Noisy prices and inference regarding returns

Time series and cross-sectional average returns are computed for a wide variety of purposes in investment and financial analysis. Further, it is common to estimate conditional (on the outcomes of various explanatory variables) mean returns by implementing regression analysis. The computation and comparison of mean rates of return for securities and portfolios may well be the single most common empirical method used in the field of Finance.

In our paper “Noisy Prices and Inference Regarding Returns” we show that these simple calculations can give misleading results if prices contain “noise,” by which we mean temporary deviations of observed prices from underlying security values. Noise can arise from microstructure frictions, such as the temporary price impact of large orders. Noise can also enter prices due to traders' behavioral biases, if arbitrage is imperfect.

In particular, the simple mean return based on noisy prices provides an upward biased estimate of the rate at which the true values and observed prices trend upward (e.g., Blume-Stambaugh (JFE 1983)). A simple example illustrates the issue. Suppose that fundamental value is constant at $10, implying that the mean true return (i.e., that computed from fundamental values) is zero. Due to market imperfections, trades occur at noisy prices of $9 or $11, equally likely. The possible returns computed from trade prices are 22.22% (if the price moves from $9 to $11), –18.18% (if the price moves from $11 to $9), or zero (if the price moves remains at $9 or $11). In a large sample, the mean observed return will be 1.35%, even though value and prices are not trending upward at all! If this example were repeated with more dispersion attributable to noise in prices, the mean observed return would be greater, and vice versa. (Our observations regarding the effect of noisy prices are related to, but distinct from, the well-known relation between arithmetic and geometric mean returns. The arithmetic mean return exceeds the geometric mean when there is any variability in returns. The preceding statement holds for both true and observed returns. In contrast, the bias between the (arithmetic) mean observed and true returns arises only if prices contain temporary deviations from value.)

Importantly, the bias in returns is always positive when prices contain noise (even though the noise itself is zero mean), implying that the bias is not diversified in portfolios. (Simple means of observed returns do have an economic interpretation. In particular, they give the outcome to a hypothetical active trading strategy that involves selling securities that have appreciated and purchasing securities that have depreciated, so as to reestablish equal weights, while assuming that the transactions can be completed at observed prices. However, because only a subset of traders can engage in any active trading strategy, the returns to this hypothetical rebalancing strategy are generally not the same as (and if prices contain noise on average exceed) the returns to investors in aggregate.) If securities are sorted into portfolios based on an attribute correlated with the amount of noise (e.g. firm size, illiquidity, or volatility) then the mean return to the portfolio containing noisier securities is upward biased by a greater amount. In an earlier paper (Asparouhova-Bessembinder-Kalcheva (JFE 2010)), we showed that the problem of noisy prices pertains not just to average return computations-it also leads to biases in coefficients estimated in any ordinary least squares (OLS) regression with returns as the dependent variable.

We consider two key issues. First, we assess three methods to correct for the bias, each of which amounts to computing weighted average returns. The methods considered are (i) weight each return by prior-period market capitalization (value weight, or VW), (ii) weight each return by the accumulated gross return from a specified formation period until the end of the prior period (initial-equal-weight, or IEW), or (iii) weight each return by the gross return in the prior period (return weight, RW). We show by theory and simulation that each is effective in removing the biases attributable to noisy prices, under reasonable assumptions. (While all three methods are effective in removing bias, their interpretation differs, because value-weighted returns place more emphasis on large firms, which can dominate a sample. A researchers preference for VW vs. RW or IEW corrections may depend on the relative importance of the information contained in large vs. small sample securities. Also, because it is common to form portfolios on an annual basis, we consider the effect of weighting by firm value measured as of the prior December. We show that the resulting mean returns remain upward biased.) Each is effective because it weights observed returns by a variable proportional to the prior-period observed price. If this price contains positive noise then the weight increases exactly as the observed return for the current period decreases (and vice versa when the noise is negative), resulting in weights and observed returns counteracting each other to offset the original upward bias in returns.

