The Fundamental Concept Behind Modern Portfolio Theory - Lab Report Example

FINANCIAL MODELING By of the of the School Briefly explain the conceptsof expected return, standard deviation, and correlation (in the context of share prices), and discuss their importance in portfolio management. The fundamental concept behind Modern Portfolio Theory (MPT) is the fact assets in an investment portfolio should generally not be selected individually, each and every one on its own merits. It is very necessary to consider how each and every asset changes in price relative to how every other asset in the portfolio changes in price. The process of investment is basically a tradeoff between expected return and risk. Generally, it is thought that assets with much higher expected returns are more risky. For any given amount of risk, MPT tends to describe and explain how one can select any given portfolio with the highest possible value of expected return. For a given value of expected return, MPT tends to explain how one can select a portfolio with the least possible risk. Standard deviation is the most commonly and widely used measure of spread and thus it measures the potential variability, volatility and risk. Standard deviation (σi ) can be used as a good measure of relative risk between two investments that have the same expected rate of return. It can be calculated for each and every individual shares, portfolios of shares and for the market as a whole. A larger value of σi implies a lower probability that actual returns will be closer to the expected returns. Covariance a. Measures the extent at which two or more variables move together b. For two assets, i and j, the covariance of rates of return is defined as: The values of the correlation coefficient ranges from The portfolio standard deviation is normally a function of; The variances of the individual separate assets that make up the portfolio The covariance matrix between all of the assets in the given portfolio The larger the portfolio, the more the impact of covariance and the lower the impact of the individual security variance 2. Procedures for finding efficient portfolios and deriving efficient frontier; Deriving efficient frontier We first calculated standard deviation, covariance matrix and expected return. The standard deviation and expected return were calculated by applying the Excel STDEV and AVERAGE functions to the historic monthly percentage returns data. Table 1 below shows the correlation matrix, standard deviations, and the average returns for the rates of return on the stock index. After we input Table 1 into our spreadsheet as shown, we created the covariance matrix in Table 2 using the relationship . Table 1: Portfolio expected return and Standard deviation Table 2: covariance matrix Variance-covariance matrix 0.09 0.04 0.02 -0.01 0.01 0.03 0.03 -0.01 0.02 0.02 0.04 0.37 0.09 0.04 0.02 0.09 0.08 0.00 0.03 0.05 0.02 0.09 0.02 0.01 0.00 0.03 0.01 0.00 0.01 0.00 -0.01 0.04 0.01 0.04 0.00 -0.01 -0.01 0.01 -0.01 0.00 0.01 0.02 0.00 0.00 0.02 -0.01 -0.01 0.00 0.00 0.00 0.03 0.09 0.03 -0.01 -0.01 0.14 0.07 0.00 0.00 0.02 0.03 0.08 0.01 -0.01 -0.01 0.07 0.09 -0.01 0.01 0.02 -0.01 0.00 0.00 0.01 0.00 0.00 -0.01 0.04 -0.01 -0.01 0.02 0.03 0.01 -0.01 0.00 0.00 0.01 -0.01 0.07 -0.01 0.02 0.05 0.00 0.00 0.00 0.02 0.02 -0.01 -0.01 0.06 Using Solver we can summarize the procedure as follows; The steps with Solver are: a We invoked the Solver by choosing Tools then Options then Solver. b We Specified in the Solver parameter Dialog Box: the Target cell that needs to be optimized specify max or min c We Chose Add to specify the constrains then OK d We lastly clicked on Solve and got the results in the spreadsheet. 3. Introduce a risk-free asset into the model and find the optimal portfolio of shares. Discuss what happens to this portfolio as the risk-free rate of return changes. The curved line represents the return values and risks that result from combination of various shares. It is also known as the efficient frontier and it represents efficient portfolios of shares that is, portfolios that give the minimum risk for a given level of return or maximum return for a given level of risk. On the other hand, the straight line is known as capital allocation line and it represents the expected return and standard deviation from various combinations of the risk-free asset and the optimal risky portfolio. It starts at the risk-free return of 4% and is perpendicular to the curved line. It represents the highest ratio of risk premium to standard deviation (Sharpe Ratio). For our computations if we invest in a free risk with 10% portfolio we get an expected return is 4% with a zero standard deviation. By investing 100% in the optimal combination of the 2 risky shares, the expected return is 10.0% and the standard deviation is 13.51%. 