The efficient frontier only considers the investments in risky assets. However, investors may choose to invest in a risk-free asset, which is assumed to have an expected return commensurate with an asset that has no standard deviation (i.e., zero variance) around the expected return. That is, a risk-free asset's expected return is entirely certain; it is known as the risk-free rate of return (RFR). Therefore, a risk-free asset lies on the vertical axis of a portfolio graph.
When a risk-free asset is combined with a risky portfolio, a graph of possible portfolio risks-return combinations becomes a straight line between the two assets.
Assume the proportion of the portfolio the investor places in the tangency portfolio P is wP:
The introduction of a risk-free asset changes the efficient frontier into a straight line. This straight efficient frontier line is called the Capital Market Line (CML) for all investors and the Capital Allocation Line (CAL) for one investor.
Investors will choose the highest CAL (i.e., the CAL tangent to the efficient frontier). This portfolio is the solution to the optimization problem of maximizing the slope of the CAL.
Now, the line rf-P dominates all portfolios on the original efficient frontier. Thus, this straight line becomes the new efficient frontier.
Investors make different financing decisions based on their risk preferences. The separation of the investment decision from the financing decision is called the separation theorem. The portfolio choice problem can be broken down into two tasks:
Optimal Investor Portfolio
We can combine the efficient frontier and/or capital allocation line with indifference curves. The optimal portfolio is the portfolio that gives the investor the greatest possible utility.
This is portfolio selection without a risk-free asset:
This is portfolio selection with a risk-free asset: