Securities, Futures and Options (SFO)

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Assignment

ICM 107 Securities, Futures and Options (SFO)

Question A)

1.1 Data

To implement the analysis, the data was downloaded from the Bloomberg. Firstly, the 20 French IT companies were selected from CAC All-Tradable Index in descending order of market capitalization. Secondly, the 121 dividend-adjusted monthly stock prices from 30/09/2004 to 30/09/2014 were collected per stock to obtain the 120 monthly total return data series from 29/10/2004 to 30/09/2014. Here, the total return means a return that is comprised of the rate of capital gain/loss plus dividend yield (Bodie, et al., 2014). Rather than using the monthly return, more precise analysis and performance measurement could be achieved by using the monthly total return. Lastly, 1 year T-bill was selected as a risk-free rate. As all monthly data such as return and standard deviation were annualized, it should be justifiable to matching the time horizon of the risk free rate with the annualized return and risk.

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Table 1: List of 20 French IT Companies

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Table 2: List of the Data Set

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1.2 Efficient frontier construction

The efficient frontier is constructed by finding the minimum variance for a given target expected return. This can be done using the Solver in Excel. Here, the assumptions and constraints for creating the efficient frontier were made as follows.

Table 3: Description of Assumptions and Constraints for Efficient Frontier

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As a result, the efficient frontier was created and plotted in Figure 1.

1.3 Market portfolio construction

The market portfolio is found by choosing the portfolio that maximizes the Sharpe ratio on the minimum variance frontier. Here, the assumptions and constraints for finding the market portfolio were made as follows.

Table 4: Description of Assumptions and Constraints for Market Portfolio

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Based on the calculation, the market portfolio with the highest Sharpe ration of 1.19 was found. Figure 1 shows that the green triangle mark on the efficient frontier represents the market portfolio.

Figure 1: Efficient Frontier and Market Portfolio

(2004-2014, No Short Selling)

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Question B)

The market portfolio shown in Figure 1 is a desirable one in terms of both a portfolio theory and a performance evaluation. Under the theory, all portfolios lying on the efficient frontier are said to be efficient from the perspective of a risk-return characteristic as the efficient frontier represents the minimum risk for a given return and the maximum return for a given risk. If the portfolio that does not lie on the efficient frontier, it could not be regarded as an optimal risky portfolio. Furthermore, the tangency portfolio of a straight line (i.e. Capital Market Line) from a risk free rate to the efficient frontier is called the market portfolio, which has the highest Sharpe ratio (Bodie, et al., 2014). As the Sharpe ratio represents a ratio to measure the investment efficiency, the market portfolio with the highest Sharpe ratio could be considered to be most efficient. Hence, all investors wish to hold the market portfolio.

Additionally, the performance evaluation confirms that the market portfolio is a practical one. Based on the performance analysis between the market portfolio and CAC All-Tradable Index, the market portfolio outperformed the stock index significantly. Specifically, the return of the market portfolio (18.2%) was more than five times higher than that of the index (3.3%). Moreover, the Sharpe ratio of the market portfolio (1.19) was considerably better than that of the stock index (0.1). On the other hand, the standard deviation, or risk of the market portfolio (13.9%) was smaller than that of the index (16.6%). Also, as shown in Figure 2-a, the market portfolio experienced a relatively moderate drop during the financial crisis in 2008 to 2009, while it saw a much stronger recovery after 2009 as compared to the index. In short, the market portfolio has a higher return and lower risk characteristic. Therefore, it can be concluded that the market portfolio is a desirable one.

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Table 5: Performance Summary of Market Portfolio and Stock Index

(2004-2014, No Short Selling)

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Figure 2-a: Cumulative Return of Market Portfolio and Stock Index

(2004-2014, No Short Selling)

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Figure 2-b: Cumulative Return of Market Portfolio and Stock Index

(2004-2014, No Short Selling)

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Question C)

3.1 Minimum variance portfolio construction

The minimum variance portfolio is obtained by finding the portfolio that minimizes the portfolio variance among the minimum variance frontier. Here, the below assumptions and constraints were made.

