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PS3Solutions - Finance 100: Problem Set 3 Alternative...

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Unformatted text preview: Finance 100: Problem Set 3 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Statistical Notation This section outlines the statistical notation that is used in this problem set and the course. The expectation of a random variable, x , is denoted by E ( x ) and cor- responds to the average or expected value. The variance of a random variable, x , is denoted by V ar ( x ) = 2 x . Sometimes I will write 2 1 and 2 2 to distinguish between the variance of asset 1s and asset 2s returns. The standard deviation of a random variable, x , is denoted by SD ( x ) = x and is simply the square root of the variance. Again, I will write 1 and 2 to distinguish between the standard deviations of asset 1s and asset 2s returns. The covariance of a pair of random variables, x and y , is denoted by Cov ( x,y ) = xy . When I write 12 , I am referring to the covariance between asset 1s and asset 2s returns. The correlation of a pair of random variables, x and y , is denoted by Corr ( x,y ) = xy . When I write 12 , I am referring to the correlation between asset 1s and asset 2s returns. 1 II. Formulas This section contains the formulas you will need for this homework set: The derivations of these formulas are included in a technical appendix at the end of these solutions. You do not need to understand the discussion in the technical appendix and may skip it . It is provided by request. 1. Expected Return on a Two-Asset Portfolio: E ( r p ) = w 1 E ( r 1 ) + w 2 E ( r 2 ) (1) where r p is the return on the portfolio, r 1 and r 2 are the returns on assets 1 and 2, w 1 and w 2 = 1- w 1 are the weights on the first and second assets (i.e. the fraction of the total dollar investment in each asset). This is a special case of the more general result that the expected return of a portfolio of N-assets is the allocation weighted average of the individual asset expected returns. 2. Return Variance of a Two-Asset Portfolio: 2 p = w 2 1 2 1 + w 2 2 2 2 + 2 w 1 w 2 1 2 12 , (2) where w 1 and w 2 = 1- w 1 are the weights on the first and second assets (i.e. the fraction of the total dollar investment in each asset). 3. Weights for the Minimum Variance Portfolio: w * 1 = 2 2- 1 12 2 1 + 2 2- 2 1 2 12 (3) and w * 2 = 1- w * 1 are the portfolio weights such that the variance of the two-asset portfolio is minimized. 4. Variance of the Minimum Variance Portfolio: 2 2 2 1- 2 2 2 1 2 12 2 1 + 2 2- 2 1 2 12 (4) 5. Correlation of Two Random Variables: 12 = 12 1 2 (5) 2 III. Problems 1....
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PS3Solutions - Finance 100: Problem Set 3 Alternative...

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