Portfolio Management

[Name of Student]

[Name of Professor]

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Introduction

In the following report, a portfolio consisting of nine stocks from different industries have been selected. These all stocks are selected from the S&P 500 index and belong to diversified industries which makes them favorable to construct portfolios given that diversification is one of the important characteristics of the portfolio. If a portfolio is not effectively constructing a composite of diversified assets, then an optimum portfolio cannot be constructed. This report presents a critical discussion on the characteristics of the stocks returns, volatility clustering, normality of returns, mean variance optimization and also calculation of alpha and beta values. In the end, a discussion is conducted to provide major findings for the investors.

Descriptive Statistics

The descriptive statistics of the stocks reveals the mean return showing an average return on stock has generated for the investors over the selected sample period and standard deviation indicating the value of deviation by which the average return may increase or decrease. Furthermore, minimum and maximum returns are also provided for each asset over the sample period, whereas the kurtosis and skewness values are also provided. The core purpose of the kurtosis and skewness is to show the tail and symmetry of the data based on which an idea about the distribution of data is made.

Table 1 Descriptive Statistics of the Stocks

There are two important aspects that can be considered; one that stock returns are averages and average does not guarantee the future performance and standard deviation reflects a possible fluctuation into the return in future based on the past performance. Furthermore, referring to the mean returns of the all assets, it can be determined that ITOB has highest mean return of 1.221% with standard deviation of 5.063% suggesting that mean return of the company may increase or decrease by the value of standard deviation. Meanwhile, the lowest mean return belonged to ICAR with mean return of 0.137% with standard deviation of 11.76% indicating mean return could vary by the value of standard deviation. Based on the return and standard deviation of this asset, it can be determined that this stock carries a significant factor for the investment because the return of the stock is very low but its standard deviation is very highly showing that it carries a significant for the investor. 

The lowest or most appropriate risk-return association can be seen in the GFBT which has a return of 0.983% with standard deviation of 3.87%, and this association and composition of return and return is better than all other stocks risk-return composition based on the fact that each of the stock’s higher risk is higher or return is lower or if it has lower risk then return is higher which is an unusual. Therefore, it can be determined that there are various complications that needs to be considered in the construction of portfolio and based on the theory of risk and return, the GDBT better reflects the theory that a higher risk will yield a higher return and lower risk will yield a lower return. Hence, higher return cannot be achieved without higher risk.

Stock’s Returns

Volatility Clustering

Volatility clustering refers to the condition of financial markets which states that large changes are followed by the large changes and small changes are followed by small changes. This implies that if there are small changes into the prices of stocks then fluctuations will also be small in intensity (Tseng and Li, 2011). However, if there has been a large change into the price of the stock then fluctuations also tend to be large. Hence, it has been defined to term that nature of the change determines change to be followed by (Tseng and Li, 2012). Meanwhile, order to determine presence of volatility clustering into the selected assets, line chart has been constructed for all stocks together as follows

Figure 1 Volatility Clustering

Referring to the figure 1, it shows the volatility clustering is present into the stocks’ return as it can be observed during the financial crises 2008 to 2009, where a small change into the price is followed by smaller but as the intensity of the price reduction increases then another higher wave is followed. Meanwhile, when market starts to normalizes then it can be observed then intensity of the fluctuations declines. Hence, there is sufficient evidence of presence of volatility clustering.

Normality of Returns

The normality refers to the distribution of the data which if comes from the normal distribution then assumption of normality is met and is stated that data has no issues. However, if the normality is absent from the data then treating the data only remains an option to followed (Karoglou, 2010). Meanwhile, in following report the normality of the stock’s return is determined by the histograms of all stocks. See appendix for the histograms of stocks return. The histogram of all stocks have approximately bell-curve shape indicating that data of the stock returns comes from the normal distribution. However, another important factor that can be highlighted here and the reason why stock’s return is normally distributed. The core reason is that return of the stock is not normally calculated as current minus previous divided by previous. But in contrast the firstly the stock prices were transformed into the natural log or also known as natural logarithm and then returns were calculated based on the natural log (Ling, 2017). Thus, the returns of the stocks come from the normal distribution.

