# Par Yield Curves vs Zero Curves

Does it make sense to look at par yield curve for German bonds in the current environment? Because low rates mean that a lot of bonds are trading above much above par (even around 150!).

I would have thought a zero curve makes more sense, as it takes out the coupon effect. Use of the curve could be anything from pricing/hedging/calculating spreads (z/ASW).

The important thing to know is that the par curve, the zero curve, the forward curve, and the discount curve are just transformations of each other; they contain exactly the same information (see What is the Swap Curve?).

I think the confusion arises because many books tell you to connect the yields to maturity of benchmark bonds and call it the par yield curve. It's ok as an approximation (since most benchmark bonds trade close to par), but that doesn't give you the "real" par curve. If a bond is trading at 150, it's certainly not a par bond and its yield is not a par yield.

The way a proper par curve is constructed is as follows: you start with actual bonds traded in the market, and then create a best-fit curve using a curve fitting technique. From this best fit curve, you can compute the theoretical par yield, which are yields of bonds trading at par.

I would put it a bit differently. You can do 2 things:

• Either you apply an optimization/fitting procedure that has all the bond prices as inputs and zero rates for the chosen maturities as outputs. The objective function is the deviation between the discounted (by the to-be-found zero-rates) cashflows of each bond and the traded bond prices. To find a stable set of parameters that don't fluctuate too much and give a good fit is a hard thing to do.

• Another approach is to derive a yield curve first. It is a construction of time-to-maturity at the x-axis and yield-to-maturity on the y-axis. As explained for example here a bond that has (theoretical) coupons equal to its yield is priced at par (100). That's why a yield-curve constructed in this way is also called a coupon-curve. In order to derive a zero-rates curve from this you can apply a bootstrapping procedure.

I also strongly recommend Overview of Forward Rate Analysis by Antti Ilmanen.