Heterogeneous Beliefs and Overvaluation

Heterogeneous Beliefs and Overvaluation (Miller 1977)

Miller’s (1977) insight is that two ingredients together produce overvaluation: investors disagree about an asset’s value, and short-selling is constrained. Heterogeneity alone is not enough — without short-sale constraints, optimists and pessimists average out and the price equals the consensus value. Constraints alone are not enough — if investors agree, the price equals the common value. The two ingredients only generate overvaluation when combined: pessimists are forced out of the market because they cannot short, leaving the price determined by the optimists alone.

The model. A unit mass of CARA-mean-variance traders splits into optimists ($j=1$, mass $1/2$) and pessimists ($j=0$, mass $1/2$). They disagree about the mean payoff of a single risky asset in fixed supply (one unit), with ${\mu }^1={\mu }^{\text{true}}+\Delta$ and ${\mu }^0={\mu }^{\text{true}}-\Delta$. They agree on the variance ${\sigma }^2$ and have common risk-aversion $\theta$. Each trader’s optimal demand at price $p$ is
$x^j=\frac{{\mu }^j-p(1+r)}{\theta {\sigma }^2}.$

Without short-sale constraints, market clearing gives
$p=\frac{{\mu }^{\text{true}}-2\theta {\sigma }^2}{1+r}=p^{\text{true}}.$
The optimist’s positive demand and the pessimist’s negative demand cancel exactly. Heterogeneous beliefs leave the price unchanged.

With short-sale constraints ($x^j\ge 0$), the pessimist’s demand becomes $x^0=\max (0,({\mu }^0-p(1+r))/\theta {\sigma }^2)=0$ at the equilibrium price (her unconstrained demand is negative). Since ${\mu }^0-p(1+r)<0$, the pessimist is out of the market. The optimist must absorb the entire supply: $x^1=2$ (since the optimist’s mass is $1/2$ and total supply is $1$, so per-capita demand is $2$). Solving the optimist’s first-order condition with $x^1=2$ gives
$p=\frac{{\mu }^1-2\theta {\sigma }^2}{1+r}=\frac{{\mu }^{\text{true}}+\Delta -2\theta {\sigma }^2}{1+r}>p^{\text{true}}.$
The price is above the rational consensus value by exactly $\Delta /(1+r)$ — the per-period optimism premium discounted to today.

Empirical hook. Diether, Malloy & Scherbina (2003) test the prediction that more disagreement should produce more overvaluation by sorting stocks on the dispersion of analyst earnings forecasts. The highest-dispersion quintile underperforms the lowest by about $0.79\%$ per month — confirming that disagreement plus short-sale constraints predicts both overpricing and subsequent low returns. The effect is strongest in small stocks where short-sale constraints bind hardest.

Intuition

In a normal market, prices reflect a consensus of opinion: optimists buy, pessimists sell, and the marginal trader sets the price. The pessimist’s selling pressure pulls the price down, balancing the optimist’s buying pressure. But “selling” for a pessimist who does not already own the asset means short-selling — borrowing shares and selling them. If short-selling is impossible or expensive, the pessimist has no way to express her view in the market. Her only option is to abstain. Once she abstains, the price is set entirely by the people who do want to buy, namely the optimists. The price reflects the optimist’s view, not the average view.

The geometry of the result is sharp: heterogeneity is irrelevant unless it interacts with an asymmetry in market access. The asymmetry can be a literal short-sale ban, a high cost of borrowing shares, an institutional rule against short positions (mutual funds, pension funds), or a fear of unlimited losses on the short side. Any of these forces filters out one tail of opinion and lets the other tail set the price. Bubble episodes therefore tend to concentrate in asset classes where short-selling is hardest — IPO lockups, hard-to-borrow stocks, illiquid real estate, cryptocurrencies before the first futures markets opened.

Properties

(Two Ingredients Required) Heterogeneity alone or constraints alone are not enough.

In the unconstrained equilibrium, optimists and pessimists submit symmetric demands and the market clears at the consensus price $p^{\text{true}}$. The disagreement washes out by aggregation. In the constrained equilibrium with no disagreement ($\Delta =0$), all traders have the same demand and the constraint never binds. Only when both heterogeneity and constraints are present does the price deviate from fundamentals. This is what makes Miller’s mechanism a joint condition, not an additive one.

(Optimism Premium) The overvaluation equals $\Delta /(1+r)$ per period.

The price gap between the constrained and unconstrained equilibria is exactly $\Delta /(1+r)$ — the optimist’s mean above the true mean, discounted one period. This says the size of the bubble scales linearly with the spread of beliefs. Doubling the disagreement doubles the overvaluation. Empirically this is the testable prediction Diether-Malloy-Scherbina exploit: the cross-section of overpricing should track the cross-section of disagreement.

(Selection of Holders) Only optimists hold the asset in equilibrium.

