Lesson 13 of 25 · 7 min

Inversion, mechanically

A short rate above a long one is a solved equation about expected cuts, and it puts a short-funded borrower at the curve's dearest point.

An inverted curve is one where a shorter rate sits above a longer one. Most descriptions treat the configuration as an omen. Mechanically it is not an omen at all; it is an equation with one unknown, and the unknown can be solved for.

If a long rate is approximately the average expected short rate across its horizon plus a term premium, then a long rate below today's short rate can mean only one thing: the market's expected average short rate across that horizon is below today's short rate, by at least the term premium. Inversion is that sentence, stated as a price.

Solve one. An illustrative curve prints 5.38 percent at three months, 4.42 at two years and 4.16 at ten, so the three-month to ten-year inversion is 1.22 points. Taking an illustrative two-year term premium of 0.10, the expected average overnight rate across two years is 4.42 − 0.10 = 4.32 percent. Today's is 5.38. If the decline from here were even, the rate two years out would be 2 × 4.32 − 5.38 = 3.26 percent. That is the path the curve is quoting, not forecasting.

For a borrower funded at the short end this is the least comfortable shape the curve takes. You are charged off the highest point on it while your collateral is discounted off a much lower one. On a $1,355,000 balance, paying 5.38 plus a 1.95 spread rather than 4.16 plus the same spread costs $16,531 a year — 1.22 points of curve shape, with no change in credit, collateral or conduct.

An inverted curve charges a short-funded borrower today for a level the market expects will not persist. The expectation costs nothing to hold and the accrual runs regardless.

Key terms

Curve inversion: A shorter tenor yielding more than a longer one. An arithmetic consequence of expected declines, not an independent signal.
Implied path: The sequence of future short rates consistent with the curve once a term premium has been assumed. Sensitive to the assumption.
Curve spread: The difference between two tenors, such as three months against ten years. Negative when the curve is inverted.

Worked example

Backing a path out of an inversion

An illustrative curve at 5.38 percent for three months, 4.42 at two years and 4.16 at ten. Balance $1,355,000, spread 1.95 percent.

Three-month to ten-year spread: −1.22 points
Assumed two-year term premium: 0.10 points
Expected average overnight rate over two years: 4.32%
Implied rate two years out, if the decline is even: 3.26%
Rate charged today: 7.33%
Annual cost of the 1.22 points of inversion: $16,531

The curve says the short rate should be near 3.26 percent in two years. The facility accrues at 7.33 percent now, and a forecast does not settle an accrual.

Check yourself

A curve prints 5.12 percent at six months and 4.20 at five years, and the term premium at five years is taken as 0.25 points. What average overnight rate is the market pricing across five years, and how far below today's is it?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.