A Look at Forward Rates

Practitioners have highlighted the use of the cubic spline approach to model forward curves and their limitations. We will use our empirical data to highlight typical forward rate behavior under the cubic spline technique. Our sample data does not reflect actual market conditions and is an extreme data set, to say the least. However, it does highlight a point with regard to forward rates that can often be observed sometimes under normal market conditions. To this end, we isolate the last subset of the data, as shown previously, and plot the forward rates for that data set in FIGURE 5.14. 

From data that was interpolated using the linear method versus data interpolated using the cubic spline, a comparison of forwards shows how the forwards in a cubic spline environment can oscillate. As expected, the relatively minor oscillations observed first in the zero rates curve are compounded excessively in the forward rate calculation. The linear interpolation approach eliminates much of the oscillation but of course is not a smooth curve, which is as undesirable. The user is confronted with the need to balance the conflicting requirements a trade-oflF is called for, and 

Using the actual United Kingdom 10-year zero curve for January 2, 2000, the forward rates have been calculated using cubic spline and linear interpolation and compared in FIGURE 5.15 and FIGURE 5.16 respectively. There is no observed reason to favor the latter approach over the former. 

FiRMRF g.l6 Forward Rates Calculated Usine Linear Interpolation 

We have presented an accessible account of how the cubic spline methodology of term structure estimation could be implemented by users involved in any area of the debt capital markets. The technique is straightforward and quick, and is valid for a number of applications, most of which are normal or conventional yield curves. For example, users are recommended to use it when curves are positively sloping, or when the long end of the curve is not downward sloping. The existence of humps along the short or medium terms of the curve can cause excessive oscillation in the forward curve, but the zero curve may still be used for valuation or relative value purposes. 

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