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Unformatted text preview: TUTORIAL 11– WEEK 12 ECON3107/ECON5106 – Economics of Finance Solutions 1. Suppose an investor decides to construct a portfolio consisting of a riskfree asset that pays 6 percent and a stock index fund that has an expected rate of return of 12 percent and a standard deviation of 20 percent. Let x denote the proportion invested in the stock index fund. This implies that the proportion (1 x ) is invested in the riskfree asset. (In your answers use 6 for 6 percent etc). (a) Find the equation for the efficient frontier. Graph the efficient frontier in ( e p ,s p ) space where e p is the expected return on the portfolio and s p the standard deviation of the return on the portfolio. Let e IF and s IF denote the expected rate of return and the standard deviation of the stock index fund. Let e RF be the riskfree rate. Then, the expected return of the portfolio is e p = xe IF + (1 x ) e RF = 12 x + 6(1 x ) . (1) Notice, that the variance of a linear combination of two random variables e y and e z is given by V ar ( a e y + b e z ) = a 2 V ar ( e y ) + b 2 V ar ( e z ) + 2 abCov ( e y, e z ) . The variance of the return on the riskfree asset is zero. The covariance of the return on the riskfree asset with any risky return is zero as well. Therefore, the variance of the portfolio is v p = x 2 ( s IF ) 2 while its standard deviation is s p = xs IF = 20 x. (2) Solving (2) for x and substituting it into equation (1) yields the following equation for the efficient frontier: e p = 6 + 0 . 3 s p . The graph of the frontier is the blue solid line in the figure above....
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 valentyn
 Economics, EFFICIENT FRONTIER

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