Mean-Variance Optimization - Case Study Example

1 mean-variance optimization framework using Matlab Mean-variance optimization is a model that helps in choosing portfolio that has highest expected returns at certain a variance after considering assumptions made on the means, covariance and variances of assets used. It is sufficient for maximizing expected return for portfolio if at least one of two conditions prevails. To achieve this, the Matlab code for the solution is as R - expected return , Var – covariance, RA- Risk Aversion, PR- Portfolio Risk; PRet- Portfolio return, OR- Option Risk, ORet- Option Return, PW- Portfolio Weights, OW- option weight The results for the output of is as follows ; Determine the optimum weight for each stock to be held in the portfolio is 0.121, 0.2831 and 0.5959. the figure above shows risk-return combination The minimum variance ignores completely the expected return on the portfolio, which is inconsistent with the mean variance framework. In the situations where the investor are avoiding risks, or the prices of the futures follow specific patterns, then the mean variance ratio of the hedge will be equal to the minimum variance ratio. This is true for the natural gas futures since it is usually common that changes in the prices of the futures is normally zero, therefore it follows a certain process. This also means that, given the properties of natural gas markets and futures, the challenge of the minimum variance can be dealt with. The formula to compute a portfolio variance with seven assets is a bit cumbersome. Indeed, amount of calculation increases greatly as the number of assets in the portfolio grows. The calculations are not difficult, but using a computer is highly recommended for portfolios consisting of more than three assets. Where the subscript denotes efficient portfolio, the term (rm-rf) can be thought of as the extra return that can be gained by increasing the level of risk (standard deviation) on an efficient portfolio by one unit. The entire second term on the right of the equation is thus the market price of risk times the amount of risk in the portfolio. The expression Rf is the price of time; that is, it is the price paid for delaying consumption for one period. The expected return on an efficient portfolio is: (Price of time) + (price of risk) (amount of risk) Although this equation sets the return of an efficient portfolio, we need to go beyond to deal with returns on non efficient portfolios or individual securities. For well-diversified portfolios, non-systematic risk tends to go to zero, and the only relevant risk is systematic risk measured by beta. Because we assume that investors are concerned only with expected return and risk the only dimensions of a security that need be of concern are expected return and beta 2 MATLAB codes to perform numerical experiments We begin by designing a mat lab code The output of the above code had the following results • Starting temperature- 10 • Generating method- • Number of iterations- 200 • Temperature variations- • Termination- 1.0000e-007 The changes in initial temperature lead chnges in time taken for inteation as well terminal temperature changing 3. Black-Scholes theory Black & Scholes option pricing model to assess the management flexibility will illustrate on how strategic acquisitions affect the value of growth opportunities, divesture options and operational flexibility, which will in the end affect acquisition decision making. By the fact that they have credibly make their case, in the proposal of a new area of application for real options; they can be logically described as one of the pioneers. Lower volatility is associated with deep in the money situation, whereas upper volatility is associated with out of the money situations. This volatility is usually estimated in the calculations mostly because it cannot be observed. There are two kinds of volatility used in the Black-Scholes option pricing model, they include: implied volatility that illustrates option’s price and historical volatility showing the value of that option. There also exists the exponentially-weighed historical volatility, which uses moving averages and helps to incorporate recent stock prices in the model’s equation; whereas the Equally-weighed historical volatility helps incorporate historical prices.The MATLAB code for 50 asset paths is as The results reveal that there is considerable variation amongst the use of sample sizes as the value weighed return and the equal weighed return. The issue becomes quite risky when reliance is placed on the frequency related to the use of data. It is possible to conceive that the volatility that is ascertained by using the small samples will be extra accurate as compared to the values determined with large samples. Although in some respects such results point at the success of ascertaining simple volatility estimation from high samples, they still appear to be more complex than the. It is better to confirm that such outcomes do not prove to become mere objects of the larger samples differences amongst data used. b). Assuming risk neutrality and using MATLAB, the Monte-Carlo simulation method the value of a European call option within a 95% confidence interval is 908.2619. When samples are increased the volatility increases. The issue becomes quite risky when reliance is placed on the sample size related to the use of data. It is possible to conceive that the volatility that is ascertained by using the statistics will be extra accurate as compared to the small determined from data in view of the larger frequencies pertaining to statistics for every operational day. Although in some respects such results point at the success of ascertaining simple volatility estimation from high frequency statistics, they still appear to be more complex than the volatility estimation obtained from small sample. There also exists the exponentially-weighed historical volatility, which uses moving averages and helps to incorporate recent stock prices in the model’s equation; whereas the Equally-weighed historical volatility helps incorporate historical prices. The code for the Matlab is as follows Results Value = 9.952 sterr = 0.0194 P_bseu = 8.2619 By setting conditional variance equal to a constant plus a weighted average the model elegantly capture the volatility clustering in asset returns aforementioned. The simplest specifications pertain to the conventional VAR systems whereby the residual vectors have a constant co-variance matrix. Additionally, such a process enables to use the opportunities in benchmarking the results of previous studies of global transmission of stock returns that use varied specifications. Since volatility is being tested, the mean squared error will tend to enhance the performance innovations towards the 4th powers in making the loss functions quite responsive to the outliers. References Huynh, H., Lai, V.S., Soumare, 2008. Stochastic Simulation and Applications in Finance with MATLAB Programs Jia, Q., 2009. Pricing American Options using Monte Carlo Methods Read More