How Par, Zero, and Forward Yield Curves Relate to Curve Shape
Summary
The document explains why par yields can sit below zero-coupon yields when the yield curve slopes upward. Zero rates over longer maturities reflect compounded averages of shorter-period forward rates. When those forward rates rise with maturity, averaging them produces a lower long-term zero rate than the latest forward rates. Par yields also reflect the discounting of multiple coupon cash flows, so their relationship to zero rates depends on the curve’s shape.
The answer describes the opposite ordering for a downward-sloping curve: par yields are highest, zero rates lie between them and forward rates, and forwards are lowest. With a flat curve, all three coincide. These comparisons are general intuitions, not universal rankings independent of curve shape or conventions. A second response characterizes par yield as a single rate discounting all bond cash flows to par and contrasts it with cash-flow-specific zero rates, but its conclusion that the maturity zero rate must exceed par is not generally valid; the curve-shape qualification in the first explanation matters.
Key ideas
- A long-maturity zero rate reflects compounded forward rates across shorter periods.
- On an upward-sloping curve, par yields are generally below zero rates, which are below relevant forward rates.
- On a downward-sloping curve, the described ordering reverses, with par yields highest and forwards lowest.
- When the curve is flat, par, zero, and forward rates coincide.
- Yield rankings depend on curve shape, so a maturity zero rate does not always exceed par.
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# Par and Zero Coupon Yield Curves
# 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.
## Answer by Helin (score 5)
https://quant.stackexchange.com/a/15664
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.
## Answer by JPI (score 0)
https://quant.stackexchange.com/a/69878
The par yield curve is calculated by solving for the single rate that discounts ALL of the bond's cashflows back to par value.
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.