Portfolio Optimization with Normally Distributed Assets: Expected Return, Standard Deviation, and Sharpe Ratio

a) The formula for the expected return of a portfolio is:\n\nE(Rp) = wE(Rx) + (1-w)E(Ry)\n\nWhere:\nE(Rp) = Expected return of the portfolio\nw = Weight of asset X in the portfolio\nE(Rx) = Expected return of asset X\nE(Ry) = Expected return of asset Y\n\nGiven that the weight of asset X is 50% and the weight of asset Y is also 50%, we can substitute these values into the formula:\n\nE(Rp) = 0.5 * 3% + 0.5 * 2%\nE(Rp) = 1.5% + 1%\nE(Rp) = 2.5%\n\nTherefore, the expected return of the portfolio containing 50% of asset X and 50% of asset Y is 2.5%.\n\nb) The formula for the standard deviation of a portfolio is:\n\nσp = √(w^2 * σx^2 + (1-w)^2 * σy^2 + 2w(1-w) * ρxy * σx * σy)\n\nWhere:\nσp = Standard deviation of the portfolio\nw = Weight of asset X in the portfolio\nσx = Standard deviation of asset X\nσy = Standard deviation of asset Y\nρxy = Correlation between asset X and asset Y\n\nGiven that the weight of asset X is 50%, the standard deviation of asset X is 18%, the standard deviation of asset Y is 16%, and the correlation between asset X and asset Y is -0.3, we can substitute these values into the formula:\n\nσp = √(0.5^2 * 0.18^2 + (1-0.5)^2 * 0.16^2 + 2 * 0.5 * (1-0.5) * -0.3 * 0.18 * 0.16)\nσp = √(0.002025 + 0.002025 + (-0.016416))\nσp = √(0.003634)\nσp = 0.0603 (rounded to four decimal places)\n\nTherefore, the standard deviation of the portfolio is 0.0603.\n\nc) To compare the options in terms of excess return per unit of risk, we need to calculate the Sharpe Ratio for each option. The Sharpe Ratio is calculated as:\n\nSharpe Ratio = (E(R) - Rf) / σ\n\nWhere:\nE(R) = Expected return\nRf = Risk-free rate\nσ = Standard deviation\n\nFor option (1), only asset X:\nSharpe Ratio (X) = (3% - 1%) / 18% = 0.11\n\nFor option (2), only asset Y:\nSharpe Ratio (Y) = (2% - 1%) / 16% = 0.06\n\nFor option (3), the portfolio:\nSharpe Ratio (Portfolio) = (2.5% - 1%) / 0.0603 = 24.91\n\nThe investor would rationally pick option (3), the portfolio, as it has the highest Sharpe Ratio. This means that the portfolio offers the highest excess return per unit of risk compared to the individual assets. The higher the Sharpe Ratio, the better the risk-adjusted performance of the investment option.