Size | Dec 10 | Dec 1 | Dec 10–1 | (Std.Err.) |

EW | 0.462 | 1.888 | –1.425^{***} | (0.32) |

RW | 0.448 | 1.410 | –0.961^{***} | (0.31) |

IEW | 0.450 | 1.194 | –0.743^{***} | (0.31) |

VW | 0.371 | 0.888 | –0.517^{**} | (0.30) |

Book-To-Market | Dec 10 | Dec 1 | Dec 10–1 | (Std.Err.) |

EW | 1.517 | 0.148 | 1.369^{***} | (0.23) |

RW | 1.301 | 0.024 | 1.277^{***} | (0.22) |

IEW | 1.331 | 0.033 | 1.298^{***} | (0.22) |

VW | 1.031 | 0.214 | 0.816^{***} | (0.26) |

Inverse Price | Dec 10 | Dec 1 | Dec 10–1 | (Std.Err.) |

EW | 1.832 | 0.579 | 1.252^{***} | (0.38) |

RW | 1.214 | 0.569 | 0.645 | (0.37) |

IEW | 1.035 | 0.583 | 0.452 | (0.36) |

VW | 0.570 | 0.405 | 0.164 | (0.41) |

Volume | Dec 10 | Dec 1 | Dec 10–1 | (Std.Err.) |

EW | 0.407 | 1.605 | –1.198^{***} | (0.27) |

RW | 0.396 | 1.243 | –0.846^{***} | (0.26) |

IEW | 0.414 | 1.098 | –0.684^{***} | (0.25) |

VW | 0.354 | 0.713 | –0.359^{**} | (0.21) |

Illiquidity | Dec 10 | Dec 1 | Dec 10–1 | (Std.Err.) |

EW | 1.580 | 0.441 | 1.139^{***} | (0.29) |

RW | 1.211 | 0.430 | 0.780^{***} | (0.28) |

IEW | 1.108 | 0.433 | 0.675^{**} | (0.28) |

VW | 0.769 | 0.356 | 0.413 | (0.29) |

Standard errors are in parentheses. One, two, and three stars mark statistical significance at the 10%, 5%, and 1% level, respectively.

The table reports time-series means of monthly returns to the extreme of the 10 attribute-sorted portfolios and to the corresponding hedge portfolio. Portfolio returns for month t are measured on an equal-weighted (EW), return-weighted (RW, weight is period t – 1 gross return), equal-initial-weighted (IEW, weight is cumulative gross return from portfolio formation through month t – 1), and prior-month-value-weighted (VW, weight is month t – 1 market capitalization). Firms are assigned to portfolios based on attributes measured in July.

Second, we assess how important the biases attributable to noisy prices are in real world data. Table 1 shows mean monthly returns to portfolios of U.S. equities from 1966 to 2009. We report the upward biased equally-weighted (EW) returns, as well as returns corrected for bias by use of the RW, IEW, and VW methods. The portfolios are created by sorting on firm characteristics, including firm size, book-to-market ratio, (inverse) share price, trading volume, and stock illiquidity. For each attribute, we sort stocks into ten portfolios, and report returns to the extreme (first and tenth) portfolios, as well as to the “hedge portfolio” that is long the tenth and short the first decile portfolios. Hedge portfolio returns are often interpreted as the mean return premium associated with investing in stocks with greater levels of the sorting characteristic.

The key observation to be gleaned from Table 1 is that the absolute value of the estimated return premium associated with each of these characteristics is larger when focusing on the (biased) equal-weighted (EW) returns than when considering the corrected estimates obtained by any of the IEW, RW, or VW methods. The bias is quite relevant for firm size, share price, trading volume, and illiquidity, but less so for the book-to-market ratio. Focusing in particular on the differential between EW and RW mean returns, noise in prices explains about one third of the apparent size effect in monthly returns, as the corrected size premium is –0.96% per month, as compared to the uncorrected estimate of –1.43% per month.

Results regarding the relation between mean returns and (inverse) share price are particularly striking, as they illustrate that inference regarding the existence, and not just the magnitude, of a return premium can be altered by bias attributable to noisy prices. Here, the estimated return premium is 1.25% per month based on EW returns, versus 0.65% per month (RW), 0.45% (IEW), or 0.16% (VW). None of the corrected estimates of the return premium associated with share price is statistically significant. That is, the biased (EW) estimate implies the existence of a return premium associated with share price, but none of the bias-corrected estimates do.

The importance of the bias in mean returns depends on how noisy the prices are. Some researchers exclude low-priced securities (e.g. those trading below $5) from their studies, reasoning that these will be most affected by noise. We show that excluding low priced securities is indeed quite effective in reducing the bias. However, statistically significant bias persists. More importantly, excluding noisy securities from a study reduces statistical power and discards important information regarding the magnitude and form (e.g. linear in firm characteristics) of return premia.

In summary, our study considers five specific firm characteristics, using monthly return data for U.S. equities since 1966, and we document significant biases in return premia estimates obtained by comparing EW returns across attribute-sorted portfolios, as well as estimates obtained by cross-sectional OLS “Fama-MacBeth” regressions. We anticipate that many of the return premium estimates that have been reported in the literature as being associated with additional firm characteristics and/or with factor exposures are also biased. We caution that biases are likely to be greater in some other applications, e.g. in studies of mean returns to corporate bonds or international equities, if prices in those markets contain more noise. Further, biases of the type we study are likely to be relatively more important in daily or higher-frequency returns, because true returns are smaller at shorter horizons, while the biases attributable to noise are not.

Hieronymus Bosch (1450–1516): The Garden of Earthly Delights. Flemish Renaissance, 1505.. Bosch probably meant for this complex painting as a caution against hazards of sinful delights. What Bosch could not imagine was that a music student from Oklahoma would transcribe music she found in this work, “written upon the posterior [butt] of one of the many tortured denizens of the rightmost panel of the painting” into modern notation. She calls it the “600 year old butt song from hell.”

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