Table 6: Description of assumptions and constraints for minimum variance portfolio

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As a result of the calculation, the minimum variance portfolio with the lowest variance of 0.0159 was found, which corresponds to the standard deviation of 12.6%. Figure 1 presents the red square mark on the efficient frontier which represents the minimum variance portfolio.

3.2 Compositions of market portfolio

As shown in Table 6, the composition of the market portfolio was comprised of 6 out of 20 stocks. The largest weight of the portfolio was 30.3% of PHARMAGEST INTERACTIVE, whilst the smallest one was 3.8% of UBISOFT ENTERTAINMENT. This portfolio was constructed by the stocks that have a relatively high Sharpe ratio. According to 0.55 of the third quartile of the Sharpe ratio, approximately 96% of the market portfolio can be explained by the top 25% stocks with high Sharpe ratio. In addition, the two smallest weight stocks such as PHARMAGEST INTERACTIVE and DASSAULT SYSTEMS SA accounted for around 55% of the entire weights. Furthermore, the negative correlation between those two stocks was found although it was a significantly small one. Therefore, it could be said that the definition of the market portfolio is met.

Table 6: Compositions of Market Portfolio

(2004-2014, No Short Selling)

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3.3 Compositions of minimum variance portfolio

The minimum variance portfolio was composed of 8 out of 20 stocks. The largest weight of the portfolio was 37% of PHARMAGEST INTERACTIVE, while the smallest one was 1.3% of LECTRA and UBISOFT ENTERTAINMENT. As the minimum variance portfolio can be built by minimizing the portfolio variance, it is an acceptable result that the top two lowest risk stocks, PHARMAGEST INTERACTIVE and DASSAULT SYSTEMS SA occupied about 67% of the portfolio weights, contributing to lowering the overall portfolio variance. Moreover, it is found out that roughly 99% of the portfolio weights lied within the first quartile risk of 37.7%.

Table 7: Compositions of Minimum Variance Portfolio

(2004-2014, No Short Selling)

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Question D)

4.1 Efficient frontier construction

The efficient frontier is constructed by finding the minimum variance for a given target expected returns. This can be done using the Solver in Excel. Here, the assumptions and constraints for finding the efficient frontier were made as follows.

Table 8: Description of Assumptions and Constraints for Efficient Frontier

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Based on the calculation, the efficient frontier was created and plotted in Figure 3.

It can be concluded that the shape of the efficient frontier is dependent on the constraints. If the constraints are released, say a short selling is allowed, the efficient frontier moves to the left. Therefore, the efficient frontier that is completely free of constraints remains at the most left. On the contrary, the efficient frontier moves more to the right and also the shape becomes smaller in size as the constraints are stricter.

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4.2 Market portfolio construction

The market portfolio is found by choosing the portfolio that maximizes the Sharpe ratio on the minimum variance frontier. Here, the assumptions and constraints for finding the market portfolio were made as follows.

Table 9: Description of Assumptions and Constraints for Market Portfolio

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As a result of the calculation, the market portfolio with the highest Sharpe ratio of 1.53 was found. As shown in Figure 3, the blue cross mark on the efficient frontier represents the market portfolio.

Figure 3: Efficient Frontier and Market Portfolio

(2004-2014, Short Selling)

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Figure 4: Efficient Frontier with No Short Selling and Short Selling

(2004-2014)

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4.3 Minimum variance portfolio construction

The minimum variance portfolio is obtained by finding the portfolio that minimizes the portfolio variance among the minimum variance frontier. Here, the below assumptions and constraints were made.

Table 10: Description of Assumptions and Constraints for Minimum Variance Portfolio

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Based on the calculation, the minimum variance portfolio with the lowest variance of 0.0126 was found, which corresponds to the standard deviation of 11.2%. As shown in Figure 3, the red square mark on the efficient frontier represents the minimum variance portfolio.

