The Efficient Frontier With Riskless Lending And Borrowing
Up to this point we have been dealing with portfolios of risky assets. The introduction of a riskless asset into our portfolio possibility set considerably simplifies the analysis. We can consider lending at a riskless rate as investing in an asset with a certain outcome (e.g.,
33Mer£on [19] has shown that the efficient set is the upper half of a hyperbola.
Figure 5.12 The efficient set when short sales are allowed.
a shortterm government bill or savings account). Borrowing can be considered as selling such a security short, thus borrowing can take place at the riskless rate.
We call the certain rate of return on the riskless asset RF. Since the return is certain, the standard deviation of the return on the riskless asset must be zero.
We first examine the case where investors can lend and borrow unlimited amounts of funds at the riskless rate. Initially assume that the investor is interested in placing part of the funds in some portfolio A and either lending or borrowing. Under this assumption we can easily determine the geometric pattern of all combinations of portfolio A and lending or borrowing. Call X the fraction of original funds that the investor places in portfolio A. Remember that X can be greater than 1 because we are assuming that the investor can borrow at the riskless rate and invest more than his initial funds in portfolio A. If X is the fraction of funds the investor placcs in portfolio A, (1  X) must be the fraction of funds that were placed in the riskless asset. The expected return on the combination of riskless asset and risky portfolio is given by
The risk on the combination is
<rc = [(1  X)2 <J} + X2o\ + 2X(1  X)oAoPpFA f Since we have already argued that aF is zero,
Solving this expression for X yields
Substituting this expression for X into the expression for expected return on the combination yields rrjl^rr+^r,
Rearranging terms,
Note that this is the equation of a straight line. All combinations of riskless lending or borrowing with portfolio A lie on a straight line in expected return standard deviation space. The intercept of the line (on the return axis) js Rp and the slope is (RA — RF)/vA. Furthermore, the line passes through the point (crA, RA). This line is shown in Figure 5.13. Note that to the left of point A we have combinations of lending and portfolio A, whereas to the right of point A we have combinations of borrowing and portfolio A.
The portfolio A we selected for this analysis had no special properties. Combinations of any security or portfolio and riskless lending and borrowing lie along a straight line in expected return standard deviation of return space. Examine Figure 5.14. We could have combined portfolio B with riskless lending and borrowing and held combinations along the line RfB rather than RpA. Combinations along RpB are superior to combinations along RpA since they offer greater return for the same risk. It should be obvious that what we would like to do is to rotate the straight line passing through Rp as far as we can in a counterclockwise direction. The furthest we can rotate it is through point G.14 Point G is the tangency point between the efficient frontier and a ray passing through the point Rp on the vertical axis. The investor cannot rotate the ray further because by the definition of the efficient frontier there are no portfolios lying above the line passing through RF and G.
All investors who believed they faced the efficient frontier and riskless lending and borrowing rates shown in Figure 5.14 would hold the same portfolio of risky assets—portfolio G. Some of these investors who were very riskaverse would select a portfolio along the
Figure 5.13 Expected return and risk when the riskfree rate is mixed with portfolio A.
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