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Quantitative Method and Lagrange Optimization

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  • 1. FIN4814 [1/2012]Minimum variance portfolio (two assets)Min0.0225W A2 + 0.0289W B2 + 0.0255W AW BS.t.WA + WB = 1W A + W B = 1 1 W A W B = 0L = 0.0225W A2 + 0.0289W B2 + 0.0255W AW B + (1 W A W B )Take partial derivatives w.r.t. each choice variable and each Lagrange multiplier L = 0.045W A + 0.0255W B = 0W A L = 0.0578W B + 0.0255W A = 0W BL = 1 WA WB = 0Rearrange in Matrix form 0.045 0.0255 1 W A 0 0.0255 0.0578 1 W B = 0 1 1 0 1W A 0.6236 W B = 0.3764 0.0377

2. FIN4814 [1/2012]Minimum variance portfolio (three assets)Min0.0225W A2 + 0.0289W B2 + 0.04WC2 + 0.0255W AW B + 0.0204W BWC + 0.024W AWCS.t.W A + W B + WC = 1W A + W B + WC = 1 1 W A W B WC = 0L = 0.0225W A2 + 0.0289W B2 + 0.04WC2 + 0.0255W AW B + 0.0204W BWC + 0.024W AWC + (1 W A W B WC )Take partial derivatives w.r.t. each choice variable and Lagrange multiplier L = 0.045W A + 0.0255W B + 0.024WC = 0W A L = 0.0578W B + 0.0255W A + 0.0204WC = 0W B L= 0.08WC + 0.0204W B + 0.024W A = 0WCL = 1 W A W B WC = 0Rearrange in Matrix form 0.045 0.0255 0.024 1 W A 0 0.0255 0.0578 0.0204 1 W B = 0 0.024 0.0204 0.08 1 WC 0 1 1 1 0 1W A 0.4793 W B = 0.3125WC 0.2082 0.0345 3. FIN4814 [1/2012]Minimum variance portfolio (two assets with additional constraints)Min0.0225W A2 + 0.0289W B2 + 0.0255W AW BS.t. WA + WB = 1 0.1W A + 0.14WB = 0.11W A + W B = 1 1 W A W B = 00.1WA + 0.14WB = 0.11 0.11 0.1WA 0.14WB = 0L = 0.0225W A2 + 0.0289W B2 + 0.0255W AW B + 1 (1 W A W B ) + 2 (0.11 0.1W A 0.14W B )Take partial derivatives w.r.t. each choice variable and each Lagrange multiplier L= 0.045WA + 0.0255WB 1 0.12 = 0WA L = 0.0578W B + 0.0255W A 1 0.14 2 = 0W BL= 1 WA WB = 01L= 0.11 0.1W A 0.014W B = 02Rearrange in Matrix form 0.045 0.0255 1 0.1 W A 0 0.0255 0.0578 1 0.14 W B = 0 1 1 0 0 1 1 0.1 0.14 00 2 0.11W A 0.75 W B = 0.25 1 0.0565 2 0.16375 4. FIN4814 [1/2012]Maximize portfolio return (three assets with additional constraints)Max0.11WA + 0.15WB + 0.08WcS .t.W A + 1.2WB + 0.9WC 1.1 0 WA 1 0 WB 1 0 WC 1 W A + WB + WC = 1Reduce the function to two variables by using the last constraintW A + WB + WC = 1WC = 1 W A WBMax0.11WA + 0.15WB + 0.08(1 WA WB )w.r.t. W A + 1.2WB + 0.9(1 W A WB ) 1.1 0 WA 1 0 WB 1 0 (1 W A WB ) 1OrMax0.03W A + 0.07WB + 0.08w.r.t.0.1W A + 0.3WB 0.2 0 WA 1 0 WB 1 0 (W A + WB ) 1Using the graphical methodDraw graph and locate the corner pointsFind the optimum solution Vesarach Aumeboonsuke