# Example of Calculations of the Forward Yield Curves

The spot yield curve provides all yields by maturity. Forward yield curves derive from the set of spot yields for various combinations of dates. The calculations below provide an example of the calculation of several forward yield curves one year from now, as well as two years from now and subsequent maturities. In the first case, we plot the forward rates starting one year from now and for one year (date 2), two years (date 3), three years (date 4), and so on. In the second case, the forward rates are two years from now, for the maturities one year (date 3), two years (date 4), and so on. The general formulas use the term structure relationships. For instance, the forward rates between dates 1 and 3 (one year from now for two years) and between dates 1 and 4 (one year from now for three years) are such that:

TABLE 7.2 The spot and forward yield curves (one year forward rates)

 Dates, t 0 1 2 3 4 5 Spot yields 3.50% 4.35% 5.25% 6.15% 6.80% 7.30% Forward, F 5.21% 6.14% 7.05% 7.64% 8.08% Forward, F 7.98% 8.28% 8.57% 8.82% FIGURE 7.9 Spot and forward yield curves If we deal with yields two years from now, we divide all terms such as (1 + i) by (1 + i2)2 to obtain 1 +f2i, when t > 2.

All forward rates are above the spot rates since the spot yield curve is upward sloping. In addition, the further we look forward; the lower is the slope of the forward curve. Forward yield curves start later than the spot yield curves. Figure 7.9 shows the forward rates embedded in spot rates.

Both Table 7.2 and Figure 7.9 provide the broad picture of investment opportunities, for investors who have recurring positive excess liquidity. When spot yield curves are upward sloping the forward rates are above the spot yield curve. Conversely, with an inverted spot yield curve, the forward rates are below the spot yield curve. When the spot yield curve is fiat, both spot and forward yields become identical.