8.tangent

8.tangent - University of California Los Angeles Department...

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University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Short sales allowed, risk-free lending and borrowing Point of tangency - solution See also, Modern Portfolio Theory and Investments Analysis by Elton, Gruber, Brown, Goetzmann, Wiley 6th Edition, 2003. When the investor faces the efficient frontier and riskless lending and borrowing the combi- nations of the risk-free asset with the risky portfolio lie on the line: ¯ R p = R f + ¯ R A - R f σ A ! σ p (1) The solution is to find the point of tangency of this line to the efficient frontier. Let’s call this point G . To find this point we want to maximize the slope of the line in (1) as follows: max θ = ¯ R p - R f σ p Subject to n X i =1 x i = 1 Since, R f = ( n X i =1 x i ) R f = n X i =1 x i R f we can write the maximization problem as max θ = n i =1 x i ( ¯ R i - R f ) ± n i =1 x 2 i σ 2 i + n i =1 n j =1 ,j 6 = i x i x j σ ij ² 1 2 or max θ = " n X i
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This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.

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8.tangent - University of California Los Angeles Department...

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