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Suppose you want to invest $ 1 million and you have two assets to invest in: Risk free asset with...

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?

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Let the weight allocated to Risk free asset be w1 and risky asset be w2 = 1 - w1

Standard deviation of risk free asset = σ1 = 0

Standard deviation of risky asset = σ2 = 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 = [ w12σ12 + w22σ22 + 2w1w2Cov(1, 2) ]1/2 = [ w1202 + (1-w1)20.402 + 2w1(1 - w1)*0 ]1/2 = 0.30

=> (1-w1)*0.40 = 0.30

=> w1 = 1 - 0.30/0.40 = 0.25 or 25%

=> w2 = 1 - w1 = 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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