Appendix

Chapter 7

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?

Was this article helpful?

0 0
Your Retirement Planning Guide

Your Retirement Planning Guide

Don't Blame Us If You End Up Enjoying Your Retired Life Like None Of Your Other Retired Friends. Already Freaked-Out About Your Retirement? Not Having Any Idea As To How You Should Be Planning For It? Started To Doubt If Your Later Years Would Really Be As Golden As They Promised? Fret Not Right Guidance Is Just Around The Corner.

Get My Free Ebook


Post a comment