4.4 Compositions of market portfolio

The market portfolio was composed of 11 long stocks and 9 short stocks (see Table 11). The total long position of the portfolio was 159.4%, while the total short position was -59.4%, resulting in the net 100% long portfolio. The top long weight was 36% of DASSAULT SYSTEMS SA, whilst the top short was of -16% of CEGEDIM SA, which was the only stock with a negative Sharpe ratio of -0.03. As compared with the performance between the market portfolio with short selling and no short selling, it is obvious that the former portfolio was a portfolio with higher return and lower risk. Hence, it can be concluded that the market portfolio with short selling could benefit more from the portfolio diversification.

Table 11: Compositions of Market Portfolio

(2004-2014, Short Selling)

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4.5 Compositions of minimum variance portfolio

As shown in Table 13, the composition of minimum variance portfolio was a combination of 11 long stocks and 9 short stocks. The total long position of the portfolio was 136.2%, while the total short position was -36.2%, resulting in the net 100% long portfolio. The positively largest weight was 39% of DASSAULT SYSTEMS SA, whilst the negatively largest weight was of -7.7% of GFI INFORMATIQUE. The first quartile of risk (28%) can explain 100% of the total long position of 136.2%. Similar to the market portfolio, the minimum variance portfolio could enjoy the portfolio diversification by allowing for a short selling. Hence, the minimum variance portfolio with short selling achieves a higher return and a lower risk one.

Table 12: Performance Statistic of Market Portfolio with Short Selling and No Short Selling

(2004-2014)

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Table 13: Compositions of Minimum Variance Portfolio

(2004-2014, Short Selling)

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Table 14: Performance Statistic of Minimum Variance Portfolio

with Short Selling and No short Selling (2004-2014)

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Question E)

5.1 Assumption of time separation

Formula 1: (2004+2014)/2=2009

Because the question (e) required dividing the whole period into 2 parts which is the first half of years and the remaining, 2009 was found by the formula 1. In this case, the whole period (2004-2014) was divided to 2 equal parts, which is from 2004 to 2009 and from 2009 to 2014.

5.2 Re-estimate the market portfolio and minimum variance portfolio using the first half of the data only.

The minimum variance portfolio means the portfolio of risky assets with the lowest variance. Based on the performance analysis of the market portfolio and minimum variance portfolio in the first five years, it is clear that the market portfolio was a higher return and lower risk portfolio with higher Sharpe ratio compared with the minimum variance portfolio. Table 15 shows that the risk of the market portfolio was 68.3%, while that of the minimum variance portfolio was 11.3%. Additionally, the return of the market portfolio (122.0%) was significantly higher than that of the minimum variance portfolio, which was only 5.0%. Moreover, the Sharpe ratio of the market portfolio (1.74) was considerably better than that of the minimum variance portfolio (0.19). In conclusion, the market portfolio is a better risk-return trade-off portfolio.

Table 15: Performance Statistic of Market Portfolio and Minimum Variance Portfolio

(2004-2009, Short Selling)

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Table 16: Portfolio Weight of Market Portfolio and Minimum Variance Portfolio

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5.3 Performance comparison between market portfolio and minimum variance portfolio using the data from 2009 to 2014 (Portfolio weight is identical to that of 2004-2009)

Figure 5-a shows that the minimum variance portfolio performed better than the market portfolio during the second five years. Specifically, the minimum variance portfolio was almost 0% at the beginning and increased until the end period (125%). Meanwhile, the market portfolio started from around -60% at 30/10/2009 and reached the top at around 23% at 28/2/2014 and at the end of this period. Additionally, the return volatility of the minimum variance portfolio was very stable as the portfolio has the least variance. On the other hand, the return of the market portfolio was considerably volatile. Moreover, it underperformed although it was constructed by the portfolio weights with the highest Sharpe of 2004-09. In conclusion, it could be considered that the market portfolio was less efficient and riskier than the minimum variance portfolio in terms of a risk-return characteristic.

