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Portfolio is a term used in finance to refer to a combination of securities or investment projects that is held by an investor (Hagin 2004). When investing, it is wise for an investor to combine one or two assets or securities in order to reduce risk. This is called diversification of risk because it helps investors to eliminate or reduce diversifiable risks. Diversifiable risks are those asset or security specific risks that can be avoided by holding and asset in a portfolio. The rationale behind holding a portfolio is the proverbial saying that you should not put all your eggs in one basket, since if the basket falls you lose all the eggs._x000D_

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Investment management is the process of ensuring that the securities or the assets an investor has invested in meets the objectives of the investor. It is the management of the investment which can be done by the investor or by professional management (Grinold & Kahn, 2000). Investment management is composed of a set of processes which are finding investment options, analyzing them, selecting the best option(s), carrying out the actual investment, monitoring the investment and lastly taking the necessary action depending on the performance of the investment. If the investment management is done by a professional investment manager, his role will be to advise the investor on the above processes._x000D_

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Investment portfolio management can therefore be defined as the process of collecting information about potential investment securities or assets, identifying the best investment based on reliable criteria, investing in the investment and carrying out a continuous evaluation to know the best action to take on the investment (Hagin, 2004). The most important step is the decision on the portfolio to make since it is the basis of success of any portfolio. An investor must analyze all the information available in order to determine whether holding the portfolio will reduce risk. Some of the methods employed to determine the viability of an investment are serial correlation, auto correlation, runs, and distribution of returns, among others._x000D_

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In this paper, the above named statistical methods of evaluating investment options are discussed and an example done. The purpose is to show how investors should make management decisions based on the results of the statistical analysis tools._x000D_

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Tests on indices_x000D_

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The indices chosen for the tests are those of USA stock exchange indices and the Japan stock Exchange indices._x000D_

Auto and Serial Correlation_x000D_

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Auto correlation means that a change in a variable causes a corresponding change in the same variable but at a different time. It signifies repeating patterns such that a change in the variable will cause a similar change but at a different time (Evans, 2002). Autocorrelation is a violation of one of the basic assumptions of linear regression analysis that a variable should be independent. It is also called serial correlation or lagged correlation. Autocorrelation can be negative or positive. Negative autocorrelation shows that a movement in the variable at one time is followed by a movement to the opposite direction at another time while positive autocorrelation mean that an increase in a variable at on point of time is followed by an increase in the same variable at another time._x000D_

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A positive Autocorrelation shows a kind of persistence in a variable that it will always keep on moving in the same direction. It is also an indication that the behavior of a variable can be estimated by forecasting techniques. Testing for autocorrelation can be done through three methods namely time series plot, lagged scatter plot and the autocorrelation function._x000D_

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Time series plot_x000D_

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This is a plot of the variable against time and a mean of the variable calculated and plotted. A positive correlation will show positive variations from the mean being followed by positive variations from the mean (Grinold & Kahn, 2004). Data with negative correlation will show positive variation from the mean being followed by a negative variation from the mean. To draw conclusions on autocorrelation of data from a time series plot is difficult and subjective._x000D_

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Lagged scatter plot_x000D_

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This is a scatter plot of time series against itself. According to Knight & Satchel, if the points are randomly scattered, there is no autocorrelation while alignment from lower left to upper right shows positive correlation. Alignment of points from upper left to lower right shows negative autocorrelation (2001)._x000D_

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Autocorrelation function/correlogram_x000D_

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This involves the calculation of autocorrelation coefficients of observations of a variable at different times. Autocorrelation is determined by calculating the Durbin-Watson statistic (d) which is calculated as follows:_x000D_

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d = ∑nt=2(et-et-1)2_x000D_

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∑nt=1 et2 _x000D_

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The Durbin Watson statistic for the US stock index is 0.00568 while that of the Japan stock index is 0.001214. The rule of thumb used in interpreting the Durbin Watson Statistic states that if the d is less than 2, then there is evidence of positive serial correlation or autocorrelation (Evans 2002). Therefore, movement in indices of both US and Japan stock exchange are positively correlated meaning that an increase in the index at one time is followed by an increase at another time._x000D_

