8.4.4. The Noise-Trader Risk Model

Noise Trader Risk Model (DSSW)

The noise trader risk model (De Long, Shleifer, Summers, Waldmann, 1990) explains how mispricing can persist even when arbitrageurs are rational, because the act of betting against noise traders exposes arbitrageurs to noise-trader risk that can worsen before it corrects. Noise traders can even earn higher expected returns by bearing the risk they create themselves. 

The model has two groups. A fraction $\mu$ are noise traders who think they have special information and therefore misperceive the expected price of the risky asset by a stochastic belief shock ${\rho }_t$, while the remaining $1-\mu$ are arbitrageurs with rational expectations who trade contrarian against noise. 

To isolate noise-trader risk, the model shuts down fundamentals:

• There are two assets that pay the same fixed dividend $r$ (equal to the required return). This eliminates fundamental risk, because cash-flows are fixed.
• The safe asset price is fixed at 1 with a perfectly elastic supply, and the risky asset has a fixed supply of 1 with a variable price $p_t$.
• All uncertainty comes from the unpredictable evolution of ${\rho }_t$ – a term that captures noise traders’ misperception of the mean of the risky asset’s price distribution.

Preferences are CARA, which makes optimal portfolio choice mean-variance. In the first period, arbitrageurs choose risky-asset holdings ${\lambda }_t^A$ (in number of shares) based on the expected excess payoff, and noise traders use the same rule but with an extra “sentiment shock” term ${\rho }_t\sim \mathcal{N}(p^{\ast },{\sigma }_{\rho }^2)$ in perceived expected payoff at time $t+1$:
${\lambda }_t^A=\frac{r+\mathbb{E}[p_{t+1}]-(1+r)p_t}{2\gamma \mathrm{Var}(p_{t+1})},{\lambda }_t^N=\frac{r+\mathbb{E}[p_{t+1}]-(1+r)p_t+{\rho }_t}{2\gamma \mathrm{Var}(p_{t+1})}.$
Here $\gamma$ is risk aversion and ${\rho }_t$ is the noise traders’ belief error. 

Market clearing averages these demands. With unit supply of the risky asset, $\mu {\lambda }_t^N+(1-\mu ){\lambda }_t^A=1$, which implies the equilibrium risky-asset price satisfies
$p_t=\frac{1}{1+r}(r+\mathbb{E}[p_{t+1}]-2\gamma \mathrm{Var}(p_{t+1})+\mu {\rho }_t).$

The key feature is the $-2\gamma \mathrm{Var}(p_{t+1})$ term. Even though dividends are certain, price risk is endogenous because the sentiment term ${\rho }_t$ moves demand and therefore prices. 

When ${\rho }_t$ is i.i.d. over time, the model pins down the steady-state mean and variance of prices (i.e., $\mathbb{E}[p_t]=\mathbb{E}[p_{t+1}]$). The price variance comes purely from sentiment shocks passed through market clearing,
$\mathrm{Var}(p_{t+1})={\left(\frac{\mu {\sigma }_{\rho }}{1+r}\right)}^2,$
and the steady-state price can be written as a decomposition,
$p_t=1+\underset{\begin{array}{c}\text{Temporary}\\ \text{Mispricing}\end{array}}{\underbrace{\frac{\mu ({\rho }_t-{\rho }^{\ast })}{1+r}}}+\underset{\begin{array}{c}\text{Price}\\ \text{Shift}\end{array}}{\underbrace{\frac{\mu {\rho }^{\ast }}{r}}}-\underset{\begin{array}{c}\text{Additional}\\ \text{Risk Premium}\end{array}}{\underbrace{\frac{2\gamma }{r}{\left(\frac{\mu {\sigma }_{\rho }}{1+r}\right)}^2}},$
where ${\rho }^{\ast }=\mathbb{E}[{\rho }_t].$ Here:

1. The first term is the fundamental value ($1$) of the risky asset.
2. The second term is the temporary mispricing from the current noise-trader sentiment shock. When noise-traders are more bullish (${\rho }_t>{\rho }^{\ast }$), prices rise, and vice-versa.
3. The third term is the average price shift if noise traders are bullish on average (i.e., if average sentiment ${\rho }^{\ast }>0$).
4. The last term is the risk premium that depends on agents’ risk aversion created from sentiment volatility.

In the DSSW model, the question is not just whether noise traders misprice assets, but whether they necessarily earn lower returns than arbitrageurs. The return gap over $t$ to $t+1$ can be written as
$\mathbb{E}[\Delta R_{N-A}]=({\lambda }_t^N-{\lambda }_t^A)[r+\mathbb{E}[p_{t+1}]-p_t(1+r)]={\rho }^{\ast }-\frac{{\sigma }_{\rho }^2+(p^{\ast }{)}^2}{2\gamma \mu {\sigma }_{\rho }^2}(1+r{)}^2,$
so relative performance depends on two objects: (1) how much more the noise traders hold of the risky asset $({\lambda }_t^N-{\lambda }_t^A)$, and (2) the risky asset’s expected excess return. In the rightmost side:

1. ${\rho }^{\ast }$ contributes a “Hold More” effect, because if noise traders are bullish on average they hold more of the risky asset on average.
2. The ${\sigma }_{\rho }^2$ term in the numerator is a “Friedman” or buy-high sell-low effect, since noise traders increase exposure precisely when sentiment is high and are more exposed to subsequent reversals.
3. The $({\rho }^{\ast }{)}^2$ term is “Price Pressure,” because persistent bullishness raises the price level and lowers subsequent expected returns.
4. Finally, the denominator captures a “Create Space” effect, since higher sentiment volatility raises price risk and makes risk-averse arbitrageurs scale back, which weakens correction and reduces how much the other effects hurt noise traders.

The punchline is that the return difference can go either way. Even though noise traders distort prices and can have poor timing, the risk they create can reduce arbitrage intensity enough that noise traders are partly compensated for bearing that risk, so they may not underperform rational arbitrageurs in expectation.

See Also

• 8.4. Noise — parent hub on noise trading and limits to arbitrage.
• 8.4.2. No-Trade Theorem — the Milgrom-Stokey theorem that motivates why noise traders are needed in the first place.
• 8.1. Principles of Behavioral Finance — the limits-to-arbitrage pillar that DSSW formalizes.
• 8.3. Challenges to Rational Pricing — the excess-volatility and predictability puzzles that noise-trader risk helps explain.
• 8.7.4. Overconfidence and Trading Volume — alternative microfoundation for trading volume via cognitive bias rather than exogenous sentiment shocks.
• 8.7.2. Heterogeneous Beliefs and Overvaluation — the Miller bubble model where short-sale constraints replace finite arbitrage capital as the limit to arbitrage.
• 8.7. Bubbles — the bubble application area where noise-trader risk and DSSW logic are widely invoked.