Using these zero-coupon products it becomes possible to derive par swap rates (forward and spot) for all maturities by making a few assumptions (including linear interpolation). The term structure of spot returns is recovered from the bond yields by solving for them recursively, by forward substitution. This iterative process is called the Bootstrap Method.
Given that, in general, we lack data points in a yield curve (there are only a fixed number of products in the market) and more importantly these have varying coupon frequencies, it makes sense to construct a curve of zero-coupon instruments from which we can price any yield, whether forward or spot, without the need of more external information.
A generic "algorithm" is described below; for more detail see Yield curve: Construction of the full yield curve from market data.
- Define set of yielding products, these will generally be coupon-bearing bonds
- Derive discount factors for all terms, these are the internal rates of return of the bonds
- 'Bootstrap' the zero-coupon curve step-by-step.
- is the coupon rate of the n-year bond
- is the length, or day count fraction, of the period, in years
- is the discount factor for that time period
- is the discount factor for the entire period, from which we derive the zero-rate.