Journal of Financial Economics 12 (1983) 387-404. North-Holland BIASES IN C O M P U T E D RETURNS An Application to the Size Effect* Marshall E. BLUME University of Pennsylvania, Philadelphia, PA 19104, USA Robert F. STAMBAUGH University of Pennsylvania, Philadelphia, PA 19104, USA Received February 1983, final version received August 1983 Previous estimates of a 'size effect' based on daily returns data are biased. The use of quoted closing prices in computing returns on individual stocks imparts an upward bias. Returns computed for buy-and-hold portfolios largely avoid the bias induced by closing prices. Based on such buy-and-hold returns, the full-year size effect is half as large as previously reported, and all of the full-year effect is, on average, due to the month of January. 1. Introduction Recent empirical work in finance reports that average risk-adjusted returns on stocks of small firms exceed those of large firms, where size is measured by the market value of outstanding common equity. 1 Using daily returns for stocks on both the New York and American Stock Exchanges, Reinganum (1982) finds that, during the 1964-1978 period, the average return for firms in the lowest market-value decile exceeds the average return for firms in the highest decile by more than 0.1 percent per day - - over 30 percent per year. He also finds that various methods of risk adjustment contribute little towards explaining such impressive differences. 2 Keim (1983) reports that almost half of the annual difference between returns on small and large firms occurs in January. The 'size effect' is particularly pronounced in the studies that use daily returns data, but we show that, due to a statistical bias, these studies significantly overstate the magnitude of the size effect. Although we *We are grateful to Edwin Elton, Donald Keim, Jay Ritter, Hans Stoll, participants in workshops at New York University and Yale University, and the referee, Allan Kleidon, for comments and suggestions. The research assistance of Tzivia Kandel is gratefully acknowledged. 1See Blume and Friend (1974) and Banz (1981) for evidence on the size effect in addition to that discussed in the text. 2See also Reinganum (1981a, b). 0304~05x/83/$3.00 �� 1983, Elsevier Science Publishers B.V. (North-Holland)

388 M.E. Blume and R.F. Stambaugh, Return computation and the size ~'[l~'et empirically analyze the bias in the context of the size effect, the same bias could potentially occur in any study using closing prices to compute returns, particularly daily returns. Using daily returns for NYSE and AMEX stocks, we find that (1) the average size effect over the entire year is about 0.05 percent per day only half as large as reported by Reinganum and Keim - - and (2) virtually all of this full-year average is attributable to January. In other words, the size effect averages about 0.60 percent per day in January and roughly zero in the remainder of the year. The sample contains all firms listed on the New York and American Stock Exchanges, and the time period covers 1963 through 1980. Thus, our study uses essentially the same data as prior studies. The difference in results arises from the method used to compute average returns. Section 2 shows that single-period returns on individual stocks computed with recorded closing prices are upward biased. This bias arises from a 'bid ask' effect in closing prices and can be non-trivial for daily returns on stocks of small firms. Reinganum and Keim use arithmetic averages of daily returns to estimate the size effect. Since the arithmetic average of computed returns contains the average bias for the individual stocks, their estimates of the size effect are upward biased. The portfolio strategy implicit in arithmetic averaging is one of daily rebalancing to equal weights. This paper shows that the returns on an alternative buy-and-hold strategy are virtually unbiased. Buy-and-hold portfolios contain a "diversification" effect, not present in rebalanced portfolios. This 'diversification" effect removes virtually all bias from the computed returns on a buy-and-hold portfolio. Section 3 presents empirical results for both rebalanced portfolios and buy-and-hold portfolios. The differences between returns on the two strategies are negligible for large-firm portfolios. In contrast, for the portfolio of the lowest-market-value firms, the rebalanced return exceeds the buy-and- hold return by an average of 0.05 percent per day, which is approximately half of the average size effect reported in previous studies. The analysis in section 4 finds that the difference between the rebalanced and buy-and-hold returns varies inversely with share price, holding market wdue constant, and bears no significant relation to market value, holding share price constant. This finding is consistent with the analysis of the bias presented in section 2. 2. Computing returns with closing prices 2.1. A model o f closing prices Define the true price at time t of stock i as Pi,,, the price at which, aside from transactions costs, a share of stock can be both bought and sold at a given time by placing a market order. On the Exchange, the true price can

M.E. Blume and R.F. Stambaugh, Return computation and the size q~]ect 389 be viewed as the price at which (nearly) simultaneous public market buy and sell orders would 'cross' on the floor. The Center for Research in Security Prices (CRSP) provides daily returns for stocks listed on the New York and American Stock Exchanges, and these returns are computed with 'closing' prices. The closing price is the price at which the last transaction occurred prior to the close of trading. 3 Let P~,, denote the reported closing price of stock i for the period ending at time t. The closing price, P~.3, can deviate from the true price, Pz,,, if, for example, the last transaction reflects a public market order on only one side. For example, a market sell order might be matched with a limit buy order or bought by the specialist on his own account. Denote the price recorded for such transactions as a 'bid' price, and note that such a price is most likely less than the true price. Similarly, a market buy order that is not crossed on the floor results in the recording of an 'ask' price, probably greater than the true price. We refer to this property of closing prices as the bid-ask effect. The bid-ask effect is modeled as or Pi,, = [ 1 +(~i,t]Pi,,, (1) Pi,t = Pi,t-4-si,t, (1') where E{6i,,}=0, 61,~ is independently distributed across t, and 6i,t is independent of Pi,�� for all r. At some points in the discussion, it will be convenient to use (1'), which is restated with an additive disturbance ei,t. It is well-known that the bid-ask effect produces negative first-order autocovariance in recorded price changes for individual stocks. 4 We show here that the bid-ask effect also imparts an upward bias to computed rates of return for individual stocks. 5 We analyze single-period returns primarily because most empirical studies employ single-period returns, often averaged either cross-sectionally or over time. 2.2. The bias in computed returns The true return for security i for period t is defined, assuming no dividends 3If there are no trades in a day, CRSP uses as the quoted closing price the average of the bid and ask prices. To the extent that the bid and ask prices are kept up to date, this practice of CRSP could help reduce the differences between true and quoted prices. 4Niederhoffer and Osborne (1966) explain how the bid-ask effect leads to 'reversals', or negative autocorrelation in price changes. SOur work is not without precedent, however. Although he does not consider the bid ask effect, Fisher (1966) discusses how deviations of closing prices from 'true' prices can bias computed returns.