Extrapolative Beliefs

Extrapolative Beliefs

Extrapolative beliefs are the second-generation behavioral framework for asset pricing. Where the first-generation models (BSV, Hong-Stein) built around discrete cognitive biases, extrapolative-beliefs models posit a single mechanism: investors form expectations about future returns or future fundamentals as a recency-weighted average of past returns or past fundamentals.

The canonical functional form (Greenwood & Shleifer, 2014; Barberis, Greenwood, Jin & Shleifer, 2015) writes the extrapolator’s expected price change as
${\mathbb{E}}_t^e[P_{t+1}-P_t]=(1-\theta ){\sum }_{j=0}^{\infty }{\theta }^j(P_{t-j}-P_{t-j-1}),$
a geometrically-declining weighted average of past one-period returns with decay parameter $\theta \in (0,1)$. The same functional form applies to earnings growth (Nagel & Xu, 2022): ${\mathbb{E}}_t[g_{t+1}]=(1-\theta ){\sum }_{j=0}^{\infty }{\theta }^j{g}_{t-j}$. The single parameter $\theta$ controls how much weight the investor places on the recent past relative to the deep past.

Empirical hook. Greenwood & Shleifer (2014) document that survey-based expectations of future stock returns are positively correlated with past returns and negatively correlated with subsequent realized returns. Both correlations are inconsistent with rational expectations and both are exactly what extrapolation predicts: investors are most optimistic just when the market is most overvalued.

Asset-pricing applications. Barberis, Greenwood, Jin & Shleifer (2015) build the X-CAPM, a continuous-time equilibrium model with a fraction $1-\mu$ of extrapolators and $\mu$ of rational traders. The model jointly produces excess volatility, return predictability from the price-dividend ratio, momentum at short horizons, and reversal at long horizons — all from a single extrapolation parameter, with no need for separate conservatism, representativeness, or noise-trader stories. Earnings-extrapolation versions (Nagel & Xu, 2022; Bordalo, Gennaioli, La Porta & Shleifer, 2020) replace returns with earnings growth and obtain similar results, with analyst long-term-growth (LTG) forecasts negatively predicting subsequent returns.

The cognitive root of extrapolative beliefs is representativeness — a streak of high returns resembles a trending process — though the second-generation literature treats the mechanism as primitive and reads the cognitive story off the data.

Intuition

Extrapolation captures the most basic intuition in behavioral finance: when stocks have been going up, investors expect them to keep going up, and when they have been going down, investors expect them to keep going down. This is empirically true in survey data on individual investors, in survey data on professional analysts and managers, and in the implied beliefs that match observed asset prices. It is the opposite of what rational expectations would imply at the aggregate level — high prices today should mean low expected returns tomorrow if dividends are stationary — yet the survey data repeatedly find that high prices coincide with high expected returns.

The mechanism is mathematically simple. Each investor takes a weighted average of recent return observations, with more weight on the most recent. After a string of high returns, that weighted average is high, so the extrapolator’s expected return is high. She bids up the price further, generating still more high returns, which feed back into still higher expectations. The feedback loop is what produces excess volatility and bubbles. Eventually fundamentals fail to validate the optimistic expectations, returns disappoint, the weighted average falls, and the process reverses — generating long-run reversal.

The same logic transfers to fundamentals: if investors extrapolate earnings growth rather than returns, a string of strong earnings makes them too optimistic about future earnings, so prices rise too much; subsequent earnings growth disappoints, and prices fall back. The two stories — return extrapolation and earnings extrapolation — are formally close cousins. Bordalo et al. (2020) show that the data favor earnings extrapolation as the deeper mechanism, with return extrapolation being a downstream consequence of earnings extrapolation acting on prices.

Properties

(Geometric Recency Weighting) The weighted average has a single parameter $\theta$.

In the canonical form ${\mathbb{E}}_t^e[P_{t+1}-P_t]=(1-\theta ){\sum }_{j=0}^{\infty }{\theta }^j(P_{t-j}-P_{t-j-1})$, the parameter $\theta \in (0,1)$ controls the half-life of memory. Small $\theta$ means the extrapolator only cares about the last few observations; large $\theta$ means the deep past matters too. Empirical fits to survey data put $\theta$ in the range $0.5$–$0.9$, implying half-lives of one to several years. The $(1-\theta )$ normalization ensures the weights sum to one, so the expected return equals a true weighted average of past observations.

(Greenwood-Shleifer Survey Evidence) Survey expectations correlate positively with past returns and negatively with future returns.

Greenwood & Shleifer (2014) collect six different surveys of investor return expectations spanning several decades — Gallup, the American Association of Individual Investors, the UBS Index of Investor Optimism, and others. All six show the same two patterns. First, expectations of future market returns are positively correlated with returns over the previous $12$ months (correlations around $+0.5$). Second, those same expectations are negatively correlated with subsequently realized returns (correlations around $-0.3$). The combination is impossible under rational expectations — if expectations were rational, the second correlation should be positive — and is exactly what extrapolation predicts. Investors are most optimistic when the market is most overvalued, and least optimistic when it offers the highest expected returns.

(X-CAPM Mechanism) Excess volatility from extrapolator-rational interaction.

In the X-CAPM of Barberis, Greenwood, Jin & Shleifer (2015), a positive dividend shock raises today’s price. Extrapolators see the price rise and update their expected future return upward — because they form expectations from past price changes. Their increased demand pushes prices still higher. Rational traders with limited risk-bearing capacity only partially counteract the overshoot, because they understand the eventual reversal but cannot bet against it without bound. The resulting price jump exceeds the fundamental shock — the excess volatility observed in the data — and the subsequent slow reversal generates predictability of returns from the price-dividend ratio.

