Yield Curve Inversion: Definitions and Forward-Rate Models
Summary
An inverted yield curve describes short-term borrowing rates that exceed longer-term rates. Since an entire curve can have a complex shape, common practical definitions compare two selected maturities: the three-month rate with the ten-year rate, or the two-year rate with the ten-year rate. Under either convention, inversion means the short-minus-long yield spread is negative.
The discussion relates zero-coupon bond yields to the average of instantaneous forward rates over a maturity period. It states that a yield curve that is not increasing with maturity is equivalent, in this setup, to a forward curve that is not increasing. For modeling, it suggests working directly with forward curves, such as in the Heath-Jarrow-Morton framework, which can produce an inversion. The answer also cautions that simple interest-rate models with monotonic zero-coupon yields cannot generate inversion under the stated comparisons. The exchange offers a conceptual explanation rather than a model specification or empirical test.
Key ideas
- Yield curve inversion is commonly identified by comparing a short maturity yield with a longer maturity yield.
- A negative short-minus-long yield spread indicates inversion under the selected convention.
- A yield is represented as the average of instantaneous forward rates across its maturity period.
- In the described setup, a non-increasing forward curve corresponds to a yield curve that is not increasing.
- Forward-curve models such as HJM can represent inverted curves, while monotonic-yield models cannot.
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# Mathematical meaning of an inverted yield curve
# Mathematical meaning of an inverted yield curve
I am currently working on rates model. I would like to understand, mathematically, what does it mean to have an inverted yield curve? And I am asking myself for a certain model, how can I generate an inverted yield curve? Any content/document is welcome. Thank you
## Answer by nbbo2 (score 1)
https://quant.stackexchange.com/a/78040
Although the entire shape of the yield curve should perhaps be taken into account, in practice just two maturities are chosen for comparison.
Campbell Harvey in 1985 defined inverted yield curve as: the 3 month yield is larger than the 10 year yield. I.e. the curve is said to be inverted when the difference between these two rates is negative. (See chart)
https://fred.stlouisfed.org/series/T10Y3M
An alternative definition some people use compares the 2 year yield to the 10 year yield.
https://fred.stlouisfed.org/series/T10Y2Y
The simplest interest models produce monotonic yield curves for ZCB's (This is apparent when you look at the formulas for yields of ZCB's in these models). So an inversion can never occur in these models by either definition.
## Answer by Rylan (score 1)
https://quant.stackexchange.com/a/78041
My thoughts:
Using the definition found on Wikipedia:
> In finance, an inverted yield curve is a yield curve in which short-term debt instruments (typically bonds) have a greater yield than longer term bonds.
At time $t$ we observe yields $Y(t, T) = \frac{1}{T-t}\int_t^Tf(t, s)ds$, where $f(t, s)$ is the instantaneous forward rate for time $s$, observed at time $t$. In this setup, an inverted yield curve observed at time $t$ is a yield curve $Y(t, T)$ that is not increasing in $T$. I can provide a proof if desired, but a forward curve $f(t, T)$ that is not increasing in $T$ is an equivalent condition.
For your second question, my thought would be that a good place to start is with a model that works on forward curves directly, such as HJM-- this model can indeed generate an inverted yield curve by generating a forward curve that is not non-decreasing.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.