In this study, I would like to optimize the portfolio return by using the Markowitz Mean-Variance (MV) Optimization and its efficient frontier, which the MV optimization is a practical tool of institutional asset management. However, although Mean-Variance optimization is a convenient and useful theoretical framework for defining a portfolio optimally, but it is still not enough in practice as it often results in maximizing error. As a result, the findings and investment strategy will be misinterpreted and vitally wrongly which can lead to serious lost in portfolio. Moreover, Mean-Variance efficiency model use the absolute risk measure variance to find out the efficient portfolio, in practice however, benchmark relative portfolio optimization is widely used. This is due to the fact that investors are more likely to know how well do their portfolios perform for a given amount of relative risk, and what kind of risk their portfolio carrying relative to the benchmark.
Thus, the benchmark is becoming an important standard to evaluate the portfolio managers' performances, and also brings more questions to the portfolio construction process at the same time. However, besides using the Mean-Variance efficiency to optimize the return, to be perfect, I will examine the Value-at-Risk (VaR) for the potential loss in my portfolio.
Hence, in this study, I will focus on all the sectors in the Bursa Malaysia; namely, finance, properties, technology, services, and etc which will choose according to the most volume traded in January 2011. Moreover, I will use 5 years of historical monthly data from January 2005 to December 2010. All the stocks are listed in the KLSE (Kuala Lumpur Stock Exchange).
As a result, besides the MV efficiency frontier, the correlation of the assets, covariance matrix of returns, and the Sharpe ratio (determine the excess return per unit of total risk in portfolio) are derived and implemented as well.
3.2 Sampling Procedures
3.2.1 Correlations of assets
Correlation plays an important role in the portfolio management. Why I said so is because one of the basic aspects of building a portfolio is to include an assets which are negatively or have a small positive correlation between each other. It is due to it is thought to be safe as they do not fluctuate as much when the assets in a portfolio do not move in the same direction.
However, the correlation is computed into the correlation coefficient, which is a range in between of -1 to +1. The correlation coefficient of -1 is defined as perfect negative correlation which implies that if one security moves in either direction while the other one security moves in the opposite direction, then we can conclude that the two securities are perfectly negatively correlated. On the contrary, the perfectly positive correlation, a correlation coefficient of +1, for example, as security A moves either going up or going down, the other security will moves in the same direction as the security A. Alternatively, if the correlation coefficient is 0, it means that there is no correlation between the movement of two security, which implies that they are completely random.
Thus, in the chapter 4, correlation matrix will be implemented to identify the assets moment correlation between the securities by using the software EViews 6 which it is a statistical, modeling, and forecasting tool. Moreover, in order to interpret the value of r perfectly, the coefficient of determination, r2, will be computed as well to identify the proportion of variance in common between two variables.
3.2.2 The Markowitz Mean-Variance Efficiency
Mean-Variance (MV) optimization approach which introduced by Harry Markowitz (1959) is a classic paradigm of modern finance for allocating the capital among risky assets. The purpose of Markowitz model is to construct an efficient portfolio and the selection of portfolio is represented as the following optimization problem:
Subject to
Where n is the number of available securities, Xi is the faction of the portfolio held in security i; E(Ri) is the expected return of security i while E(Rp) is a portfolio's expected return; σi² is the variance of security i's return; σij is the covariance of return of security i and j; and σp² is the variance of return of portfolio.
In portfolio, the goal is to minimize the variance of portfolio, σp², while to achieve the maximizing of the acceptable expected portfolio return, E(Rp). Normally, portfolio risk is represented by the square root of variance, which equal to the standard deviation, σp.
In this study, the MV objective function is given as above, which to minimize the variance, will be implemented by using the MS Excel. The steps are as below:
Step 1: Calculate the each stock return for each period by dividing the difference of the stock price of 2nd period and 1st period to the 1st period of stock price.
Step 2: Calculate the average returns of each asset, which also known as the mean, for monthly (since the monthly data was used), and the mean for annual return which is because the mean of return for a portfolio is determined by annul return.
Step 3: Calculate the excess returns by subtracting each asset's return to the monthly mean return.
Step 4: Transpose the excess return matrix. When come to this step, it unlike the steps before which is just a simple subtracting but it needs to use the "Transpose" function in the MS Excel.
Step 5: Calculate the variance-covariance matrix (VCM). VCM is a compact way to present data for the securities by using the "MMULT" function, which is a function to return the matrix product of two arrays that we generated before.
Step 6: Create the efficient frontier by using the portfolios expected return and portfolio's standard deviation that computed.
After found the efficient frontier, the Sharpe ratio will be computed. It is to measures the ratio of return to volatility. It is useful in comparing two portfolios or stocks in term of risk-adjusted return. The Sharpe ratio is calculated as follows:
Where
As the higher the Sharpe ratio, the more sufficient the returns for each unit of risk.
3.2.3 Monte Carlo Simulation-To Identify the Value-at-Risk
Value-at-Risk (VaR) is one of the important concepts in the risk management especially in constructing a portfolio. It is because portfolio face the risk of its value decreasing, and sometimes drastically. Hence, VaR try to statistically quantify the significance of the loss.
However, for risk management, one of the methods to quantify the VaR is using the Monte Carlo simulation method which it was performs much better especially for the portfolios that containing the options in identified the potential loss.
Therefore, in this study, I am used the MS Excel to measure the VaR by using the Monte Carlo simulation method.
The computation by using the Monte Carlo simulation first requires to generate T number of scenarios as each scenario was comes from a model of future assets price that are randomly computed using the Monte Carlo's random outcome principle of variables. In addition, the confidence level is assumed and fixed at given time horizon, while the confidence interval is the percentage of the portfolio price changes in the VaR calculation. And the VaR is found by finding the loss of all the simulations which corresponds to the one that satisfies the percent desired.
During the Monte Carlo simulation, the variance-covariance matrix and the effect of the portfolio will be implemented as well.
3.3 Sources of Data
This study focused on all the sectors of the Bursa Malaysia: namely, finance, properties, technology, services, and etc. In this study, all the data are secondary data and 20 stocks will be selected from the main board of KLSE. All the stocks are chosen according to the most volume traded in January 2011 (considered the month of that i want to analyze), it is appropriate to assume that the conclusions of this study are true reflection of the real investment. On the other hand, the study would cover the period from January 2005 to December 2010 and monthly data will be taken. Hence, there would have 72 observations to be examined.
All the data are downloaded from http://finance.yahoo.com/.
Besides that, Modern portfolio theory assumes that there must at least one risk-free asset inside the portfolio. Due to the risk-free rate is lower than the expected return on the risky asset (since the risk-free rate is also representing the rate of return on the risk-free asset). Hence, I would use the 3-month Malaysia Treasury Bill as my risk-free rate. In fact, the 3-month Treasury Bill is a risk-free asset in Malaysia and the rate that stated at 3rd of January 2011 will be used in this study.
The rate is available at the Bank Negara Malaysia Government Securities Yield's website, http://www.bnm.gov.my/statistics/govtsecuritiesyield.