(Joint Explanation) One parameter, four facts.

The extrapolation framework jointly delivers (i) excess volatility relative to dividend news, (ii) negative correlation of price-dividend ratios with subsequent returns, (iii) short-horizon momentum, and (iv) long-horizon reversal — all from the same recency-weighted-average mechanism. First-generation models needed separate stories for each fact (conservatism for momentum, representativeness for reversal, noise traders for excess volatility). The unification is what makes extrapolation the modern leading framework.

(Earnings Extrapolation) Bordalo et al. (2020) show analyst LTG forecasts predict returns negatively.

Bordalo, Gennaioli, La Porta & Shleifer (2020) examine analyst long-term growth (LTG) forecasts for individual stocks. They find that revisions in LTG are partly driven by past earnings growth — analysts extrapolate. Crucially, the component of LTG forecasts driven by past growth is negatively correlated with subsequent realized returns: stocks with the most upward-revised growth forecasts subsequently underperform. The same pattern that Greenwood-Shleifer find for aggregate-market expectations shows up for individual stocks via professional-analyst expectations. Earnings extrapolation is therefore not just a household-investor phenomenon.

(Cognitive Root) Representativeness is the leading psychological microfoundation.

Why do investors extrapolate? Because a streak of high returns resembles a trending process, and representativeness judges probabilities by resemblance. The hot-hand bias from basketball ($\mathrm{Pr}(\text{hit}\mid \text{streak})>\mathrm{Pr}(\text{hit})$) is the same psychological phenomenon transposed into financial markets. The cognitive story is consistent with the second-generation literature though the models themselves treat the recency-weighted-average mechanism as a primitive and let the data validate it.

(Limits) Extrapolation does not by itself explain which asset is extrapolated.

The framework takes “the asset being extrapolated” as given, but in reality not every asset class produces extrapolation feedback loops at the same intensity. Bubbles tend to emerge in specific asset classes — internet stocks in the late 1990s, U.S. housing in 2005–07, cryptocurrencies in 2017 and 2021 — and extrapolation alone does not tell you why those asset classes and not others. The companion ingredients are usually short-sale constraints, retail investor enthusiasm, and a salient narrative that rationalizes the extrapolation. See 8.7. Bubbles for the broader picture.

Examples

Return extrapolation and the dot-com bubble

Through the late $1990$s, U.S. tech stocks delivered annualized returns above $30\%$ for several years running. Survey expectations of future market returns reached their highest levels in recorded history in early $2000$ — exactly when the price-dividend ratio was at its highest and expected fundamental returns were therefore at their lowest. An extrapolator with $\theta =0.7$ and $5$ years of recent $30\%$ returns in her memory window would have an expected return well above the historical average, leading her to demand more equity exposure precisely when rational expectations said to demand less. The subsequent crash — annualized returns below $-15\%$ from $2000$–$2002$ — caused the same extrapolator to become pessimistic at precisely the bottom. Survey expectations bottomed out in late $2002$ and early $2003$, just as forward returns were highest.

The X-CAPM equilibrium (Barberis, Greenwood, Jin & Shleifer, 2015)

The model has two assets: a risk-free asset paying $r$ and a risky asset (the stock market) paying a continuous dividend whose level evolves as $d{D}_t=g_{D}dt+{\sigma }_{D}d\omega$. There is a continuum of rational traders (fraction $\mu$) and extrapolators (fraction $1-\mu$). Extrapolators form a sentiment variable
$S_t=\beta {\int }_{-\infty }^t{e}^{-\beta (t-s)}d{P}_{s-dt},$
which is an exponentially-weighted average of past price changes, and forecast next-period price change as ${\mathbb{E}}_t^e[d{P}_t]/dt={\lambda }_0+{\lambda }_1{S}_t$. Rational traders forecast correctly. The equilibrium price is determined by market clearing. After a positive dividend shock, the price jumps; extrapolators update $S_t$ upward and demand more; rational traders, knowing the overshoot, sell but only partially because of risk aversion and limited capital. The net result: the price overshoots the fundamental, then reverts as $S_t$ slowly decays. Calibrating $\beta$, $\mu$, and the dividend process to match historical moments gives a model that simultaneously fits excess volatility, return predictability from the price-dividend ratio, and the survey-expectation correlations from Greenwood-Shleifer.

Bordalo-Gennaioli-La Porta-Shleifer (2020) earnings extrapolation

The authors collect analyst long-term growth (LTG) forecasts for individual stocks from $1981$–$2015$ and decompose them into a component driven by past earnings growth and a residual. They find that the past-growth-driven component is positively related to current valuations (high-LTG stocks trade at high price-earnings ratios) and negatively related to subsequent realized returns (high-LTG stocks underperform over the next $1$–$5$ years). A present-value model in which prices are formed by extrapolating earnings — rather than by rational forward expectations — fits the time series of the S&P $500$ index quite well, while a rational present-value model overstates volatility and misses the bubble periods. The conclusion is that extrapolation of earnings growth, not just of returns, is doing the work.

See Also

• 8.6. Belief-Based Biases — parent hub.
• 8.6.5. Representativeness Heuristic — the cognitive root of extrapolation.
• 8.6.8. Experience Effects — sibling: extrapolation with cohort heterogeneity.
• 8.9.3. X-CAPM — the equilibrium asset-pricing model built on extrapolative beliefs.
• 8.9.1. Barberis-Shleifer-Vishny Model — the first-generation model that extrapolation generalizes.
• 8.8. Momentum — the cross-sectional anomaly that extrapolation jointly explains alongside reversal.
• 8.7. Bubbles — the natural application of extrapolation to bubble episodes.
• 8.3. Challenges to Rational Pricing — the excess-volatility and predictability puzzles that extrapolation addresses.