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Using Smooth Forward Curves to Assess Bond Richness and Cheapness

Article Quant Q&A · Author: Timothy Ng
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Summary

The document explains why a forward curve above a par yield curve does not, by itself, show that bonds are cheap. For an upward-sloping yield curve, forwards tend to sit above par yields; the reverse relationship holds for a downward-sloping curve. The example under discussion uses a nearly flat par curve to highlight a different point: irregular forward rates can exaggerate local pricing differences.

The proposed valuation method compares observed forwards with a smoother curve treated as a fair-value reference. If forward rates in a maturity sector are unusually high, discounting that sector’s bond cash flows at those rates produces lower prices than discounting them using the smoother reference curve. That relative price difference can indicate cheapness. The explanation supports evaluating a long-maturity bond against a shorter one when the longer sector appears unusually rich in forwards, but it gives no empirical performance evidence. Its conclusion depends on the assumption that a smooth forward curve is a sensible benchmark, and the document suggests calculating spreads against such a curve directly.

Key ideas

  • The position of the forward curve relative to the par curve depends on the slope of the yield curve.
  • A forward curve above the par curve alone is not evidence that a bond sector is cheap.
  • Unusually high forwards imply lower bond prices when used to discount cash flows.
  • Comparing observed forwards with a smooth reference curve can reveal relative richness or cheapness.
  • The valuation signal depends on the smooth curve being a reasonable fair-value benchmark.

Tags

Fixed incomeStatistics
Full text
# Forward Curves and Par Yield Curves


# Forward Curves and Par Yield Curves












I'm recently reading a research paper on the yield curve by Salomon brothers and in it it states that when the forward curve is above the par yield curve, it is seen as cheaper. If for example, the years 9-12 of the forward rate curve lie above the par yield curve with the forward 12 year rate above the 9 year rate as well, it is recommended to buy the 12 year bond while selling the 9 year bond.

Unfortunately, I am unable to accurately grasp the concept behind this in relation to the par yield curve. Please help! Thank you!

## Answer by Helin (score 1)

https://quant.stackexchange.com/a/16761

First, it's not true that a market sector is cheap whenever the forward curve lies above the par curve. In fact, whenever the yield curve is upward sloping, the forward curve will always lie above the par curve. Conversely, when the yield curve is downward sloping, forwards will always lie beneath the par curve. In the example you quoted, Ilmanen chose a day on which the par curve is virtually flat to make a very specific point: forwards can amplify small misplacing.

To understand the rich/cheap signals, let's think through the shape of the yield curve first: the forward curve Ilmanen provided (we'll call it "Curve A") is not particularly smooth. But it can be argued that a sensible forward curve SHOULD be very smooth; after all, market participants can't possible have very differing views regarding short-term interest rate 10-years from today vs 11-years from today. So imagine that there is a "fair value" version of the forward curve that's super smooth (we'll call it "Curve B").

Now let's price bonds with the two curves. Because the forward rates on curve A (the wavy one) in the 12-year sector are particularly high, bond cash flows, when discounted with these high forward rates, will result in lower prices. By contrast, the forward rates on the fair value curve B will be much lower – because of the smoothness constraint, the curve can't swing up like that. Accordingly, the fair value of 12-year bonds, when priced using these lower forward interest rates, will be higher. This is why the 12-year sector can be perceived as cheaper than fair value.

In practice, it's probably easier to build a smooth forward curve and calculate spreads relative to this smooth curve, instead of depending on a wavy forward curve to detect rich/cheap signals.

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.