X-CAPM

The X-CAPM (Barberis, Greenwood, Jin & Shleifer, $2015$ ) is a continuous-time equilibrium model of the aggregate stock market in which a fraction $1 - μ$ of investors form beliefs about future returns by extrapolating past price changes while a fraction $μ$ hold fully rational beliefs. It is the canonical second-generation behavioral asset-pricing model: where BSV (1998) and Hong-Stein (1999) required separate conservatism and representativeness stories calibrated to different horizons, X-CAPM derives a family of aggregate-market anomalies from a single extrapolation parameter.

Setup. Two assets: a risk-free asset with constant rate $r$ , and a risky asset (the aggregate stock market) in fixed per-capita supply $Q$ . The risky asset pays a continuous dividend stream evolving as arithmetic Brownian motion,
$d D_{t} = g_{D} d t + σ_{D} d ω_{t},$
and the equilibrium price $P_{t}$ is pinned down by market clearing between the two investor types.

Extrapolator beliefs. Define the sentiment variable as an exponentially-weighted average of past price changes,
$S_{t} = β \int_{- \infty}^{t} e^{- β (t - s)} d P_{s^{-}}, β > 0.$
Extrapolators forecast future price changes as a linear function of sentiment,
$E_{t}^{e} [d P_{t}] / d t = λ_{0} + λ_{1} S_{t}, λ_{1} > 0,$
so past price increases raise their expected future returns one-for-one through $S_{t}$ . The discrete-time form of the same belief is $E_{t}^{e} [P_{t + 1} - P_{t}] = (1 - θ) \sum_{k \geq 0} θ^{k} (P_{t - k} - P_{t - k - 1})$ with $θ \in (0, 1)$ playing the role of $e^{- β}$ . Rational traders, by contrast, know the true equilibrium price process and trade on it optimally, but have finite CARA risk tolerance and cannot absorb arbitrary extrapolator flows.

Headline results.

Excess volatility. A positive dividend shock raises the fundamental value; extrapolators then observe the price rise, mark up their expected future return, and buy more, amplifying the initial jump. Return volatility exceeds dividend volatility — the Shiller ( $1981$ ) excess-volatility puzzle emerges as an equilibrium output.

Predictability from the price-dividend ratio. The transient sentiment wedge decays slowly. Elevated $P_{t} / D_{t}$ forecasts low subsequent returns, matching the Campbell-Shiller ( $1988$ ) evidence on aggregate-market predictability.

Consistent with survey data. Investor expectations in the model are positively correlated with past returns (by construction) but negatively correlated with subsequent realized returns (because extrapolators are systematically wrong on average). This is exactly the joint pattern documented in US investor surveys by Greenwood & Shleifer ($2014$) — a pattern impossible under rational expectations.

Unification. One extrapolation parameter generates excess volatility, $P / D$ predictability, aggregate bubbles, and a time-series momentum-then-reversal pattern. No separate under- and over-reaction biases are required.

Domain and limits. X-CAPM targets the aggregate stock market and is silent on the cross-section. It cannot by itself explain why small or low-analyst-coverage stocks show stronger momentum — those patterns remain Hong-Stein’s territory. It also treats extrapolative beliefs as primitive rather than deriving them from deeper psychology (representativeness, hot-hand bias, availability). Later extensions (Barberis et al. $2018$ ; Nagel & Xu $2022$ ) move the framework toward the cross-section and earnings-based extrapolation.

Intuition

Any rational-expectations model of the aggregate stock market predicts that investors’ stated expectations of future returns should line up with subsequent realized returns — high expected returns should forecast high realized returns. The survey evidence runs the other way: Greenwood and Shleifer (2014) show that investor expectations are positively correlated with past returns and negatively correlated with what actually happens next. This single fact is devastating for a pure rational story of the aggregate market, because no rational updating rule can generate it. X-CAPM is the minimum model that takes the fact seriously: put some investors who extrapolate past returns (“extrapolators”) into the same market as investors who don’t (“rational traders”), let limited rational arbitrage keep the system from exploding, and solve for equilibrium. What falls out is a coherent joint explanation of excess volatility, $P / D$ predictability, bubbles, and the Greenwood-Shleifer survey pattern — from one behavioral parameter.

