The Efficient Frontier with Risk-Free Lending and Borrowing

In the previous discussion of portfolios of risky assets, the availability of a risk-free asset has been ignored. The introduction of a risk-free asset into the portfolio possibility set considerably simplifies the analysis. Assuming that an investor is interested in placing part of his funds in some portfolio A and either lending or borrowing, we can define

the expected return on the combined risk-free asset and risky portfolio as

The risk on the combination is

Since the return on the risk-free asset is known with certainty, the standard deviation
of the risk-free asset must be zero. The above simplifies to

Solving for X yields

Substituting this expression for X into the expression for expected return on the combination provides 

Upon rearranging the terms

Thus all combinations of risk-free lending or borrowing with portfolio A lie on a straight line in expected return standard deviation space. The intercept of the line is RF and the slope is. Therefore, the existence of a risk-free lending and borrowing rate implies that there is a single portfolio of risky assets that is preferred to all other portfolios. Furthermore, in return standard deviation space, this portfolio plots on the ray connecting the risk-free asset and a risky portfolio that lies furthest in the direction. In figure 3.6, for example, the portfolios on the ray RF - B are preferred to both those on the ray RF - A, and all other portfolios of risky assets. The efficient frontier is the entire length of the ray extending through RF - B, with different points along the ray RF - B representing different amounts of borrowing and/or lending in combination with the optimum portfolio - B - of risky assets.

Investors can lend at the risk-free rate by buying government bonds. However, they will probably pay a higher borrowing rate. Figure 3.7 plots the efficient frontier with different borrowing and lending rates. As shown, there are a small range of risky portfolios that investors may opt to hold. If RF and RB are close, the assumption of risk-free lending and borrowing at the same rate may provide a good approximation of
the optimal range of risky portfolios that investors could hold.