Runs_x000D_

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A run can be defined as a series of similar observations showing the number of expected observations against the given number of observations. If the number of actual observations is more than the expected observations, there is negative correlation and if the actual observations are less than expected observations, there is positive correlation. To develop data for a single variable, one must choose a cut off point and all points below the cut off should be given a value while points above the cut off point are given different value._x000D_

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For the USA stock exchange index, let us choose a cut off point of 10,978.87 which is the mean of the observations. All the values above 10,978.87 are 89 while those below are 106. If we expected the index to be less than 10, 987.87, then there is positive serial correlation since the actual outcome (Above 10,978.87) is more than expected. In the Japan stock exchange index, the mean is 12,279.16 which will serve as the cut off point. 63 points are above the cut off point while 132 points are below. Assuming that we expected the index to be less than 12, 279.16, then there is positive correlation since the actual observations are less than the expected observations. Runs confirm that there is positive autocorrelation in the two indices._x000D_

Distribution of Returns_x000D_

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Distribution of returns refers to the probability of a set of returns that an investor expects from an asset or investment. Coming up with the distribution of returns enable calculation of the risks associated with an investment (Grinold & Kahn, 2004). When the distribution of returns takes the normal distribution, it is possible to calculate the riskiness of the investment by calculating the standard deviation and the riskiness of the investment._x000D_

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The distribution of the US stock exchange index is negatively skewed, that is, more returns are distributed in the upper ends._x000D_

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From the figure above, the Japan Stock Excahnge index is negatively skewed meaning that more observations are in the lower ends._x000D_

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DISCUSSION_x000D_

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The efficiency market hypothesis proposes that a market is efficient if it is able to process information released by a company and reflect the information in the stock price of that market. There are three forms of market efficiency which are strong form, semi-strong form and the weak form of market efficiency. In a market with strong form of efficiency, stock prices react to all publicly available information about a company, past and present (Saleh, 2010). In the semi-strong, the stock prices reflect past public information while in the weak form, only past public information is reflected. The issue that the efficiency market hypothesis tries to address is whether stock prices reflect the true value of the securities of a firm._x000D_

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Due to different evaluation of information released by a firm and different expectations of investors, the efficient market should be characterized by randomness of stock prices. There should be no autocorrelation in the prices of securities in an efficient market. Therefore, it should not be possible to estimate future returns or prices of stocks. Existence of autocorrelation in the index of stock exchange show that it is possible to estimate future movements in the stock exchange and this violates the assumption of Efficiency Market Hypothesis of equity in information (Hagin, 2004)._x000D_

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In the US and Japan stock exchanges, there is positive autocorrelation in the movement of the index which is an indicator that the market is not efficient. The market values of the stocks do not represent the value of the firm despite the information released to the market. Investors and traders value the stocks depending on their own assessments but not depending on the information available such that the information released is not reflected in the stock prices._x000D_

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The recent 2009/2010 financial crisis was partly contributed by failure of financial markets (Saleh, 2010). The prices of the stocks traded in the financial markets do not reflect the intrinsic value of the firm and this has made investors to invest in non viable projects. This has caused many firms to fail when the investors were expecting them to perform better thus investors have been losing money even when the markets are purported to be efficient. However, this does not make the efficiency market hypothesis irrelevant since investors view information from different perspectives. Investors also make their investment decisions based on alternative criteria like future expectations and emotions._x000D_

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In markets where stock prices do not represent the intrinsic value of the firm, investors can play the market and make arbitraging profits. They can do so by buying under priced securities and selling them to others at a higher price. Similarly, in a market with autocorrelation in stock price movements, investors can buy and hold stocks when their prices are in an upward trend because the upward trend is expected to continue in the future._x000D_

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Conclusion_x000D_

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In making investment decision, it is important for an investor to carry out efficiency market studies to establish whether the stock prices reflect the intrinsic value of the company. This can be done by use of determining whether there is evidence of auto correlation or serial correlation in the stocks of a company. The reason why investors continue to lose their money despite the efficiency market hypothesis is that investors interpret the information in different ways thus making the stock prices not to reflect the real value of the companies. This is partly the cause of the financial crisis in the year 2009/2010.