Mean-Variance Optimization

10232012 p kolm 23 another way of calculating the

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Unformatted text preview: n by μp = w ′μ + (1 − w ′ι)rf 2 σ p = w ′Σw Note that w = γΣ−1 (μ − rf ⋅ ι ) w 0 = 1 − w ′ι says that the weights of the risky assets of any minimum variance portfolio are proportional to the vector Σ−1(μ − rf ι) , with a proportionality constant γ RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 22 • This risky portfolio is called the tangency portfolio • Fama demonstrated that under the assumptions of that all investors are M- V optimizers, (1) the tangency portfolio must consist of all assets available to investors, and (2) each asset must be held in proportion to its market value relative to the total market value of all assets. In other words, under these assumptions he showed that the tangency and market portfolios are the same. We will drive this resulted in the lecture about CAPM Let us determine the weights of the tendency portfolio, w Tang It must hold that w Tang′ι = 1 from which it follows that w Tang = 1 ⋅ Σ−1(μ − rf ι) ι ′Σ(μ − rf ι) We can t...
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