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Suppose there are two risky assets, call them C and D
Suppose there are two risky assets, call them C and D. The tangency portfolio is 60% C and 40% D. The expected yearly returns are 4% and 6% for assets C and D. The standard deviations of the yearly returns are 10% and 18% for C and D and the correlation between the returns on C and D is 0.5. The risk-free yearly rate is 1.2%.
a) If you want an efficient portfolio with a standard deviation of the yearly return equal to 3%, what proportion of your equity should be in the risk-free asset? If there is more than one solution, use the portfolio with the higher expected yearly return.
b) If you want an efficient portfolio with an expected yearly return equal to 4%, what proportions of your equity should be in asset C, asset D, and the risk-free asset?
Expert Solution
a) 85% should be in risk free assets.
The target standard deviation is 3%, and the variance is 0.09%. The covariance between the two stocks = 0.5∗10%∗18%=0.9%.0.5∗10%∗18%=0.9%. With only these two stocks, the portfolio variance = 0.6∗(10%)2+0.4∗(18%)2+2∗0.9%=3.696%0.6∗(10%)2+0.4∗(18%)2+2∗0.9%=3.696%
Let pp be weight on risk free assets, which has variance zero. The portfolio variance now becomes (1−p)2∗3.696%(1−p)2∗3.696%. To achieve a standard deviation of 3%, we must have (1−p)2∗3.696%=0.09%(1−p)2∗3.696%=0.09%, which yields p=0.85p=0.85.
b) 13.3% should be in risk free asset.
To ensure efficiency, the relative value between stock C and Stock D should remain at 60-40. Let pp be the fraction of equity on risk free assets, the expected return on the portfolio = (1−p)∗(60%∗4%+40%∗6%)+p∗1.2%=4%(1−p)∗(60%∗4%+40%∗6%)+p∗1.2%=4%, which yields p=13.3%.
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