Now assume there is a risk-free asset available and the annual risk-free rate is 2%. Using mean-variance portfolio theory, determine the 4 portfolio weightings and the portfolio variance for the unique fund F defined by the tangent portfolio and the one-fund theorem.

Note that the portfolio weights sum to 1 and short selling (negative weights) are permissible.

What is the portfolio weight for SPX in the unique fund F?

Consider the following table of annual rates of return, in percentage, for four common risky assets over the time period 2010 to 2019 Berkshire Hathaway (ticker: BRK/A) S&P 500 Index (ticker: SPX) NASDAQ 100 Index (ticker: NDX) Russell 2000 Index (ticker: RUT) YEAR BRK/A SPX NDX RUT 2010 21.4 15.1 19.22 26.85 2011 -4.7 2.1 2.7 -4.18 2012 16.8 16 16.82 16.35 2013 32.7 32.4 34.99 38.82 2014 27 13.7 4.89 17.94 2015 -12.5 1.4 8.43 -4.41 2016 23.4 12 21.31 5.89 2017 14.65 21.9 21.8 31.52 2018 -11.01 2.8 -4.4 -1.04 37.96 25.52 2019 31.5 Assuming there is no risk-free asset available, suppose you desire to invest in a portfolio of these 4 risky assets such that you minimize the portfolio variance, subject to the constraint that the portfolio weights sum to 1. Note that short positions (negative weights) are permissible. Using mean-variance portfolio optimization, determine the 4 minimum variance portfolio weightings and the minimum portfolio variance. Consider the following table of annual rates of return, in percentage, for four common risky assets over the time period 2010 to 2019 Berkshire Hathaway (ticker: BRK/A) S&P 500 Index (ticker: SPX) NASDAQ 100 Index (ticker: NDX) Russell 2000 Index (ticker: RUT) YEAR BRK/A SPX NDX RUT 2010 21.4 15.1 19.22 26.85 2011 -4.7 2.1 2.7 -4.18 2012 16.8 16 16.82 16.35 2013 32.7 32.4 34.99 38.82 2014 27 13.7 4.89 17.94 2015 -12.5 1.4 8.43 -4.41 2016 23.4 12 21.31 5.89 2017 14.65 21.9 21.8 31.52 2018 -11.01 2.8 -4.4 -1.04 37.96 25.52 2019 31.5 Assuming there is no risk-free asset available, suppose you desire to invest in a portfolio of these 4 risky assets such that you minimize the portfolio variance, subject to the constraint that the portfolio weights sum to 1. Note that short positions (negative weights) are permissible. Using mean-variance portfolio optimization, determine the 4 minimum variance portfolio weightings and the minimum portfolio variance