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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 risk-free 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 risk-free 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 risk-free 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 risk-free asset is zero. The covariance of the return on the risk-free 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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- Economics, EFFICIENT FRONTIER