# 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: $$(1 + y_{10})^{10} = (1 + f_{0,1})(1 + f_{1,2})\ldots(1 + f_{9,10}).$$

So whenever the forward curve is upward sloping, the zero curve will be a bit lower because of the averaging. Similarly, par yields are (somewhat complex) average of the zero coupon rates, so when the curve is upward sloping, they're even lower that the zero curve. (I wrote about this topic here http://hungrydummy.com/blog/bond-risk-premium-part-i-a-review-of-different-yield-curves/).

When the yield curve is downward sloping, however, the par curve is actually the highest, zero curve in the middle, and the forward curve is the lowest.

When the yield curve is flat, the three curves will be identical.

• Your image in your blog doesn't show anymore. Jan 27, 2015 at 14:20

The par yield curve is calculated by solving for the single rate that discounts ALL of the bond's cashflows back to the current market price.

The Zero curve is calculated by solving for the INDIVIDUAL rates that discounts EACH cash flow of the bond (coupons and maturity). The shorter term cash flows need a lower rate than the average to be discounted whereas the final cashflow (maturity payment) will require a higher rate to discount it back.

Therefore the Zero-coupon rate at maturity will be higher than the par yield.