2009 ABM 1,217 £1.8900 £2,300.13 £11.50 £2,311.63 £0.00 £10.00 £2,301.63

17/04/2009 ADS 212,146 £0.0045 £944.05 £4.72 £948.77 £0.00 £10.00 £938.77

17/04/2009 ANR 5,510 £0.0352 £193.95 £0.97 £194.92 £0.00 £10.00 £184.92



77

A.6.3. Portfolio Valuation at the end of Holding Period

Table 22: Portfolio valuation of Contrarian Portfolio at the end of the holding period before selling

Security Name Ticker Quantity Market Price Holding Value Profit / Loss

Beazley Group BEZ 15,341 £0.8800 £13,500.08 -£38.35

Elementis ELM 40,990 £0.2500 £10,247.50 £307.42

GETECH Group GTC 53,731 £0.1650 £8,865.61 -£1,074.62

KCOM Group KCOM 56,801 £0.1725 £9,798.17 -£142.00

Kazakhmys KAZ 3,768 £3.6850 £13,885.08 £3,946.98

Raymarine RAY 77,963 £0.1125 £8,770.84 -£1,169.44

Royal Dutch Shell 'A' RDSA 657 £15.6800 £10,301.76 £374.49

Titan Europe TSW 104,147 £0.1350 £14,059.85 £520.74

Vitec Group VTC 6,098 £1.5500 £9,451.90 -£487.84

Yell Group YELL 65,182 £0.1375 £8,962.53 -£977.73

Totals

£107,843.32 £1,259.65

Table 23: Portfolio valuation of Value Investing Portfolio at the end of the holding period before selling


Accident Exchange Group ACE 77,667 £0.1175 £9,125.87 -£3,300.85

Aero Inventory AI. 7,815 £1.5375 £12,015.56 -£410.29

Blacks Leisure BSLA 31,460 £0.3450 £10,853.70 -£1,573.00

Bloomsbury Publishing BMY 9,205 £1.2000 £11,046.00 -£1,380.75

Delta DTLA 10,756 £1.1600 £12,476.96 £914.26

Jersey Electricity 'A' JEL 172 £70.0000 £12,040.00 -£344.00

Lonmin LMI 813 £14.8500 £12,073.05 -£349.59

Man Group EMG 5,773 £2.4675 £14,244.88 £1,818.50

Totals

£93,876.02 -£4,625.72

Table 24: Portfolio valuation of Zulu Principle Portfolio at the end of the holding period before selling


Cash Converters CCVU 87,225 £0.1800 £15,700.50 -£872.25

Dewhurst DWHT 8,722 £1.8600 £16,222.92 -£348.88

Jersey Electricity 'A... JEL 220 £70.0000 £15,400.00 -£1,100.00

Prologic PGC 41,432 £0.3600 £14,915.52 -£1,657.28

Visonic VSC 36,828 £0.4350 £16,020.18 -£552.42

Zytronic ZYT 12,947 £1.2700 £16,442.69 -£129.47

Totals

£94,701.81 -£4,660.30

78

Table 25: Portfolio valuation of Optimal Portfolio 1 at the end of the holding period before selling


Abcam ABC 44 £6.0750 £267.30 £6.35

ADL AD. 327,127 £0.2050 £67,061.04 -£8,565.87

Advent Capital (Holdings) ADV 11 £1.2500 £13.75 -£11.73

AEC Education AEC 671 £0.1850 £124.14 -£20.74

Regenesis Group RGN 3,779,278 £0.0043 £16,250.90 -£7,689.10

Totals £83,717.12 -£16,281.09



Abcam ABC 79 £5.8900 £465.31 -£52.82

Access Intelligence ACC 424,543 £0.0025 £1,061.36 -£68.64

ADL AD. 395,178 £0.2050 £81,011.49 -£10,345.40

Advanced Computer Software ASW 7,342 £0.2150 £1,578.53 -£36.34

Advanced Medical Solutions Group AMS 10,652 £0.3625 £3,861.35 £345.38

Advent Capital (Holdings) ADV 1,316 £1.4250 £1,875.30 £13.69

Totals £89,853.34 -£10,144.14



ADL AD. 383,848 £0.1950 £74,850.36 -£13,887.61

Admiral Group ADM 44 £9.1700 £403.48 £1.03

Advanced Computer Software ASW 9,518 £0.2475 £2,355.71 £121.71

Advent Capital (Holdings) ADV 1,608 £1.4000 £2,251.20 -£304.06

AEC Education AEC 8,079 £0.1750 £1,413.83 -£220.05

Albemarle and Bond Holdings ABM 1,217 £1.9200 £2,336.64 -£168.45

Alexander David Securities GRP ADS 212,146 £0.0062 £1,315.31 -£389.69

Altona Energy ANR 5,510 £0.0362 £199.46 -£16.53

Totals £85,125.98 -£14,863.67

79

A.6.4. Market, Bid and Offer prices

Table 28: Market, Bid and Offer prices at the time of sale of stocks for the Contrarian Portfolio

