business report 8.1

xxxxxxxxxxxxxxxxxxxxxxxxWord Count 12,500Modelling Risk Reduction in Equity Portfoliosusing Exchange-Traded Funds with MultipleVariance-Covariance Matrices

1 Acknowledgements My appreciations to xxxxxx as my dissertation supervisor for his advice and guidance throughout this dissertation. I would also like to thank my girlfriend xxxxxx and my family for their support and encouragement.

2 Abstract This research paper investigates two core pillars of Modern Portfolio Theory literature; Harry Markowitz’s seminal paper on ‘Portfolio Selection’ in 1952 and William Sharpe’s 1963 and 1964 papers that created the Capital Asset Pricing Model. These papers compute the historical portfolio risks and returns to produce the expected portfolio risk and return values. We wanted this research to have a strong literature underpinning, therefore we kept both these models in our research, as well as expanding with alternative Shrinkage variance-covariance (VCV) matrices from Ledoit and Wolf (2004). This research paper uses 25 S&P 500 index stocks with four exchange-traded funds (ETFs) to build 12 Efficient Frontiers to address four research objectives using 10 hypotheses. This was conducted to test whether the addition of ETFs in equity portfolios lowered expected portfolio risk using historical data. We state that there are two main variables in modelling expected portfolio risk and return; Firstly, the length of the data set, as risks and returns vary over time. Secondly, the type of VCV matrix that is used in the portfolio optimisation process, we test four VCV matrices in this research to challenge this statement. Our results show that portfolio risk can be lowered without sacrificing portfolio return when including U.S. fixed income (AGG) and gold bullion (GLD) exchange-traded funds in equity portfolios. When comparing long-short equity only portfolios with long-short equity portfolios including ETFs portfolio risk is reduced by 44% using the Markowitz (1952) VCV matrix and 42% using the Single Index Model VCV matrix, (Sharpe, 1963). This research shows that the Sharpe ratio (1993) increases when using AGG and GLD exchange-traded funds in long-short equity portfolios. Thus, demonstrating that the inclusion of ETFs in equity portfolios create higher risk adjusted returns. We state that the most efficient method to achieve a higher optimised Global Minimum Variance Portfolio (GMVP) is to use the Tangency Portfolio of long-short equity portfolios that include AGG and GLD ETFs coupled with a weighting in the risk-free rate. This research illustrates that Long-Short equity portfolios with AGG and GLD ETFs have a lower portfolio risk of 72% and a higher portfolio return of 33% when compared to the S&P 500 equity index (SPX) when using the Markowitz’s VCV matrix and the past ten years of trading data. To conclude, we outline the strengths and limitations of the research and finally, we state recommendations for future research extensions such as; using Merton’s (1973) quantitative technique for depicting the Efficient Frontier and research replications using a larger sample size of exchange-traded funds.

3 List of Figures 1. Figure 1: Map of Research Structure 2. Figure 2: Edited from Solnick (1974) Showing Portfolio Domestic Diversification Lowering Specific Risk and Portfolio Internationalisation Lowering Systematic Risk (Edited from Solnick, 1974) List of Tables 1. Table 1: Base Portfolio – 25 mega-cap U.S. S&P 500 stocks (Bloomberg, 2017) 2. Table 2: ETFs to be included into Portfolios 4, 5, 7, 8, 9, 10, 11 and 12 3. Table 3: List of Portfolios 1 to 12 4. Table 4: Markowitz Variance-Covariance Matrix 5. Table 5: Single Index Model (SIM) Variance-Covariance (VCV) Matrix 6. Table 6: Shrinkage Variance-Covariance (VCV) Matrix 7. Table 7: List of ETF Weightings for the GMVP of Portfolios 4,5,7,8,9,10,11 &12 8. Table 8: Sharpe Ratio of SPX and OEX 9. Table 9: Comparing the Tangency Portfolio of Portfolio 5 with a combination of Tangency Portfolio of Portfolio 3 and the Risk-Free Rate using the CAL 10. Table 10: Comparing the Minimum Variance on the GMVP of Portfolio 5 with a combination of Tangency Portfolio of Portfolio 3 and the Risk-Free Rate using the CAL 11. Table 11: Comparing the GMVP for Portfolio 5 to SPX, OEX and SML Equity Indexes List of Formulas 1. Formula 1: Number of Covariances in a matrix (Bodie, et al., 2014) 2. Formula 2: Covariance of Assets (Wilmott, 2007) 3. Formula 3: Capital Asset Pricing Model (CAPM) (Sharpe, 1963) 4. Formula 4: Portfolio’s Variance using CAPM (Bodie, et al., 2014) 5. Formula 5: SIM VCV Matrix VBA Formula (Benninga, 2014) 6. Formula 6: Shrinkage Variance-Covariance Matrix (Ledoit and Wolf, 20042) 7. Formula 7: Markowitz Variance Covariance Matrix Microsoft Excel Formula (Benninga, 2014) 8. Formula 8: Diagonal Variance Matrix Microsoft Excel Formula (Benninga, 2014) 9. Formula 9: Shrinkage Variance Covariance Matrix Microsoft Excel Formula (Benninga, 2014) 10. Formula 10: Expected Portfolio Return (Kennedy, 2010) 11. Formula 11: Expected Portfolio Return Microsoft Excel Formula (Benninga, 2014) 12. Formula 12: Variance of an asset (Kennedy, 2010) 13. Formula 13: Expected Portfolio Risk (Kennedy, 2010) 14. Formula 14: Microsoft Excel Formula (Benninga, 2014) 15. Formula 15: Sharpe Ratio (Sharpe, 1994) 16. Formula 16: Portfolio Return between Tangency Portfolio and a Risk-Free Rate (Bodie, et al., 2014) 17. Formula 17: Portfolio Risk between Tangency Portfolio and a Risk-Free Rate (Bodie, et al., 2014) List of Charts 1. Chart 1: Weighting of Sectors in the Base Portfolio (Bloomberg, 2017) 2. Chart 2: Markowitz Bullet and the Efficient Frontier 3. Chart 3: Portfolio 1 – Equal Weight Long (Markowitz VCV Matrix) Portfolio Return and Risk 4. Chart 4: Portfolio 1 –Equal Weight Long Portfolio (Markowitz VCV Matrix) Asset Weightings 5. Chart 5: Portfolio 2 – Efficient Frontier for Weighted Long-Only (Markowitz VCV Matrix) 6. Chart 6: Portfolio 2 – Weighted Long-Only (Markowitz VCV Matrix) GMVP Asset Weightings 7. Chart 7: Portfolio 3 – Efficient Frontier for Long-Short (Markowitz VCV Matrix) 8. Chart 8: Portfolio 3 – Long-Short (Markowitz VCV Matrix) GMVP Asset Weightings 9. Chart 9: Portfolio 4 – Efficient Frontier for Long-Short including ETFs (Markowitz VCV Matrix) 10. Chart 10: Portfolio 4 – Long-Short including ETFs (Markowitz 11. Chart 11: Portfolio 5 – Efficient Frontier for Overweight Bond ETF 60% (Markowitz VCV Matrix) 12. Chart 12: Portfolio 5 – Overweight Bond ETF 60% (Markowitz VCV Matrix) GMVP Asset Weightings 13. Chart 13: Portfolio 6 – Efficient Frontier for Long-Short (SIM VCV Matrix) 14. Chart 14: Portfolio 6 – Long-Short (SIM VCV Matrix) GMVP Asset Weightings 15. Chart 15: Portfolio 7 – Efficient Frontier for Long-Short including ETFs (SIM VCV Matrix) 16. Chart 16: Portfolio 7 – Long-Short including ETFs (SIM VCV Matrix) GMVP Asset Weightings 17. Chart 17: Portfolio 8 – Efficient Frontier for Overweight Bond ETF 60% (SIM VCV Matrix) 18. Chart 18: Portfolio 8 – Overweight Bond ETF 60% (SIM VCV Matrix) GMVP Asset Weightings 19. Chart 19: Portfolio 9 – Efficient Frontier for Long-Short including ETFs (Shrinkage Markowitz VCV Matrix) 20. Chart 20: Portfolio 9 – Long-Short including ETFs (Shrinkage Markowitz VCV Matrix) GMVP Asset Weightings 21. Chart 21: Portfolio 10 – Efficient Frontier for Overweight Bond ETF 60% (Shrinkage Markowitz VCV Matrix) 22. Chart 22: Portfolio 10 – Overweight Bond ETF 60% (Shrinkage Markowitz VCV Matrix) GMVP Asset Weightings 23. Chart 23: Portfolio 11 – Efficient Frontier for Long-Short including ETFs (Shrinkage SIM VCV Matrix)

4 24. Chart 24: Portfolio 11 – Long-Short including ETFs (Shrinkage SIM VCV Matrix) GMVP Asset Weightings 25. Chart 25: Portfolio 12 – Efficient Frontier for Overweight Bond ETF 60% (Shrinkage SIM VCV Matrix) 26. Chart 26: Portfolio 12 – Overweight Bond ETF 60% (Shrinkage SIM VCV Matrix) GMVP Asset Weightings 27. Chart 27: Objective 1 - Reducing Portfolio Risk with Markowitz Mean-Variance Efficiency and including Short-Sells 28. Chart 28: Objective 2 – Reducing Expected Portfolio Risk with International Equity and Other Asset ETFs using the Markowitz VCV matrix 29. Chart 29: Objective 3 – Reducing Expected Portfolio Risk with International Equity and Other Asset ETFs using the Single Index Model VCV matrix 30. Chart 30: Objective 4 (Hypothesis 7) - Demonstrating the Difference in Estimating the Efficient Frontiers using the Shrinkage Markowitz VCV Matrix and the Markowitz VCV Matrix 31. Chart 31: Objective 4 (Hypothesis 8) - Demonstrating the Difference in Estimating the Efficient Frontiers from the Shrinkage Markowitz VCV Matrix and the Markowitz VCV Matrix 32. Chart 32: Objective 4 (Hypothesis 9) - Demonstrating the Difference in Estimating the Efficient Frontiers using the Shrinkage SIM VCV Matrix and the SIM VCV Matrix 33. Chart 33: Objective 4 (Hypothesis 10) - Demonstrating the Difference in Estimating the Efficient Frontiers from using the Shrinkage SIM VCV Matrix over the SIM VCV Matrix 34. Chart 34: Objective 1- Showing the Reduction in Expected Portfolio Risk for the Same Level of Expected Portfolio Return of 0.97% using Portfolios 1 and 3 35. Chart 35: Showing the Weightings of Portfolio 3 for an Expected Portfolio Return of 0.97% and an Expected Portfolio Risk of 2.58% 36. Chart 36: Objective 1- Showing the Reduction of Expected Portfolio Risk for a Higher Expected Portfolio Return using Portfolios 2 and 3 37. Chart 37: Showing the Reduction in Expected Portfolio Risk for the Same Level of Expected Portfolio Return of 0.91% using Portfolios 3 to 5 38. Chart 38: Objective 2 - Showing the Weightings of Portfolio 5 for an Expected Portfolio Return of 0.91% and an Expected Portfolio Risk of 1.46% 39. Chart 39: Objective 3- Showing the Reduction in Portfolio Risk for the Same Level of Portfolio Return of 0.82% using Portfolios 6 to 8 40. Chart 40: Objective 3 - Showing the Weightings of Portfolio 8 for an Expected Portfolio Return of 0.82% and an Expected Portfolio Risk of 1.24% 41. Chart 41: Objective 4 - Showing the GMVP Range for Long-Short including ETFs Equity Portfolio using Multiple Variance-Covariance 42. Chart 42: Objective 4 - Showing the GMVP Range for an Equity portfolio using an Overweight Bond ETF of 60% using Multiple Variance-Covariance 43. Chart 43: Increasing the Sharpe ratio by Moving the CML to the Left (Portfolio 3 and 5) 44. Chart 44: Sharpe Ratio Calculation for Portfolio 3 45. Chart 45: Sharpe Ratio Calculation for Portfolio 5 46. Chart 46: Tangency Portfolio for Portfolio 5 47. Chart 47: Weighting in Tangency Portfolio and Risk-Free Rate for Portfolio 3 48. Chart 48: Comparing the GMVP of Portfolio 5 with a Combination of the Risk-Free Rate and the Tangency Portfolio of Portfolio 3 49. Chart 49: Comparing the GMVP of Portfolios 5 with Equity Market Indexes 50. Chart 50: Efficient Frontiers for Portfolios 1 to 5 using the Markowitz VCV matrix 51. Chart 51: Risk to Return for Risk-Free Asset, AGG Bond ETF and the GMVP for Portfolio 5 (Markowitz VCV matrix) and 12 (Shrinkage SIM VCV matrix) Abbreviations 1. AGG, U.S. government and corporate investment grade bonds ETF (10-30 years maturity) 2. CAL, Capital Allocation Line 3. CAPM, Capital Asset Pricing Model 4. CML, Capital Market Line 5. EFA, International non-U.S. mega-cap equity ETF 6. ETF(s), exchange-traded fund(s) 7. GLD, gold bul ion ETF 8. GMVP, Global Minimum Variance Portfolio 9. OEX, S&P 100 equity index 10. MPT, Modern Portfolio Theory 11. Shrinkage, a method of shrinking the VCV matrix 12. SIM, Single Index Model 13. SML, S&P 600 small cap equity index 14. SPX, S&P 500 equity index 15. T, tangency Portfolio 16. VCV, variance-covariance 17. VNQ, U.S. commercial and residential real estate ETF 18. σ, expected portfolio risk using historical data 19. μ, expected portfolio return using historical data

