You are working for a fund that invests actively in different investment sectors. Analysts at your firm have produced the following estimates of sector portfolio (annualized) return variance-covariance matrix from historical data:

Assume that the risk-free rate is 2% per annum and that your firm can short-sell as well as buy any sector portfolios and the risk-free asset. Your firm does not trust numerical optimizers such as Excel Solver and demands analytical solutions. 

Analysts at your firm have also estimated the following annualized historical average returns: Consumer sector 10% Industrial sector 10% Technology sector 10% Services sector 10% Average return p.a.

John believes that the historical estimates above are representative of the forward- looking estimates of return covariances and expected returns. John wishes to get an expected return of 8% by forming a portfolio of the four sectors and the risk-free asset. What is the minimum return standard deviation that John must bear given his beliefs? Show your workings

Consumer sector Industrial sector Technology sector Services sector Consumer sector 0.3 0.2 0.3 0.1 Industrial sector 0.2 0.4 0.3 0.2 Technology sector 0.3 0.3 0.6 0.2 Services sector 0.1 0.2 0.2 0.3