A Whole Line That Lifts Is a Missing Axis, Not a Discount

An estimated line of expected return against market sensitivity that sits entirely above the theoretical one is not a market on sale, because pricing errors scatter above and below instead of lifting everything by the same amount. In a two-factor world where every asset carries the same 0.75 exposure to the second risk, the fitted single-factor line comes out exactly parallel to the theoretical one and exactly 3 percentage points above it, with residuals of zero, while a mispricing world engineered to have the same average lift leaves errors of both signs as large as 6.5 points. Let the second exposure grow with sensitivity and the slope moves too, at which point the two lines can cross inside an ordinary sample.

Put every asset on one chart. Expected return up the vertical axis, sensitivity to the market along the horizontal one. A single-factor theory says the points lie on one straight line that leaves the vertical axis at the risk-free rate and climbs at the market premium. Now estimate that line from data and suppose it comes out entirely above the theoretical one, at every level of sensitivity in the sample. Which direction is a coherent story, and which assumption has given way?

The answer is that the upward case is the coherent one and the assumption that fails is the count of factors. It is worth working out why, because the reflex reading of the same picture is wrong in a way the picture itself refutes.

Reading the gap as free money

The tempting interpretation of a gap between a fitted line and a theoretical one is mispricing. Every asset earns more than the theory says it should, so every asset is cheap, so buy the lot. Written out, that sentence is already odd. It claims that a whole market is underpriced by the same amount at the same time, which is not a thing mispricing does. Bad prices are individual mistakes, and individual mistakes point in both directions.

That objection can be measured rather than asserted. Take seven assets with sensitivities from 0.20 to 1.70, give each one a pricing error, and choose the errors so that they average exactly 3 percentage points: 7.5, then $-2.0$ , then 6.0, then $-3.5$ , 7.0, $-1.0$ and 7.0. The average lift is identical to the case we are about to build. Fit a line by least squares and it comes out with an intercept of 5.80 percent and a slope of 6.21 percent, close to the target, and it leaves residuals of both signs as large as 6.5 percentage points. Four assets sit above the fitted line and three below it.

Fig. 1 — The mispricing story, given the same average lift of three points. The fit is no longer exact and the errors have both signs, which is what a cloud of individual mistakes looks like.

Nothing here rules out mispricing as a cause of some of the gap. What it rules out is mispricing as an explanation of a gap that is constant across the whole cross-section.

What does lift a whole line

Here is a generating process that produces exactly the picture in question. Suppose returns are compensated for two priced risks rather than one:

\mathbb{E}[R_i] = r_f + \beta_i \lambda_m + \beta_{2,i}\,\lambda_2

(1)

with a 3 percent risk-free rate, $\lambda_m = 6\%$ per unit of market sensitivity and $\lambda_2 = 4\%$ per unit of exposure to the second risk. Take the second exposure to be the same for every asset in the sample, $\beta_{2,i} = c = 0.75$ . The exposure does not have to be common in general; taking it common is what makes the arithmetic transparent. Substituting and grouping terms:

\mathbb{E}[R_i] = \underbrace{(r_f + \lambda_2 c)}_{\text{lifted intercept}} + \beta_i\,\lambda_m

(2)

Equation (2) is still a straight line in market sensitivity, and it has the theoretical slope untouched. Only the intercept has moved, by $\lambda_2 c = 0.04 \times 0.75 = 0.03$ , which is exactly 3 percentage points. So the theoretical line runs from 3 percent at zero sensitivity to 9 percent at sensitivity one, and the line the data will hand you runs from 6 to 12. Three points apart at both ends, and at every point in between.

Fig. 2 — The two-factor world, projected onto one factor. Same slope, higher intercept, and every asset exactly on the fitted line.

The gap does not depend on sensitivity at all. Differentiate it and you get $\partial(\lambda_2 c)/\partial\beta_m = 0$ , which is the algebraic content of the word parallel. And because (2) is an identity rather than an approximation, a least-squares fit to this world returns the intercept and the slope exactly, with residuals below $10^{-15}$ . The mispricing world engineered to have the same average lift leaves a largest residual more than ten orders of magnitude bigger. That contrast is the whole diagnostic.

