Discuss about the Dissertation For Financial Portfolio Analysis.
The construction of a financial portfolio is a key issue in the current economies for different nations. Having a financial portfolio plays a paramount role in the theoretical and practical application of the sustainable financial economics. In a portfolio composition analysis, an investor selects a set of individual assets. This paper seeks to study 5 assets obtained from Yahoo finance over a 2-year period from 01/09/2013 to 31/08/2015 so as to make investments. The portfolio can accommodate various shares, unit trusts, bonds, PEPS, ISAS, options, cash, foreign currency or other financial instruments. The portfolio concerns itself more over how to spread the investment funds among the chosen assets as opposed to the criteria when choosing the assets. Each asset has a unit price at the time of purchase and can be subsequently updates. It has a price and return at the purchase time and at any future time. Return mainly measures the increase in price of the asset as calculated on a daily basis as either a fraction or a percentage. It is given by
Ri (t) =
When the portfolio does an analysis over a lengthier period of time, for instance, this paper does a research on a two-year period, the log return is considered much more convenient. It is, therefore, obtained by,
LRi (t) = =
Where P is the price vector, L =length(P) and LR=
The assets chosen for this paper are analyzed for returns rather than prices so as to avoid the issues that would result due to currency different. Using returns makes the problem-solving process purely equity based. The profit consists of return from the portfolio of investments and its capital appreciation. Associated is the task of limiting the risk of losing money. A risk is considered an opportunity in the financial markets. The financial portfolio analysis seeks to develop a theory based on getting the best return from the portfolios over a single period of time. The Nasdaq share prices of MSFT are obtained from finance.yahoo.com.
Objectives Of The Study
- To determine the equivalent mean, variance, standard deviation of the shares returns annually, and the annual covariance matrix V of the returns.
- To analyze the covariance and share price behavior over a given time interval and the efficient frontier and explicitly plot the outcome.
- To use the utility functions to find the ideal investment portfolio at the beginning of the given time period (Year 1- construct portfolio, Year 2- future return validation).
- To investigate the relationship between SPY and VIX.
Different companies do their stock market valuation or share prices based on the performance of the company over a long-term period only. Short term valuations can beat down the record held by companies valued highly over longer periods as problems are bound to occur over long period appraisals than when short period appraisals are done. Interest can be viewed from the lender’s and borrower’s end. Interest is the gain from investing for a lender and the reward for trouble of lending a borrower advanced capital. The amount of initial outlay and the length of time the money is lent decides how much the interest is due. There are three frequently used financial rates namely the AER, EAR, and APR. Investment is perceived as the utilization of funds on assets with a focus on earning income or capital appreciation. The main investment strategy has two key attributes: time and risk. These two attributes indicate the risk factor. The risk is undertaken in a view to reap some return from the investment. The financial investment ensures the distribution of money of assets that are expected to yield some gain over a period of time. It is an exchange of financial claims for money. They are expected to yield returns and experience capital growth over the years. The main aim of the construction of a financial portfolio is to ensure the maximization of return, minimization or risk, and hedge against the inflation (Kasilingam, n.d.).
The paper uses the dataset of the historical data obtained from Nasdaq. It is the first electronic exchange that provides a platform for the investors to buy and sell stock. Nasdaq is a statistical measure of a slice of the market. The Dow 30 shows a dataset comprised of prices for different institutions and how the compare to each other. The DJIA tracks the routine of thirty different companies that are considered key players in the market. The Nasdaq tracks about four thousand stocks that trade on the Nasdaq exchange. Nasdaq deals with organizations such as CISCO, Microsoft, intel and apple. The two market indices are a representative of a mathematical average used to make sense of the stock market. Assets at origin are funds that give no return and have no risk. The efficient frontier is defined by portfolio that no point has high return with the risk of H and no point with the return of H has lower risk. It is measured by the expected utility function.
The financial portfolio is concerned about risk, return, safety, and liquidity. The return is a random variable that is obtained as the mean of given data and is referred to as the expected return and the associated risk. It is a standard deviation denoted as σi. The empirical random variable model of the Ri is obtained by
Where ej is a random variable term.
The data obtained was evaluated at the closing price. All the shares trade in USD for uniformity on the currency and to avoid complicated addendum based on risk of using different currencies.
The figure above show a monthly based analysis with data obtained for the third day of each month. There is a significant improvement in the share returns recorded from the year 2013 to the year 2015. Statistical analysis was conducted on the closing share indices of the returns for the monthly data set in use.
All the calculations are done based on the adjusted close values in USD.
The results of the statistical evaluation to achieve objective 1 are discussed below. These are the resulting values for the entire period for the adjusted close. The MATLAB code snippet for the statistical evaluation is as shown below
Annual covariance matrix V of the share returns
The same amount for variance is obtained for covariance. The covariance matrix is discussed below.
Efficient frontier portfolio for the multiple risky assets
The statistical mean on the data is obtained using the equation shown below,
It can be denoted in vector form as,
Where both w and μ are sets raised to the power of T. secondly, for the variance, there is a well-known result. The standard deviation is the square root of the variance. This is denotated in summation form as,
This can be denoted in matrix terms as the neater expression of the variance
The covariance matrix is similar to the variance value obtained. Note that,
And that the covariance of the matrix is purely concerned with the statistically linear relation aspects. These are the aspects of the relation between the returns Ri and Rj from the assets Ai and Aj. The var(Ri) is associated with the variability of the return Ri from Ai. For efficient frontier, the mean-standard deviation plots for portfolio optimization is completely specified by using a portfolio object when the two conditions are met. MATLAB code uses the finance toolbox to create a portfolio object and set the required parameters. A covariance matrix is needed to achieve the asset mean and asset covariance for analysis for efficient frontier.
When variables are positively related, they move in the same direction but when the relationship is negative, the variables are inversely related. The covariance and correlation indicates this relationship. The trend observed on the relationship of the share returns and prices over the two-year period, in the first year the data was used to construct a portfolio. In the second year, the remaining data set was used to test the portfolio and determine if the portfolio works well on the test data. The economic growth and Nasdaq share returns are evaluated to show a positive covariance. The formula used, assuming that x-economic growth and y-Nasdaq share returns, is,
The economic growth of the country is evaluated as the independent variable, the values used are mean values of the data set in the first section. As a result, one can determine if the units of measure are increasing or decreasing. Correlation come in to aid the covariance measure the degree to which a relationship is positive or negative. It is the statistical dependency between returns which controls the sought feature of obtaining high return for low risk. The efficient frontier is decided by portfolios which minimize the risk for a required expected return that maximizes the expected return for a given level of risk.
From a given study, the relationship between the closing prices of the SPY and VIX shows that the SPY has a higher trend line than the VIX. The SPY needs to go down a few indices so as to move in the same direction as the VIX. This would ascertain the rule of thumb for positive correlation and covariance. According to the black Scholes formula for options, the MATLAB function used has arguments such as the current price for the underlying asset, exercise price of the option, annualized, continuously compounded risk-free rate of return over the life of the option, time taken by option to expire (years), and the continuously compounded yield of the underlying asset over the life of the option.
In a nutshell, the financial portfolio analysis attains very substantial information from the dataset used. The data set was obtained from the Nasdaq stock exchange and the share returns and share prices were evaluated at the adjusted close price for each quarter or even on monthly basis. Several graphical representations were obtained using the MATLAB software such as the efficient frontier and the statistical calculations.
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