Noise Trader Risk in Financial Markets

The paper identifies a source of risk to short-horizon arbitrageurs distinct from fundamental (dividend) risk: “the risk that noise traders’ beliefs will not revert to their mean for a long time and might in the meantime become even more extreme.” Earlier work (Figlewski 1979; Shiller 1984; Campbell and Kyle 1987) had shown that arbitrageurs’ aversion to fundamental risk alone can severely limit arbitrage; this paper’s contribution is to show that even when there is no fundamental risk whatsoever, the mere unpredictability of noise traders’ own future opinions creates a risk in the asset’s resale price that deters arbitrage – so that, in the authors’ phrase, “noise traders thus create their own space,” and “all the main results of our paper come from the observation that arbitrage does not eliminate the effects of noise because noise itself creates risk.”

The model is a stripped-down two-period-lived overlapping-generations economy (following Samuelson 1958) with a safe asset s (fixed price of one, paying riskless dividend r) and a fixed-supply risky asset u that pays the identical dividend r but whose price, in the presence of noise traders, need not equal one. Noise traders (present in measure μ) misperceive the asset’s expected price by an i.i.d. normally distributed amount ρ_t ~ N(ρ*, σ²); sophisticated investors (measure 1-μ) perceive returns correctly. Because the price at which this period’s young will be able to sell the asset next period depends on next period’s noise traders’ as-yet-unknown beliefs, both types of agents face genuine price risk even with zero fundamental risk – which is why, the authors note, “if the price next period were certain…investors would therefore try to take infinite bets against each other [and] an equilibrium would not exist.”

Solving the model yields a pricing equation for the risky asset with three variable components beyond its fundamental value of one: a variance term reflecting the volatility of noise traders’ shifting misperceptions, a “price pressure” term reflecting the average level of noise trader bullishness (ρ), and a risk term, “the heart of the model,” reflecting the compensation sophisticated investors require for bearing the risk that noise traders will become more bearish before they can sell.* The price converges to its fundamental value of one only as the distribution of noise traders’ misperceptions collapses to a point mass at zero; otherwise, “even though asset u is not subject to any fundamental uncertainty and is so known by a large class of investors, its price varies substantially as noise traders’ opinions shift,” and more numerous noise traders relative to sophisticated investors produce more volatile prices.

Noise traders earn higher expected total returns only if they are, on average, moderately bullish (ρ positive but not too large) and the risky asset is priced below its fundamental value.* Two effects raise noise traders’ relative expected returns: the “hold more” effect (noise traders hold more of the risky asset on average and so capture more of the reward to risk-bearing) and the “create space” effect (greater variability of noise traders’ beliefs raises the price risk that sophisticated investors must bear to exploit mispricing, limiting how hard they bet against noise traders). Two effects lower noise traders’ relative returns: the “price pressure” effect (bullish noise traders bid up the price they themselves pay, reducing the return to risk-bearing) and the “Friedman” or buy-high-sell-low effect (noise traders’ stochastic beliefs give them, on average, the worst possible market timing). Noise traders cannot earn higher average returns if they are on average bearish (no hold-more effect) or if they are too bullish (the price-pressure effect, which rises with the square of ρ, dominates); the larger investors’ risk aversion, the wider the intermediate range of bullishness over which noise traders out-earn sophisticated investors.*

No: noise traders’ higher expected returns come from holding riskier portfolios and so correspond to lower expected utility computed under the true probability distribution, while sophisticated investors are “necessarily better off” in the presence of noise traders than without them, since noise trading simply expands their opportunity set. This undercuts Friedman’s (1953) argument that “destabilizing speculation is…largely equivalent to saying that speculators lose money”: the paper shows a trading strategy can earn a higher mean return without being more rational, as long as it carries more risk, so “it is not possible to make any blanket statement that noise traders lose money and eventually become unimportant” through market selection on average wealth, though the authors caution that a selection criterion based on the probability of ending up poor, rather than on expected wealth, might still work against noise traders – a point they say they could not tractably formalize.

