De Long–Shleifer–Summers–Waldmann Noise Trader Model

De Long–Shleifer–Summers–Waldmann noise trader model (commonly abbreviated as the DSSW model) is the foundational behavioral-finance asset-pricing model in business and economics theory demonstrating how the unpredictable sentiment of irrational "noise traders" creates a non-diversifiable source of resale price risk—termed noise trader risk—that prevents risk-averse rational arbitrageurs with finite horizons from driving asset prices to fundamental value. Published by economists J. Bradford De Long, Andrei Shleifer, Lawrence H. Summers, and Robert J. Waldmann in their August 1990 Journal of Political Economy article "Noise Trader Risk in Financial Markets", the DSSW model directly refuted the classic Milton Friedman (1953) and Eugene Fama (1965) market-selection hypothesis, which asserted that irrational investors who trade on noise rather than information must consistently buy high and sell low against rational arbitrageurs, losing wealth until they are driven to extinction. Employing a Samuelson-style two-period overlapping-generations (OLG) model with Constant Absolute Risk Aversion (CARA) utility, De Long, Shleifer, Summers, and Waldmann proved two landmark theorems: first, because rational arbitrageurs must liquidate their portfolios at a finite future date, the risk that noise traders might become even more pronounced tomorrow deters full arbitrage today even when the asset has zero fundamental dividend risk; second, because noise traders hold a disproportionately large share of the risky asset on average when they are bullish, they earn the risk premium created by their own sentiment volatility (the "create space" effect) and can earn higher expected returns than rational arbitrageurs.

Prior to De Long, Shleifer, Summers, and Waldmann (1990), mainstream financial economics relied on two pillars to defend the Efficient Market Hypothesis against behavioral psychology:

1. Immediate Price Correction via Arbitrage: Whenever irrational traders ("noise traders" in the sense of Fischer Black, 1986) bid an asset's price p_t above or below its fundamental discounted dividend value, rational arbitrageurs take opposing positions and restore fundamental pricing.
2. Evolutionary Wealth Extinction (Friedman, 1953): Even if arbitrage is temporarily incomplete, noise traders who trade at distorted prices transfer wealth to rational arbitrageurs on average and must eventually lose their capital and vanish from the market.

DSSW (1990) identified the hidden assumption underlying both pillars: infinite arbitrage horizons. If an arbitrageur lives forever and holds an asset strictly for its perpetual stream of dividends, future resale price fluctuations are irrelevant. In reality, professional fund managers, corporate treasurers, and individual life-cycle savers have short investment horizons: they buy an asset at Date t and must sell it at Date t + 1 to finance consumption, pay retirees, or settle periodic performance evaluations (later extended in the Shleifer–Vishny limits of arbitrage model). Consequently, an arbitrageur who shorts an overpriced asset at t bears the hazard that noise traders will become even more irrationally exuberant at t + 1, forcing the arbitrageur to buy back the short position at a steep capital loss.

Asset Structure: Safe Asset (s) vs. Unsafe Asset (u)

The DSSW economy is a discrete-time overlapping-generations model in which each generation lives for two periods: youth (Period t, when agents purchase a portfolio with an exogenous labor endowment) and old age (Period t + 1, when agents collect dividends, sell their asset holdings to the next young generation, and consume all wealth w_t+1). There is no bequest motive and no labor supply choice. Two assets pay identical real risk-free fundamental dividends r > 0 in every period:

1. Safe Asset (s): In perfectly elastic supply at a fixed unit price p_s = 1, paying a certain real dividend r per unit in every period. Notice that r is the risk-free real interest rate.
2. Unsafe Asset (u): In fixed aggregate net supply normalized to 1 unit. The unsafe asset pays the same certain real dividend r in every period as the safe asset (s), so its fundamental intrinsic value is identically equal to 1 (V_u = r/r = 1) with zero fundamental cash-flow risk. However, unlike the safe asset, the unsafe asset is traded by noise traders and therefore has an endogenous market-clearing resale price p_t at Date t.

Rational Sophisticated Investors (i) vs. Irrational Noise Traders (n)

In each young cohort t (of total population measure 1), a fraction 1 − μ are sophisticated rational investors (i) and a fraction μ ∈ (0, 1) are noise traders (n). Both types maximize expected CARA utility U(w) = −exp(−2γw) over Period-(t + 1) final wealth w, where γ > 0 is the coefficient of absolute risk aversion:

• Sophisticated Investors (Fraction 1 − μ): Hold rational expectations (Grossman–Stiglitz noisy rational expectations without asymmetric dividend information). They accurately perceive the true conditional distribution of tomorrow's resale price p_t+1 ∼ N(E_t[p_t+1], σ_pt+1²) and choose their demand λ_tⁱ for the unsafe asset to maximize mean-variance expected utility:

• Noise Traders (Fraction μ): Misperceive the expected resale price of the unsafe asset at t + 1 by an independent and identically distributed (i.i.d.) normal sentiment shock S_t ∼ N(ρ*, σ_ρ²):

Here, ρ* represents the average bullishness (mean optimism) of noise traders, and σ_ρ² > 0 is the variance of noise trader sentiment from period to period.

