Short Rates

Another way of approaching this problem involves working with the term structure's short rates. Short rates are one period implied forward rates, and because they will become so important later in asset valuation problems, we develop them now. The short rates are given in the first column of our forward rate analysis—this column is highlighted in Table 3.2 and italicized.

Table 3.2 Estimating Short Rates.

To reiterate, 6.605 percent is the one period rate implied by the current spot curve to hold one year from now. Likewise, 7.489 percent is the one period rate implied two years from now and so on. Geometrically linking these short rates will reproduce the original spot curve. For example: [(1.0571)(1.06605)]^0.5 – 1 = 0.06088. These are useful for discounting single-period cash flows. For the current problem under consideration, we are expecting five years of cash flows, each in the amount of $100. The present value of the last $100 received in the last year is therefore $100. The present value of the project at the end of the fourth year is the discounted PV of this amount plus the $100 received in year four. The PV of the project at the end of the third year is the discounted value of this new amount plus, again, the $100 cash flow in period three. This is called a running present value and it is a convenient way to present value cash flows on the run. The problem is set up in Table 3.3:

Table 3.3 Running PV.

The discount rates are the reciprocals of the short rates, that is, 0.947 = 1/(1.05571) and 0.918 = 1/(1.08893). These two solutions are equivalent (up to a rounding error).

Thus, we see that the current Treasury spot curve implied a term structure of forward rates, which become the discount factors that we use to solve present value problems. The short rates become important because any set of short rates can be used to build the entire term structure. Thus, all one needs are the short rates to solve any term structure problem. With running PV methods, we can present value projects and securities on the run using only short rates. The intuition is this—the standard present value method uses the discount factors d_i from the spot structure as we did in our first solution, that is:

We can rewrite this basic discounting relation however as:

In the limit, any problem can be written as:

We therefore work backward to get the solution using the one period short rates d_t,t+1.

Finally, we show the entire short rate structure implied by the Treasury yield curve as of January 6, 2010. The spot rates for years 4, 6, 8, and 9 are interpolated since there are no Treasuries issued for these maturities.

Table 3.4 The Treasury Curve on January 6, 2010.

The short rate structure is given in Table 3.5.

Table 3.5 Implied Short Rates on January 6, 2010.

Each row in this table can be used to present value future cash flows beginning in the current year (row 1), beginning one year from now (row 2), and so on.