# Par and Zero Coupon Yield Curves

The government par yield curve shows a marginally lower yield than the Government zero coupon curve.

What is the reason for this in general.

This is actually only true when the yield curve is upward sloping. Intuitively, zero rates are average forward rates; e.g., the 10-year zero coupon yield is the geometric average of the 0y forward 1y rate, 1y forward 1y rate, 2y forward 1 year rate, ..., and 9y forward 1y rate: $$(1 + y_{10})^{10} = (1 + f_{0,1})(1 + f_{1,2})\ldots(1 + f_{9,10}).$$