activeportfoliocompute

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.

Company	Alpha	Beta	Residual Variance
X	1%	0.8	8%
Y	3%	1.5	9%
Z	-2%	1.2	8%

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} \,
? i t ~ N ( 0 , s i ) {\displaystyle \epsilon _{it}\sim N(0,\sigma _{i})\,} \epsilon_{it} \sim N(0,\sigma_i) \,

where:

rit is return to stock i in period t
rf is the risk free rate (i.e. the interest rate on treasury bills)
rmt is the return to the market portfolio in period t
a i {\displaystyle \alpha _{i}} \alpha _{i} is the stock's alpha, or abnormal return
ß i {\displaystyle \beta _{i}} \beta _{i} is the stocks's beta, or responsiveness to the market return
Note that r i t - r f {\displaystyle r_{it}-r_{f}} r_{it} - r_f is called the excess return on the stock, r m t - r f {\displaystyle r_{mt}-r_{f}} r_{mt} - r_f the excess return on the market
? i t {\displaystyle \epsilon _{it}} \epsilon _{{it}} are the residual (random) returns, which are assumed independent normally distributed with mean zero and standard deviation s i {\displaystyle \sigma _{i}} \sigma _{i}

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.

Computation of the initial weight of each company in the active portfolio, based on the contribution of its Alpha and residual variance.

Company X wa = 0.01/0.08 = 0.125 Company Y = 0.333 Company Z= -0.25