The line the theory draws

In a single-factor equilibrium, expected excess return is proportional to the covariance of the asset with the market portfolio, so $\mathbb{E}[R_i] = r_f + \beta_i \lambda_m$ with $\beta_i = \mathrm{Cov}(R_i, R_m)/\mathrm{Var}(R_m)$ . The relation is a straight line by construction, so any systematic departure from a straight line, or from this particular straight line, is information about the model rather than about the assets.

When the second exposure varies with the first

A common exposure is a convenience. The general version lets the second exposure grow with market sensitivity, $\beta_2 = c + k\,\beta_m$ , which is closer to what happens in practice because the same underlying leverage tends to show up in both. Substituting into (1) and grouping again:

\mathbb{E}[R] = \underbrace{(r_f + \lambda_2 c)}_{\text{intercept}} + \underbrace{(\lambda_m + \lambda_2 k)}_{\text{slope}}\,\beta_m

(3)

Still a straight line, still fitted exactly, but now the slope has moved too. With $c = 0.5$ and $k = 0.3$ the intercept is 5 percent and the slope is 7.2 percent, and the fitted line stays above the theoretical one across every sensitivity from 0 to 2. The single-factor regression has absorbed the second risk into two coefficients that were supposed to mean something else, and it has done so without leaving a trace in the residuals.

This is what makes the misspecification hard to spot from a regression summary. A missing variable that happens to be collinear with the one you kept does not announce itself as poor fit. It announces itself as a coefficient that will not replicate on the next sample.

Why up rather than down

The algebra in (2) is indifferent to sign. The lift is $\lambda_2 c$ , so the fitted line sits above the theoretical one when that product is positive and below it when the product is negative. Nothing in the derivation prefers one case.

The asymmetry comes from the economics of the omitted term. A misspecified market proxy leaves out risk that investors bear and expect to be paid for, and compensation for bearing risk is positive, so $\lambda_2 > 0$ is the default reading. With a positive common exposure the product is positive and the line lifts. Getting a line entirely below the theoretical one requires either a negative premium on the omitted risk, which means investors pay to hold it because it hedges something they fear more, or a negative common exposure across the whole sample. Both are possible and neither is the ordinary case.

Where the argument stops working

A few limits are worth stating, because each one is a way of being wrong while quoting the result correctly.

The phrase entirely above is range-dependent as soon as the slopes differ. In (3) the gap is $\lambda_2(c + k\beta_m)$ , which vanishes at $\beta_m = -c/k$ . Take $c = 0.5$ and $k = -0.3$ and the two lines cross at a sensitivity of 1.67, well inside the range of a normal cross-section. Only the common-exposure case is genuinely parallel and therefore genuinely never crossing.

The exact fit is a feature of the construction and not a prediction about data. Real estimates of this relation have residuals, because exposures are not identical and returns are measured with noise. What survives is the shape of the departure: a level shift is evidence about the model, and scatter is evidence about the assets.

Finally, the theoretical line is not observable. Comparing an estimate against it requires the true market portfolio, and any tradeable index is a proxy for that portfolio rather than the thing itself. The point has been made forcefully in the literature on testability: a rejection of the single-factor relation may be a rejection of the proxy instead. Which is the same conclusion this article reached from the other end. A whole line that lifts is a statement about the axis you chose, not about the assets you priced.

Sources and further reading

The single-factor relation and the line it predicts — Capital asset pricing model
Why the choice of market proxy makes the relation hard to test — Roll's critique
What a second and third priced factor look like empirically — Fama-French three-factor model
The statistical name for the effect in equation (3) — Omitted-variable bias
The estimator used for both fits — Ordinary least squares

Both fits in this article were computed by least squares written out by hand rather than called from a library. The common-exposure world returns a slope equal to the theoretical one and a gap of exactly 3 points at sensitivities 0, 0.5, 1, 1.5, 2 and 5, with residuals below $10^{-15}$ ; the affine world is fitted exactly and stays above across sensitivity 0 to 2; the mispricing world with the same average lift leaves a largest residual of 6.5 percentage points, with four positive and three negative.