The central tension the model formalizes is between short-run momentum seekers and long-run contrarians. Extrapolators buy after price rises and sell after price declines, which amplifies every shock. Rational traders understand this but cannot fully counteract it. While the stock market is overvalued at time $t$ , the extrapolators will still be bullish tomorrow because their sentiment $S_{t}$ decays only slowly, so the near-term expected return on a short position is close to zero even though the long-run return is negative. A risk-averse rational trader balances the modest expected profit from shorting against the interim volatility and only partially offsets the overpricing. Prices therefore overshoot, and the slow mean-reversion that follows is exactly the predictability rational traders were betting on all along.

Properties

(Extrapolator Belief Equation) Expected returns are an exponentially-weighted average of past returns.

The extrapolator’s expected price change per unit time is $E_{t}^{e} [d P_{t}] / d t = λ_{0} + λ_{1} S_{t}$ , with sentiment $S_{t} = β \int_{- \infty}^{t} e^{- β (t - s)} d P_{s^{-}}$ . The decay parameter $β$ controls how quickly past observations fade — higher $β$ means extrapolators weight only very recent returns, lower $β$ means they weight a long history. The sign restriction $λ_{1} > 0$ is the behavioral content: recent price increases raise expected future returns. This is the continuous-time formalization of “put too much weight on the recent past,” and it is the only behavioral departure in the model — everything else (risk aversion, market clearing, dividend process) is standard.

(Two-Agent Market Clearing) The equilibrium price balances rational and extrapolator demands.

Each investor type solves a CARA portfolio problem against her own beliefs about the distribution of $d P_{t}$ . The equilibrium price is determined by the condition that aggregate risky-asset demand equals the fixed per-capita supply $Q$ . Rational traders (fraction $μ$ ) use the true, endogenously-determined price process; extrapolators (fraction $1 - μ$ ) use the sentiment-driven process. When $μ = 1$ the model collapses to a standard rational-expectations Lucas-tree equilibrium; when $μ \to 0$ the system loses its anchor and prices can diverge without bound. The interesting region is interior $μ \in (0, 1)$ , where rational traders discipline extrapolators just enough to keep the system stationary but not enough to eliminate the anomalies.

(Excess Volatility Mechanism) Extrapolator feedback amplifies dividend shocks.

When a positive Brownian shock $d ω_{t} > 0$ hits the dividend stream, three things happen. First, the rational-only fundamental value rises to reflect the shock. Second, extrapolators observe the resulting price jump, update sentiment $S_{t}$ upward, mark up their expected future return via the $λ_{1} S_{t}$ channel, and increase their demand — pushing the equilibrium price further. Third, rational traders only partially counteract the overshoot because their optimal short position is bounded by their risk tolerance. The equilibrium price jump exceeds what the dividend shock alone would justify, and the stationary return volatility is strictly greater than the dividend volatility $σ_{D}$ . Shiller’s ( $1981$ ) excess-volatility puzzle thus emerges as an equilibrium output rather than being imposed by assumption.

(Limits-to-Arbitrage Subtlety) Rational traders under-bet overvaluation because reversal is slow.

The most subtle piece of the model is why rational traders do not aggressively short the overvalued market. At time $t$ , the stock is above fundamentals and the long-run expected return to a short position is negative. But the near-term expected return is close to zero, because extrapolators will remain bullish for as long as sentiment $S_{t}$ stays elevated — which is typically many periods since $β$ is small. A rational trader holding a short position faces the full variance of the price process for a long holding period in exchange for a modest expected gain per unit time. With finite CARA risk tolerance, the optimal short is bounded and small. This is the formal content of “limits to arbitrage” in X-CAPM: rational traders know the market is mispriced and trade on it, but only cautiously, because reversal is slow and interim volatility is high. The mispricing persists in equilibrium.

( $P / D$ Predictability) Elevated price-dividend ratios forecast low subsequent returns.

Because extrapolator-driven overshooting raises the price above the dividend-justified level, the price-to-dividend ratio $P_{t} / D_{t}$ is high precisely when sentiment is high. As sentiment decays (exponentially with rate $β$ ) and rational-trader flows eventually dominate, prices revert toward fundamentals and realized returns over the subsequent years are low. The model therefore generates a negative correlation between current $P / D$ and future returns — the Campbell-Shiller ( $1988$ ) aggregate-market predictability result. The strength and horizon of predictability are tied to the extrapolator fraction $1 - μ$ and the sentiment decay $β$ : more extrapolators or slower decay produce stronger and longer-dated predictability.

(Survey-Expectations Diagnostic) The model matches the Greenwood-Shleifer ( $2014$ ) joint pattern.