5 Table of Contents 1. Introduction1.1. Initiation of Research 7 1.2. Construction of Portfolios 9 1.3. Dissertation Question 13 1.4. Research Objectives and Hypotheses 14 1.4.1. Objective 1 1.4.2. Objective 2 1.4.3. Objective 3 1.4.4. Objective 4 1.5. Dissertation Overview 16 2. Literature Review2.1. Theoretical Literature Review 17 2.2. Empirical Literature Review 213. Methodology3.1. Data Collection & Ethics 23 3.2. Building the Variance-Covariance (VCV) Matrices 24 3.2.1. Markowitz VCV Matrix 3.2.2. Single Index Model VCV Matrix 3.2.3. Shrinkage VCV Matrix 3.3. Expected Portfolio Return and Risk 31 3.3.1. Expected Portfolio Return 3.3.2. Expected Portfolio Risk 3.4. Computing the Global Minimum Variance Portfolio 33 3.5. Building the Markowitz Bullet and the Efficient Frontier 34 3.6. Efficient Frontier for Portfolios 1 to 12 35 3.6.1. Portfolios 1: Equal Weight Long (Markowitz VCV matrix) 3.6.2. Portfolio 2: Weighted Long-Only (Markowitz VCV 3.6.3. Portfolio 3: Long-Short (Markowitz VCV matrix) 3.6.4. Portfolio 4: Long-Short incl. ETFs (Markowitz VCV matrix) 3.6.5. Portfolio 5: Overweight Bond ETF 60% (Markowitz VCV matrix) 3.6.6. Portfolio 6: Long-Short (SIM VCV matrix) 3.6.7. Portfolio 7: Long-Short incl. ETFs (SIM VCV matrix) 3.6.8. Portfolio 8: Overweight Bond ETF 60% (SIM VCV matrix) 3.6.9. Portfolio 9: Long-Short incl. ETFs (Shrinkage Markowitz VCV matrix) 3.6.10. Portfolio 10: Overweight Bond ETF 60% (Shrinkage Markowitz VCV matrix) 3.6.11. Portfolio 11: Long-Short including ETFs (Shrinkage SIM VCV matrix) 3.6.12. Portfolio 12: Overweight Bond ETF 60% (Shrinkage SIM VCV matrix) 3.7. Methodology Reflections 47 4. Results4.1. Objective 1 48 4.2. Objective 2 49 4.3. Objective 3 50 4.4. Objective 4 51

6 5. Discussion of Results5.1. Objective 1 Analysis 55 5.2. Objective 2 & 3 Analysis 57 5.3. Objective 4 Analysis 60 5.4. Risk Adjusted Returns with Sharpe Ratio 62 5.5. CAL for Portfolio Risk Comparison 65 5.6. Benchmark Analysis 68 6. Conclusions6.1. Research Recapitulation 68 6.2. Strengths and Limitations of Research 71 6.2.1. Research Strengths 6.2.2. Research Limitations 6.3. Research Extensions and Recommendations 73 7. References 75 8. Appendices 8.1. Converting Daily Adjusted Dividend Share Price to Monthly Total Returns 77 8.2. Monthly Adjusted Dividend Percentage Change 78 8.3. Excess Returns 79 8.4. VBA Code for the SIM VCV Matrix 80 8.5. Solver Parameters in Microsoft Excel 81 8.5.1. Portfolio 3 (Long-Short) 8.5.2. Portfolio 4 (Long-Short incl. ETFs) 8.5.3. Portfolio 5 (Overweight Bond ETF 60%)8.6. Efficient Frontier Data Table (Portfolios 1 to 4) 84 8.7. Efficient Frontier Data Table (Portfolios 5 to 8) 85 8.8. Efficient Frontier Data Table (Portfolios 9 to 12) 86 8.9. VBA Code for Computing Merton’s (1973) GMVP 87 8.10. Comparing the GMVP Portfolio 5 & 8 Weightings (Markowitz and SIM VCV matrices) 88

7 1. Introduction 1.1. Initiation of Research Since the first half of 1900’s the U.S. economy has collided with multiple financial recessions that harmed the economy, including the Great Depression of 1929-1933 where U.S. unemployment hit 25%, (Margo, 1993). It is therefore not surprising that academics such as Allais, DeFinetti and Hicks started talking about equity risks in the 1930’s to 1950’s, (Board, et al., 1999). However, it was not until Harry Markowitz’s seminal paper in 1952 on “Portfolio Selection” and his more extensive book on “Portfolio Selection: Efficient Diversification” (1959), when it was mathematically proven that portfolio risk could be lowered when increasing the number of uncorrelated stocks (ρ < 1) within that portfolio, and thus the Mean-Variance model was born. In 1964, fellow Nobel Prize winner, William Sharpe developed the Capital Asset Pricing Model (CAPM) that expanded on Markowitz’s work and gave further insights into the risks on portfolios. Markowitz and Sharpe’s contributions to Modern Portfolio Theory in the 1950’s and 1960’s created an explosion in demand for portfolio management. In the 1950’s to 1990’s portfolio diversification was mostly conducted by fund managers using active portfolio management strategies. In the 1990’s, mutual funds and exchange-traded funds (ETFs) started to gain popularity, however it is not until recently when ETFs have seen a dramatic increase in tradable volume. In 2005, the global ETF market was approximately $425 billion, (Hill, 2015) and in 2015 global assets under management (AUM) using ETFs was approximately $3 trillion, (Madhavan, 2016). This academic paper will investigate using ETFs as a proxy for other trading asset products, such as; futures, options and other derivatives that have been previously been used by the fund management world for diversification. The attraction of using ETFs in equity portfolios is that ETFs are traded very much like cash equities and therefore ETFs can be held in traditional cash equity portfolios. We investigate multiple ways to compute the variance-covariance (VCV) matrix for portfolio optimisation including; the Markowitz VCV matrix, the Single Index Model VCV matrix, the Constant Correlation VCV matrix and the Shrinkage VCV matrix. These four methods of computing the VCV matrix for Mean-Variance analysis uses ex-post data and thus is addressing historical volatility as opposed to discussing future implied volatility. We looked at using option data for an ex-ante analysis of implied volatility and thus a forward looking VCV matrix, however we wanted to first understand historical volatility and thus kept Markowitz’s Mean-Variance model for this research. We will use the Shrinkage VCV matrix as it has been previously proven to generate a lower Global Minimum Variance Portfolio (GMVP) as stated

8 by Ledoit and Wolf (2004) and Benninga (2014). This is due to a claimed superior estimation in the VCV matrix and not due to expected portfolio risk reducing, this is a subtle but salient point. This research will use historical volatility as a measure of risk. We used historical data to conduct this research and will therefore use portfolio risk, expected portfolio risk and historical risk interchangeably. This research will use monthly data ranging from July 2007 to July 2017, therefore our portfolios risk and return values are monthly. The Sharpe ratios are also monthly, (Sharpe, 1993). Furthermore, we noted that academia sometimes refers to the Markowitz variance-covariance matrix as the ‘simple variance-covariance’ matrix. To avoid confusion, we will use the name, ‘Markowitz variance-covariance’ matrix.

1.2. Construction of Portfolios Firstly, we investigated the optimal number of stocks to diversify most of the specific risk away within the portfolios. Wagner and Lau (1971) argued that 15 stocks are enough, Solnick (1974) states the number to be approximately 20 stocks. Statman (1987) argues the figure to be 30 stocks and Odegaard (2009) states that a portfolio holding 25 equities is optimal as the reduction in standard deviation from holding more equities is negligible after this point. Using this theoretical literature, we used an average number of equities from Solnick (1974) and Statman (1987) research and the exact number of stocks from Odegaard (2009), therefore we used 25 securities for our ‘base portfolio’. We state that any further reduction in portfolio risk can be viewed as reducing systematic risk, (Table 1).

Ticker

Name

Market Cap ($m) -as of (July 2017)

Sector

AAPL

Apple Inc.

781,293.96

Information Technology

MSFT

Microsoft Corp.

562,092.08

Information Technology

AMZN

Amazon.Com Inc.

488,366.68

Consumer Discretionary

JNJ

Johnson & Johnson

360,273.25

Health Care

XOM

Exxon Mobil Corp.

341,438.86

Energy

GOOGL

Alphabet Inc. (GOOGLE-C)

334,568.25

Information Technology

JPM

JP Morgan Chase & Co.

323,645.16

Financials

WFC

Wells Fargo & Co.

273,603.16

Financials

BAC

Bank of America Corp.

237,452.31

Financials

GE

General Electric Co.

232,860.47

industrials

WMT

Wal-Mart Stores Inc.

229,810.34

Consumer Staples

PG

Procter & Gamble Co.

226,374.45

Consumer Staples

BERK

Berkshire Hathaway Inc. Class B

223,229.60

Financials

ORCL

Oracle Corp.

208,013.05

Information Technology

PFE

Pfizer Inc.

198,580.03

Health Care

CVX

Chevron Corp.

196,447.10

Energy

KO

Coca-Cola Co.

191,410.65

Consumer Staples

CMCSA

Comcast Corp.

185,065.20

Consumer Discretionary

C

Citigroup Inc.

184,041.52

Financials

HD

Home Depot Inc.

182,214.54

Consumer Discretionary

UNH

Unitedhealth Group Inc.

178,430.42

Health Care

VZ

Verizon Communications Inc.

176,657.09

Tele Com Services

MRK

Merck & Co. Inc.

171,239.63

Health Care

DIS

Walt Disney Co.

166,784.81

Consumer Discretionary

PEP

PepsiCo Inc

164,161.79

Consumer Staples

Total Market Cap.

