A Second Hump: Svensson and the Curve Central Banks Actually Fit

Lars Svensson was a monetary economist, not a bond quant, and that matters for understanding this paper. He was not primarily trying to fit a curve. He was trying to read the market's mind, and he needed a better instrument to do it with.

The paper is nominally about Sweden in the early 1990s, a country that had just been forced off its currency peg and was in the middle of adopting inflation targeting. But its contribution was general, and the formula it introduced is now what most central banks in the world use to publish their yield curves.

The problem: yields blur, forwards separate

Start with why Svensson cares about forward rates at all.

A yield on a ten-year bond is an average. It blends together the market's view of what will happen next year, the year after, and every year out to ten. If the ten-year yield rises, you cannot tell whether the market has changed its mind about next year's policy rate or about inflation a decade from now. Everything is mixed together.

A forward rate unmixes it. The forward rate for year 8 is, roughly, what the market thinks the short-term interest rate will be in eight years' time. So the forward curve reads like a timeline: here is what the market expects for the next few months (which is mostly about the current policy stance), here is what it expects in the medium term (the expected path of the recovery), and here is what it expects far out (which, for a central bank, is mostly about long-run credibility: has the market accepted the inflation target, or does it think the central bank will let inflation drift?).

For a policymaker, the long forward rate is the single most valuable number in the market. It is an unfiltered read on whether anyone believes you.

But you can only extract it if your fitted curve is trustworthy, and here Svensson hits the wall. Forward rates are the slope of the fitted curve, which means they magnify every defect in the fit. A curve that looks fine can produce a forward curve full of nonsense. And Nelson and Siegel's formula, elegant as it is, has a specific shortcoming for this purpose: it can only produce one hump.

The key idea via analogy: two bumps, not one

Nelson-Siegel gives you three sliders: a level, a tilt, and one hump. That was enough for the calm US Treasury curves of the 1980s.

It was not enough for Sweden in 1993. The forward curve there had genuine structure: a sharp feature at the short end, driven by expectations of imminent policy moves after the currency crisis, and then a separate feature further out as the market worked through the medium-term inflation path. Two distinct bumps. Nelson-Siegel, with one hump available, had to choose which to fit, and would then smear the other one away or, worse, distort the long end trying to compromise.

Svensson's fix is exactly as simple as it sounds: add a second hump term, with its own height and its own decay parameter, so the curve can bulge in two places independently.

The analogy is a graphic equaliser again. Nelson and Siegel gave you bass, treble and one mid-range band. Svensson adds a second mid-range band. You can now shape the middle of the curve in a way that leaves the ends alone, and you can put a feature at eighteen months and a different feature at five years without the two fighting each other.

That is essentially the whole technical contribution, and it is why the model is often called Nelson-Siegel-Svensson rather than getting a name of its own. But the payoff is real, because the flexibility buys you exactly what you need: a forward curve at the long end that is not being contorted to compensate for a misfit at the short end. And the long end is where the policy information lives.

The other half of the paper, and the half Svensson cared most about, is the interpretation. He lays out how to read the fitted forward curve: how to decompose it into expected real rates, expected inflation, and expected currency depreciation; how to see whether an inflation target has been believed; how to watch credibility being won or lost, week by week, in the market's own prices. This is the paper that taught central banks to treat the forward curve as a live instrument panel rather than an academic curiosity.

Why it mattered

• It is the standard, in production, at central banks. The Federal Reserve's published US Treasury yield curve dataset (constructed by Gurkaynak, Sack and Wright) uses the Svensson form. So do many others. When someone quotes "the ten-year zero-coupon yield," there is a very good chance it came out of this formula.
• It made monetary policy legible. Reading expected policy paths, breakeven inflation and credibility out of the forward curve is now completely routine at every central bank in the developed world. Svensson wrote the manual.
• It fixed Nelson-Siegel where it hurt. More flexibility in the middle of the curve, without sacrificing the crucial property that the model stays rigid enough not to fit noise or explode when extrapolated. That balance is delicate, and Svensson got it right.
• It is a case study in fit-for-purpose modelling. The extension exists because a specific, real problem (reading the Swedish curve after a currency crisis) exposed a specific limitation. Not elegance for its own sake; a tool built because the old tool broke.

The honest limitations

• Still not arbitrage-free. Like Nelson-Siegel, this is a curve-fitting formula with no financial theory inside it. Nothing guarantees the fitted curve does not imply arbitrage opportunities. Fine for describing the curve; not fine for pricing derivatives off it without further work.
• More parameters, more instability. Two decay parameters, both entering nonlinearly, both hard to pin down. The estimation becomes a nasty optimisation with multiple local optima, and it is common for the fitted parameters to jump around dramatically from one day to the next even when the actual curve barely moved. Different bumps can trade off against each other and produce near-identical curves, so the individual parameters have little stable meaning.
• The extra flexibility can be abused. The second hump exists to capture genuine structure. It will just as happily capture a couple of stale quotes. The whole virtue of the Nelson-Siegel family is that it is too rigid to fit noise, and every parameter you add erodes that.
• The interpretation rests on the expectations hypothesis. Reading the forward rate for year 8 as "what the market expects the short rate to be in year 8" assumes the term premium is zero or constant. It is not. Fama and Bliss, Campbell and Shiller, and Cochrane and Piazzesi all showed that bond risk premia move around a lot and are predictable. So the forward curve is a mixture of expectations and risk compensation, and the naive reading over-attributes to expectations. Every central bank knows this and still, in practice, half-ignores it, because separating the two requires a model that nobody fully trusts.
• Practitioners cheat on the decay parameters. Because estimating them properly is painful, they are frequently fixed at conventional values. That is a hidden assumption baked into a lot of published curves.
• It smooths away real information. The formula deliberately erases bond-specific deviations. Those deviations are precisely what a relative-value trader is hunting for, so this is a policymaker's tool, not a trader's.

The one-line takeaway

Svensson added a second hump to the Nelson-Siegel formula so the forward curve could bend at the short end and the medium term independently, and then showed central bankers how to read that forward curve as a live report on whether the market believes them.