## n volatility, higher excess return is achieved. Formula: (Rx

n this Portfolio Optimization Project, I selected ten stocks traded from S&P500  and gathered  these stocks’ historical data from Yahoo Finance 1. Then I analyze this data in order to obtain the optimal weights of our initial portfolio. To maintain our investment in a current tangency portfolio, we recalculated the optimal weights . I also computed the risk of each asset and portfolio beta and also I computed estimates of each asset’s expected returns and return variances of ten stocks for each of our factor models. Furthermore, I computed estimates of the covariances among our asset returns. In order to find which model performs best, we compared each portfolio’s actual return with its corresponding estimated portfolio return.  Mean, standard deviation,  summarize the statistical behavior of the achieved return achieved from each trading strategy is reported in a monthly basis in order to examine and compare the absolute performance of the returns. Sharpe-ratio defined that is the excess return of the portfolio (from risk free rate) divided by its standard deviation. The higher Sharpe-ratio, the better the performance of the portfolio, since for each unit of volatility, higher excess return is achieved. Formula: (Rx – Rf) / StdDev(x) Here are the individual components: x = the investment rx = the average rate of return Rf = the best available rate of return of a risk-free security StdDev = the standard deviation of the return   Moreover, after taking the difference of the Sharpe ratio of the portfolios and the benchmark, we test whether the mean difference is greater than zero. Alpha That is the same as Jensen alpha, defined  the estimated intercept of the regression runs from portfolio excess return over markets’s excess return. As the alpha is  positive and significant, it means that, the portfolio generates higher return. On the other hand, If alpha is statistically non-zero and negative, it means that the portfolio is overpriced relative to the benchmark. The S&P 500 . In this portfolio Beta is a measure of  risk relative to the S&P 500 ,nine  assets have negative Beta that shows investment that moves in the opposite direction, when the S&P 500 market increases negative_beta investment would decrease, although negative beta indicate a risk free but it may have volatile in over time   less than one  I estimated beta by formula:   In this  portfolio I use Capital Asset Pricing model to analyze expecting return of ten stocks  .and S&P 500 index is used as market portfolio and the stock “monthly adjusted closing price are form 1/11/2012 to 1/10/2017.   ra = rrf + Ba (rm-rrf)  Where: rrf = the rate of return for a risk-free security  rm = the broad market’s expected rate of return  Ba = beta of the asset  Alpha is compared with market index and it measures active return in investment.Hence in this portfolio the investment’s return is 0.24% over a period of 60 months  better than market  in during same  time.

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