Mean Variance Optimization

The mean variance optimization refers to the process of achieving a portfolio with minimum variance or at least offers a best compensated return at given level of risk. In modern theory, the return is associated with the risk, which can be translated as that if the stock has higher return then it would also have a higher risk (Bruni et al., 2015). However, if the risk is lower then return also tend to be lower. As the efficient market hypothesis states that investor cannot beat the market which means the investor cannot be earn more than market until and unless if the investment is made into the risky assets. This means investors cannot outperform the market if they make investment into the risky assets (Baitinger and Papenbrock, 2017). Meanwhile, in order to achieve a mean variance optimization portfolio, we used the excel solver with optimization option which allows to achieve a desired return at given level of risk and also allows to achieve optimal risky portfolio at which portfolio can achieve highest possible return at lowest possible risk level. This means it would be level at which the Sharpe ratio would be maximized given that a higher Sharpe ratio means that portfolio effectively compensate each risk point in terms of returns (Omisore, Yusuf and Christopher, 2011). Meanwhile, calculations of the mean variance optimization as follows

Table 2 Stock average returns and risks

The table 2 provides the risk and returns of all stocks calculated for the construction of portfolio. Furthermore, based on the calculations, the firstly the Pearson’s correlation was calculated between all stocks and then covariance and variance matrix was computed with the help excel functions and this helped to develop the optimal portfolio as the portfolio returns cannot be computed until and unless we have correlation and variance-covariance matrix (Kim and Francis, 2013). As per the requirements, two different portfolios were constructed; (1) long-constraint portfolio and (2) short-selling allowed portfolio. The results of the portfolio as follows

Table 3 Portfolios

Based on the table 3, it can be determined that either long or short, the weights of portfolios are same although during the calculations of solver, the in short portfolio the constraint of all weights greater than zero was removed to let excel decide which stock to make short and which not to. But the results of the both portfolios were same and this indicates that this is only optimal solution and minimum variance portfolio that can provide investor with highest possible return with lowest possible return. Furthermore, the based on the constructed portfolios, the efficient frontier was also constructed with help of excel solver where multiple portfolios were constructed based on trial and error and efficient frontier was constructed and is present as follows

Figure 2 Efficient Frontier

The efficient frontier shows the minimum variance portfolio highlighted in red dot exactly at the curve where an optimal portfolio is achieved. Therefore, it can be determined if the investment is made as per the optimal portfolio then a better return can be achieved after managing the risk.

Calculation of Alpha and Beta

The alpha refers to the excess return over an investment after adjusting for the market related risks to the investment and beta refers to the systematic risk associated with the stock. Meanwhile, to analyze the stocks and portfolio the alpha and betas of all stocks were calculated as follows

Table 4 Alpha and Beta

The table 4 provides the alpha and beta of the stocks where it is evident that ITOB provides highest excessive return as it has highest alpha and riskiest stock is SLHI since it has highest beta of 1.97 which is near to 2. This implies that if market losses 1% then this stock will almost loose double than market due to higher fluctuations.

Discussion and Conclusion

Based on the findings of the portfolio optimization, it can be determined that for construction of portfolio the diversified stocks must be selected from different industries but their selection should also be systematic rather than random. Thus, stocks should be chosen based on their past performance, portfolio objectives and risk-return expectations. This would allow investor to have higher returns based on the fact risk would be managed more effectively. Thus, diversified portfolio would enable us to manage the risk easily. However, it is critical that short-selling should not be used while constructing a portfolio which may affect overall performance of the portfolio as investor may not be able to meet his expectations.

References

•  Baitinger, E. and Papenbrock, J., 2017. Interconnectedness risk and active portfolio management. Journal of Investment Strategies.

• Bruni, R., Cesarone, F., Scozzari, A. and Tardella, F., 2015. A linear risk-return model for enhanced indexation in portfolio optimization. OR spectrum, 37(3), pp.735-759.

• Karoglou, M., 2010. Breaking down the non-normality of stock returns. The European journal of finance, 16(1), pp.79-95.

•  Kim, D. and Francis, J.C., 2013. Modern portfolio theory: Foundations, analysis, and new developments. John Wiley & Sons.

• Ling, X., 2017. Normality of stock returns with event time clocks. Accounting & Finance, 57, pp.277-298.

•  Omisore, I., Yusuf, M. and Christopher, N., 2011. The modern portfolio theory as an investment decision tool. Journal of Accounting and Taxation, 4(2), pp.19-28.

• Tseng, J.J. and Li, S.P., 2011. Asset returns and volatility clustering in financial time series. Physica A: Statistical Mechanics and its Applications, 390(7), pp.1300-1314.

• Tseng, J.J. and Li, S.P., 2012. Quantifying volatility clustering in financial time series. International Review of Financial Analysis, 23, pp.11-19.

Appendix