Optimal Portfolio Sharpe Single Index Model | |||||||||||||||||
A three company active portfolio is to be constructed then optimised. A brief word is required here regarding active and passive portfolios. An active portfolio management strategy is aimed at outperforming the market, using as proxy some specific index. A passive portfolio strategy aims to reproduce the performance of a particular index. Clearly the active portfolio carries more risk. The Single Index Model measures both the risk and return of a company's value.
r_{it} - r_f = \alpha_i + \beta_i(r_{mt} - r_f) + \epsilon_{it} \, r i t - r f = a i + ß i ( r m t - r f ) + ? i t {\displaystyle
r_{it}-r_{f}=\alpha _{i}+\beta _{i}(r_{mt}-r_{f})+\epsilon _{it}\,} r_{it}
- r_f = \alpha_i + \beta_i(r_{mt} - r_f) + \epsilon_{it} \, where: rit is return to stock i in period t These equations show that the stock return is influenced by the market (beta), has a firm specific expected value (alpha) and firm-specific unexpected component (residual). Each stock's performance is in relation to the performance of a market index (such as the All Ordinaries). Security analysts often use the SIM for such functions as computing stock betas, evaluating stock selection skills, and conducting event studies. |
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Computation of the initial weight of each company in the active portfolio, based on the contribution of its Alpha and residual variance.
Scale the weights so that they total to 1
Computing the weighted Alpha of the active portfolio.
Calculate the residual variance
Calculate the weighted Beta of the active portfolio
Calculate the initial weight of the active portfolio
Calculate the final weight of the active portfolio *adjusting for Beta)
Computation of the expected risk premium using a weighted alpha and beta based on active and passive portfolios
Variance calculation for the optimum active portfolio
Sharpe Ratio computation using the expected risk premium and the standard deviation
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