Excerpt

## Contents

1 Introduction

2 Investment Decision

3 Portfolio Selection

3.1 Return and Risk Level Calculations

3.2 Correlation and Covariance Calculations

3.3 The Efficient Frontier

3.4 Optimal Portfolio

4 Performance Measurement

4.1 Risk Adjusted Performance Metrics

5 Investments across Asset Classes

5.1 International Diversification / Emerging Markets

5.2 Commodities

5.3 Corporate Bonds / Government Bonds (‘‘Gilts‘‘)

5.4 Real Estate

5.5 Hedge Funds

6 Conclusion - Analysis Limitation

7 References

8 Appendix

List of Figures

Figure 1: Investment decision process

Figure 2: Investment Portfolio under Examination

Figure 3: Average Risk and Return Levels per individual portfolio security

Figure 4: Performance levels of two individual portfolio securities

Figure 5: Efficient Frontier - Relevant constrains

Figure 6: Left - Unrestricted frontier theoretical model / Right - Restricted and unrestricted frontiers

Figure 7: Selected portfolios along the eff. frontier plus optimal portfolio in blue (Short sales not allowed)

Figure 8: Selected portfolios along the eff. frontier plus optimal portfolio in blue (Short sales allowed)

Figure 9: Sharp Ratio

Figure 10: M2 Measure

Figure 11: Jensen’s Alpha Short Sales not allowed (left) Short Sales allowed (right)

Figure 12: Treynor Ratio

Figure 13: Risk Adjusted Performance Metrics - Calculation Basis

Figure 14: International Stock Markets - Correlation Coefficients

Figure 15: Gains from International Portfolio Diversification

Figure 16: Correlation of Commodity Futures vs. Stocks & Bonds

Figure 17: UK gilts and corporate bonds correlations with other asset classes

Figure 18: EPRA/NAREIT Global Real Estate Index

Figure 19: Comparison of Hedge Fund Returns with Major Asset Classes

Figure 20: Portfolio risk in relation to the number of stocks in the portfolio

## List of Abbreviations

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## 1 Introduction

According Warren Buffett, financial investors should never purchase a security, if they cannot accept that the value might be cut in half (Schömann-Finck, 2010).

One of the most successful global investors has illustrated with this quote the risks involved in the business of financial investments. In order to optimize risk-return tradeoffs, scientific research has developed efficient diversification techniques. This paper examines the process of portfolio diversification based on a sample of 10 randomly selected securities. First the optimal portfolio is identified in order to evaluate its performance against the market trend via industry accepted benchmarking comparison tools in a second step. Finally, potential portfolio gains, achieved via diversification across additional asset classes, are discussed and evaluated.^{1}

## 2 Investment Decision

According to Bodie et al. (2008) the investment decision process can be separated into three major steps (see figure 1): Capital Allocation, Asset Allocation and Security Selection. For the purpose of this paper 100% of the available funds are assumed to be allocated into stocks. The portfolio created (see figure 2) consists of 10 randomly selected securities taken from the FTSE 100 index (see enclosed CD, files 1 & 2).^{2}

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Figure 1: Investment decision process (Bodie et al., 2008)

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Figure 2: Investment Portfolio under Examination

## 3 Portfolio Selection

The Markowitz portfolio selection process can be divided into several steps.^{3} First the efficient frontier can be calculated in order to detect the optimal portfolio afterwards.^{4} Portfolio selection as a mean-variance optimization process is seeking for the optimal combination of average investment return and risk levels, delivering the highest risk premium per unit of risk. Several measures have to be calculated in order to define the efficient frontier in a first step (Bodie et al., 2008).

### 3.1 Return and Risk Level Calculations

The calculation of monthly return levels for each security is based on the continuously compounded return formula (see equation 1 and enclosed CD, file 2). In order to generate the average return per security across the observation period the arithmetic mean is applied (Time Weighted Returns, see figure 3, p.8).

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Figure 3: Average Risk and Return Levels per individual portfolio security (Data Base: Yahoo Finance, 2011)

### 3.2 Correlation and Covariance Calculations

The portfolio theory is based on the concept of combining securities with a contradicting performance. Based on the average monthly returns across the monitoring period the portfolio asset covariance and correlation matrix were calculated (see appendix a) & b) and enclosed CD, files 4 & 5)^{5}. Both measures describe the degree of similarity between two random variables. Figure 4 displays two securities out of the examined portfolio which tend to move similar and consequently provide only limited diversification potential (correlation coefficient: 0,62).