6,818,054.40

9 Table 1: Base Portfolio – 25 mega-cap U.S. S&P 500 stocks (Bloomberg, 2017)

10 We created our base portfolio of 25 stocks that are all in the top 30 of the S&P 500 equity index by market capitalisation, thus capturing a sample of the U.S. mega-cap equity sentiment in a diversified portfolio, (Table 1). We used the last 10 years of historical data (July 2007 to July 2017) and thus we did not include stocks that had insufficient data for reasons such as; recent initial public offerings (IPOs) or de-listings and re-listings in the last 10 years. Therefore, we replaced Facebook (FB), Philip Morris International (PMI) and Visa (V) with Merck (MRK), Walt Disney (DIS) and PepsiCo (PEP). Additionally, we excluded Google and Berkshire Hathaway Class A shares and only kept Google Class C shares (GOOG) and Berkshire Hathaway Class B shares (BERK). We did this as we did not want to create two entries for the same stock in our portfolio, furthermore these Class A stocks are less liquid and thus are less attractive to us. We wanted to have an overweighting to information technology, financials and consumer discretionary sectors as these sectors dominate the S&P 500 equity index price movement, (Chart 1). Our 25 stocks represent approximately $7 trillion in market capitalisation or 35% of the total market capitalisation of the S&P 500 index, which is currently valued at over $20 trillion, (Bloomberg, 2017; see Table 1). We note that we could diversify away more of the portfolio risk if we had an equal weighted sector balance base portfolio. Secondly, we state that if we increased our number of stocks in our base portfolio we would de-risk further, however the question is, whether the additional reduction in portfolio risk is worth the trading costs and management time of holding a higher number of securities in a portfolio? Chart 1: Weighting of Sectors in the Base Portfolio (Bloomberg, 2017) 28%18%15%12%10%8% 6%3% Information TechnologyFinancialsConsumer DiscretionaryConsumer StaplesHealth CareEnergyHealth CareTele Com Services

11 Table 2 gives the ticker1, name and market exposure for the relevant exchange-trade funds to be included into Portfolios 4, 5, 7, 8, 9, 10, 11 and 12. We used iShares MSCI EAFE ETF (EFA) for international non-U.S. mega-cap equity exposure as it has been proven by (Solnick, 1974), that international equity portfolios have a lower portfolio risk than domestic portfolios with Home-Bias2. Secondly, we used iShares Core U.S. Aggregate Bond ETF (AGG), which mimics the movement of long-term U.S. government and corporate debt with maturities of 10 to 30 years. We used AGG as an extreme diversifier as the bond market and the equity market tends to have an inverse correlation, (Shahzad, 2017). Thirdly, we used SPDR Gold Shares ETF (GLD) as a proxy to the precious metals market and more specifically, a safe-haven alternative to the U.S. bond market. GLD holds physical gold bullion and has a correlation 0.9 with the gold spot price over the last 10 years, (www.spdrgoldshares.com, 2017). Lastly, Vanguard REIT ETF (VNQ) is designed to get exposure to the U.S. commercial and residential real estate market. The inclusion of real estate assets could have diversification benefits resulting in a lower portfolio risk. EFA, AGG, GLD and VNQ were selected for two reasons; firstly, all four ETFs have over 10 years of data and secondly, all four ETFs are extremely liquid thus giving the ETFs market credibility. Ticker ETF Name ExposureEFA iShares MSCI EAFE ETF World Equity ex U.S.AGG iShares Core U.S. Aggregate Bond ETF U.S. Debt MarketsGLD SPDR Gold Shares ETF World - Gold BullionVNQ Vanguard REIT ETF U.S. Real EstateTable 2: ETFs to be included into Portfolios 4, 5, 7, 8, 9, 10, 11 and 12 We changed the weights of the portfolio via Markowitz’s portfolio optimisation, added constraints, added exchange traded funds (ETFs) and made alterations to the method of calculating the variance-covariance matrix. Table 3 shows a list of all the portfolios used to conduct this research to address research objectives 1 to 4. Unless stated otherwise we used a maximum of 25% portfolio weighting (W) in any one stock or ETF for long positions and a maximum short position of -25% of the portfolio weighting, (-25% ≤ W ≤ 25%). We used 25% limits for the majority of our portfolio constituents as we did not want to limit the diversification benefits from having lower limits and conversely, we did not want to have a disproportionate weighting in anyone asset apart from when we test portfolios that have an overweighting to the bond ETF (AGG). This ensures us that we did not take unrealistic long or short positions unless intended.1 The ticker is the asset symbol which is used in an abbreviation 2 ‘Home bias’ in investment literature has been proven in many papers. See Mishra (2015) for explanations on ‘home bias’ in a 42-country observation study.

Table 3: List of Portfolios 1 to 12Portfolio Portfolio Name Portfolio Constituents, Constraints and InformationPortfolio 1 Equal Weight Long (Markowitz) Base portfolio of long only 25 stocks. Weightings of the stocks are [W(Equities)] = 4%.This portfolio uses the Markowitz VCV MatrixPortfolio 2 Weighted Long (Markowitz) Base portfolio of long only 25 stocks. Weightings of the stocks are 0 ≤ W(Equities) ≤ 25%.This portfolio uses the Markowitz VCV MatrixPortfolio 3 Long-Short (Markowitz) Base portfolio of long-short 25 stocks. Weightings of the stocks are -25% ≤ W(Equities) ≤ 25%.This portfolio uses the Markowitz VCV MatrixPortfolio 4 Long-Short incl. ETFs (Markowitz) Base portfolio of long-short 25 stocks including 4 ETFs (EFA, AGG, GLD, VNQ).Weighting of these assets are; [-25% ≤ W(Equities) ≤ 25%], [0% ≤ W(ETFs) ≤ 25%].This portfolio uses the Markowitz VCV matrix.Portfolio 5 Overweight Bond ETF 60% (Markowitz) Base portfolio of long-short 25 stocks including 4 ETFs (EFA, AGG, GLD, VNQ).Weighting of these assets are; [(-25% ≤ W(Equities) ≤ 25%)], [(0% ≤ W(EFA, GLD, VNQ) ≤ 25%)], [(0% ≤ W(AGG) ≤ 60%)].This portfolio uses the Markowitz VCV matrix.Portfolio 6 Long-Short (SIM) Replication of Portfolio 3 but uses the Single Index Model VCV matrix.Portfolio 7 Long-Short incl. ETFs (SIM) Replication of Portfolio 4 but uses the Single Index Model VCV matrix.Portfolio 8 Overweight Bond ETF 60% (SIM) Replication of Portfolio 5 but uses the Single Index Model VCV matrix.Portfolio 9 Long-Short incl. ETFs (Shrinkage Markowitz) Replication of Portfolio 4 but uses the Shrinkage method on the Markowitz VCV matrix.Portfolio 10 Overweight Bond ETF 60% (Shrinkage Markowitz) Replication of Portfolio 5 but uses the Shrinkage method on the Markowitz VCV matrix.Portfolio 11 Long-Short incl. ETFs (Shrinkage SIM) Replication of Portfolio 4 but uses the Shrinkage method on the Single Index Model VCV matrix.Portfolio 12 Overweight Bond ETF 60% (Shrinkage SIM) Replication of Portfolio 5 but uses the Shrinkage method on the Single Index Model VCV matrix.

18 The optimal number of securities to hold in a diversified portfolio to minimise specific risk has been in question ever since the late 1960’s. Using standard deviation as a measure of risk, Evans and Archer (1968) empirically proved that holding 10 assets was sufficient enough to diversify portfolio risk and reduce specific risk. Since then the number of stocks to hold in a diversified portfolio has been in question from 15 stocks to 30 stocks (see; Wagner and Lau, 1971; Solnick, 1974; Bloomfield et al., 1977; Bird and Tippett, 1986; Statman, 1987) Grubel (1968) was the first academic to state that cross-border diversification could increase the payoff to risk ratio, however, Grubel’s (1968) paper addresses welfare gains and capital flows. It was not until Solnick (1974) when academia started to take notice that international diversification lowers portfolio risk, (Figure 2). Since Solnick (1974) there has been a vast amount of literature confirming that international diversification lowers overall portfolio risk by reducing systematic risk, (Errunza and Senbet, 1981; Adler and Duma, 1983; Curcio and Ziobrowski, 1991). This is because when a portfolio is internationally diversified the portfolio benefits in a reduction of systematic risk as the possibility of a country’s recession or political instability does not affect the whole portfolio. Figure 2 shows that national diversification of portfolios reduces idiosyncratic risk to near zero and international diversification reduces some of the systematic risk, therefore resulting in overall lower expected portfolio risk. Tesar and Werner (1995) and Mai and Santa-Clara (2012) extended Sharpe (1964), Linter (1965) and Mossin (1966) CAPM by proposing the International Capital Asset Pricing Model (ICAPM). Figure 2: Edited from Solnick (1974) Showing Domestic Portfolio Diversification Lowering Specific Risk and Portfolio Internationalisation Lowering Systematic Risk (Edited from Solnick, 1974) Specific RiskSystematic Risk

19 Using Markowitz’s (1952; 1959) and Tobin’s (1958; 1965) cornerstone research Robert Merton (1973) mathematically derived the Efficient Frontier and proved that in certain conditions the “classic graphical technique for deriving the efficient portfolio frontier is incorrect.” Merton (1973) suggested that the Global Minimum Variance Portfolio (GMVP) and the Efficient Frontier should be calculated in a quantitative and not a qualitative process, therefore disregarding the Markowitz Bullet as an exact measure of risk to reward. Furthermore, Merton (1973) explained and detailed a quantitative method for computing the Global Minimum Variance Portfolio (GMVP). After academia proved that international diversification lowered systematic risks, many academics turned their focus towards diversifying a portfolio using other assets. Curcio and Ziobrowski (1991) empirically proved that multinational and multi-asset portfolios benefited from further risk reduction. Complementing Coeurdacier and Guibaud (2011), which states that diversification is most efficient when selecting low correlations with multi-assets internationally. We state that too much diversification can have adverse effects, as the more portfolio constituents one has, the less one knows about all the constituents’ behaviour, (Huij and Derwall, 2011). If a portfolio manager holds all the equities on the S&P 500 equity index, the manager would essentially be mirroring the index, however, the returns would be lower than holding the SPY3 ETF due to transaction costs, (Odegaar, 2009). It is important to finally state that not all systematic risks can be lowered to zero as the global economy has an infinite number of correlations and thus risks, (Raffestin, 2014). To discuss the variance covariance matrix for computing historical portfolio risk we must first list the most well-known methods for computing this matrix; the Markowitz Variance-Covariance (VCV) matrix (Markowitz, 1952; 1959), the Single Index Model matrix (Sharpe, 1963), the Shrinkage VCV matrix (Ledoit and Wolf, 2003; 20041; 20042), the Constant Correlation matrix and a matrix based from option pricing for implied portfolio risk. Markowitz (1952; 1959) stated that the risk of a portfolio is made up from all the variances and covariances within the portfolio. Sharpe (1964) states that all specific risk can be diversified away, therefore a security’s risk is simply its beta or systematic risk. The SIM VCV matrix does contain specific risks in the diagonal variance in the matrix but Sharpe (1964) states that an investor should only be compensated for systematic risks, (see Section 3.2.2.). Here the Single Index Model matrix is made up from all the assets betas. The shrinkage VCV matrix was first introduced by Ledoit and Wolf (2003) in their paper, “Improved Estimation of the Covariance Matrix of Stock returns with an Application to Portfolio Selection.” This academic work was extended in 2004 with their paper, “A-well Conditioned Estimator for Large-Dimensional Covariance Matrices,” which states a higher weighting to the diagonal variances in the variance-covariance matrix. This was tested in Ledoit and Wolf’s second 2004 paper, “Honey, I Shrunk the Sample Covariances Matrix,” which states the shrinkage formula (see Section 4.2.2.). Benninga (2014) suggests the use of 30% weighting to the Markowitz VCV matrix and a 70% weighting dedicated to the diagonal variances. The exact number for3 SPY, S&P 500 equity index tracker exchange traded fund

