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

When a risk-free asset is combined with a risky portfolio, a graph of possible portfolio risks-return combinations becomes a

Assume the proportion of the portfolio the investor places in the tangency portfolio P is w

- The expected rate of return for the new portfolio is the weighted average of the two returns: E(R) = (1 - w
_{P}) R_{f}+ w_{P}E(R_{T}) - The standard deviation of the new portfolio is the linear proportion of the standard deviation of the risky asset portfolio P: σ
_{portfolio}= w_{P}σ_{P}

The introduction of a risk-free asset changes the efficient frontier into a straight line. This straight efficient frontier line is called the

- Investors at point r
_{f}have 100% of their funds invested in the risk-free asset. - Investors at point P have 100% of their funds invested in portfolio P.
- Between r
_{f}and P, investors hold both the risk-free asset and portfolio P. This means investors are lending some of their funds (buying the risk-free asset). - To the right of P, investors hold more than 100% of portfolio P. This means they are borrowing funds to buy more of portfolio P. This represents a levered position.

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 r

Investors make

- Choosing P, a technical matter (can be done by the broker)
- Deciding on the proportion to be invested in P and in the riskless asset.

We can combine the efficient frontier and/or capital allocation line with indifference curves. The

- Two investors will select the same portfolio from the efficient set only if their utility curves are identical.
- Utility curves to the right represent less risk-averse investors; utility curves to the left represent more risk-averse investors.

This is portfolio selection without a risk-free asset:

This is portfolio selection with a risk-free asset: