Question

# You invest \$10,000 in portfolio XYZ. The portfolio XYZ is composed of a risky asset with...

You invest \$10,000 in portfolio XYZ. The portfolio XYZ is composed of a risky asset with an expected rate of return of 15% and a standard deviation of 20% over the one year time period. The risk free asset has a rate of return of 5% over the same time period. How much money should be invested in the risky asset so that the standard deviation of returns on XYZ portfolio is 10% over the one year time horizon

standard deviation for a two-asset portfolio σp = (w12σ12 + w22σ22 + 2w1w2Cov1,2)1/2

where σp = standard deviation of the portfolio

w1 = weight of Asset 1

w2 = weight of Asset 2

σ1 = standard deviation of Asset 1

σ2 = standard deviation of Asset 2

Cov1,2 = covariance of returns between Asset 1 and Asset 2

Cov1,2 = ρ1,2 * σ1 * σ2, where ρ1,2 = correlation of returns between Asset 1 and Asset 2

For a risk free asset, the standard deviation is zero, and the covariance with the risky asset is zero

σp = (w12σ12 + w22σ22 + 2w1w2Cov1,2)1/2

0.10 = (w120.202 + w220.002 + (2w1w2 * 0)/)1/2

0.10 = (w120.202)1/2

0.10 = w1 * 0.20

w1 = 0.50

Proportion of portfolio to invest in risky asset = 0.50, or 50%

Money to invest in risky asset = \$10,000 * 50%

Money to invest in risky asset = \$5,000

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