1. Quadratic Portfolio Management Variance - Covariance Analysis EMIS 8381 Prepared by Carlos V Caceres SID: 27957167

2. Agenda Objectives Portfolio Management Investment In Action – The Return – The Volume Weighted Average Price (VWAP) – The Risk – Variance and Covariance Optimization Of The Portfolio: Quadratic Programming

3. 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 Solver and the variance-covariance procedure to resolve Markowitz quadratic model

4. 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

5. 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

6. The Volume Weighted Average Price Investment in Action – VWAP

7. The portfolio

8. 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

9. 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 =

10. Probability Of Losing A Stock The Z-Value =

11. 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 =

12. 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

13. 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

14. 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

15. Covariance and Correlation

16. The Model. Portfolio Optimization

17. The Model. Minimum Risk

18. The Efficient Frontier How to balance maximum return and minimum risk?

19. Questions?