Optimal Portfolio Demands and the Stationary Equilibrium Price Formula

Differentiating each agent's perceived mean-variance objective with respect to portfolio holdings gives the linear asset demands at Date t:

Note that short sales are completely unrestricted for both agent types. Imposing market clearing across the unit supply of the unsafe asset—(1 − μ)λ_tⁱ + μλ_tⁿ = 1—and solving recursively for the stationary steady-state rational-expectations equilibrium where σ_pt+1² = σ_p² is constant over time yields the DSSW Equilibrium Pricing Equation:

where the unconditional and conditional one-step-ahead price variance of the unsafe asset is:

The closed-form DSSW pricing equation decomposes the deviation of the market price p_t from its fundamental value V = 1 into four distinct economic terms:

Observe the most counterintuitive implication of Term 4: even if noise traders have zero average bias (ρ* = 0), the unconditional expected price of the unsafe asset is less than its fundamental value 1:

Simply by existing and trading on unpredictable sentiment (σ_ρ² > 0), noise traders inject resale price risk σ_p² into an otherwise riskless asset, depressing its equilibrium price below fundamental value and generating an endogenous equity risk premium (closely linked to the Mehra–Prescott equity premium puzzle and Robert Shiller's excess volatility puzzle).

To test Milton Friedman's (1953) evolutionary extinction hypothesis, De Long, Shleifer, Summers, and Waldmann compute the expected difference in total portfolio returns ΔR_tⁿ⁻ⁱ earned by noise traders relative to rational arbitrageurs between t and t + 1:

Substituting λ_tⁿ − λ_tⁱ = S_t / (2γ σ_p²) and taking unconditional expectations across S_t ∼ N(ρ*, σ_ρ²) gives the DSSW Relative Return Theorem:

This formula decomposes the relative return of noise traders versus rational arbitrageurs into two competing forces:

1. The "Hold More" / "Create Space" Effect (+ρ*): Whenever noise traders are optimistic on average (ρ* > 0), they hold a larger average portfolio share of the unsafe asset than rational arbitrageurs (E[λ_tⁿ] > E[λ_tⁱ]). Because noise trader risk depresses the asset's price and creates a positive expected excess return over the safe asset, holding more of the unsafe asset earns noise traders a positive risk-premium reward (+ρ*).
2. The "Friedman" + "Buy High, Sell Low" Effect (Negative Second Term): Because noise traders increase their buying precisely when S_t is high (driving p_t up) and sell when S_t is low, their market timing is systematically mistimed (contributing the penalty −(1 + r)²σ_ρ²/(2γμσ_ρ²)), and excessive optimism ((ρ*)² large) causes them to overpay on average.

Crucially, for any moderate positive average bullishness 0 < ρ* < 2γμσ_ρ²/(1 + r)², the positive "Hold More" effect can outweigh the negative Friedman timing effect, so noise traders earn strictly higher expected returns than rational arbitrageurs (E[Rⁿ] > E[Rⁱ]) and never vanish from the market.

In Section IV of their 1990 paper and a companion 1991 Journal of Finance study by Lee, Shleifer, and Thaler, DSSW applied the noise trader risk model to resolve the closed-end mutual fund puzzle:

• A closed-end fund holds a publicly disclosed portfolio of liquid traded securities, so its fundamental Net Asset Value (NAV) is known exactly every day.
• Nevertheless, closed-end funds are launched at a 5–10 percent premium (when noise traders are optimistic, S_t > 0), trade on average at a persistent 10–15 percent discount below NAV (explained by Term 4 of the DSSW formula: even if ρ* = 0, noise trader sentiment volatility σ_ρ² requires closed-end shares to trade at a permanent discount to compensate rational buyers for resale price risk), and fluctuate wildly in discount width across time before converging to NAV only upon fund liquidation.

What is "noise trader risk" in the DSSW (1990) model?

Noise trader risk is the risk borne by a finite-horizon arbitrageur that noise traders' irrational sentiment S_t+1 will widen even further in the future when the arbitrageur needs to liquidate the position. Because of noise trader risk, an asset with a completely riskless dividend stream r acquires an endogenous resale price variance σ_p² = μ²σ_ρ²/(1 + r)² > 0, deterring risk-averse arbitrageurs from taking unlimited positions to eliminate mispricing.

Why can noise traders earn higher expected returns than rational investors in the DSSW model?

Although noise traders suffer from bad market timing ("buy high, sell low" as S_t fluctuates), their unpredictable trading makes the unsafe asset risky and forces its equilibrium price down so that it pays a positive expected risk premium above the risk-free rate. When noise traders are moderately bullish on average (ρ* > 0), they allocate a larger share of their portfolios to the risky asset than cautious rational arbitrageurs do, capturing the risk premium that their own noise trading created.

How does the DSSW (1990) model relate to the Shleifer–Vishny (1997) limits of arbitrage model?

The DSSW (1990) model establishes limits to arbitrage through individual risk aversion (γ > 0) and finite OLG horizons (resale price risk at t + 1). The Shleifer–Vishny limits of arbitrage model (1997) shows that even if specialized arbitrageurs are risk-neutral (γ = 0), agency separation between fund managers and outside investors creates performance-based capital redemptions when noise trader sentiment worsens at t = 2.

1. De Long, J. B., Shleifer, A., Summers, L. H., & Waldmann, R. J. (1990). "Noise Trader Risk in Financial Markets." Journal of Political Economy, 98(4), 703–738. https://doi.org/10.1086/261703
2. Black, F. (1986). "Noise." The Journal of Finance, 41(3), 528–543. https://doi.org/10.1111/j.1540-6261.1986.tb04513.x
3. Lee, C. M. C., Shleifer, A., & Thaler, R. H. (1991). "Investor Sentiment and the Closed-End Fund Puzzle." The Journal of Finance, 46(1), 75–109. https://doi.org/10.1111/j.1540-6261.1991.tb03746.x
4. Shleifer, A., & Vishny, R. W. (1997). "The Limits of Arbitrage." The Journal of Finance, 52(1), 35–55. https://doi.org/10.1111/j.1540-6261.1997.tb03807.x
5. Campbell, J. Y., & Kyle, A. S. (1993). "Smart Money, Noise Trading and Stock Price Behaviour." Review of Economic Studies, 60(1), 1–34. https://doi.org/10.2307/2297810