Suppose you want to invest $ 1 million and you have two assets to invest in: Risk free asset with return of 12% per year and a risky asset with expected return of 30% and standard deviation of 40%. If you want a portfolio with standard deviation of 30% how much do you invest in each of the assets?
Let the weight allocated to Risk free asset be w_{1} and risky asset be w_{2} = 1 - w_{1}
Standard deviation of risk free asset = σ_{1} = 0
Standard deviation of risky asset = σ_{2} = 0.40
Covariance between risky asset and risk free asset = Cov(1, 2) = 0 (since the covariance/correlation with risk free security is always zero)
Standard deviation of the portfolio = P = 0.30
P = [ w_{1}^{2}σ_{1}^{2} + w_{2}^{2}σ_{2}^{2} + 2w_{1}w_{2}Cov(1, 2) ]^{1/2} = [ w_{1}^{2}0^{2} + (1-w_{1})^{2}0.40^{2} + 2w_{1}(1 - w_{1})*0 ]^{1/2} = 0.30
=> (1-w_{1})*0.40 = 0.30
=> w_{1} = 1 - 0.30/0.40 = 0.25 or 25%
=> w_{2} = 1 - w_{1} = 1 - 0.25 = 0.75 or 75%
Hence, Investment in risk free asset = 0.25*1000000 = $250000
Investment in risky asset = 0.74*1000000 = $750000
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