X-CAPM’s most distinctive empirical test comes from survey data. By construction, extrapolators’ stated expected return is a linear function of past returns, so the cross-sectional correlation between surveyed expectations and past realized returns is positive. But extrapolators are systematically wrong: the rational-trader fraction pulls the true price process back toward fundamentals, so the correlation between extrapolators’ stated expectations and subsequent realized returns is negative. This joint pattern — positive with the past, negative with the future — is exactly what Greenwood & Shleifer ( $2014$ ) found in US investor surveys covering the Gallup, Shiller, Michigan, and CFO samples. No rational-expectations model can reproduce this joint pattern without internal contradiction, which is why X-CAPM was received as a sharp empirical win for extrapolation as the organizing principle for aggregate-market behavior.

(Unification Advantage) One parameter generates a family of aggregate-market anomalies.

The unification X-CAPM offers is a strictly stronger claim than any first-generation reaction model makes. BSV requires separate conservatism and representativeness parameters plus a Markov-switching probability. Hong-Stein requires an information-diffusion speed and a momentum-trader risk tolerance. X-CAPM derives excess volatility, $P / D$ predictability, aggregate bubbles, and time-series momentum followed by reversal from a single extrapolation parameter $λ_{1}$ (equivalently the discrete-time weight $θ$ ). When the empirical target is a collection of aggregate-market stylized facts that used to need multiple stories, one parameter now does the work of several.

(Failure Modes) Where X-CAPM does not apply.

X-CAPM is an aggregate-market model and has no cross-section. It cannot explain why small stocks, stocks with fewer analysts, or stocks with lower trading volume exhibit stronger momentum — those remain Hong-Stein targets. It assumes a continuous-time diffusion for dividends and a smooth extrapolation kernel, which makes it ill-suited to discrete earnings surprises and the post-earnings-announcement drift that BSV directly targets. It is silent on the psychological source of extrapolation — the model takes extrapolative beliefs as primitive rather than deriving them from representativeness, the hot-hand fallacy, or the availability heuristic. And it assumes extrapolators are homogeneous within their type, ignoring the cross-sectional heterogeneity by cohort that experience-effects models emphasize. Subsequent work extends X-CAPM to earnings-based extrapolation (Nagel & Xu, $2022$ ) and to the cross-section (Barberis et al., $2018$ ), partially closing these gaps.

Examples

Why extrapolator feedback amplifies a $\$ 1$ dividend shock

Consider a simplified discrete-time sketch of the model with rational-trader fraction $μ = 0.5$ and a single extrapolator weight $θ = 0$ on old returns — so sentiment equals the most recent one-period return. Suppose the market has been flat, so $S_{t} = 0$ and $P_{t} = P^{*}$ (the fundamental value). At time $t$ , a dividend shock raises the fundamental value by $\$ 1 $, so t h er a t i o na l - t r a d er s - o n l y p r i ce w o u l d j u m pt o$ P^* + 1$.

Round 1. Rational traders would clear at $P^{*} + 1$ . Extrapolators, however, see the price rise of $\$ 1 $a s a s i g na l o f a f u t u r e p r i cer i seo f$ \lambda_1 \cdot 1 $(s a y$ \lambda_1 = 0.3 $), so t h e i r e x p ec t e d r e t u r nin cr e a ses b y$ 0.30 $p er u ni tt im e . T h e i r in cr e a se dd e man d p u s h es t h ee q u i l ib r i u m p r i ce hi g h er . I na sy mm e t r i c C A R A se t u pw i t h r a t i o na l - t r a d er f r a c t i o n$ \mu = 0.5 $, t h ee q u i l ib r i u m p r i ceo v er s h oo t s b y r o ug h l y ha l f t h ee x t r a p o l a t or - in d u ce d w e d g e — c a l l i t$ $0.15 $. S o t h e p r i ce a f t er r o u n d 1 i s$ P^* + 1.15 $, n o t$ P^* + 1 $. R e t u r n v o l a t i l i t y e x cee d s t h e$ $1 $d i v i d e n d v o l a t i l i t y b y$ 15%$.

Round 2. At $t + 1$ , extrapolators see the round-1 price change of $\$ 1.15 $(r a t h er t han$ $1 $) an d u p d a t e t h e i r e x p ec t e d r e t u r nb y a f u r t h er$ \lambda_1 \cdot 0.15 = 0.045$. This feeds a small additional push on the equilibrium price, amplifying the overshoot slightly further.