24 3.2. Building the Variance-Covariance (VCV) Matrices 3.2.1. Markowitz VCV Matrix To create the Markowitz variance-covariance matrix we must first find out the number of covariances the matrix has. We used the formula 1 below. Where; N = Number of Assets in the Matrix Formula 1: Number of Covariances in a matrix (Bodie, et al., 2014) A covariance is described as the standard deviation or risk of two assets multiplied together and then multiplied with the correlation of those two assets. Formula 2 shows the computation of the covariance. Where; σ = Standard Deviation, Risk ρ = Correlation Formula 2: Covariance of Assets (Wilmott, 2007) This can be done in Microsoft Excel by using the Function (COVARIANCE.S) and selecting all the excess returns data for those two assets. We used COVARIANCE.S over COVARIANCE.P as we are using a sample (.S) and not the whole population (.P) of data. Table 4 shows the Markowitz variance-covariance matrix, where; B2 = COVARIANCE.S (AAPL Excess Returns, AAPL Excess Returns) B3 = COVARIANCE.S(AAPL Excess Returns, MSFT Excess Returns) C4 = COVARIANCE.S(MSFT Excess Returns, JNJ Excess Returns) E10 = COVARIANCE.S(JNJ Excess Returns, BAC Excess Returns)

26 3.2.2. Single Index Model Variance Covariance Matrix We used Sharpe’s (1964) Capital Asset Pricing model to compute the Single Index Model variance-covariance (VCV) matrix, (Formula 3). The Single Index Model (SIM) assumes that the only sources of risk are the assets variances and the betas of the securities against a benchmark, (Sharpe, 1963), however, Sharpe states that an investor should only be reward for the systematic risks a portfolio manager is taking and not specific risks within the portfolio. Formula 4 shows the systematic risk component and the specific risk component in Sharpe’s portfolio risk, (Sharpe, 1963). Sharpe (1963) states that σ2(εp) can be diversified away and thus has a value of zero when number of assets goes infinitely larger, (see Section 2.1.). To compute the portfolio risk, we simply square root the portfolio variance. The Single Index Model is a method of simplifying some of the calculations in the Markowitz VCV matrix. The idea of the Single Index model is that the returns of each security can be linearly regressed. Where; Ri = Return on asset αi = Alpha of asset (unexplained movement) βi = Beta of asset (movement compared to the market) RM = Return on market εi = 0, error term Formula 3: Capital Asset Pricing Model (CAPM) (Sharpe, 1963) Formula 4: Portfolio’s Variance using CAPM (Bodie, et al., 2014) The SIM matrix requires changes to the estimation of the covariances but not the diagonal variances in the matrix. We constructed the SIM variance-covariance matrix using Visual Basic Application (VBA) code in Microsoft Excel, (to see this code please view Appendix 8.4). We generated a function called “sim_test”, which was then used in the matrix with the excess returns data and the returns data for the S&P 500 index. We used Formula 5 to compute the SIM variance-covariance matrix. Table 5 shows the SIM variance-covariance matrix. {sim_test(excess returns data, S&P total returns data} Formula 5: SIM VCV Matrix VBA Formula (Benninga, 2014) Specific RiskSystematic Risk

30 Diagonal V Matrix Shrinkage VCV Matrix Table 6: Shrinkage Variance-Covariance (VCV) Matrix Enter a Varaince-Covaraince Matrix HERE either Sample VCV or SIM VCV

32 3.3.2. Expected Portfolio Risk We must first state what type of risk we will be using and how we evaluate what risk is in an equity portfolio. We use the variance and standard deviation of an asset as a measure of risk. The variance of a security is simply the historical share price fluctuation or volatility and the standard deviation of an asset is the square root of its variance. The variance of an asset can be described as the expected return minus expected mean squared. Formula 12 shows the equation for computing the variance of a security. Formula 12: Variance of an asset (Kennedy, 2010)To calculate the expected portfolio risk, we square root the weighted covariance for all the assets in the portfolio, Formula 2. Cov (ri,rj) is the covariance of returns between i and j and can be expressed as the product of the correlation between the two returns (ρi,j) and the standard deviations of the assets. We used Formula 13 as a mathematically base to create Formula 14 shows the Microsoft Excel equation to compute the expected portfolio risk. Formula 13: Expected Portfolio Risk (Kennedy, 2010){=SQRT(MMULT(TRANSPOSE(List of Asset Weightings),MMULT(VCV Matrix, List of Asset Weightings)))} Formula 14: Microsoft Excel Formula (Benninga, 2014)

34 3.5. Building the Markowitz Bullet and Efficient Frontier Now that we have the expected risk and return figures for the GMVP with the GMVP asset weighting we put all the data into an Efficient Frontier table, (Appendix 8.6, 8.7 and 8.8). In the returns column, we create a decreasing return list. We start with a monthly expected return of 2%, dropping the figure by 25 basis points until we hit 0% expected monthly returns. The GMVP figure will vary from portfolio to portfolio therefore the position of where the GMVP is from 2% to 0% return range will change. To build the Efficient Frontier we need to add in one more constraint, ‘Target Return = Expected Portfolio Return’. We then run Solver at 2% target return and move down the monthly returns target list in increments of 25 basis points. We follow this process until 0%. We choose this base level as a matter of logic, as no investor would be content with a negative return when they could be in cash. Once all the data is in the Efficient Frontier table we graphically map the returns and risk columns, this creates the Efficient Frontier. Chart 2 shows the Markowitz Bullet and the Efficient Frontier in red, with the GMVP being the closest point to the y-axis and the lowest point on the Efficient Frontier. Portfolio managers want to be in the top left corner in Chart 2, low risk and high return. See Appendix 8.5, 8.6, 8.6 for Solver’s parameter settings for Portfolios 3, 4 and 5. This creates the Efficient Frontier data. Chart 2: Markowitz Bullet and the Efficient Frontier Expected Portfolio Risk (σ)

35 3.6. Efficient Frontier for Portfolios 1 to 12 In the section, we will graphically depict the Efficient Frontier and show the Global Minimum Variance Portfolio (GMVP). In addition, we will show the weights that are needed to achieve the GMVP for 12 portfolios. 3.6.1. Portfolios 1: Equal Weight Long (Markowitz VCV matrix) Chart 3 shows that Portfolio 1, which is an equal long portfolio, has a historic portfolio risk of 4.48% and a historic portfolio return of 0.97%, (Appendix 8.6). This portfolio is an equal long with 25 assets in the portfolio, therefore each asset has a weighting of 4%. Chart 4 shows the portfolio weightings. Chart 3: Portfolio 1 – Equal Weight Long (Markowitz VCV Matrix) Portfolio Return and Risk Chart 4: Portfolio 1 –Equal Weight Long Portfolio (Markowitz VCV Matrix) Asset Weightings 0.00%0.50%1.00%1.50%3.00% 3.50% 4.00% 4.50% 5.00%Monthly Expected Portfolio Risk (σ)0.00%5.00%10.00%15.00%20.00%25.00%

36 3.6.2. Portfolio 2: Weighted Long-Only (Markowitz VCV matrix) Chart 5 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 2. The GMVP has a historic portfolio risk of 3.06% and a historic portfolio return of 0.75%, (Appendix 8.6). Chart 6 shows the asset weightings for the GMVP. Chart 5: Portfolio 2 – Efficient Frontier for Weighted Long-Only (Markowitz VCV Matrix) Chart 6: Portfolio 2 – Weighted Long-Only (Markowitz VCV Matrix) GMVP Asset Weightings 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%3.00% 3.25% 3.50% 3.75% 4.00% 4.25% 4.50% 4.75% 5.00% 5.25% 5.50% 5.75%Monthly Expected Portfolio Risk (σ)0.00%2.00%4.00%6.00%8.00%10.00%12.00%14.00%16.00%18.00%20.00%

37 3.6.3. Portfolio 3: Long-Short (Markowitz VCV matrix) Chart 7 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 3. The GMVP has a historic portfolio risk of 2.62% and a historic portfolio return of 0.91%, (Appendix 8.6). Chart 8 shows the asset weightings for the GMVP. Chart 7: Portfolio 3 – Efficient Frontier for Long-Short (Markowitz VCV Matrix) Chart 8: Portfolio 3 – Long-Short (Markowitz VCV Matrix) GMVP Asset Weightings 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%2.50% 2.75% 3.00% 3.25% 3.50% 3.75%Monthly Expected Portfolio Risk (σ)-25.00%-20.00%-15.00%-10.00%-5.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%

38 3.6.4. Portfolio 4: Long-Short incl. ETFs (Markowitz VCV matrix) Chart 9 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 4. The GMVP has a historic portfolio risk of 1.84% and a historic portfolio return of 0.72%, (Appendix 8.6). Chart 10 shows the asset weightings for the GMVP. Chart 9: Portfolio 4 – Efficient Frontier for Long-Short including ETFs (Markowitz VCV Matrix) Chart 10: Portfolio 4 – Long-Short including ETFs (Markowitz VCV Matrix) GMVP Asset Weightings0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50%Monthly Expected Portfolio Risk (σ)-20.00%-15.00%-10.00%-5.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%

39 3.6.5. Portfolio 5: Overweight Bond ETF 60% (Markowitz VCV matrix) Chart 11 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 5. The GMVP has a historic portfolio risk of 1.26% and a historic portfolio return of 0.51%, (Appendix 8.6). Chart 12 shows the asset weightings for the GMVP. Chart 11: Portfolio 5 – Efficient Frontier for Overweight Bond ETF 60% (Markowitz VCV Matrix) Chart 12: Portfolio 5 – Overweight Bond ETF 60% (Markowitz VCV Matrix) GMVP Asset Weightings 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50%Monthly Expected Portfolio Risk (σ)-10.00%-5.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%55.00%60.00%

3.6.6. Portfolios 6: Long-Short (SIM VCV matrix) Chart 13 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 6. The GMVP has a historic portfolio risk of 2.12% and a historic portfolio return of 0.82%, (Appendix 8.7). Chart 14 shows the asset weightings for the GMVP. Chart 13: Portfolio 6 – Efficient Frontier for Long-Short (SIM VCV Matrix) 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%2.00% 2.20% 2.40% 2.60% 2.80% 3.00% 3.20% 3.40% 3.60%Monthly Expected Portfolio Risk (σ)000%2.50%5.00%7.50%10.00%12.50%15.00%17.50%20.00%22.50%

Chart 14: Portfolio 6 – Long-Short (SIM -10.00%-7.50%-5.00%-2.50%.

