Level, Slope, Curvature: Litterman and Scheinkman Find the Three Factors

Ask a bond trader in 1990 to hedge a portfolio and they would compute its duration, that is, its sensitivity to a parallel shift in the yield curve, and offset it.

Ask the same trader what happens if the curve does not shift in parallel, and they would look uncomfortable, because everybody knew it usually does not. Short rates and long rates go their own ways all the time. The curve steepens, flattens, twists, humps. Duration says nothing about any of that, and yet duration was the tool.

Robert Litterman and Jose Scheinkman, then at Goldman Sachs, did something almost embarrassingly straightforward. They took the data and asked it how the curve moves. The answer they got has organised the fixed income world ever since.

The problem: too many yields, and they all move together

The trouble with the yield curve as a risk object is that it is high-dimensional and hopelessly redundant at the same time.

High-dimensional: there are yields at dozens of maturities, and in principle each could do its own thing, so you have dozens of risks to worry about.

Redundant: they emphatically do not do their own thing. When the 5-year yield rises, the 7-year yield almost always rises too. These things are so tightly correlated that treating them as dozens of independent risks is absurd. You would be double-counting the same underlying movement over and over.

So how many risks are there, really? Is the yield curve a fifty-dimensional object, or a three-dimensional one wearing a fifty-dimensional costume? Nobody had answered that empirically.

The key idea via analogy: find the crowd's real dance moves

Litterman and Scheinkman apply a statistical technique called principal components analysis. Strip the jargon and here is what it does.

Imagine watching a crowd of people from above. Everyone is moving, and the raw data is thousands of individual positions changing every second. Overwhelming. But you notice that most of the movement is not independent: the whole crowd surges forward together, then a wave ripples from left to right, then people bunch in the middle. Three simple patterns account for almost everything, and the "individual" movements are mostly just people participating in those three patterns.

Principal components analysis is the algorithm that finds those patterns automatically. It searches for the single movement that explains the most variation, then the movement that explains the most of what is left, and so on, with each new pattern required to be independent of the previous ones. It is not told what to look for. It reads the answer out of the data.

Point it at US Treasury bond returns and here is what comes out, in order of importance:

One: the level. All yields move up or down together. The whole curve shifts, more or less in parallel. This is by far the biggest factor, and it accounts for the large majority of the variation in bond returns. When people talk about "rates going up," this is what they mean, and it is the factor that duration is designed to capture. Duration is not wrong. It is just incomplete.

Two: the slope. Short yields and long yields move in opposite directions, so the curve steepens or flattens. This is the second-biggest factor and it is nothing like a parallel shift. A duration-hedged portfolio, one with zero exposure to factor one, can still be badly exposed to this and lose serious money. This is the factor duration is blind to.

Three: the curvature. The middle of the curve moves one way while both ends move the other, so the curve bulges or sags in the belly. Smaller than the first two, but real, and it is what a butterfly trade is designed to isolate.

Add these three together and you have accounted for the overwhelming bulk of the variation in bond returns. Whatever is left over is small.

That is the paper. Three numbers, not fifty. The yield curve is a three-dimensional object.

Why it mattered

This result is now so embedded in fixed income that people forget it was ever a discovery.

• It rebuilt hedging. Duration hedges factor one. Litterman and Scheinkman showed you also need to hedge factor two, and probably factor three. Modern bond risk management is explicitly organised around exposure to level, slope and curvature, and every risk system reports them.
• It defined the trading vocabulary. A steepener or flattener is a bet on factor two. A butterfly is a bet on factor three. Relative value desks are organised around these trades. Litterman and Scheinkman named the axes that the market now trades along.
• It told model builders how many factors they need. Every one-factor short rate model, Vasicek, CIR, Hull-White, can only produce level movements. It is structurally incapable of representing a curve that twists. This paper is the empirical indictment: a one-factor model is missing a substantial fraction of what actually happens, and it explains precisely why multi-factor term structure models exist.
• It confirmed Nelson and Siegel from a completely different direction. Nelson and Siegel had built a curve-fitting formula out of a level, a tilt and a hump, chosen because those shapes fit observed curves. Litterman and Scheinkman found the same three objects in the return data, with no assumptions at all. Two independent routes to the same three factors is strong evidence the factors are real, and Diebold and Li later fused the two insights into a forecasting model.
• It gave portfolio managers a language for their bets. "We are long duration and short the steepener" is a complete description of a bond portfolio's directional risk, in three numbers. That compression is what makes fixed income portfolio management possible.

The honest limitations

• The factors are statistical, not economic. Principal components analysis finds patterns of co-movement. It does not tell you what they mean. "Level" is not an economic force; it is a mathematical direction in yield space that happened to explain the most variance. It probably has something to do with inflation expectations, and "slope" probably has something to do with the business cycle and monetary policy, but the technique itself is silent, and reifying these factors as economic objects is a leap the paper does not license.
• The factors are not unique or stable. They are extracted from a covariance matrix estimated over a sample. Change the sample, change the currency, change the set of maturities, and the factors shift, sometimes materially. They are not laws of nature.
• The three components are forced to be uncorrelated. Reality is not. PCA constructs factors that are orthogonal by mathematical construction. Actual level and slope movements are correlated in the real world, and the technique cannot represent that, so it produces a clean picture of a messier reality.
• Small factors are not harmless factors. The residual after three components is small in variance terms, but it is exactly where the money is for some strategies. A relative-value trader making a living on tiny dislocations between individual bonds cares intensely about the residual that this analysis throws away. "Explains most of the variance" and "explains most of the P&L" are very different claims.
• It is linear, and crises are not. PCA is a linear technique on a covariance matrix. It assumes the relationships hold uniformly. In a crisis, correlations lurch, the curve does things it has never done, and a hedge built on three historically estimated factors can fail exactly when you need it. The tail is where the linear approximation breaks, and hedging is mostly about tails.
• Hedging factors is not the same as hedging risk. A portfolio with zero level, slope and curvature exposure is not riskless. It is merely immunised against the three most common movements, which is a narrower promise than it sounds, and one that history has repeatedly punished people for over-reading.

The one-line takeaway

Litterman and Scheinkman let the data speak and found that the entire yield curve, in all its apparent complexity, essentially only does three things: it shifts, it tilts, and it bulges, a discovery that rebuilt bond hedging, named the trades the market makes, and told model builders how many factors they were missing.