The Efficient Frontier with RiskFree Lending and Borrowing
In the previous discussion of portfolios of risky assets, the availability of a riskfree asset
has been ignored. The introduction of a riskfree 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 riskfree asset and risky portfolio as
_{
}
The risk on the combination is
Since the return on the riskfree asset is known with certainty, the standard deviation
of the riskfree 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 riskfree lending or borrowing with portfolio A lie on a straight line in expected return standard deviation space. The intercept of the line is R_{F} and the slope is. Therefore, the existence of a riskfree 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 riskfree asset and a risky portfolio that lies furthest in the direction. In figure 3.6, for example, the portfolios on the ray R_{F}  B are preferred to both those on the ray R_{F}  A, and all other portfolios of risky assets. The efficient frontier is the entire length of the ray extending through R_{F}  B, with different points along the ray R_{F}  B representing different amounts of borrowing and/or lending in combination with the optimum portfolio  B  of risky assets.
Investors can lend at the riskfree 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 R_{F} and R_{B} are close, the assumption of riskfree lending and borrowing at the same rate may provide a good approximation of
the optimal range of risky portfolios that investors could hold.