Long run. The amplification cascade converges geometrically (ratio $λ_{1} (1 - μ)$ in this sketch), so the total price overshoot is bounded — approximately $\$ 1 \times \lambda_1 (1-\mu) / [1 - \lambda_1(1-\mu)] = $1 \times 0.15 / 0.85 \approx $0.176 $. O v er t im e, o n ce t h es h oc k s t o p s a r r i v in g an d se n t im e n t d ec a y s, e x t r a p o l a t or ssc a l e ba c k t h e i r e x p ec t e d r e t u r n s, an d t h er a t i o na l t r a d er s^{'} l o n g - r u n p u l l r e a sser t s i t se l f : t h e p r i ces l o w l y r e v er t s f r o m$ P^* + 1.176 $t o w a r d$ P^* + 1 $, p r o d u c in g an e g a t i v er e t u r n p r e d i c t ab l e f r o m t h ee l e v a t e d$ P/D $r a t i o . T h es h oc k ha s b ee nam pl i f i e d o nim p a c t * an d * g e n er a t e d a p r e d i c t ab l er e v er s a l — b o t h f r o m t h es am e$ \lambda_1 > 0$.

This arithmetic is a toy version of the continuous-time equilibrium BGJS derive analytically. The qualitative features — on-impact amplification, geometric convergence of the feedback loop, and slow mean-reversion afterward — are exactly what the full continuous-time model produces, and what the data on aggregate returns show.

How X-CAPM produces the Greenwood-Shleifer survey pattern

Start from a period of high past returns — say the S&P has been up $20%$ over the last year. Extrapolators’ sentiment $S_{t}$ is elevated, so their surveyed expected return is above its long-run mean. This produces a positive correlation between past returns and surveyed expectations, just as Greenwood & Shleifer ( $2014$ ) find.

But the extrapolators are wrong. The elevated price-dividend ratio signals that the market is already overvalued. As sentiment decays over the next several years and rational traders’ long-run pull reasserts itself, actual realized returns are below the historical mean — exactly the realized reversal that makes elevated $P / D$ a predictor of low future returns. Surveying extrapolators at the peak therefore catches them at precisely the moment their expectations are most optimistic and most disconnected from what will actually happen.

The net result: surveyed expectations and subsequent realized returns are negatively correlated, even though surveyed expectations and past returns are positively correlated. A rational-expectations model cannot produce this joint pattern without contradicting itself — rational expectations, by definition, must equal the conditional expectation of realized returns, so the correlation between expectations and realized returns must be $+ 1$ , not negative. X-CAPM produces the pattern mechanically: extrapolators’ expectations look backward at prices while realized returns look forward at the fundamentals that sentiment has temporarily detached them from.

tags	domain/finance, type/concept, level/intermediate, status/complete
aliases	Extrapolative CAPM, Barberis-Greenwood-Jin-Shleifer Model, BGJS 2015, X-CAPM Model

Project Alexandria

Explorer

X-CAPM

Intuition

Properties

(Extrapolator Belief Equation) Expected returns are an exponentially-weighted average of past returns.

(Two-Agent Market Clearing) The equilibrium price balances rational and extrapolator demands.

(Excess Volatility Mechanism) Extrapolator feedback amplifies dividend shocks.

(Limits-to-Arbitrage Subtlety) Rational traders under-bet overvaluation because reversal is slow.

( $P / D$ Predictability) Elevated price-dividend ratios forecast low subsequent returns.

(Survey-Expectations Diagnostic) The model matches the Greenwood-Shleifer ( $2014$ ) joint pattern.

(Unification Advantage) One parameter generates a family of aggregate-market anomalies.

(Failure Modes) Where X-CAPM does not apply.

Examples

See Also

Graph View

Table of Contents

Backlinks

Project Alexandria

Explorer

X-CAPM

Intuition

Properties

(Extrapolator Belief Equation) Expected returns are an exponentially-weighted average of past returns.

(Two-Agent Market Clearing) The equilibrium price balances rational and extrapolator demands.

(Excess Volatility Mechanism) Extrapolator feedback amplifies dividend shocks.

(Limits-to-Arbitrage Subtlety) Rational traders under-bet overvaluation because reversal is slow.

(P/D Predictability) Elevated price-dividend ratios forecast low subsequent returns.

(Survey-Expectations Diagnostic) The model matches the Greenwood-Shleifer (2014) joint pattern.

(Unification Advantage) One parameter generates a family of aggregate-market anomalies.

(Failure Modes) Where X-CAPM does not apply.

Examples

See Also

Graph View

Table of Contents

Backlinks

( $P / D$ Predictability) Elevated price-dividend ratios forecast low subsequent returns.

(Survey-Expectations Diagnostic) The model matches the Greenwood-Shleifer ( $2014$ ) joint pattern.