Overview

When markets deteriorate, the reported conditional pricing kernel differs most from an unconditional average in negative-return states. In the preliminary Federal Reserve working paper, lower VIX or higher term spread corresponds to a higher and steeper estimated left-tail kernel, while crisis estimates shift equity-premium attribution toward negative returns. The conditional estimate gives a 0.63% mean monthly equity risk premium versus 0.76% under the unconditional kernel, compared with a reported 0.67% data counterpart, and performs better on reported call and delta-hedged put tests. Outright put Euler equations remain rejected, so the projected one-month market-return kernel is not a complete option-pricing model.

Learning outcomes

The conditional kernel varies most strongly in negative-return states across different crisis information environments.
In the reported sample, conditioning lowers the mean monthly equity risk premium estimate from 0.76% to 0.63%, closer to the 0.67% data counterpart.
Reported tests favor the conditional kernel for calls and delta-hedged puts, but outright put Euler equations remain rejected under both kernels.

Keywords

pricing kernelleft tailequity premiumVIXdelta hedgingoutright puts

When the Market Turns Bad, the Left Tail Takes Over

This chapter interprets what the conditional pricing kernel does when market conditions deteriorate. The focus is not whether the VIX is high in the abstract, but whether conditional information changes the prices assigned to negative-return states, the inferred equity premium, and the performance of tail-sensitive option tests. That focus supports a concrete judgment: a kernel that averages across all conditions can put risk compensation in the wrong part of the return distribution precisely when the left tail matters most.

The distinction becomes visible in the Federal Reserve paper’s comparison between its conditional kernel and an otherwise similar unconditional estimate. The unconditional estimate uses one average relationship between future returns and prices. The conditional estimate allows that relationship to move with the VIX and term spread. In ordinary periods the two can look broadly similar. In crises, their difference becomes an economic result rather than a technical detail.

Why the left tail carries the economic meaning

A pricing-kernel curve assigns a relative price to a future return state. The central part of the curve describes ordinary market movements, but the far-left part describes what happens to the price of payoffs when the market falls sharply. That region matters because a payoff delivered during a market collapse can be unusually valuable to an investor whose other holdings are losing value. The price of protection is therefore not determined only by how much the market moves on average.

The Federal Reserve estimates show that the kernel varies most strongly in negative-return states. When the VIX is relatively low or the term spread is relatively high, the estimated kernel in the left tail is higher and steeper. When the VIX is high or the term spread is low, it is lower and less steep. The paper’s interpretation is that a severe negative state may be less expected in the first environment, making resources delivered there especially valuable. In the second environment, adverse outcomes are already part of the information set, and the estimated state-price pattern changes.

This does not mean that a high VIX makes a crash less important. It means that the price assigned to a given negative-return state is a relationship between the state, the information available before it occurs, and the compensation attached to bearing it. A high-volatility environment can contain a large amount of perceived risk while still producing a lower estimated kernel in some negative-return regions. “Risk is high” and “the state price is high” are not interchangeable statements.

The distinction is also visible in the timing of the kernel’s shape. In the paper’s bivariate estimate, the kernel is especially low in negative-return regions during the 2008 financial crisis and the 2020 COVID-19 crisis; at a future monthly return of -10%, the estimated value can fall below one. That number should not be read as a universal threshold or a direct preference parameter. It is an implication of the projected empirical estimate. Its importance is comparative: the same future return is mapped differently when the current information environment is different.

The average shape is not a simple textbook curve everywhere. The paper finds that monotonicity is generally not rejected through about 8% of the monthly market return, roughly the central range of the distribution, but is rejected in deep right tails.

Published by DeepMox

FAQ

Why does the left tail matter for a pricing kernel?

The left tail matters because payoffs delivered during a market collapse can be especially valuable when other holdings are losing value, so their state prices and compensation need not match average volatility.

How did the conditional and unconditional equity-premium estimates differ?

In the reported sample, mean monthly equity risk premium was 0.63% under the conditional kernel and 0.76% under the unconditional kernel, compared with a reported 0.67% data counterpart.

Did the conditional kernel price outright puts successfully?

No. Euler equations for outright put returns were rejected under both kernels in the reported 30-day tests, although the conditional kernel performed better for calls and delta-hedged puts.

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Conditional Pricing Kernel: Left Tail in Crises

When the Market Turns Bad, the Left Tail Takes Over

Why the left tail carries the economic meaning

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