AIM:  Describe the capital market line (CML) and the construction of the efficient frontier both with and without a risk‐free asset.

The capital market line (CML) is given by:

The capital market line (CML) is a straight line that intersects the y-axis at the risk-free rate (i.e., the expected return for an investment with no volatility is the risk-free rate of return) and carves a tangent line to the efficient frontier (i.e., the nonlinear efficient frontier that consists only of risky assets). The capital market line “re-expresses” the efficient frontier as an allocation choice between the portfolio of risky assets and the risk-free asset. It slopes upward to reflect the idea that we can switch (progressively reallocate) risk free allocations in favor of risky allocation for higher expected returns (at the cost of higher volatility, of course).

The upward-sloping part of the curve is called the efficient frontier. The investor prefers portfolios on the efficient frontier because, for any given level of volatility, a portfolio on the efficient frontier has the highest level of expected return.

Asset Allocation with a Risk-Free Asset

By combining the risk-free asset and a portfolio on the frontier, the investor can obtain all the expected return and volatility combinations on the straight line that meets the frontier at the portfolio of risky assets chosen to form these combinations.

In regard to the graph above:

• Without the riskless asset, the green curve describes the risk/return trade-off. But the efficient frontier is always preferred to the lower part of the curve
• With the riskless asset, the asset allocation decision refers to a choice on the blue-line which is the capital market line (CML): allocation to more (less) riskless asset implies lower (higher) return and less (more) risk

The Risk of a Security in a Diversified Portfolio

The risk premium is the extra expected return of a security (or of a portfolio) over the risk-free rate. The risk premium is given by,

Where R_m is the return of portfolio m and R_F is the risk-free rate.

The variance of the return of the market portfolio is given by,

In words, this variance is the return covariance of the market portfolio with itself.

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