Overview

The 2009 ECB model uses one affine stochastic discount factor to price government bonds and a broad stock index together. Inflation, payout yield, and two latent real-rate factors form a state whose evolution makes expected equity premia vary by horizon and date, while the equity benchmark remains the model-implied real bond yield at the matching horizon. The model allows upward, downward, or hump-shaped conditional curves, but unconditional expected stock returns are flat across horizons; its conclusions remain tied to factor, pricing, and reinvestment restrictions.

Learning outcomes

A shared pricing kernel makes the stock premium and risk-free benchmark part of one no-arbitrage system.
Current inflation, payout yield, and latent real-rate factors can reshape conditional premia differently across horizons.
A flat unconditional expected stock-return term structure does not contradict a curve that changes shape with the current state.

Keywords

Equity risk premiumPricing kernelBondsStocksInflationPayout yield

How One Pricing Kernel Prices Bonds and Stocks

This chapter examines the second route to an equity-premium term structure: the 2009 ECB model that prices government bonds and a broad stock index with one arbitrage-free pricing kernel. Its purpose is to show how current economic conditions can make expected equity compensation differ by horizon, even when the model’s unconditional average stock return is flat across horizons. The result is a structural time-series view of the same subject that the cross-sectional cash-flow method approaches through differences among firms.

A common pricing system for two asset classes

A pricing kernel, also called a stochastic discount factor, is a rule for translating an uncertain future payoff into a price today. If the payoff is large in states of the world when investors value money less, it need not be worth as much as the same payoff delivered in bad states. The kernel captures that state-dependent discounting. An arbitrage-free model requires the same rule to price assets that can be combined into trades; otherwise, inconsistent prices could create an apparent riskless profit.

The ECB model uses one such rule for both government bonds and equity. That choice matters because the equity premium is measured relative to a risk-free investment. Rather than estimating the stock market in isolation and attaching a bond yield afterward, the model makes the bond and stock sides part of the same system. The risk-free benchmark is not an external number added at the end; it is generated by the same underlying state and pricing assumptions.

The model’s state vector has four components. Inflation connects real values to nominal bond payments. A payout-yield factor records the payments made to equity holders relative to the stock price. Two latent factors determine the one-period real interest rate. “Payout” is broader than cash dividends: it can include stock buybacks and other ways companies distribute profits. The paper often uses dividend-yield language for convenience, but the model’s theoretical object is the total payout yield.

The two real-rate factors are latent, meaning they are not observed directly in the data. They are inferred from the joint behavior of observed bond yields, stock returns, payout yields, and inflation. Together with the payout-yield and inflation factors, they form the state that summarizes the model’s information at a given date. The factors follow a stationary vector autoregression of order one: today’s state helps forecast tomorrow’s state, while new shocks move the system.

The pricing kernel’s market prices of risk are themselves affine functions of the state. In plain language, the amount of compensation assigned to a shock can change as inflation, payout yield, or the latent real-rate factors change. This is the part of the model that creates time variation. A factor can affect the short real rate, affect expected future states, and carry a changing price of risk at the same time.

That shared architecture imposes useful cross-asset discipline.

Published by DeepMox

FAQ

What is a pricing kernel in the ECB model?

It is the state-dependent rule that converts uncertain future payoffs into today’s prices, and the model uses the same rule for government bonds and equity.

Why can the ECB model’s conditional curve change shape?

Its conditional curve can change shape because current inflation, payout yield, and latent real-rate factors affect expected returns and risk prices across horizons.

How can unconditional stock returns be flat while premia vary by horizon?

Unconditional expected stock returns are flat across horizons in the model because horizon-dependent terms cancel at the long-run mean, while conditional premia still respond to the current state and bond yields.

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