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Unformatted text preview: H AAS S CHOOL OF B USINESS U NIVERSITY OF C ALIFORNIA AT B ERKELEY UGBA 103 A VINASH V ERMA P ORTFOLIO T HEORY : T WO-S ECURITY P ORTFOLIOS [ R EAD THIS NOTE TOGETHER WITH THE E XCEL F ILE CALLED “2- SEC PORTFOLIOS . XLS ” ] 1. The main question that we are concerned with in portfolio theory is that of asset allocation: how much should we invest in what asset or security 1 given our twin goals of maximizing the expected return on the portfolio and minimizing the portfolio risk? We shall get basic insights into the problem by looking at it in the simplified context of two securities before we go on to the case of many securities. Let us start analyzing two- security portfolios by recalling from the previous lecture note ( Uncertainty and Risk in Finance ) the formula for the expected return on a portfolio: E R x E R p i i i n ( ~ ) ( ~ ) = = ∑ 1 . Setting n=2 , and denoting E R p ( ~ ) more concisely as E p , and E R ( ~ ) 1 and E R ( ~ ) 2 as E 1 and E 2 , we get: E x E x E p = + 1 1 2 2 .  2. The previous note concluded with the formula for the variance of the returns on the portfolio: σ σ p i j ij j n i n x x 2 1 1 = = = ∑ ∑ . Setting n=2 , and expanding the double sum in terms of the matrix in the last section of the previous note, we get: SECURITY 1 SECURITY 2 SECURITY 1 x x x 1 1 11 1 2 1 2 σ σ = x x x x 1 2 12 1 2 21 σ σ = SECURITY 2 x x x x 2 1 21 1 2 12 σ σ = x x x 2 2 22 2 2 2 2 σ σ = Summing the four boxes would lead to (the two “off-diagonal 2 ” boxes are identical): σ σ σ σ p x x x x 2 1 2 1 2 2 2 2 2 1 2 12 2 = + + , Observing that 2 1 12 12 σ σ ρ σ = , we can restate the equation above as: 2 1 12 2 1 2 2 2 2 2 1 2 1 2 2 σ σ ρ σ σ σ x x x x p + + = .  1 A security is a homogenized asset. A house is an asset, because one house differs from another, and the difference is relevant to financial valuation. For the purposes of financial valuation, one share of Microsoft Common Stock is exactly the same as any other share. Thus, Microsoft Common stock is a security. We shall be using the words “asset” and “security” interchangeably distinguishing between the two only if the distinction matters in the specific context. 2 The diagonal of a matrix runs from top left to bottom right. Two-Security Portfolios 1 H AAS S CHOOL OF B USINESS U NIVERSITY OF C ALIFORNIA AT B ERKELEY UGBA 103 A VINASH V ERMA 3. We shall analyze two-security portfolios by plotting the reward (as measured by the expected return) on the vertical axis against the risk (as measured by the standard deviation of returns) on the horizontal axis. Since we are interested in exploring how combining two securities in a portfolio might affect the risk of the portfolio, we shall keep fixed all attributes of the two securities other than the correlation between them, which will be gradually varied from its highest possible value of +1 to its lowest possible value of –1. We can think of changes in correlation as follows: Suppose there are two value of –1....
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This note was uploaded on 04/30/2010 for the course L&S 101 taught by Professor Chow during the Spring '10 term at Berkeley.
- Spring '10