## Inclusion Of A Riskfree Asset

In the previous few sections we have implicitly assumed that the // assets available are all risky; that is, they each have a > 0, A risk-free asset has a return that is deterministic (that is, known with certainty) and therefore has a — 0 In other-words, a risk-free asset is a pure interest-bearing instrument; its inclusion in a portfolio corresponds to lending or borrowing cash at the risk-free rate Lending (such as the purchase of a bond) corresponds to the r isk-free asset having a positive weight, whereas borrowing corresponds to its having a negative weight,

The inclusion of a risk-free asset in the list of possible assets is necessary to obtain realism. Investors invariably have the opportunity to borrow or lend Fortunately, as we shall see shortly, inclusion of a risk-free asset introduces a mathematical degeneracy that greatly simplifies the shape of' the efficient frontier.

To explain the degeneracy condition, suppose that there is a risk-free asset with a (deterministic) rate of return rj. Consider any other risky asset with rate of return /, having mean 7 and variance a2 Note that the covariance of these two returns must be zero. This is because the covariance is defined to be E[(/ — 7)(if - ty)] — 0.

Now suppose that these two assets are combined to form a portfolio using a weight of a for the risk-free asset and 1 — a for the risky asset, with a < 1 The mean rate of return of this portfolio will be aif -f- (1 — a)r. The standard deviation of the return will be >/(! — a)2er2 — (1 — a)a. This is because the risk-free asset has no variance and no covariance with the risky asset. The only term left in the formula is that due to the risky asset

If we define, just for the moment, cry ~ 0, we see that the portfolio rate of return has mean = aiy 4- (1 — <x)7

These equations show that both the mean and the standard deviation of the portfolio vary linearly with a. This means that as a varies, the point representing the portfolio traces out a straight line in the F-ct plane

Suppose now that there are n risky assets with known mean rates of return f, and known covariances a( l. In addition, there is a risk-free asset with rate of return iy The inclusion of the risk-free asset in the list of available assets has a profound effect on the shape of the feasible region. The reason for this is shown in Figure 6, 13(a) First we construct the ordinary feasible region, defined by the n risky assets, (This region may be either the one constructed with shorting allowed or the one constructed without shorting.) This region is shown as the datkiy shaded region in the figure. Next,

FIGURE 6,13 Effect of a risk-free asset. Inclusion of a risk-free asset adds lines to the feasible region (a) If both borrowing and lending are allowed, a complete infinite triangular region is obtained (b) If only lending is allowed, (he region will have a triangular front end, but will curve for larger er

FIGURE 6,13 Effect of a risk-free asset. Inclusion of a risk-free asset adds lines to the feasible region (a) If both borrowing and lending are allowed, a complete infinite triangular region is obtained (b) If only lending is allowed, (he region will have a triangular front end, but will curve for larger er for each asset (or portfolio) in this region we form combinations with the risk-free asset, In forming these combinations we allow borrowing or lending of the risk-free asset, but only purchase of the risky asset. These new combinations trace out the infinite straight line originating at the risk-free point, passing through the risky asset, and continuing indefinitely. There is a line of this type for every asset in the original feasible set. The totality of these lines forms a triangularly shaped feasible region, indicated by the light shading in the figure

This is a beautiful result. The feasible region is an infinite triangle whenever a risk-free asset is included in the universe of available assets,

If borrowing of the risk-free asset is not allowed (no shorting of this asset), we can adjoin only the finite line segments between the risk-free asset and points in the original feasible region. We cannot extend these lines further, since this would entail borrowing of the risk-lree asset The inclusion of these finite line segments leads to a new feasible region with a straight-line front edge but a rounded top, as shown in Figure 6 13(b).

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