Because none of the price variance in the model is justified by changes in fundamentals, the price of the risky asset is “excessively volatile” in the sense that “it moves more than can be explained on the basis of changes in fundamental values,” and if noise traders’ misperceptions follow a mean-reverting (AR(1)) process, asset returns inherit a mean-reverting, negatively-autocorrelated component consistent with the evidence in Fama and French (1988b) and Poterba and Summers (1988). The paper shows that unconditional price variance can grow rapidly as misperceptions become more persistent, so that a high unconditional variance of prices can coexist with only a small exploitable opportunity for arbitrage, since sophisticated investors facing serially correlated misperceptions won’t bet heavily even if they know the underlying process. The authors extend the logic to the term structure: treating the short bond as asset s and the long bond as asset u, the same mean-reverting mispricing explains the well-documented anomaly that when long rates exceed short rates, long rates tend to fall rather than rise as the (noise-free) expectations hypothesis predicts.

Because a closed-end fund’s fundamental value – the market price of the stocks it holds – is directly observable, the authors treat its persistent, fluctuating discount relative to net asset value (Malkiel 1977; Herzfeld 1980) as a cleaner test of mispricing than ordinary stock-price “excess volatility,” which requires an unobservable fundamental benchmark. Mapping the underlying stocks to the safe asset s and the fund itself to the risky asset u, noise trader risk in the fund’s own price produces a discount even if noise traders are not on average bearish about the fund (ρ* = 0) – the discount “arises solely because holding the fund entails additional noise trader risk.” The authors argue this fits evidence that the leading rival explanations – agency costs of fund management, and failure to account for unrealized capital-gains tax liability – cannot explain, respectively, why funds are rationally formed at all, or why fund prices rise on announcements of open-ending (which should, if anything, accelerate the tax liability).

Mehra and Prescott (1985) showed that the roughly 8 percent average realized U.S. equity return versus near-zero realized real return on safe bonds is hard to reconcile with the standard representative-consumer model without implausibly high risk aversion; the paper’s model implies that whenever noise traders earn higher expected returns than sophisticated investors, the risky asset must on average be priced below its fundamental value, generating exactly such an equity premium. The mechanism works through noise traders’ consumption, which does not satisfy the standard Euler equation with respect to true expected returns, making aggregate consumption less sensitive to the variation in true expected equity returns than the representative-consumer model assumes – so a large equity premium can coexist with a low covariance between equity returns and aggregate consumption. The authors are explicit that this logic runs in only one direction as evidence: “the fact that the Mehra-Prescott equity premium obtains in an economy is evidence for the proposition that the expected returns of noise traders are likely to be higher than those of sophisticated investors” – the model sheds light on the puzzle only to the extent that equities are in fact underpriced, which is itself the condition the model needs for noise traders to out-earn sophisticated investors in the first place.

The paper breaks the standard recursive argument that trading and long-term investing must price assets identically regardless of horizon, since an infinitely-lived, risk-averse sophisticated investor could always short the safe asset and buy the risky one to capture a riskless 1 - p_t gain with no liquidity date ever forcing a resale at an uncertain price – so with infinitely-lived agents, noise trader risk could not distort prices at all. With finite but longer-than-two-period horizons, investors’ asset demands become more price-responsive for two reasons: they collect several periods of dividend “insurance” before having to sell, and they have multiple chances to liquidate whenever the price happens to revert toward the mean, which “reduces his risk.” The authors connect the overlapping-generations device explicitly to real institutional features – such as triennial performance evaluations of pension-fund managers – that can give even fully rational agents effectively short horizons, and conclude that “the claim that short horizons are bad for the economy is both coherent and true in our model.”

The authors note that in a model with noise traders, the Modigliani-Miller irrelevance theorem “does not necessarily apply”: its standard homemade-leverage proof shows a rational investor can undo the effects of a firm’s capital structure, but “it does not suggest that less than rational traders will do so,” so unless noise traders happen to trade in a way that exactly offsets a leverage change, capital structure can affect value. If noise traders are more likely to become confused about long-duration, fundamentally risky payouts than about certain, immediate cash flows, a firm can reduce the noise trader risk borne by its securities – and so raise their price – by paying dividends, swapping debt for equity, or repurchasing shares, each of which makes the firm’s capital structure “look” more like the safe asset to noise traders; the authors tie this to Jensen’s (1986) evidence that constraining a firm’s free cash flow (debt-for-equity swaps, dividend increases, buybacks) raises market valuation, while dividend cuts or new share issues lower it.