VC

V

Ma

t

rix

) G

MVP Asset Weightings

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3.6.7. Portfolio 7: Long-Short incl. ETFs (SIM VCV matrix) Chart 15 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 7. The GMVP has a historic portfolio risk of 1.51% and a historic portfolio return of 0.69%, (Appendix 8.7). Chart 16 shows the asset weightings for the GMVP. Chart 15: Portfolio 7 – Efficient Frontier for Long-Short including ETFs (SIM VCV Matrix) 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.00% 1.50% 2.00% 2.50% 3.00% 3.50%Monthly Expected Portfolio Risk (σ)000%2.50%5.00%7.50%10.00%12.50%15.00%17.50%20.00%22.50%25.00%

Chart 16: Portfolio 7 – Long-Short inc -10.00%-7.50%-5.00%-2.50%.

luding ETFs (SIM VCV Matrix) GMVP Ass

e

41 t Weightings

3.6.8. Portfolio 8: Overweight Bond ETF 60% (SIM VCV matrix) Chart 17 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 8. The GMVP has a historic portfolio risk of 1.04% and a historic portfolio return of 0.51%, (Appendix 8.7). Chart 18 shows the asset weightings for the GMVP. Chart 17: Portfolio 8 – Efficient Frontier for Overweight Bond ETF 60% (SIM VCV Matrix) 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50%Monthly Expected Portfolio Risk (σ)000%5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%55.00%60.00%

Chart 18: Portfolio 8 – Overwei -10.00%-5.00%.

g

ht Bond ETF 60% (SIM VCV Matrix) GM

V

P Asset Weightings

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43 3.6.9. Portfolio 9: Long-Short incl. ETFs (Shrinkage Markowitz VCV matrix) Chart 19 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 9. The GMVP has a historic portfolio risk of 1.50% and a historic portfolio return of 0.71%, (Appendix 8.8). Chart 20 shows the asset weightings for the GMVP. Chart 19: Portfolio 9 – Efficient Frontier for Long-Short including ETFs (Shrinkage Markowitz VCV Matrix) Chart 20: Portfolio 9 – Long-Short including ETFs (Shrinkage Markowitz VCV Matrix) GMVP Asset Weightings 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50% 3.75%Monthly Expected Portfolio Risk (σ)-5.00%-2.50%0.00%2.50%5.00%7.50%10.00%12.50%15.00%17.50%20.00%22.50%25.00%

3.6.10. Portfolio 10: Overweight Bond ETF 60% (Shrinkage Markowitz VCV matrix) Chart 21 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 10. The GMVP has a historic portfolio risk of 1.06% and a historic portfolio return of 0.52%, (Appendix 8.8). Chart 22 shows the asset weightings for the GMVP. Chart 21: Portfolio 10 – Efficient Frontier for Overweight Bond ETF 60% (Shrinkage Markowitz VCV Matrix) 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50% 3.75%Monthly Expected Portfolio Risk (σ)5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%55.00%60.00%

44 Chart 22: Portfolio 10 – Overweight Bond ETF 60% (Shrinkage Markowitz VCV Matrix) GMVP Asset Weightings -5.00%0.00%

45 3.6.11. Portfolio 11: Long-Short including ETFs (Shrinkage SIM VCV matrix) Chart 23 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 11. The GMVP has a historic portfolio risk of 1.36% and a historic portfolio return of 0.70%, (Appendix 8.8). Chart 24 shows the asset weightings for the GMVP. Chart 23: Portfolio 11 – Efficient Frontier for Long-Short including ETFs (Shrinkage SIM VCV Matrix) Chart 24: Portfolio 11 – Long-Short including ETFs (Shrinkage SIM VCV Matrix) GMVP Asset Weightings 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%Monthly Expected Portfolio Risk (σ)-5.00%-2.50%0.00%2.50%5.00%7.50%10.00%12.50%15.00%17.50%20.00%22.50%25.00%

3.6.12. Portfolio 12: Overweight Bond ETF 60% (Shrinkage SIM VCV matrix) Chart 25 shows the Markowitz Bullet and the Efficient Frontier for Portfolio 12. The GMVP has a historic portfolio risk of 0.97% and a historic portfolio return of 0.52%, (Appendix 8.8). Chart 26 shows the asset weightings for the GMVP. Chart 25: Portfolio 12 – Efficient Frontier for Overweight Bond ETF 60% (Shrinkage SIM VCV Matrix) 0.00%0.25%0.50%0.75%1.00%1.25%1.50%1.75%2.00%2.25%0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%Monthly Expected Portfolio Risk (σ)5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%55.00%60.00%

46 Chart 26: Portfolio 12 – Overweight Bond ETF 60% (Shrinkage SIM VCV Matrix) GMVP Asset Weightings -5.00%0.00%

48 4. Results 4.1. Objective 1 Objective 1 focuses on reducing expected portfolio risk by using Markowitz’s Portfolio Selection (1952; 1959) analysis and includes short-sells. Hypothesis 1 has a null hypothesis that states, ‘Portfolio 1 produces a lower portfolio risk than Portfolio 2,’ and has an alternative hypothesis that states that, ‘Portfolio 1 does not produce a lower portfolio risk than Portfolio 2,’ Portfolio 1’s GMVP has an expected portfolio risk of 4.48% and an expected portfolio return of 0.97%. Portfolio 2’s GMVP has an expected portfolio risk of 3.06% and an expected portfolio return of 0.75%. We therefore reject the null hypothesis and accept the alternative hypothesis, ‘Portfolio 1 does not produce a lower portfolio risk than Portfolio 2’, for Hypothesis 1. Hypothesis 2 has a null hypothesis that sates, ‘Portfolio 2 produces a lower portfolio risk than Portfolio 3’, and has an alternative hypothesis that states, ‘Portfolio 2 does not produce a lower portfolio risk than Portfolio 3’. Portfolio 3’s GMVP has an expected portfolio risk of 2.62% and an expected portfolio return of 0.91%. We therefore, reject the null hypothesis and accept the alternative hypothesis, ‘Portfolio 2 does not produce a lower portfolio risk than Portfolio 3,’ for Hypothesis 2. Hypothesis 1 and 2 are stated below. Chart 27 shows the expected portfolio risk reduction from moving from Portfolio 1’s GMVP to Portfolio’s 3 GMVP via Portfolio’ 2 GMVP. It is important to note that in general when one reduces risk they make a compromise with a lower expected return. However, Chart 27 shows that Portfolio 3 has a lower expected portfolio risk and a higher expected portfolio return when looking at the GMVP, than Portfolio 2. This shows the importance of including shorts into the portfolios. . Hypothesis 1: H0: Portfolio 1 produces a lower portfolio risk than Portfolio 2. Reject H1: Portfolio 1 does not produce a lower portfolio risk than Portfolio 2. Accept Hypothesis 2: H0: Portfolio 2 produces a lower portfolio risk than Portfolio 3. Reject H1: Portfolio 2 does not produce a lower portfolio risk than Portfolio 3. Accept Chart 27: Objective 1 - Reducing Portfolio Risk with Markowitz Mean-Variance Efficiency and including Short-Sells 0.00%0.50%1.00%1.50%2.00%2.50%2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00% 5.50% 6.00% 6.50% 7.00%Monthly Expected Portfolio Risk (σ) Portfolio 1: EqualWeight Long(Markowitz)Portfolio 2: WeightedLong (Markowitz)Portfolio 3: Long-Short(Markowitz)

49 4.2. Objective 2 Objective 2 investigates whether the inclusion of international equity and other assets ETFs into equity portfolios reduces expected portfolio risk using the Markowitz variance-covariance matrix. Hypothesis 3 has a null hypothesis that states, ‘Portfolio 3 produces a lower portfolio risk than Portfolio 4’, and has an alternative hypothesis that states that, ‘Portfolio 3 does not produce a lower portfolio risk than Portfolio 4’. Portfolio 3’s GMVP has an expected portfolio risk of 2.62% and has an expected portfolio return of 0.91%. Portfolio 4’s GMVP has an expected portfolio risk of 1.84% and an expected portfolio return of 0.72%. For Hypothesis 3, we therefore reject the null hypothesis and accept the alternative hypothesis, ‘Portfolio 3 does not produce a lower portfolio risk than Portfolio 4’. Hypothesis 4 has a null hypothesis that states, ‘Portfolio 4 produces a lower portfolio risk than Portfolio 5,’ and has an alternative hypothesis that states, ‘Portfolio 4 does not produce a lower portfolio risk than Portfolio 5’. Portfolio 5’s GMVP has an expected portfolio risk of 1.26% and an expected portfolio risk of 0.51%. For Hypothesis 4, we reject the null hypothesis and accept the alternative hypothesis, ‘Portfolio 4 does not produce a lower portfolio risk than Portfolio 5.’ Hypotheses 3 and 4 are noted below. Chart 28 shows that Portfolio 5 has a lower expected portfolio risk, this is shown by moving from Portfolio 3’s GMVP to Portfolio 4’s GMVP to Portfolio 5’s GMVP. Here the laws of risk to return are more commonly depicted, lower expected portfolio risk for lower expected portfolio return. From Portfolio 3 to 5 the expected portfolio risk reduction is 136 basis points and the expected portfolio return reduction is 40 basis points. Hypothesis 3: H0: Portfolio 3 produces a lower portfolio risk than Portfolio 4. - Reject H1: Portfolio 3 does not produce a lower portfolio risk than Portfolio 4. – Accept Hypothesis 4: H0: Portfolio 4 produces a lower portfolio risk than Portfolio 5. - Reject H1: Portfolio 4 does not produce a lower portfolio risk than Portfolio 5. – Accept Chart 28: Objective 2 – Reducing Expected Portfolio Risk with International Equity and Other Asset ETFs using the Markowitz VCV matrix 0.00%0.50%1.00%1.50%2.00%2.50%0.75% 1.25% 1.75% 2.25% 2.75% 3.25% 3.75% 4.25% 4.75%Monthly Expected Portfolio Risk (σ) Portfolio 3: Long-Short(Markowitz)Portfolio 4: Long-Shortincl. ETFs (Markowitz)Portfolio 5: OverweightBond ETF 60%(Markowitz)

50 4.3. Objective 3 Objective 3 analyses whether the inclusion of international equity and other assets ETFs into equity portfolios reduces expected historical portfolio risk using the Single Index Model variance-covariance matrix. Hypothesis 5 has a null hypothesis that states, ‘Portfolio 6 produces a lower GMVP than Portfolio 7’, and has an alternative hypothesis that states that, ‘Portfolio 6 does not produce a lower GMVP than Portfolio 7’. Portfolio 6’s GMVP has an expected portfolio risk of 2.12% and has an expected portfolio return of 0.82%. Portfolio 7’s GMVP has an expected portfolio risk of 1.51% and has an expected portfolio return of 0.69%. For Hypothesis 5, we therefore reject the null hypothesis and accept the alternative hypothesis, ‘Portfolio 6 does not produce a lower GMVP than Portfolio 6’. Hypothesis 6 has a null hypothesis that states, ‘Portfolio 7 does not produce a lower GMVP than Portfolio 8’, and has an alternative hypothesis that states, ‘Portfolio 7 does not produce a lower GMVP than Portfolio 8’. Portfolio 8’s GMVP has an expected portfolio risk of 1.04% and an expected portfolio risk of 0.51%. For Hypothesis 6, we reject the null hypothesis and accept the alternative hypothesis, ‘Portfolio 7 does not produce a lower GMVP than Portfolio 8.’ Hypotheses 5 and 6 are noted below. Chart 29 shows that. Portfolio 8 has a lower expected portfolio risk, this is shown by moving from Portfolio 6’s GMVP to Portfolio 7’s GMVP to Portfolio 8’s GMVP. Here the laws of risk to return are more commonly depicted, lower expected portfolio risk for lower expected portfolio return. From Portfolio 6 to 8 the expected portfolio risk reduction is 108 basis points and the expected portfolio return reduction is 31 basis points. Hypothesis 5: H0 Portfolio 6 produces a lower GMVP than Portfolio 7. Reject H1: Portfolio 6 does not produce a lower GMVP than Portfolio 7. Accept Hypothesis 6: H0 Portfolio 7 produces a lower GMVP than Portfolio 8. Reject H1: Portfolio 7 does not produce a lower GMVP than Portfolio 8. Accept Chart 29: Objective 3 – Reducing Portfolio Risk with International Equity and Other Asset ETFs using SIM VCV matrix 0.00%0.50%1.00%1.50%2.00%2.50%1.00% 1.50% 2.00% 2.50% 3.00% 3.50%Monthly Expected Portfolio Risk (σ)Portfolio 6: Long-Short (SIM)Portfolio 7: Long-Short incl. ETFs (SIM)Portfolio 8: Overweight Bond ETF 60% (SIM)