Name Ticker Bid Mid Offer Spread

Beazley Group BEZ £0.8800 £0.8800 £0.8825 £0.0025

Elementis ELM £0.2500 £0.2500 £0.2525 £0.0025

GETECH Group GTC £0.1500 £0.1650 £0.1700 £0.0200

Inchcape INCH £0.6775 £0.6775 £0.6800 £0.0025

Kazakhmys KCOM £0.1700 £0.1725 £0.1800 £0.0100

KCOM Group KAZ £3.6800 £3.6850 £3.6875 £0.0075

Raymarine RAY £0.1100 £0.1125 £0.1150 £0.0050

Royal Dutch Shell 'A' RDSA £15.6800 £15.6800 £15.6900 £0.0100

SIG SHI £1.0375 £1.0400 £1.0500 £0.0125

Titan Europe TSW £0.1350 £0.1350 £0.1550 £0.0200

Vitec Group VTC £1.5500 £1.5500 £1.5750 £0.0250

Yell Group YELL £0.1375 £0.1375 £0.1400 £0.0025

Table 29: Market, Bid and Offer prices at the time of sale of stocks for the Value Investing Portfolio


Accident Exchange Group ACE £0.1050 £0.1175 £0.1350 £0.0300

Aero Inventory AI. £1.5000 £1.5375 £1.5450 £0.0150

Blacks Leisure BSLA £0.3300 £0.3450 £0.3650 £0.0350

Bloomsbury Publishing BMY £1.2000 £1.2000 £1.2250 £0.0250

Delta DTLA £1.1500 £1.1600 £1.1525 £0.0025

Jersey Electricity 'A' JEL £65.0000 £70.0000 £75.0000 £10.0000

Lonmin LMI £14.8600 £14.8500 £14.8700 £0.0100

Man Group EMG £2.4650 £2.4675 £2.4625 £0.0025

Ocean Wilsons OCN £5.2000 £5.2750 £5.3500 £0.1500

Table 30: Market, Bid and Offer prices at the time of sale of stocks for the Zulu Principle Portfolio


Cash Converters CCVU £0.1750 £0.1800 £0.1900 £0.0250

Dewhurst DWHT £1.8150 £1.8600 £1.9050 £0.0900

Jersey Electricity 'A' JEL £65.0000 £70.0000 £75.0000 £10.0000

Prologic PGC £0.3200 £0.3600 £0.4000 £0.0800

Visonic VSC £0.4200 £0.4350 £0.4400 £0.0200

Zytronic ZYT £1.2550 £1.2700 £1.2850 £0.0300

80



Abcam ABC £5.7750 £6.0750 £6.1700 £0.3950

ADL AD. £0.1800 £0.2050 £0.2300 £0.0500

Advent Capital ADV £1.1000 £1.2500 £1.4000 £0.3000

AEC Education AEC £0.1700 £0.1850 £0.2000 £0.0300

Regenesis Group RGN £0.0023 £0.0043 £0.0630 £0.0040



Abcam ABC £5.5900 £5.8900 £5.9850 £0.3950

Access Intelligence ACC £0.0024 £0.0025 £0.0099 £0.0075

ADL AD. £0.1800 £0.2050 £0.2300 £0.0500

Advanced Computer Software ASW £0.2025 £0.2150 £0.2325 £0.0300

Advanced Medical Solutions AMS £0.3250 £0.3625 £0.3350 £0.0100




ADL AD. £0.1700 £0.1950 £0.2200 £0.0500

Admiral Group ADM £9.1650 £9.1700 £9.1800 £0.0150

Advanced Computer Software ASW £0.2350 £0.2475 £0.2650 £0.0300


AEC Education AEC £0.1600 £0.1750 £0.1900 £0.0300

Albemarle & Bond ABM £1.8900 £1.9200 £1.9500 £0.0600

Alexander David Securities Group ADS £0.0045 £0.0062 £0.0080 £0.0035

Altona Energy ANR £0.0352 £0.0362 £0.0372 £0.0020

81

A.6.5. Stocks Returns from Portfolios

Table 34: Stock Returns after transaction costs for the Contrarian Portfolio

Company Name Ticker Profit/Loss 𝑹𝒊 (%)

