The yield curve: shape, slope and inversion

The horizontal axis is time to maturity, running from a few months at the left to twenty or thirty years at the right, and out to fifty in a handful of markets. The vertical axis is the annual yield: the single rate that makes the bond's remaining payments add up to today's price. That is what you earn if you hold the bond to maturity and the borrower pays, and on a coupon bond it also assumes each coupon is put back to work at that same rate, which is an assumption rather than a promise. One dot per maturity, joined into a line, is the yield curve.

Two rules keep the picture meaningful. Every point comes from the same issuer, almost always a government borrowing in its own currency, so credit risk is roughly constant across the line and what changes from left to right is mainly time. And every point is taken at the same moment, because yields move all day. A yield curve is a snapshot, not a history.

Nobody publishes the curve as an opinion. Each point starts as a market price converted into a rate, so the line is anchored to what buyers and sellers actually agreed to, with a curve fitted through those points to fill the gaps between the maturities that trade. In the United States the Treasury publishes a daily par yield curve built from quotes on the most recently issued security at each maturity. Analysts then convert it into a zero-coupon curve, which strips out the effect of coupons paid along the way, and into a forward curve, which reads the same prices as a path of future short-term rates.

Most days the whole line shifts up or down together. The moves worth naming are the ones that change its shape: steepening, when long yields rise relative to short ones, and flattening, when the gap between the two ends closes.

Suppose you want to lend for two years. You can buy a two-year bond, or you can buy a one-year bond and buy another one when the first matures. Competition keeps those two routes roughly in line, because if one were widely expected to pay more, buyers would move towards it until its price rose and its yield came back down. It is not a riskless arbitrage, though. Nobody knows today what the second year will pay, and the two routes leave you exposed to different things: the rolling route to the rate you get on the second bond, the long route to the price you would take if you had to sell early. That gap is why the link between a long yield and expected short rates is an approximation with a premium attached rather than an identity:

$$y_n \approx \frac{r_1 + r_2 + \cdots + r_n}{n} + tp_n$$

Here $y_n$ is the yield on a bond with $n$ years to run, $r_1$ to $r_n$ are the one-year rates the market expects in each of those years, and $tp_n$ is the term premium. The averaging is written the plain way here. The exact version compounds the rates rather than adding them, which is what the worked examples below do, and it moves the answer by less than a hundredth of a percentage point over two years and by more as the horizon lengthens.

The term premium is the extra yield a lender asks for taking on the risks that come with time itself: that inflation runs hotter than expected and eats the repayment, that rates move against you while your money is committed, that you have to sell early at whatever price is on offer that day. A nominal yield only helps if it beats inflation, which is what the real return calculator works out. The term premium is normally positive, and that is the main reason the curve slopes up even when nobody expects rates to change at all.

It is not printed anywhere. The term premium is the residual left over once a model has estimated what the market expected short rates to do, so estimates differ, and they have at times been negative. Steady demand for long bonds from pension funds matching long promises, or from a central bank buying them, pushes it down.

Four descriptions cover almost everything you will see, and the middle two are transitions between the outer two.

A fifth shape turns up often enough to name: a humped curve rises to a peak somewhere in the middle, often around one to three years, then falls away. That is the market pricing in a little more tightening now and cuts later.

Which end moves matters as much as the shape. A curve can steepen because long yields rose, which usually means the market has raised its view of future growth, inflation or the term premium. It can also steepen because short yields fell, which usually means it has brought forward the date of the first cut. The line ends up looking similar and the message is not the same, so read the two ends separately before reading the slope.

Because the two-year yield has to be consistent with rolling one-year money twice, the curve implies a rate for the second year that you can solve for. That implied number is the forward rate:

$$(1 + y_2)^2 = (1 + y_1)(1 + f)$$

Rearranged, the one-year rate the curve implies for a year from now is:

$$f = \frac{(1 + y_2)^2}{1 + y_1} - 1$$

Run it on the first worked example below. A one-year yield of 4.0 percent alongside a two-year yield of 4.3992 percent implies a one-year rate of 4.7999 percent starting twelve months out, which is the 4.8 percent the example was built from once the rounding in that 4.3992 is allowed for. The same arithmetic extends across the whole curve, which is how a forward curve is drawn.