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Figure 4: Performance levels of two individual portfolio securities

### 3.3 The Efficient Frontier

Efficient portfolios provide investors the highest possible expected return for a given level of risk. The set of efficient portfolios is called the efficient frontier, which is generated in this paper via the Microsoft Excel Solver calculation tool. Based on the required constraints (see figure 5, p.8) the starting point of the efficient frontier (Minimum Variance Portfolio) can be obtained by running the solver variance minimization exercise^{6} (see enclosed CD, file 6). Repeating this exercise while changing the expected return levels delivers the individual efficient portfolios in order to draw the efficient frontier (short sales not allowed, see figure 6).

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Figure 5: Efficient Frontier - Relevant constrains

‘’Short selling is defined as selling a stock which the seller does not own at the time of trade’’ (Bombay Stock Exchange Limited, 2007). Short-sellers speculate on decreasing security prices in order to buy later on a lower price level than the asset was sold.^{7} The approval of short sales extends the efficient frontier, leading to an infinite upper bound as displayed in the theoretical model (see figure 6). Running the solver variance minimization exercise again delivers the unrestricted frontier (short sales allowed, see figure 6 and enclosed CD, file 7).

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Figure 6: Left - Unrestricted frontier theoretical model (Gruber, 2010) / Right - Restricted and unrestricted frontiers

### 3.4 Optimal Portfolio

Each portfolio located on the Capital Market Line (CML) is characterized by the highest possible Sharpe Ratio, which represents the CML slope.

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The CML intercept is defined by the monthly return on a risk free asset (see equation 2) which is calculated based on the annual return level of UK government bonds averaged across the observation period. Considering semiannual compounding, returns are transformed from an annual onto a monthly basis in order to match the stock analysis methodology (see enclosed CD, file 8). Using Excel-Solver in order to maximize the sharp ratio delivers the tangency point of the CML with the efficient frontier representing the optimal portfolio^{8} (see figure 7 / 8 and enclosed CD, files 6 & 7).

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Figure 7: Selected portfolios along the eff. frontier plus optimal portfolio in blue (Short sales not allowed)

In order to achieve the minimum variance portfolio under restricted short sales conditions even assets with a slightly negative return (BP, Red Elsevier) are considered due to the low overall correlation level to the additional shares (see figure 7). As the return level has to be increased in order to obtain the optimal portfolio, funds are progressively taken out of shares with negative weights and are concentrated on assets with a low coefficient of variation (see figure 3) and overall correlations (Centrica, Smith & Nephew).

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Figure 8: Selected portfolios along the eff. frontier plus optimal portfolio in blue (Short sales allowed)

Under unrestricted short sales conditions the total number of assets is included throughout all efficient portfolios. Than higher the expected return in order to obtain the optimal portfolio than more intensively assets with a negative weight will be short sold and vice versa (see figure 8).

**[...]**

^{1} The observation period for this analysis commenced in March 2006 and ended in March 2011. Individual security key performance indicators across the observation period were obtained from Yahoo Finance on a monthly basis.

^{3} Theory is based on several assumptions which are discussed by Brown and Reilly (2009, pp.181), however which will not be outlined in detail in this context.

^{4} The allocation of funds between the risky portfolio and the risk-free asset is not considered in this paper.

^{5} The covariance matrix delivered by MS Excel is not corrected for degrees-of-freedom bias. Consequently each individual covariance was multiplied by 60/59 to eliminate downward bias.

^{6} This exercise requires the elimination of constraint number 1 (see figure 5).

^{7} Short selling procedures are restricted to some extend: ‘’All classes of investors, viz., retail and institutional investors are permitted to short sell. Naked short selling shall not be permitted in the Indian securities market’’ (Bombay Stock Exchange Limited, 2007). ‘’ The US Security and Exchange Commission has traditionally held the belief that protections against abusive short selling are important for issuer and investor confidence (US Security and Exchange Commission, 2011).

^{8} Constraint number 1 (see figure 5, p.8) has to be removed from excel solver before running the maximization.

- Quote paper
- Patrick Daum (Author), 2011, Investment Portfolio Selection and Performance Measurement, Munich, GRIN Verlag, https://www.grin.com/document/193415

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