Figure 5-a: Cumulative Return of Market Portfolio and Minimum Variance Portfolio

(2009-2014, Short Selling)

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Figure 5-b: Cumulative Return of Market Portfolio and Minimum Variance Portfolio

(2009-2014, Short Selling)

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Question F)

6.1 Equally weighted portfolio and market-capitalization weighted portfolio construction

According to the description of market-capitalization in Table 2, the weight of market-capitalization weighted portfolio is on the date of 30/9/2009, while the weight of equally weighted portfolio is 1/20 (5% each stock). Figure 6 illustrates the different performance of different weighted portfolio. The expected return of market-capitalization weighted portfolio was 8.74%, which was only 4.13% lower than that of equally weighted portfolio. Meanwhile, the risk of market-capitalization weighted portfolio (23.79%) was also lower than equally weighted portfolio (24.68%). Therefore, the equally weighted portfolio could achieve a higher return and higher risk than the market-capitalization weighted portfolio. However, the equally weighted portfolio and market-capitalization weighted portfolio were inefficient as both portfolios were inside the efficient frontier.

Figure 6: Efficient Frontier and Market Portfolio

(2004-2009, Short Selling)

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6.2 Performance comparison of the four portfolios: the market portfolio, the minimum variance portfolio, the equally weighted portfolio, and the market- capitalization weighted portfolio (Portfolio weight such as market portfolio and minimum variance portfolio is identical to that of 2004-2009)

Figure 7-a shows the cumulative return of the market portfolio, minimum variance portfolio, equally weighted portfolio and market-capitalization weighted portfolio between 2009 and 2014.

It is noticeable that the market portfolio was by far the lowest profitable of the four types of portfolio over the period, while the minimum variance portfolio outperformed the most for the majority of the period. In addition, the equally weighted portfolio and market-capitalization weighted portfolio performed the same level of return with similar risk (see in Table 18).

At 30/10/2009, the cumulative return of the market portfolio was -56.2% while that of other three portfolios was approximately 0%. During the next 2.5 years, the cumulative return of the market portfolio fluctuated from -38.3% in 26/2/2010 to -149.3% in 31/5/2012. During the same period, the data of minimum variance portfolio increased steadily to around 61.1% in 30/5/2012, while the cumulative return of the equally weighted portfolio and the market capitalization weighted portfolio rose to 35.51% and 33.82% at 31/5/2011. By 29/8/2014, the cumulative return of the minimum variance portfolio increased to 122.8%, and the return of the market portfolio also reached the top which was nearly 19.4%. Furthermore, the cumulative return of the equally weighted portfolio and the market cap weighted portfolio grew slowly to 84.5% and 87.5%.

In conclusion, the minimum variance portfolio with the lowest risk achieved the best performance, whilst the market portfolio with the highest risk recorded the worst performance. In other words, the higher return the lower risk. Hence, the result could not be consistent with the portfolio theory, which specifies taking a higher risk for gaining a higher return.

Figure 7-a: Cumulative Return of Market Portfolio, Minimum Variance Portfolio, Equally Weighted Portfolio, and Market-Capitalization Weighted Portfolio

(2009-2014, Short Selling)

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Figure 7-b: Cumulative Return of Market Portfolio, Minimum Variance Portfolio, Equally Weighted Portfolio, and Market-Capitalization Weighted Portfolio

(2009-2014, Short Selling)

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Table 17: Portfolio Weight of Market Portfolio, Minimum Variance Portfolio, Equally Weighted Portfolio and Market-Capitalization Weighted Portfolio

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Table 18: Performance Statistic of Market Portfolio, Minimum Variance Portfolio, Equally Weighted Portfolio and Market-Capitalization Weighted Portfolio

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References

Archer, S. H., Francis, J.C. (1971). Portfolio Analysis. USA: Prentice-Hall, Inc.,

Bodie, Z., Kane, A., Marcus, A (2014). Investments. UK: McGraw-Hill Education

Levy, H., Sarnat, M (1984). PORTFOLIO AND INVESTMENT SELECTION: THEORY AND PRACTICE. USA: Prentice-Hall, Inc.,

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