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 $\displaystyle w_X^i$ and $\displaystyle w_Y^i$ represent individual

**i**’s holdings of the two risky assets, the two-fund theorem implies that:

$\displaystyle \frac{w_X^i}{w_Y^i}=\frac{w_X^j}{w_Y^j}$