Two cautions come with that number. First, the forward rate is what the market will trade at, not what anybody predicts: it contains the term premium, so it usually sits above the rate people genuinely expect, and it has been a biased forecast of actual future short rates. Second, annualising a multi-year total is a compounding calculation and not an average, which is the same operation as a growth rate over several years. The CAGR calculator does that step, and the compound interest calculator shows why the difference between averaging and compounding widens as the horizon gets longer.

An inversion is strange on its face. Lenders are accepting less to be paid back in ten years than to be paid back in three months, which only makes sense if they expect short-term rates to fall a long way, far enough to more than offset the term premium they are giving up.

The front of the curve is dominated by policy, so short rates mostly fall when the central bank is expected to cut, and it cuts when inflation is coming down or the economy is weakening. An inverted curve is therefore the bond market saying, in prices rather than words, that it expects conditions soft enough to bring cuts. The record behind the attention is a United States record, and it runs in one direction only. The gap between the 10-year yield and the 3-month bill has turned negative before every recession dated since the late 1960s. It has also turned negative and been followed by no recession at all, starting with an episode in the mid-1960s and not ending there, so the record describes what has preceded downturns and does not tell you which inversions will be followed by one. The Federal Reserve Bank of New York publishes a monthly recession probability fitted to that spread, which summarises the historical relationship rather than measuring a cause.

One candidate channel does look more like cause than coincidence, and it is worth stating with its caveats attached. Banks fund themselves short and lend long, so a flat or inverted curve compresses the margin available on new lending, standards tighten, weaker borrowers are turned down and credit growth slows. The caveats are real: banks hedge this exposure, the rate they pay depositors moves slowly and is not the same thing as a market short rate, and measured margins do not track the slope of the curve neatly. Treat it as a plausible transmission channel that is still argued over rather than the mechanism that settles the question.

One detail decides half the arguments about whether the curve has inverted at all: which two points you subtract. The 10-year minus the 3-month bill is the spread the research record is built on. The 10-year minus the 2-year note is quoted more often in the press and tends to invert earlier. They can disagree for months.

The honest description of the recession record is a correlation with a variable lag, not a mechanism with a schedule.

• The sample is tiny. Fewer than ten United States recessions sit in the modern data, and the same handful of episodes is quoted every time. A rule with eight or nine observations cannot carry the confidence a percentage figure implies.
• The converse has failed. Every recession having been preceded by an inversion is not the same claim as every inversion being followed by a recession, and the second one is what a reader usually wants. Curves have inverted, stayed inverted for a long stretch, and gone back to a normal slope with no dated recession behind them.
• Each downturn had its own trigger. The United States recession that began in early 2020 counts as a hit for an inversion the year before, and what actually caused it was a pandemic. A record built partly on episodes like that is a record of what the bond market worried about, not of what it saw coming.
• The lag has been all over the place. From the first inversion to the start of the recession has run from roughly six months to about two years. A signal that fires somewhere inside a two-year window is not a timing tool.
• The same shape can have two causes. A curve can invert because cuts are expected, or because heavy demand for long bonds crushed the term premium. The line looks identical and the message is not, which is why analysts decompose the curve rather than reading the slope alone.
• Un-inversion is not an all clear. In the United States, recessions have often begun after the curve had already returned to a positive slope, because the cuts the market was pricing in arrive and steepen it. The sequence usually ends with the signal switched off.
• It says nothing about depth. Even when the call is right, the curve carries no information about how deep or how long a downturn will be.

Used well, the curve is one input among several. It shows what the largest and most closely watched government bond market is currently pricing for short rates, which is worth knowing, and pricing is not the same as predicting, because the same number carries a term premium nobody can observe directly. Everything here is educational material rather than financial advice.

Why the curve takes the shape it does, and why an inversion is read as a recession signal, is a macroeconomics question: the yield curve.