4. Introduce a risk aversion factor into the model and discuss its effect upon the optimal complete portfolio. The concept of risk aversion plays a very vital role in the modern portfolio theory (MPT). From the excel analysis presented, our analysis reveals that, in the absence of complex constraints or objective functions, the optimal mean-variance portfolio, return, utility and risk of the all the companies decrease as the risk aversion increases. A risk-averse investor will have to penalize the expected rate of return of a risky portfolio by a certain percent and a greater risk will attract a greater penalty. There is a positive association between risk aversion and returns that is, a decrease in risk aversion results to a decrease the return an investor would require on stocks. As such the prices on stocks will increase since the cost of equity would have declined. Subsequently, with a decline in risk aversion, the risk premium will too decline as compared to the previous difference between returns on stocks and bonds. For instance, in our case with a risk aversion factor of 10, implies that the companies would invest 74% of the money in the risk-free asset and just 26.05% in the optimal risky portfolio. The companies’ return would be 5.56% and the standard deviation would be 3.52%. When we changed the risk aversion factor to 2 we found out that the percent we can invest has gone down in the risk free asset but increased in the optimal risky portfolio. This means we have become less averse to risk. Finally we find the expected return has gone up and the risk has also increased. 5. Discuss the limitations of this approach to portfolio selection. The mean-variance portfolio optimization makes an assumption that the parameters of the expected returns used as inputs in the model and obtained using MLE are known with perfect precision. However, in many instances, it is actually very difficult to estimate expected returns precisely. The portfolio also ignores estimation the error resulting to very poor properties for instance, the weights of the portfolio often have extreme values and that tend to fluctuate dramatically over time. MPT also assumes that risk is same as volatility. This assumption has been tested using empirical studies, Haugen and Heins (1975) observed that there is little correlation that exists between risk (volatility) and returns. Another assumption of MPT that results to its limitation is its assumption that portfolio returns can possibly be represented using the normal distribution is not always the case because of frequent swings and swifts in the market. The correlations between assets are never constant and neither are they fixed since the correlations change with changes in universal relations that exist between different assets. The theory calculates expected values mathematically based on past performance in order to measure the correlations between return and risk. Studies done by experienced investors have shown that past performance is however not a guarantee of performance in the future. If we consider only past performances we are subject to the leave out the current circumstances or scenarios that might as well not have been present when past data collection was being done. References Barry, C. B., 1974, “Portfolio Analysis under Uncertain Means, Variances, and Covariances," Journal of Finance, 29, 515-22 Black, F., and R. Litterman, 1992, “Global Portfolio Optimization," Financial Analysts Journal, 48, 28-43. Gregory Curtis, “Modern Portfolio Theory and Quantum Mechanics,” The Journal of Wealth Management, Fall 2002, pp. 1-7. Holden, Craig W. Excel Modeling: How to Build Financial Models in Excel. Pearson/Prentice Hall. Available WWW: http://www.spreadsheetmodeling.com/Portfolio%20Optimization%20-%20Dynamic%20Chart.htm Haugen and Heins. Risk and the Rate of Return on Financial Assets: Some Old Wine in New Bottles. Journal of Financial and Quantitative Analysis. Vol. 10, Iss. 5, p. 775-84. 1975. Markowitz, H.M. (March 1952). "Portfolio Selection". The Journal of Finance 7 (1): 77-91 William J. Coaker II, The Volatility of Correlation Important Implications for the Asset Allocation Decision, Journal of Financial Planning. http://www.fpanet.org/journal/articles/2006_Issues/jfp0206-art7.cfm Read More