Question 3 – leverage and diversification
Imagine you have a risk-free asset, whose return is 0, and two risky assets with μ1=0.1, σ1=0.1, μ2=0.2, σ2=0.3, ρ1,2=0. Once we compute the tangency portfolio, we get that it has μT=0.118 and σT=0.098 (you do not have to compute it).
What are the mean return and standard deviation of the portfolio on the efficient frontier as a function of the share of the risk-free asset in the portfolio?
Assume that you are rational and risk-averse, so that you choose portfolios along the efficient frontier from the question above. Moreover, assume that you have leverage in your portfolio of the following sort: you sell short (borrow) so much of the risk-free asset that its share in your portfolio is -0.5. What is the mean return and standard deviation of your portfolio?
You wake up next day and you see that the only parameter that changes is the correlation between your risky assets: it decreased to ρ1,2=-0.5. This led to the change in your tangency portfolio, which has now the following parameters μT=0.122 and σT=0.073 (notice what happened with these two parameters).
In order to earn the same return as in question 2, what share of your portfolio should be in the risk-free asset? Does your leverage increase or decrease? What happens with the variance of such portfolio?
In order to keep the risk (variance) of your new portfolio the same as in question 2, what share of your portfolio should be in the risk-free asset? Does your leverage increase or decrease? What happens with the mean return of such portfolio?
Sample Solution
