Ch4Portfolio - Introduction to Financial Econometrics...

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Introduction to Financial Econometrics Chapter 4 Introduction to Portfolio Theory Eric Zivot Department of Economics University of Washington January 26, 2000 This version: February 20, 2001 1 Introduction to Portfolio Theory Consider the following investment problem. We can invest in two non-dividend paying stocks A and B over the next month. Let R A denote monthly return on stock A and R B denote the monthly return on stock B. These returns are to be treated as random variables since the returns will not be realized until the end of the month. We assume that the returns R A and R B are jointly normally distributed and that we have the following information about the means, variances and covariances of the probability distribution of the two returns: μ A = E [ R A ] , σ 2 A = Var ( R A ) , μ B = E [ R B ] , σ 2 B = Var ( R B ) , σ AB = Cov ( R A ,R B ) . We assume that these values are taken as given. We might wonder where such values come from. One possibility is that they are estimated from historical return data for the two stocks. Another possibility is that they are subjective guesses. The expected returns, μ A and μ B , are our best guesses for the monthly returns on each of the stocks. However, since the investments are random we must recognize that the realized returns may be di f erent from our expectations. The variances, σ 2 A and σ 2 B , provide measures of the uncertainty associated with these monthly returns. We can also think of the variances as measuring the risk associated with the investments. Assets that have returns with high variability (or volatility) are often thought to be risky and assets with low return volatility are often thought to be safe. The covariance σ AB gives us information about the direction of any linear dependence between returns. If σ AB > 0 then the returns on assets A and B tend to move in the
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same direction; if σ AB < 0 the returns tend to move in opposite directions; if σ AB =0 then the returns tend to move independently. The strength of the dependence between the returns is measured by the correlation coe cient ρ AB = σ AB σ A σ B . If ρ AB is close to one in absolute value then returns mimic each other extremely closely whereas if ρ AB is close to zero then the returns may show very little relationship. The portfolio problem is set-up as follows. We have a given amount of wealth and it is assumed that we will exhaust all of our wealth between investments in the two s problem is to decide how much wealth to put in asset A and how much to put in asset B. Let x A denote the share of wealth invested in stock A and x B denote the share of wealth invested in stock B. Since all wealth is put into the two investments it follows that x A + x B =1 . (Aside: What does it mean for x A or x B to be negative numbers?) The investor must choose the values of x A and x B . Our investment in the two stocks forms a
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This note was uploaded on 09/13/2011 for the course ECON 503 taught by Professor Pujara during the Spring '11 term at Punjab Engineering College.

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Ch4Portfolio - Introduction to Financial Econometrics...

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