Beazley Group BEZ -£128.04 -0.94%

Elementis ELM £236.72 2.37%

GETECH Group GTC -£1,950.29 -19.50%

Inchcape INCH £7,710.80 77.11%

KCOM Group KCOM -£353.71 -3.54%

Kazakhmys KAZ £3,857.45 38.58%

Raymarine RAY -£1,434.05 -14.34%

Royal Dutch Shell 'A' RDSA £303.85 3.04%

SIG SHI -£67.52 -0.71%

Titan Europe TSW £431.04 3.17%

Vitec Group VTC -£557.54 -5.58%

Yell Group YELL -£1,047.43 -10.47%

Table 35: Stock Returns after transaction costs for the Value Investing Portfolio


Accident Exchange Group ACE -£4,354.82 -34.84%

Aero Inventory AI. -£787.48 -6.30%

Blacks Leisure BSLA -£2,129.03 -17.03%

Bloomsbury Publishing BMY -£1,464.88 -11.72%

Delta DTLA £726.89 6.25%

Jersey Electricity 'A' JEL -£1,287.92 -10.34%

Lonmin LMI -£425.57 -3.41%

Man Group EMG £1,719.93 13.76%

Ocean Wilsons OCN -£867.78 -6.94%

Table 36: Stock Returns after transaction costs for the Zulu Principle Portfolio


Cash Converters CCVU -£1,850.30 -11.10%

Dewhurst DWHT -£803.62 -4.82%

Jersey Electricity 'A' JEL -£2,305.50 -13.89%

Prologic PGC -£3,420.26 -20.52%

Visonic VSC -£1,210.67 -7.26%

Zytronic ZYT -£882.68 -5.30%

82

Table 37: Stock Returns after transaction costs for Optimal Portfolio 1


Abcam ABC -£16.85 -6.46%

ADL AD. -£16,755.55 -22.16%

Advent Capital (Holdings) ADV -£23.38 -91.76%

AEC Education AEC -£40.80 -28.16%

Regenesis Group RGN -£15,259.16 -63.74%



Abcam ABC -£87.52 -16.89%

Access Intelligence ACC -£132.71 -11.74%

ADL AD. -£20,236.35 -22.15%

Advanced Computer Software ASW -£138.11 -8.55%

Advanced Medical Solutions Group AMS -£63.07 -1.79%




ADL AD. -£23,169.03 -26.11%

Admiral Group ADM -£7.18 -1.78%

Advanced Computer Software ASW £3.91 0.18%


AEC Education AEC -£344.78 -21.10%

Albemarle and Bond Holdings ABM -£203.46 -8.12%

Alexander David Securities GRP ADS -£766.22 -44.94%

Altona Energy ANR -£31.08 -14.39%

83

A.7. MATLAB codes

function [R, ER, COV] = Statdata_and_find_eigenvalues(R);

% Need to transpose returns data in order to get suitable solutions. This

% is because we downloaded the data of stock prices with the weekly prices

% in columns and stock in rows. Also If we try to download data with the

stocks in columns,

% MS Excel 2003 cannot fit 1914 stocks.

R = transpose(R);

% The following calculate the mean and variance from observed data, that is

data that are not missing.

ER = nanmean(R);

COV = nancov(R);

% Decomposes the matrix into a vector of eigenvalues. This is useful for

% checking if the covariance matrix is positive semi-definite or not.

Eig_of_COV = eig(COV);

% Convert the covariance matrix into correlation matrix

[ExpSigma,ExpCorrC] = cov2corr(COV);

% Decompose correlation matrix into matrix of eigenvectors and diagonal

% matrix of eigenvalues.

[V,D] = eig(ExpCorrC);

D = diag(D); % Find the vector of eigenvalues so that we can replace

negative eigenvalues with their absolute values. This because we do not

have sufficient experience of MATLAB to write a code which can do this

automatically.

end

Figure 29: MATLAB code for Statdata_and_find_eigenvalues.m.

84

function [COVa, Eig_of_COVa] = gettranscovmatrix(D, V, ExpCorrC);

D = diag(diag(diag(D))); % Creates diagonal matrix from vector of

eigenvalues (after replacing the negative ones with their absolute values

manually).

% The following computes the new correlation matrix and transforms it into

% covariance matrix.

ExpCorrC = V*D*transpose(V);

Eig_of_ExpCorrC = eig(ExpCorrC); % We need to check manually if the matrix

is positive semi-definite or not.

COVa = corr2cov(ExpSigma, ExpCorrC); % Converts correlation matrix into

covariance matrix

% We need to check manually if the covariance matrix is positive semi-

definite or not.

