A. Proof That Minimum Portfolio Variance Occurs with Equal Weights When Securities Have Equal Variance
O port = w1(O1)2 + (1 - w02(O1)2 - 2w1(1 - w1)ru(O1)2 = (O1)2[w2 + 1 - 2w1 + w2 + 2w1r12 - 2w2r12] = (O1)2 [2w2 + 1 - 2w1 + 2w1r12 - 2w2r1,2]
For this to be a minimum,
4w1 - 2 + 2r12 - 4w1r12 = 0 4wj(1 - ru) - 2(1 - r1,2) = 0
regardless of r12. Thus, if O1 = O2, Oport will always be minimized by choosing w1 = w2 = 1/2, regardless of the value of r12, except when r12 = +1 (in which case Oport = O1 = O2. This can be verified by checking the second-order condition d ( o Port) ôwj2
Problems 1. The following information applies to Questions 1a and 1b. The general equation for the weight of the first security to achieve minimum variance (in a two-stock portfolio) is given by
(01)2 + (O2)2 -2r1,2(01)(O2) 1a. Show that w1 = 0.5 when O1 = O2.
1b. What is the weight of Security 1 that gives minimum portfolio variance when r12 = 0.5, O1 = 0.04, and O2 = 0.06?
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