51 4.4. Objective 4 Objective 4 states that in equity portfolios using ETFs, does the Shrinkage Markowitz variance-covariance matrix and the Shrinkage Single Index Model variance-covariance matrix underestimate expected portfolio risks when compared to the Markowitz variance-covariance matrix and Single Index Model variance-covariance matrix by computing a lower Global Minimum Variance Portfolio on the Efficient Frontier? Hypothesis 7 has a null hypothesis that states, ‘Portfolio 9 overestimates the GMVP on the Efficient Frontier when compared to Portfolio 4’, and has an alternative hypothesis that states, ‘Portfolio 9 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 4’. Portfolio 4 estimates that the expected historical portfolio risk is 1.84% and the expected historical portfolio return is 0.72%. Portfolio 9 estimates the expected historical portfolio risk is 1.50% and has an expected historical portfolio return of 0.71%. For Hypothesis 7 we reject the null hypothesis and accept the alternative hypothesis, see below. Chart 30 shows the Efficient Frontier for Portfolios 4 and 9. Hypothesis 7: H0 Portfolio 9 overestimates the GMVP on the Efficient Frontier when compared Portfolio 4. Reject H1: Portfolio 9 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 4. Accept Chart 30: Objective 4 (Hypothesis 7) - Demonstrating the Difference in Estimating the Efficient Frontiers using the Shrinkage Markowitz VCV Matrix and the Markowitz VCV Matrix 0.00%0.50%1.00%1.50%2.00%2.50%1.25% 1.75% 2.25% 2.75% 3.25% 3.75%Monthly Expected Portfolio Risk (σ) Portfolio 4: Long-Short incl. ETFs(Markowitz)Portfolio 9: Long-Short incl. ETFs(Shrinkage Markowitz)Shrinkage VCVunderestimating theGMVP compared toSample VCV

52 Hypothesis 8 has a null hypothesis that states, ‘Portfolio 10 overestimates the GMVP on the Efficient Frontier when compared to Portfolio 5’, and has an alternative hypothesis that states, ‘Portfolio 10 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 5’. Portfolio 5 estimates that the expected historical portfolio risk is 1.26% and the expected historical portfolio return is 0.51%. Portfolio 10 estimates the expected historical portfolio risk is 1.06% and has an expected historical portfolio return of 0.52%. For Hypothesis 8 we reject the null hypothesis and accept the alternative hypothesis, see below. Chart 31 shows the Efficient Frontier for Portfolios 5 and 10. Hypothesis 8: H0 Portfolio 10 overestimates the GMVP on the Efficient Frontier when compared Portfolio 5. Reject H1: Portfolio 10 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 5. Accept Chart 31: Objective 4 (Hypothesis 8) - Demonstrating the Difference in Estimating the Efficient Frontiers from the Shrinkage Markowitz VCV Matrix and the Markowitz VCV Matrix 0.00%0.50%1.00%1.50%2.00%2.50%1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%Monthly Expected Portfolio Risk (σ)Portfolio 5: Overweight Bond ETF 60% (Markowitz)Portfolio 10: Overweight Bond 60% (Shrinkage Markowitz)Shrinkage VCVunderestimating theGMVP compared toSample VCV

53 Hypothesis 9 has a null hypothesis that states, ‘Portfolio 11 overestimates the GMVP on the Efficient Frontier when compared to Portfolio 7’ and has an alternative hypothesis that states, ‘Portfolio 11 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 7’. Portfolio 7 estimates that the expected historical portfolio risk is 1.51% and the expected historical portfolio return is 0.69%. Portfolio 11 estimates the expected historical portfolio risk is 1.36% and has an expected historical portfolio return of 0.70%. For Hypothesis 9 we reject the null hypothesis and accept the alternative hypothesis, see below. Chart 32 shows the Efficient Frontiers for Portfolios 7 and 11. Hypothesis 9: H0 Portfolio 11 overestimates the GMVP on the Efficient Frontier when compared Portfolio 7. Reject H1: Portfolio 11 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 7. Accept Chart 32: Objective 4 (Hypothesis 9) - Demonstrating the Difference in Estimating the Efficient Frontiers using the Shrinkage SIM VCV Matrix and the SIM VCV Matrix0.00%0.50%1.00%1.50%2.00%2.50%1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50% 3.75%Monthly Expected Portfolio Risk (σ)Portfolio 7: Long-Short incl. ETFs (SIM)Portfolio 11: Long-Short incl. ETFs (Shrinkage SIM)Shrinkage VCVunderestimating theGMVP comparedto SIM VCV

54 Hypothesis 10 has a null hypothesis that states, ‘Portfolio 12 overestimates the GMVP on the Efficient Frontier when compared to Portfolio 8’ and has an alternative hypothesis that states, ‘Portfolio 12 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 8’. Portfolio 8 estimates that the expected historical portfolio risk is 1.04% and the expected historical portfolio return is 0.51%. Portfolio 12 estimates the expected historical portfolio risk is 0.97% and has an expected historical portfolio return of 0.52%. For Hypothesis 10 we reject the null hypothesis and accept the alternative hypothesis, see below. Chart 33 shows the Efficient Frontier for Portfolios 8 and 12. Hypothesis 10: H0 Portfolio 12 overestimates the GMVP on the Efficient Frontier when compared Portfolio 8. Reject H1: Portfolio 12 does not overestimate the GMVP on the Efficient Frontier when compared Portfolio 8. Accept Chart 33: Objective 4 (Hypothesis 10) - Demonstrating the Difference in Estimating the Efficient Frontiers from using the Shrinkage SIM VCV Matrix over the SIM VCV Matrix0.00%0.50%1.00%1.50%2.00%2.50%0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50% 3.75%Monthly Expected Portfolio Risk (σ)Portfolio 8: Overweight Bond ETF 60% (SIM)Portfolio 12: Overweight Bond ETF 60% (Shrinkage SIM)Shrinkage VCVunderestimating theGMVP comparedto SIM VCV

55 5. Discussion of Results 5.1. Objective 1 Analysis Chart 34 shows a reduction in expected portfolio risk of 42% for the same level of expected portfolio monthly return of 0.97% when using a long-short optimised strategy over an equal weighted long strategy, while keeping constant the number of securities in the portfolio. Chart 35 shows Portfolio 3’s weightings to achieve 0.97% of expected portfolio return for 2.58% of expected portfolio risk. We note, that not all trading or investment funds allow short-sells in the fund mandates, however if permitted, there are significant risk reduction benefits from using a long-short strategy. Chart 34: Reduction in Portfolio Risk for a Portfolio Return of 0.97% using Portfolios 1 and 3 Chart 35: Weightings of Portfolio 3 for a Portfolio Return of 0.97% and an Expected Portfolio Risk of 2.58% 0.00%0.50%1.00%1.50%2.00%2.50%2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00%Monthly Expected Portfolio Risk (σ)Portfolio 1: Equal Weight Long (Markowitz) Portfolio 3: Long-Short (Markowitz)2.58%, 0.97% 4.48%, 0.97%-25.00%-20.00%-15.00%-10.00%-5.00%0.00%5.00%10.00%15.00%20.00%25.00%

56 Chart 36 shows the difference in the Global Minimum Variance Portfolio from Portfolios 2 and 3. Chart 36 shows if an investor held a long-only weighted portfolio of our 25 stocks in the ‘base portfolio’ (Portfolio 2) and then included short-sells into the portfolio (Portfolio3) then one could increase the portfolio return for less portfolio risk. From Portfolio 2 to Portfolio 3 the portfolio risk reduction is 15% while generating an increase in portfolio return of 21%. This is contrary to the traditional risk to reward relationships of lower risk, lower return or higher risk, higher return. We state that Portfolio 3 dominates Portfolio 2, when looking at historical portfolio risk and return. Chart 36: Reduction of Portfolio Risk for a Higher Portfolio Return using Portfolios 2 and 3 0.00%0.50%1.00%1.50%2.00%2.50%2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00% 5.50% 6.00%Monthly Expected Portfolio Risk (σ)Portfolio 2: Weighted Long (Markowitz) Portfolio 3: Long-Short (Markowitz)3.06%, 0.75%2.62%, 0.91%

57 5.2. Objective 2 & 3 Analysis Chart 37 shows a reduction in expected portfolio risk of 44% for the same level of expected portfolio return of 0.91%, when including exchange-traded funds and using the Markowitz variance-covariance matrix. We note that only U.S. government and corporate bonds with maturities of 10 to 30 years (AGG) and gold bullion (GLD) exchange-traded funds were used in the portfolio and thus provided diversification benefits. This accepts Curcio and Ziobrowski, (1991), Viceira et al., (2017) and Shahzad (2017) research that states that U.S. fixed income assets diversify equity portfolios by producing an inverse correlation to U.S. equitiess. We accept Hoang, et al. (2015) research, which states that gold lowers expected portfolio risk in equity portfolios and we reject Michis (2014) research, which states that gold has the lowest variance when compared to U.S. fixed income. Michis (2014) research suggests that gold is a more diversified asset over fixed income for equity portfolios. We reject this statement and state that gold does have an impact in reducing the expected portfolio risk, however, not as much of an influence as the salient U.S. fixed income market. Evidence of this is given in our research by the amount of weighting assigned to the ETFs in the optimisation process, (Chart 38). See Appendix 8.5.1, 8.5.2 and 8.5.3 for the portfolio parameters in Solver. EFA and VNQ were not used suggesting that international equity ex-U.S. ETFs (EFA) and U.S. real estate (VNQ) do not provide diversification benefits. These results therefore reject Wagner and Lau (1971), Solnick (1974) and Huang and Lin (2011) research, that state that international diversification lowers overall expected portfolio risk. Furthermore, we reject Rees and Selcuk-Kestel’s (2014) research, which states that the U.S. real estate market can in some situations provide diversification benefits, as our results show on two portfolio GMVP’s use VNQ, (Table 7). Chart 38 shows Portfolio 5’s weightings to achieve 0.91% of expected portfolio return for 1.46% of expected portfolio risk. Chart 37: Reduction in Portfolio Risk for the Same Level of Portfolio Return of 0.91% using Portfolios 3 to 5 0.00%0.50%1.00%1.50%2.00%2.50%0.75% 1.25% 1.75% 2.25% 2.75% 3.25% 3.75% 4.25% 4.75%Monthly Expected Portfolio Risk (σ) Portfolio 3: Long-Short(Markowitz)Portfolio 4: Long-Shortincl. ETFs (Markowitz)Portfolio 5: OverweightBond ETF 60% (Markowitz)2.62%, 0.91%1.46%, 0.91%