Pessimists abstain ($x^0=0$) and optimists buy the entire supply ($x^1=2$). The portfolio composition of the marginal trader is $100\%$ optimist. This has an important corollary: surveys of who holds bubble assets should find systematically optimistic investors, not a representative cross-section. Empirically this matches the dot-com era (retail investors and momentum funds dominated tech holdings) and the housing bubble (subprime borrowers and speculator-flippers concentrated in the most overvalued markets).

(Static Model) The Miller framework is one-period and produces no trade.

In the basic setup, beliefs are fixed and trade happens once at $t=0$. The model produces overvaluation but not the high trading volume observed in real bubbles. To match the volume evidence — for instance, the Hong-Stein (2007) finding that glamour stocks have systematically higher turnover than value stocks — the model must be extended to a dynamic setting where beliefs change over time. That extension is the Harrison-Kreps (1978) dynamic version.

(Diether-Malloy-Scherbina Empirical Test) Dispersion in analyst forecasts predicts low returns.

Diether, Malloy & Scherbina (2003) sort stocks into quintiles by the standard deviation of analyst earnings forecasts (a proxy for belief disagreement). The highest-dispersion quintile underperforms the lowest by approximately $0.79\%$ per month, with $t$-statistic $2.88$. The effect is concentrated in small-cap stocks (where short-sale constraints bind hardest) and is robust to controls for size, book-to-market, and momentum. The pattern is exactly what Miller predicts: stocks investors disagree about most are overpriced today and underperform tomorrow as the optimism reverses.

(Limit on Realism) The model treats short-sale constraints as binary.

In reality, short-selling is rarely impossible — it is costly. Investors can borrow shares from their brokers at a rebate rate that varies by stock. The Miller mechanism still operates but in a graded form: stocks that are hard or expensive to borrow (high “loan fee”) behave like the constrained-equilibrium case, while liquid stocks with cheap borrowing behave like the unconstrained case. Empirical work using actual loan-fee data (Cohen, Diether & Malloy, 2007) confirms that the overpricing-and-subsequent-underperformance pattern is concentrated in hard-to-borrow stocks.

Examples

The dot-com bubble through the Miller lens

In late $1999$ and early $2000$, internet stocks traded at price-earnings ratios above $100$ and in many cases had no earnings at all. Two ingredients fit Miller’s model perfectly. First, beliefs about internet companies were extraordinarily heterogeneous: optimists projected exponential revenue growth indefinitely while pessimists saw no path to profitability. Second, short-selling was severely constrained by IPO lockups (which forbade insider sales for $180$ days), high borrowing costs on retail tech names, and institutional mandates that prevented many funds from holding short positions. The pessimists were structurally locked out, so prices reflected the optimists’ view alone. When the lockups expired in late $2000$ and $2001$, supply increased and prices collapsed — exactly the pattern Miller would predict.

Solving the constrained equilibrium step-by-step

Setup. A unit mass of traders, half optimists and half pessimists, with mean-variance preferences and risk aversion $\theta$. The risky asset is in fixed supply of $1$ unit, has variance ${\sigma }^2$, and the riskless rate is $r$. Optimists believe ${\mu }^1={\mu }^{\text{true}}+\Delta$; pessimists believe ${\mu }^0={\mu }^{\text{true}}-\Delta$.

Unconstrained. Each trader’s demand is $x^j=({\mu }^j-p(1+r))/\theta {\sigma }^2$. Market clearing: $\frac{1}{2}{x}^1+\frac{1}{2}{x}^0=1$. Substituting and solving:
$\frac{1}{2}\cdot \frac{{\mu }^1-p(1+r)}{\theta {\sigma }^2}+\frac{1}{2}\cdot \frac{{\mu }^0-p(1+r)}{\theta {\sigma }^2}=1.$
The $\Delta$ terms cancel. Solving for $p$: $p=({\mu }^{\text{true}}-2\theta {\sigma }^2)/(1+r)=p^{\text{true}}$.

Constrained. Pessimists’ unconstrained demand at $p^{\text{true}}$ is negative, so the constraint binds and $x^0=0$. Market clearing reduces to $\frac{1}{2}{x}^1=1$, giving $x^1=2$. From the optimist’s first-order condition:
$2=\frac{{\mu }^1-p(1+r)}{\theta {\sigma }^2}⟹p=\frac{{\mu }^1-2\theta {\sigma }^2}{1+r}=\frac{{\mu }^{\text{true}}+\Delta -2\theta {\sigma }^2}{1+r}.$
The overvaluation is $p-p^{\text{true}}=\Delta /(1+r)$.

See Also

• 8.7. Bubbles — parent hub.
• 8.7.1. Rational Bubbles — the rational alternative that Miller’s model replaces.
• 8.7.3. Speculative Bubbles — the dynamic extension where the same mechanism generates trading volume and a resale option.
• 8.7.4. Overconfidence and Trading Volume — the cognitive source of the belief heterogeneity Miller assumes.
• 8.3. Challenges to Rational Pricing — the empirical puzzles this model addresses.