How Par, Zero, Forward, and Discount Curves Relate
Summary
The document compares par yield curves with zero curves, especially when benchmark bonds trade far above par. It explains that par, zero, forward, and discount curves are transformations of one another and therefore encode the same information. A benchmark bond’s yield to maturity is not itself a par yield when the bond trades away from par, so simply connecting benchmark yields can be a poor approximation to a true par curve.
One described construction fits a curve to traded bond prices, then uses that fitted curve to calculate theoretical yields for bonds priced at par. Another approach fits zero rates directly so that discounted bond cash flows match observed prices; this requires choosing a stable fit. Alternatively, a yield-to-maturity curve can be constructed first and converted to zero rates through bootstrapping. The document gives no particular fitting algorithm or comparative performance evidence, and emphasizes that stability and fit quality can be difficult to balance.
Key ideas
- Par, zero, forward, and discount curves are different representations of the same underlying rate information.
- A traded bond’s yield to maturity is not a par yield when its price differs from par.
- A fitted curve can be used to calculate theoretical par yields from market bond prices.
- Zero rates can be fit to bond prices directly or derived from a yield curve through bootstrapping.
- Curve fitting must balance price fit with stability.
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Full text
# Par Yield Curves vs Zero Curves # 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). ## Answer by Helin (score 4, accepted) https://quant.stackexchange.com/a/15330 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. ## Answer by Richi Wa (score 3) https://quant.stackexchange.com/a/15333 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.
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