58 Chart 38: Weightings of Portfolio 5 for a Portfolio Return of 0.91% and a Portfolio Risk of 1.46% Chart 39 shows a reduction in expected portfolio risk of 42% for the same level of expected portfolio risk of 0.82%, when including exchange-traded funds and using the Single Index Model variance-covariance matrix. Chart 40 shows the asset weightings for Portfolio 8 to achieve 0.82% of expected portfolio return for 1.24% of expected portfolio risk. Chart 39: Reduction in Portfolio Risk for the Same Level of Portfolio Return (0.82%) using Portfolios 6 to 8 -10.00%-5.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%0.00%0.50%1.00%1.50%2.00%2.50%1.00% 1.50% 2.00% 2.50% 3.00% 3.50%Monthly Expected Portfolio Risk (σ)Portfolio 6: Long-Short (SIM)Portfolio 7: Long-Short incl. ETFs (SIM)Portfolio 8: Overweight Bond ETF 60% (SIM)2.12%, 0.82%1.24%, 0.82%

0 00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%55.00%60.00%

Chart 40: Weightings of Portfolio 8 for a Portfolio Return of 0.82% and a Portfo We state that when AGG and GLD exchange-traded funds are incluboth the Markowitz’s variance-covariance matrix and the Single Indematrix compute similar portfolio risk reductions of 44% and 42% reshort portfolios. This duplication of analysis using two differencovariance matrix confirms and gives creditability to the usage ofGLD) for the purpose of portfolio risk reduction. It must be said tvariance-covariance matrices uses different weightings to achieverisk, see Charts 38, 40 and Appendix 8.10 for a Chart comparingaccept Quinn and Voth’s (2008) research, which states that global eand international diversification may not be the most efficient metshows that AGG dominates the ETF asset weighting on all portfoliosEFA shows diversification benefits in Portfolios 9, 10 and 11. Lasleast amount of diversification with only having fractional weightings ETFs EFA AGG GLDPortfolio 4 0.00% 25.00% 15.66Portfolio 5 0.00% 60.00% 5.30Portfolio 7 0.00% 25.00% 10.73Portfolio 8 0.00% 60.00% 5.87Portfolio 9 1.62% 25.00% 9.60Portfolio 10 0.63% 60.00% 4.38Portfolio 11 1.03% 25.00% 8.74Portfolio 12 0.50% 60.00% 4.76 Table 7: List of ETF Weightings for the GMVP of Portfolios 4,5,7,8,9,10,11 &12 -15.00%-10.00%-5.00%.

ldxst eh t qh tl %%%%%%%%

59 io Risk of 1.24% ed into long-short equity portfolios Model (SIM) variance-covariance pectably when compared to long-ways to compute the variance-xchange-traded funds (AGG and ough that the Markowitz and SIM his reduction in expected portfolio the different portfolio weights. We uity markets are highly correlated od of Portfolio Selection. Table 7 listed in Table 7, followed by GLD. y, we state that VNQ creates the in Portfolios 11 and 12. VNQ 0.00%0.00%0.00%0.00%0.00%0.00%0.23%0.09%

60 5.3. Objective 4 Analysis Portfolio 4’s minimum expected portfolio risk using the Markowitz VCV matrix for long-short portfolios including ETFs is 1.84%. Portfolio 7’s minimum expected portfolio risk using the SIM VCV matrix for long-short portfolios including ETFs is 1.51%. Portfolio 9’s minimum expected portfolio risk using the Shrinkage Markowitz VCV matrix for long-short portfolios including ETFs is 1.50%. Portfolio 11’s minimum expected portfolio risk using the Shrinkage SIM VCV matrix for long-short portfolios including ETFs is 1.36%. Therefore, the minimum expected portfolio risk range using the Markowitz VCV, the SIM VCV matrix, the Shrinkage Markowitz VCV matrix and the Shrinkage SIM VCV matrix is in between 1.36% and 1.84%. This expected portfolio risk range is 48 basis points. Chart 41 show the GMVP range for Portfolios 4, 7, 9 and 11. This shows that different variance-covariance matrices compute different GMVP calculations and thus investors and traders must understand the importance of VCV matrix selection for GMVP predictions. Comparing the GMVP asset weightings in Sections 3.6.4, 3.6.7, 3.6.9 and 3.6.11 the Shrinkage VCV matrix GMVP weightings create sub-optimal portfolios as short-sell weightings are fairly insignificant when using Portfolio 9 and 11 Shrinkage VCV matrices over Portfolios 4 and 7 that use the Markowitz and Sim VCV matrices. This shows that the Shrinkage VCV matric could be understating the portfolio risk. Chart 41: GMVP Range for Long-Short including ETFs Equity Portfolio using Multiple Variance-Covariance Portfolio 5’s minimum expected portfolio risk using the Markowitz VCV matrix for long-short portfolios including ETFs and an overweighting to AGG of 60% is 1.26%. Portfolio 8’s minimum expected portfolio risk using the SIM VCV matrix for long-short portfolios including ETFs and an overweighting to AGG of 0.00%0.50%1.00%1.50%2.00%2.50%1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50% 3.75%Monthly Expected Portfolio Risk (σ)Portfolio 4: Long-Short incl. ETFs (Markowitz)Portfolio 9: Long-Short incl. ETFs (Shrinkage Markowitz)Portfolio 7: Long-Short incl. ETFs (SIM)Portfolio 11: Long-Short incl. ETFs (Shrinkage SIM)GMVP Range

61 60% is 1.04%. Portfolio 10’s minimum expected portfolio risk using the Shrinkage Markowitz VCV matrix for long-short portfolios including ETFs and an overweighting to AGG of 60% is 1.06%. Portfolio 12’s minimum expected portfolio risk using the Shrinkage SIM VCV matrix for long-short portfolios including ETFs and an overweighting to AGG of 60% is 0.97%. Therefore, the minimum expected portfolio risk range using the Markowitz VCV, the SIM VCV matrix, the Shrinkage Markowitz VCV matrix and the Shrinkage SIM VCV matrix is in between 0.97% and 1.26%. This expected portfolio risk range is 29 basis points. Chart 42 show the GMVP range for Portfolios 5, 8, 10 and 12. This confirms that different variance-covariance matrices compute different GMVP calculations and thus market participants must understand the importance of VCV matrix selection for GMVP predictions. Ledoit and Wolf’s (2004) paper states that the Shrinkage method on the VCV matrix produces superior estimates than the Markowitz or SIM VCV matrix for modelling historical portfolio risk. Our results do show a lower GMVP calculation for the Shrinkage VCV matrix, however we do not state this is due to a superior estimation matrix for portfolio risk. Ledoit and Wolf (2004) state the Shrinkage VCV matrix is an improved matrix for estimation of portfolio risk. We do not have enough evidence in this research to agree or disagree with this statement. All our research can say is that a lower figure for the GMVP is produced when using the Shrinkage VCV matrix, we assume that these lower GMVP values using the Shrinkage VCV matrix are the simplifications of underestimating all the portfolio risks. Chart 42: GMVP Range for an Equity portfolio using an Overweight Bond ETF of 60% using Multiple Variance-Covariance 0.00%0.50%1.00%1.50%2.00%2.50%0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% 3.25% 3.50% 3.75% 4.00%Monthly Expected Portfolio Risk (σ)Portfolio 5: Overweight Bond ETF 60% (Markowitz)Portfolio 10: Overweight Bond ETF 60% (Shrinkage Markowitz)Portfolio 8: Overweight Bond ETF 60% (SIM)Portfolio 12: Overweight Bond ETF 60% (Shrinkage SIM)GMVP Range

63 Chart 44, shows how to calculate the Sharpe ratio for Portfolio 3. Our excess return is the expected portfolio return minus the risk-free rate and then divided by the portfolio risk. As you can see from Chart 44, Portfolio 3 has a Sharpe ratio of 0.52. Chart 44: Sharpe Ratio Calculation for Portfolio 3 Chart 45, shows how to compute the Sharpe ratio for Portfolio 5. As you can see from Chart 53, Portfolio 5 has a Sharpe ratio of 0.56. Portfolio 5’s Sharpe ratio (0.56) is higher than Portfolio 3’s Sharpe ratio (0.52). This confirms Chart 43’s analysis, the closer the Markowitz Bullet to the y-axis, the higher the Sharpe ratio. Chart 45: Sharpe Ratio Calculation for Portfolio 5 0.00%0.50%1.00%1.50%2.00%2.50%3.00%3.50%0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00% 5.50% 6.00%Monthly Expected Portfolio Risk (σ) CMLR(f) =0.18%E(r[p])= 2.50% Sharpe Ratio2.32% /4.40% = 0.52 E(r[p]) - R(f) = 2.32%σ(p) =4.40%0.00%0.50%1.00%1.50%2.00%2.50%0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00%Monthly Expected Portfolio Risk (σ) CMLR(f) =0.18%E(r[p])= 1.60% σ(p) = 2.50%Sharpe Ratio1.42% /2.50%=0.56 E(r[p]) - R(f) = 1.42%

64 We compared the Sharpe ratios from Portfolios 3 and 5 with the S&P 500 index (SPX – largest 500 stocks in the U.S.) and the S&P 100 index (OEX – largest 100 stocks in the U.S.) to demonstrate the dominance of Portfolio 3 and 5 over the market indexes from a risk adjusted returns point of view. Table 8 shows the Sharpe ratios for the S&P 500 index (SPX) and the S&P 100 index (OEX) from the last 10 years. We state that the Sharpe ratios for both SPX and OEX are close to zero and thus demonstrates poor risk adjusted returns. Furthermore, we state that the recession of December 2007 to June 2009, (www.federalreserve.gov, 2017) significantly affected the Sharpe ratios of the SPX and OEX when compared to Portfolio 3 and 5. This shows the importance of including short-sells into an equity portfolio for hedging market volatility as well as the added benefit of including ETFs for further asset volatility4 reduction. SPX OEX Risk 4.53% 4.37%Return 0.34% 0.30%Risk Free 0.18% 0.18%Sharpe Ratio 0.04 0.03 Table 8: Sharpe Ratio of SPX and OEX4 We use risk and volatility interchangeably.

65 5.5. CAL for Portfolio Risk Comparison In this section, we discuss the use of the Capital Allocation Line (CAL) in building a two-asset portfolio, comprising of the Tangency Portfolio for Portfolio 3 and the risk-free rate. We then compare this two-asset portfolio to long-short equity portfolios including ETFs (Portfolio 5 – Overweight Bond ETF 60%) to see which option creates the higher return to risk ratio. Formula 16 shows the portfolio return of a weighting in a risky portfolio and the risk-free rate. Formula 17 shows the computation for a two-asset portfolio risk. Again, we used a weighting in the risky portfolio (Tangency Portfolio 3) and a weighting in the risk-free rate. Formula 16: Portfolio Return between Tangency Portfolio and a Risk-Free Rate (Bodie, et al., 2014) Formula 17: Portfolio Risk between Tangency Portfolio and a Risk-Free Rate (Bodie, et al., 2014) Chart 46 shows the Tangency Portfolio for Portfolio 5. The Tangency Portfolio has a portfolio return of 1.6% and a portfolio risk of 2.5%. Portfolio 5’s return and risk values are produced by having a 55% weighting in ETFs (47% in AGG and 8% in GLD) and a 45% weighting in equities. We then use Chart 47 to target a portfolio risk of 2.5%. We used a weighting in the Tangency Portfolio for Portfolio 3 of 44% and a weighting in the risk-free rate of 56%. This achieves a portfolio risk of 2.50%, however, the portfolio return is 1.50%. Showing a reduction of 10 basis points in portfolio return when using the CAL in Portfolio 3 over Portfolio 5. Using Chart 47 to target a portfolio return of 1.60%. We used a weighting in the Tangency Portfolio for Portfolio 3 of 40% and a weighting in the risk-free rate of 60%, This achieves a portfolio return of 1.60%, however, the portfolio risk is 2.80%. Showing an increase in portfolio risk of 30 basis points. We therefore state that, Portfolio 5 dominates Portfolio 3 and a combination of the risk-free rate. Table 9 shows a summary of the analysis above.

