Questions: Efficient Frontier and Capital Market Line

Question 1

According to the separation theorem, what distinguishes the tangency portfolio from all other points on the efficient frontier?

• A. It is the portfolio with the lowest variance among all possible risky portfolios
• B. It maximizes the Sharpe ratio and is the optimal risky portfolio for every investor regardless of risk aversion
• C. It is only optimal for investors with moderate risk aversion
• D. It is the portfolio with the highest expected return on the efficient frontier

Answer: (B)

The tangency portfolio is the unique point where the Capital Market Line touches the efficient frontier — this occurs at the maximum Sharpe ratio. Because all investors face the same risk-free rate and the same efficient frontier of risky assets, the tangency portfolio is optimal for all of them. Risk aversion only determines the split between the risk-free asset and the tangency portfolio, not which risky portfolio to hold.

Question 2

An individual stock can lie on the Capital Market Line.

• T. True
• F. False

Answer: False

The CML is constructed from efficient portfolios — combinations of the risk-free asset and the fully diversified tangency portfolio. Individual stocks carry both systematic risk (which cannot be diversified away) and idiosyncratic risk (which can). Since a rational investor would diversify away idiosyncratic risk, holding a single stock is never efficient: it has the same or higher variance as an efficient portfolio with the same expected return. Therefore individual stocks lie strictly below the CML.

Question 3

When a risk-free asset is introduced to the portfolio problem, why does the efficient frontier change from a curved boundary to a straight line (the Capital Market Line)?

Model answer: Combining any risky portfolio with a risk-free asset (zero variance, zero correlation with risky assets) produces portfolios that lie on a straight line from the risk-free rate through that risky portfolio's point in risk-return space. Because the risk-free asset has no variance, the portfolio variance is purely proportional to the allocation to the risky portfolio, making the risk-return tradeoff linear. The optimal such line is the one tangent to the curved risky efficient frontier — the steepest achievable Sharpe ratio.

The curvature of the risky-asset efficient frontier comes from imperfect correlations between assets. The risk-free asset has a correlation of zero with everything and a variance of zero, so the combination is just a weighted average of return and a linearly scaled variance. This eliminates the curvature. The tangency portfolio is chosen because it produces the steepest (highest Sharpe ratio) straight line — any other risky portfolio would yield a line that crosses below the tangency CML.