Mean-Variance Optimization with a Risk-Free Asset
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Catégorie :Category: nCreator TI-Nspire
Auteur Author: SPITZER2001
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 2.14 Ko KB
Mis en ligne Uploaded: 22/11/2024 - 05:34:06
Uploadeur Uploader: SPITZER2001 (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4334737
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 2.14 Ko KB
Mis en ligne Uploaded: 22/11/2024 - 05:34:06
Uploadeur Uploader: SPITZER2001 (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4334737
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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Denote the risk-free rate with 5E5S . ¶ Define 50 to be the vector of percentage weights of risky assets only, so the weight of the risk-free asset is 1 5025Ï = 1 [5d1 + 5d2 + ï + 5d5A] ¶ The portfolio return is the weighted sum of constituents returns, i.e., 5_5] = 5025+ + [1 5025Ï]5E5S = 5E5S + 502[5+ 5E5S 5Ï] If the investment universe contains both risky assets and a risk-free asset, the expected return and the variance of a portfolio are given by 55] = 5E5S + 502 5Í 525] = 5025º50 where: ¶ 50 is the vector of percentage weights in the risky assets; ¶ 5Í and 5º are the expected excess returns and the covariance matrix of the risky assets; ¶ 5E5S is the risk-free rate. The Sharpe ratio of a portfolio is the expected excess return over the risk-free rate, divided by volatility: 55] =( 55] 5E5S) /55] ¶ SR measures the expected reward per unit of risk taken. The Sharpe ratio of a portfolio is the slope of the line that connects the risk-free asset and the portfolio in the (55], 55]) diagram. The maximum Sharpe ratio (MSR) portfolio belongs to the efficient frontier with risky assets. ¶ Why? ¶ The efficient frontier is concave. ¶ Because the marginal gain in 55] for a fixed increase in 55] is decreasing. The tangency / maximum Sharpe ratio portfolio is the portfolio of risky assets only that maximizes the Sharpe ratio. Its weights are given by 50tan = (1 /5Ï25º15Í) 5º1 5Í ¶ Technical assumption required to ensure the existence of the tangency portfolio: the expected return on the GMV portfolio must be greater than the risk-free rate, i.e., 5O/5N > 5E5S where 5O = 5Ï25º15Í and 5N = 5Ï25º15Ï Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
Denote the risk-free rate with 5E5S . ¶ Define 50 to be the vector of percentage weights of risky assets only, so the weight of the risk-free asset is 1 5025Ï = 1 [5d1 + 5d2 + ï + 5d5A] ¶ The portfolio return is the weighted sum of constituents returns, i.e., 5_5] = 5025+ + [1 5025Ï]5E5S = 5E5S + 502[5+ 5E5S 5Ï] If the investment universe contains both risky assets and a risk-free asset, the expected return and the variance of a portfolio are given by 55] = 5E5S + 502 5Í 525] = 5025º50 where: ¶ 50 is the vector of percentage weights in the risky assets; ¶ 5Í and 5º are the expected excess returns and the covariance matrix of the risky assets; ¶ 5E5S is the risk-free rate. The Sharpe ratio of a portfolio is the expected excess return over the risk-free rate, divided by volatility: 55] =( 55] 5E5S) /55] ¶ SR measures the expected reward per unit of risk taken. The Sharpe ratio of a portfolio is the slope of the line that connects the risk-free asset and the portfolio in the (55], 55]) diagram. The maximum Sharpe ratio (MSR) portfolio belongs to the efficient frontier with risky assets. ¶ Why? ¶ The efficient frontier is concave. ¶ Because the marginal gain in 55] for a fixed increase in 55] is decreasing. The tangency / maximum Sharpe ratio portfolio is the portfolio of risky assets only that maximizes the Sharpe ratio. Its weights are given by 50tan = (1 /5Ï25º15Í) 5º1 5Í ¶ Technical assumption required to ensure the existence of the tangency portfolio: the expected return on the GMV portfolio must be greater than the risk-free rate, i.e., 5O/5N > 5E5S where 5O = 5Ï25º15Í and 5N = 5Ï25º15Ï Made with nCreator - tiplanet.org
>>
