Stochastic Discount Factors, No-Arbitrage Moments, and HJ Distance

A stochastic discount factor is the object that prices everything at once. If it fails, the failure shows up as a portfolio the model misprices.

The Intuition

Chapter 14 moves from factor extraction to no-arbitrage pricing. That shift is easy to miss because the notation looks similar: factors, returns, exposures, and pricing errors all appear in both worlds. The objective is different.

• A reconstruction-style model asks: can I explain return variation?
• An SDF model asks: can I price assets so that no portfolio delivers free expected value after risk adjustment?

That is why adversarial SDF estimation is more than "another deep factor model." It trains against the portfolios that expose the model's worst pricing failures.

The chapter's objective ladder helps keep the concepts straight:

• PCA asks for covariance structure
• RP-PCA asks for covariance structure with pricing relevance
• adversarial SDF estimation asks for no-arbitrage pricing directly

The Euler Equation

Let \(R_{i,t+1}\) be a gross return and \(M_{t+1}\) the stochastic discount factor. No arbitrage requires

$$ \mathbb{E}_t[M_{t+1} R_{i,t+1}] = 1 $$

for every asset \(i\).

This is the Euler equation. It says that once returns are discounted by the right state-contingent pricing kernel, every tradable payoff has fair value.

If you work with excess returns \(r_{i,t+1}^e\), the same condition becomes

$$ \mathbb{E}_t[M_{t+1} r_{i,t+1}^e] = 0. $$

This version is often easier to interpret. A valid SDF should make the conditional expected discounted excess return vanish.

From Factors to an SDF

A linear factor model can be written as

$$ M_{t+1} = a - b^\top f_{t+1}, $$

where \(f_{t+1}\) are factor realizations and \(b\) are factor prices. The exact parameterization is less important than the logic; the intercept \(a\) is pinned by normalization, that is, by the risk-free pricing condition:

• factor models describe returns through exposures and factor shocks
• an SDF summarizes those same pricing relations in one object

This is why Chapter 14 says some factor models can be represented as SDFs. But "can be represented as" is not the same as "was trained to minimize pricing errors." A CAE trained on reconstruction error may imply an SDF indirectly. An adversarial SDF model targets no-arbitrage moments directly.

Conditional Moments and Test Assets

In practice, the Euler condition is tested through moment restrictions. If \(g_t\) is an instrument measurable at time \(t\), then valid pricing implies

$$ \mathbb{E}\!\left[g_t \left(M_{t+1} r_{i,t+1}^e\right)\right] = 0. $$

The instrument \(g_t\) matters because mispricing can be conditional. A model may price average returns acceptably while failing badly in certain states, sectors, or characteristic-sorted portfolios.

That is why test assets matter. If your test assets span only easy portfolios, the model can look good while still missing economically important risk. The adversarial step in Chapter 14 is trying to construct or emphasize the portfolios that most expose those failures.

Hansen-Jagannathan Distance

The Hansen-Jagannathan distance is a geometric summary of pricing error. Informally, it asks:

how far is the candidate SDF from the set of discount factors that would exactly price the chosen assets?

In mean-variance language, it is the norm of the pricing error after accounting for the covariance structure of returns. A simple GMM-style expression is

$$ d_{HJ}^2 = \alpha^\top W \alpha, $$

where \(\alpha\) is the vector of pricing errors and \(W\) is a weighting matrix built from return second moments, in the standard HJ formulation typically the inverse second-moment matrix of excess returns.

The important lesson is not the exact formula variant. It is the interpretation:

• large raw pricing errors in low-variance directions can matter a lot
• small raw pricing errors in noisy directions may matter less
• HJ distance is about economically meaningful mispricing, not just squared prediction loss

This is one reason Chapter 14 warns that predictive metrics and no-arbitrage metrics answer different questions.

A Worked Two-Asset Example

Suppose there are two excess-return assets, \(r_1^e\) and \(r_2^e\), and you posit an SDF

$$ M_{t+1} = 1 - b f_{t+1}. $$

If the model is correct, then both moments

$$ \mathbb{E}[M_{t+1} r_1^e] = 0, \qquad \mathbb{E}[M_{t+1} r_2^e] = 0 $$

should hold.

Now imagine the first asset is priced roughly correctly but the second is not. Then some portfolio weight vector \(w\) puts more mass on asset 2 and exposes a large violation:

$$ \mathbb{E}[M_{t+1} w^\top r_{t+1}^e] \neq 0. $$

That portfolio is evidence against the SDF. The adversarial logic in Chapter 14 is to search for portfolios like this on purpose instead of hoping a fixed benchmark set happens to reveal them. In that sense, the HJ distance is the geometric summary of exactly this problem: how badly the worst-exposed portfolio is still mispriced after accounting for return covariance. Equivalently, the HJ bound ties this distance to the maximum Sharpe ratio of the pricing-error portfolio.

Why This Differs from Reconstruction Error

A reconstruction model can explain a large fraction of return covariance and still price assets poorly. High-variance factors dominate reconstruction loss. But a low-variance factor can carry a large risk premium, and missing it creates economically important mispricing.

The objectives get closer and closer to the asset-pricing question.

What Good Evaluation Looks Like

A good SDF evaluation block should report:

• pricing errors on a transparent set of test assets
• HJ-style or related distance measures
• sensitivity to the chosen test-asset span
• out-of-sample performance, not only in-sample fit

This last point matters. A low in-sample HJ distance can be manufactured by a flexible model that overfits the benchmark portfolios. That is why weighting-matrix choices, regularization, and out-of-sample testing remain load-bearing.

In Practice

Use these rules:

• keep the Euler equation in mind when reading every SDF objective
• ask what the test assets are and what span of mispricing they can reveal
• distinguish predictive fit from pricing fit
• report robustness to test-asset choice and sample period
• treat a low HJ distance as evidence of better pricing, not proof of economic truth

Common Mistakes

• Treating the SDF as just another latent factor summary.
• Confusing low reconstruction loss with good no-arbitrage pricing.
• Reporting one benchmark set of test assets as if it were decisive.
• Ignoring the weighting matrix behind the distance measure.
• Reading low pricing error as proof that the model is structurally correct.

Connections

This primer supports Chapter 14's adversarial SDF material. It connects directly to RP-PCA, GMM intuition, test-asset design, and the book's broader distinction between explaining returns and pricing returns under no-arbitrage restrictions.