Eig_of_COVa = eig(COVa);

clear Eig_of_COV D D1 Eig_of_ExpCorrC Eig_of_COVa V R % Clears all

unnecessary variables.

end

Figure 30: MATLAB code for Gettranscovmatrix.m.

function optimalportolio(ER, COVa);

% Define variables and specify the number of portfolios

ExpReturn = ER;

ExpCovariance = COVa;

NumPorts = 20;

% Compute efficient frontier.

[PortRisk, PortReturn, PortWts] = portopt(ExpReturn, ExpCovariance,

NumPorts);

% Calculate optimal portfolio with riskless rate (that is the LIBOR in our

% case) and without short sales.

RisklessRate = 0.007275;

BorrowRate = 0.007275;

[RiskyRisk, RiskyReturn, RiskyWts, RiskyFraction, OverallRisk,

OverallReturn] = portalloc(PortRisk, PortReturn,PortWts, RisklessRate,

BorrowRate);

portalloc(PortRisk, PortReturn,PortWts, RisklessRate, BorrowRate); %

Displays efficient frontier and optimal portfolio point.

end

Figure 31: MATLAB code for Optimal_Portfolio.m.

85

A.8. Glossary of Terms

Asset – property which the company owns. Tangible assets are those that have a physical substance, such as buildings, equipment and workers, whereas intangible assets are those which do not exist, such as brand names and copyrights. Book value – valuation of the company if all of its assets are sold. Includes tangible and intangible assets. Capital Employed – measures the amount of money used to operate the company. Cash flow – measures the amount of cash goes in and out of the company.

Current assets – assets of the company which is expected to be sold or otherwise used up

in the near future, say within one year. Current liabilities – debts need to be paid off by the end of the company year. Current ratio – measures the ability of the company to pay off its debts. The formula is

current ratio = Current assets ⁄ Current liabilities. Dividend – payment from the net profits of the company. The investors are basically the

owners of the company so they are entitled to take profits from a company. Dividend cover – measures the security of the dividend. The formula is dividend cover =

Net profits ⁄ Dividends = EPS ⁄ DPS. Dividend per share (DPS) – The formula is DPS = Dividends ⁄ Number of shares in issue.

Dividend yield – percentage return from net profits per share, for example, owning 100 shares at £1 and a dividend of 5p per share is paid so the yield is 5%. Earnings per share (EPS) – The formula is EPS = Net profits ⁄ Number of shares in issue.

Gearing – the amount of loan the company has taken out. It is a percentage of net assets.

This can either magnify profits and losses. Index – a measure of performance of a selection of stock in the stock market. Examples are

the FTSE 100, FTSE 250, FTSE AIM 50 and Dow Jones Industrial Average. Market Capitalisation – notional value of the company listed on the stock market.

Operating Profit – profit after costs from ongoing operations.

Price to book value (P/BV) ratio – measures the value of assets of the company relative to

the market value of the company. The formula is P/BV = Share price ⁄ Book value per share. Price to cash flow (P/CF) ratio – measures the price of the volume cash flow per share at

market value. The formula is P/CF = Share price ⁄ Cash flow per share. Price to earnings (P/E) ratio – measures of the number of year it would take the company

to earn net profits equal to the market value of the company. The formula is P/E = Share price ⁄ EPS.

86

Price to earnings growth (PEG) ratio – measures the earnings growth in terms of P/E ratio. The formula is PEG = (P/E)⁄ EPS Growth. Price to research (P/R) ratio – measures the price of total expenditure of research and

development per share. The formula is P/R = Market Capitalisation ⁄ Turnover = Share price ⁄ Research and Development (R&D) expenditure per share Price to sales (P/S) ratio – measures the price of turnover per share. The formula is P/S = Market Capitalisation / Turnover = Share price ⁄ Turnover per share. Price to tangible book value (P/TBV) ratio – same as P/BV but only includes tangible assets. The formula is P/TBV = Share price / Tangible book value per share. Relative strength – measures the price trend of a stock which indicates its performing

relative to other stocks in its industry. Return on Assets (ROA) – measure profit the company's assets in generating turnover.

Return on Capital Employed (ROCE) – measures the profit the company makes for its

capital. Sector – list of a business classification on the stock market. Each stock market listed

company is allocated a relevant sector, for example, Barclays in ‘Banks’, QinetiQ in ‘Aerospace and Defence’ and Sainsburys in ‘General Retailers’. Turnover – the total of sales in one time period.

Documents

Does the Optimal Portfolio Deliver the Best Performance in the Stock Market