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Quadratic Portfolio Management Variance - Covariance Analysis EMIS 8381 Prepared by Carlos V Caceres SID: 27957167
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Objectives To buy and invest in stocks efficiently by maximizing the return To maximize the expected return of a stock portfolio To minimize the risk with investing in different stocks To use Excel and the variance-covariance procedure applied to Quadratic Programming
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Portfolio Management Portfolio management optimization is often called mean-variance (MV) optimization. The mathematical problem can be formulated in many ways. The principal problems can be summarized as follows: 1.Minimize risk for a specified expected return 2.Maximize the expected return for a specified risk 3.Minimize the risk and maximize the expected return using a specified risk aversion factor 4.Minimize the risk regardless of the expected return 5.Maximize the expected return regardless of the risk 6.Minimize the expected return regardless of the risk
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Investment in Action – The Return The return is calculated by the following components: Where: P t – P t-1 is the price variance of the stock in the P t-1 market Pt = Price at time t Pt-1 = Price at t-1 Dt = Dividends for each stock Ct = Premium The Return: R t = P t - P t - 1 + D t + C t P t - 1
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The Volume Weighted Average Price Investment in Action – VWAP
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The portfolio
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Expected Return We first calculate the variances of the prices Where: -VWAP t is price of current week -VWAP t-1 price of prior week VARVWAP for Portfolio Expected Return = VARPRICE Average
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The Risk of A Stock To determine the risk of a stock we apply the variance/covariance method * Where i is the number of observations STDDEV =
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Probability Of Losing A Stock The Z-Value =
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The Portfolio Expected Return is the weighted sum of the expected returns for all the stocks Rp = Σ Rj * Aj where Rp: Expected return of portfolio Rj: Expected return of stock j Aj : Percent of investment in stock j The Risk of the portfolio is equal to Risk =
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Optimization of the Portfolio Maximize Return = Σ Ai * VARVWAPi subject to: Where: - Ai is the percent of investment in stock I - VARVWAP is the volume weighted average price of the stock - COVARij is the covariance between each pair of stock - B is the desired risk level
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Covariance It shows how two stock prices behave between one another with respect to the expected return of each stock Covariance (A1,A2) = (1/(n-1))*Σ(A1i-U1)(A2i-U2) Where: - A1i = Variance in price of Stock 1 - A2i = Variance in price of Stock 2 - A1 = Stock 1 - A2 = Stock 2 - U1 = Expected Return of Stock 1 - U2 = Expected Return of Stock 2
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Correlation Coefficient ( r ) Where: r = -1 the correlation is perfect and inverse r = 1 the correlation is perfect and direct r = 0 means that the two stocks are not correlated
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Covariance and Correlation
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The Model. Portfolio Optimization
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The Model. Minimum Risk
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