The Capital Allocation Line – Introducing the Risk-free Asset

The discussion of diversification benefits focused on a portfolio consisting of risky assets; when a risk-free asset is incorporated, diversification is still prevalent but a linear trade-off between risk and return is established.

The introduction of a risk-free asset does not change the construct of the minimum variance frontier graphical structure (y-axis = expected return; x-axis = standard deviation or variance).

However, the introduction of the risk-free asset does shift the optimal risk-return trade off on the efficient frontier.

Capital Allocation Line (CAL): This line shows the highest returns for each level of risk in a portfolio containing a risky and risk-free asset.

CAL Equation

E(RP) = RF + ((E(RT) – RF)/σT)*σP

E(RP) = expected return

RF = rate of return on the risk free asset

E(RT) = expected return on CAL portfolio that is tangent to the minimum variance frontier

σT = standard deviation of the tangent portfolio

σP = standard deviation of the portfolio analyzed

Note that when there is zero allocation to the risky asset, the portfolio’s return will equal the risk-free rate and the standard deviation will be zero; this is the y-intercept of the CAL.

CAL Slope Coefficient = Sharpe Ratio; This is the expected change in return for a given change in risk (where risk is defined as the standard deviation of returns).

Sharpe Ratio = (E(Rasset) – RF)/σasset

The Sharpe Ratio is a risk/return tradeoff measure; if two assets offer a similar expected return but different standard deviations, the asset with the higher Sharpe Ratio is considered superior.

Optimal Portfolio: Once the risk-free asset is introduced, the only optimal portfolio on the minimum variance frontier is the tangent portfolio.