Tangency Portfolio for Porfolio 5 (Long-Short incl. ETFs with Overwight Bond ETF[AGG])

W(ETF)

W(Equities)

R(p)

σ(p)

55%

45%

1.60%

2.50%

Portfolio 3 (Long-Short only) & Risk Free

W(Rf)

W(T)

R(p)

σ(p)

56%

44%

1.50%

2.50%

60%

40%

1.60%

2.80%

66 Table 9: Comparing the Tangency Portfolio of Portfolio 5 with a combination of Tangency Portfolio of Portfolio 3 and the Risk-Free Rate using the CAL Chart 46: Tangency Portfolio for Portfolio 5 Chart 47: Weighting in Tangency Portfolio and Risk-Free Rate for Portfolio 3 0.00%0.50%1.00%1.50%2.00%2.50%0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00%Monthly Expected Portfolio Risk (σ)Tangency Portfoliofor Portfolio 5 CALE(r[p])= 1.60%R(f) =0.18% GMVP0.00%0.50%1.00%1.50%2.00%2.50%3.00%3.50%0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00% 5.50% 6.00%Monthly Expected Portfolio Risk (σ)Compare toPortfolio 5TangencyPortfolio CALR(f) =0.18% T(1-T)

67 Next, we target the Global Minimum Variance Portfolio (GMVP) of Portfolio 5. The GMVP for Portfolio 5 has a portfolio risk of 1.26% and a portfolio return of 0.51%. Table 10 shows a two-asset portfolio which includes the risk-free rate and the Tangency Portfolio for Portfolio 3. As you can see from Table 9, using a weighting in the Tangency Portfolio for Portfolio 3 of 28% and a weighting in the risk-free rate of 72% produces a portfolio risk of 1.25% and a portfolio return of 0.85%, (Chart 48). A combination in the Tangency Portfolio for Portfolio 3 and the risk-free rate produces the same amount of portfolio risk when compared to the GMVP of Portfolio 5, however this combination in the CAL produces a 40% higher portfolio return. We state that if fund managers wanted to use the lowest portfolio risk in Portfolio 5 (the GMVP) they would be better off using a combination of the Tangency Portfolio in Portfolio 3 and the risk-free rate. In this example Portfolio 3 and the risk-free rate dominates Portfolio 5. T R(f)Average Return (M) 2.50% 0.18%Average Risk (M) 4.40% 0.00%%W in T %W in R(f) PortfolioReturn PortfolioRisk(Tan Portfolio) (1-T) 0.00% 100.00% 0.18% 0.00%1.00% 99.00% 0.20% 0.04%2.00% 98.00% 0.23% 0.09%28.00% 72.00% 0.85% 1.25% 98.00% 2.00% 2.45% 4.31%99.00% 1.00% 2.48% 4.36%100.00% 0.00% 2.50% 4.40% Table 10: Comparing the Minimum Variance on the GMVP of Portfolio 5 with a combination of Tangency Portfolio of Portfolio 3 and the Risk-Free Rate using the CAL Chart 48: Comparing the GMVP of Portfolio 5 with a Combination of the Risk-Free Rate and the Tangency Portfolio of Portfolio 3 0.00%0.50%1.00%1.50%2.00%2.50%3.00%3.50%0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 4.50% 5.00% 5.50% 6.00%Monthly Expected Portfolio Risk (σ) CALE(r[p])= 1.50%R(f) =0.18% T(1-T) Showing Higher Returns forSame Risk when Comparing tothe GMVP of Portfolio 5

5.6. Benchmark Analysis Chart 49 shows the GMVP for Portfolio 5, as well as the historical last 10-year risk and return values for three U.S. equity market indexes. SPX is the ticker5 for the S&P 500 index, which includes the top 500 stocks in the U.S. by market capitalisation. OEX represents the top 100 U.S. equities in the U.S. by market capitalisation and SML represents 600 small capitalisation stocks in the U.S. As you can see from Chart 49, the indexes show normalities in the risk to reward profiles. Large capitalisation stocks have lower returns but lower risks and smaller capitalisation stocks have higher returns but high risks, (Switzer, 2010). We state that the GMVP for Portfolio 5 dominates all three indexes by computing a higher portfolio return for a lower portfolio risk. Chart 49: Comparing the GMVP of Portfolios 3 and 5 with Equity Market Indexes One can reduce portfolio risk by holding only 25 U.S. mega-cap equities and two exchange-traded funds (AGG and GLD) when compared to the S&P 100 index (OEX), the S&P 500 index (SPX) and the S&P 600 Small Cap Index (SML), see Chart 49. We note the potential irrelevance of comparing our mega-cap equity portfolio to the SML index, however this benchmark was used purely to illustrate the different risk to reward profiles when looking at market capitalisation. Table 11 shows the reduction in portfolio risk and the increase in portfolio return when comparing Portfolio 5 to OEX, SPX and SML. 4.53%, 0.34%4.37%, 0.30% 5.75%, 0.46%0.00%0.10%0.20%0.30%0.40%0.50%0.60%0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00%Monthly Portfolio Risk Portfolio 5 (MarkowitzVCV matrix)SPXOEXSMLReducing Portfolio Risk andIncreasing Portfolio Return

OEX

SPX

SML

Portfolio 5 (GMVP)

Reduction in Risk

71.20%

72.20%

78.10%

Increase in Retun

41.20%

33.40%

9.80%

68 Table 11: Comparing the GMVP for Portfolio 5 to OEX, SPX and SML Equity Indexes5 The ticker is the asset symbol which is used in an abbreviation

69 6. Conclusions 6.1. Research Recapitulation This research addresses the use of historical volatility or standard deviation as a measure of expected portfolio risk. Our modelling is based on ex-post data predicting and determining future equity price behaviour. When including short-sells into equity portfolios one can reduce portfolio risk and increase portfolio return, therefore rejecting the convention view of lower risk, lower return or higher risk, higher return. We note that only if assets that were shorted continue to decrease in asset price. This is due to rational investing or trading expectations of being rewarded for taking addition risks. The Global Minimum Variance Portfolio for Portfolio 2 (Weighted long) has a portfolio risk of 3.06% and a portfolio return of 0.75%. The Global Minimum Variance Portfolio for Portfolio 3 (long-short) has a portfolio risk of 2.62% and a portfolio return of 0.91%. This shows that including short-sells into equity portfolios can reduction portfolio risk by 15% while increasing the return by 21%. When comparing long-short equity portfolios using the Markowitz and Single Index Model variance-covariance matrix one can reduce portfolio risk by 44% and 42% respectability by including two exchange-traded funds (ETFs); U.S. fixed income, AGG and gold bullion, GLD, (see Section 5.2.). Using AGG and GLD in long-short strategies over using a long-only equal weighted equity portfolio on can keep the portfolio return constant but reduce portfolio risk by 65%, (Chart 50). Chart 50: Efficient Frontiers for Portfolios 1 to 5 using the Markowitz VCV matrix 4.48%, 0.97%1.57, 0.97%0.00%0.50%1.00%1.50%2.00%0.75% 1.25% 1.75% 2.25% 2.75% 3.25% 3.75% 4.25%Monthly Portfolio Risk (σ)Portfolio 1: Equal Weight LongPortfolio 2: Weighted LongPortfolio 3: Long-ShortPortfolio 4: Long-Short incl. ETFsPortfolio 5: Overweight Bonds 60%

70 We note that the Markowitz and the Single Index Model (SIM) VCV matrices compute different values for the GMVP of Portfolios 1 to 8. Additionally, we added a Shrinkage VCV matrix to the Markowitz and SIM VCV matrices and again our calculations vary dramatically. We debate the accuracy for using different variance-covariance (VCV) matrices other than the Markowitz VCV matrix. Using the Markowitz VCV matrix, the SIM VCV matrix, the Shrinkage Markowitz VCV matrix and the Shrinkage SIM VCV matrix on long-short equity portfolios using ETFs, the minimum portfolio risk is in a range between 1.36% to 1.84% for portfolio 4 and 0.97% to 1.26% for Portfolio 5. Showing a portfolio risk range of 48 basis points and 29 basis points respectability. This shows the importance of VCV matrix selection for accurate portfolio risk and return modelling as the asset weighting change depending on what VCV matrix is used in the portfolio optimisation process. Hypothetically speaking, if the equity markets started to sell-off and sentiment became bearish on equities, many fund managers would look to rotate from equities into bonds (if fund mandates allow). We theorised in holding a 100% weighted U.S. bond ETF (AGG) portfolio. The portfolio risk would simply be the expected monthly standard deviation (σM) of AGG, which is 1.08% forthe expected mean return is 0.31%. The GMVP asset weightings for Portfolio 12, which uses the Shrinkage Single Index Model VCV matrix are shown in Section 4. 4. 12. Using these weightings, the portfolio risk is 0.97% and the portfolio return is 0.52%. Portfolio 12’s GMVP has an expected monthly portfolio risk of 10.2% lower than the monthly risk of AGG, while computing a 68% higher expected portfolio return. This clearly shows how using different VCV matrices can distort portfolio risk and return calculations. Chart 51 shows the risk and return for a risk-free asset, AGG bond ETF, Portfolio 5 (Markowitz VCV matrix) and Portfolio 12 (Shrinkage Single Index Model VCV matrix).Chart 51: Risk to Return for Risk-Free Asset, AGG Bond ETF and the GMVP for Portfolio 5 (Markowitz VCV matrix) and 12 (Shrinkage SIM VCV matrix) 0.00%0.10%0.20%0.30%0.40%0.50%0.60%0.00% 0.20% 0.40% 0.60% 0.80% 1.00% 1.20% 1.40%Risk (σ) Risk-free RateAGGPortfolio 5Portfolio 12

81 8.5. Solver Parameters in Microsoft Excel 8.5.1. Portfolio 3 (Long-Short) Portfolio Risk EquationPortfolio Asset WeightingsTo Minimise thePortfolio RiskPortfolio Target return =Expected Portfolio ReturnAsset weights = 100%Equity WeightingConstraints from-25% to +25%

82 8.5.2. Portfolio 4 (Long-Short incl. ETFs)Adding ETFs Weightingconstraints from 0% to +25%.ETFs are long-only instrumentsand cannot be shorted on thesame security

83 8.5.3. Portfolio 5 (Overweight Bond ETF 60%) Changing the AGG (U.S. bondETF) weighting on the longside from +25% to +60%

88 8.10. Comparing the GMVP Portfolio 5 & 8 Weightings (Markowitz and SIM VCV matrices) The chart below shows the same portfolio constituents; however, Portfolio 5 uses the Markowitz variance-covariance (VCV) matrix and Portfolio 8 uses the Single Index Model (SIM) VCV. We notice that depending on what VCV matrix one uses determines what weightings are allocated to different assets. The Markowitz VCV matrix has a larger long position in U.S. fixed income (AGG) over the SIM VCV matrix. Furthermore, the Markowitz VCV matrix has a larger aggregate portfolio short position over the SIM VCV matrix, with larger shorts in J.P. Morgan (JMP), Wells Fargo (WFC), Bank of America (BAC), General Electric (GE), Chevron (CVX) and Citigroup (C). -15.00%-10.00%-5.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%55.00%60.00% Portfolio 5 (Markowitz VCV matrix) Portfolio 8 (SIM VCV matrix)