Financial Economics: Risky assets question - involves two-fund theorem
I've gotten an answer to this from another source, but I'm in need of a second opinion. I'll first list the question, then the answer I got.
Here's the answer I got online:
Stock X has an expected return of 12% and a variance of 30%. Stock Y has an expected return of 13% and a variance of 35%. There is a risk-free asset with a return of 4%.
David and Laurie have identical mean-variance utility functions and differ only in their attitude toward bearing risk. I give David $100 and tell him to invest in stock X, stock Y, and the risk-free asset. David has a PhD in physics, so he computes the covariance between returns of X and Y, and selects the portfolio that maximizes his utility. He invests $20 in Stock X and $30 in stock Y.
I give Laurie $200 and tell her to invest it in stock X, stock Y, and the risk-free asset. Laurie maximizes her utility by investing $50 in stock X. How much does Laurie invest in the risk-free asset?
HINT: Assume the two-fund (separation property) theorem holds. Then everyone in the economy holds the same risky portfolio of assets and divides his wealth between this portfolio and the risk-free asset. If
represent individual i
’s holdings of the two risky assets, the two-fund theorem implies that:
Is this answer sufficient, or is there a better way to show it mathematically?
Disregarding one’s taste for risk, the optimal distribution of stocks in a portfolio is the same for everyone: The point at which a line starting from risk-free return is tangent to the efficient frontier, where efficient frontier is the set of maximum returns for each given level of risk obtained by compositions of risky assets.
Your taste of risk only affects the distribution of your wealth between the risk-free asset and the optimal risky portfolio (the point of tangency). So, if David selects a 20-to-30 split between stock X and stock Y, so should Laurie because that's the optimal distribution for the risky assets portfolio.
She has 200-50 = $150 to distribute between X and Y, and she will split it $60 versus $90.