Stock weighting and nontrading bias in estimated portfolio returns

2.1. Base case

The central issues in this study can be easily conveyed with two simple numerical examples. Assume that, at time 0, stock A and stock B are combined into an equally weighted portfolio, which is held without rebalancing until the end of month 4 (i.e. it is a buy-and-hold portfolio). For ease of exposition, and without loss of generality, assume that neither stock experiences a dividend or capitalisation adjustment over the holding period.

Table 1 shows the monthly prices and returns for this base case.3 Liu and Strong (2008) document that it is common practice in empirical research to simply average the monthly returns on the stocks in the portfolio. For example, by averaging month-1 returns to stock A (25 per cent) and stock B (−4 per cent), the portfolio return is 10.50 per cent. Following the same approach in each month gives the time series of portfolio returns. Taking the product of the portfolio price relatives, the estimated 4-month holding-period return is shown as 3.06 per cent.4

In reality, the true holding-period return on the portfolio is zero. Assume an initial budget of $100. Formation of an equally weighted portfolio requires the purchase of 25 shares in stock A and 10 shares of stock B (hence, a $50 investment in each stock). After 1 month, this portfolio is worth $110.50 (25 shares in A worth $2.50 and 10 shares in B worth $4.80). Consequently, the true portfolio return in month 1 is 10.50 per cent ($110.50/$100 −1). While this month-1 portfolio return is identical to the return calculated above by averaging the returns on each stock, this is where the similarities end. At the end of month 2, the portfolio is worth $109, so the portfolio return is −1.36 per cent ($109/$110.50 −1). In contrast, the averaging approach erroneously suggests that the portfolio has a positive return in month 2 (0.25 per cent). Similarly, the averaging approach generates portfolio returns in months 3 and 4 that differ from the true returns.

By tracking dollar investments in each stock, Table 1 clearly demonstrates that the averaging approach generates an incorrect time series of monthly portfolio returns and therefore a spurious holding-period return of 3.06 per cent.5 Clearly, the portfolio value is $100 at both month 0 and month 4; hence, the return over the holding period is zero. An examination of the dollar investments columns in Table 1 shows why the two approaches differ. At time 0, the portfolio is indeed equally weighted with $50 invested in each stock (this is why the averaging and dollar approaches agree in the first month). However, the month-1 stock returns tilt the portfolio away from equal weights. At the end of month 1, stock A contributes 56.56 per cent ($62.50/$110.50) and stock B contributes 43.44 per cent ($48/$110.50). These are the correct (updated) weights for calculating the month-2 portfolio return. Similarly, the correct month-3 weights are the initial time-0 weights, updated to reflect each stock's cumulative returns over the first 2 months.

Analytically, Liu and Strong 2008 Eqn 3) depict the calculation of the month-t portfolio return using correctly updated weights as follows:

(1)

where is the initial portfolio weight for stock i, is the month-t simple return on stock i, and there are N stocks in the portfolio. With reference to Table 1, the numerator of (1) measures the ‘dollar investment’ in each stock at the beginning of month t, while the denominator captures the ‘portfolio value’. The key take away from (1) is that the correct weights to apply in calculating month-t portfolio return are a time-varying function of each stock's cumulative returns from the commencement of the holding period up to the beginning of month t. Hereinafter, we refer to this as the updated weights approach.

In contrast, the averaging approach uses the same initial vector of stock weights to calculate portfolio return in every holding-period month. By averaging the monthly stock returns, the implicit assumption is that the portfolio is rebalanced to the initial equal weights each month (from this point on, we refer to this as constant weights approach). This assumption is therefore inconsistent with the buy-and-hold position that is intended in many empirical applications, and clearly, transaction costs would prohibit monthly rebalancing of portfolios containing a large number of stocks.6

While these are important issues, they are not new. Blume (1974), Roll (1983), Blume and Stambaugh (1983), Conrad and Kaul (1993) and Dissanaike (1994) have all highlighted potential methodological traps in averaging stock returns into portfolios and estimating multiperiod returns. We revisit this issue in Section 2.3. and propose a number of possible reasons why researchers continue to utilise constant-initial weights to calculate portfolio returns.

One final comment pertains to the implementation of the correctly updated weights approach to estimating portfolio returns. While Equation 1 serves to illustrate why and how the initial stock weights must be updated to reflect returns on a genuine buy-and-hold portfolio, it is notationally cumbersome and arguably lacking in intuition. A simple and more intuitive exposition of the calculation of portfolio returns follows from utilising the dollar value of the portfolio along the lines of the far right columns in Table 2.7 The Appendix provides details of this approach and shows that it is equivalent to Liu and Strong's (2008) approach.

2.2. Nontrading and missing returns

The base case illustrates the main issue: rather than averaging the monthly returns of constituent stocks, true portfolio returns follow from updating initial weights to reflect each stock's cumulative return since the beginning of the holding period. While calculating correctly updated weights was straightforward in the stylised example, complications arise when constituent stocks have missing returns. Even in highly developed markets like the UK and Australia, the incidence of nontrading is nontrivial. It is instructive, therefore, to carefully consider the alternative treatments of missing returns available to researchers. This allows a better understanding of why researchers adopt certain empirical methodologies and an assessment of the likely influence of the chosen approach on estimated returns.

Table 2 extends the example by considering an equally weighted portfolio of three stocks. The data for stocks A and B are identical to the previous example. Stock C does not trade in month 2; hence, no return can be calculated in months 2 and 3.8 Under the constant weights approach, a convenient ‘solution’ to missing returns is to simply calculate the portfolio return as the average over stocks with a valid return. While this is trivial for equal-weighted portfolios, it is unclear how to proceed for value-weighted portfolios.9 An alternative treatment of missing returns is to ‘fill the gaps’ by inserting a price relative of unity (i.e. zero return). This approach allows easy calculation of the monthly portfolio return under the constant weights approach, irrespective of whether stocks are equal or value weighted. In Table 2, the portfolio returns in month 2 (0.17 per cent) and month 3 (1.18 per cent) are calculated by replacing the missing price relatives with unity, then taking the average across the three stocks. Having filled the gaps created by missing returns, the estimated 4-month holding-period return on the portfolio is −6.06 per cent.

Under the updated weights approach, the first point to note is that the true 4-month portfolio holding-period return is again zero (by construction). If each constituent stock trades at the beginning and end of the holding period (as is the case here), calculating the portfolio dollar investments and holding-period return is straight forward. On a month-to-month basis, difficulties arise when stocks do not trade. For example, stock C's nontrading in month 2 makes it difficult to calculate the dollar investment in stock C, and therefore the month-2 portfolio return. As was the case with the constant weights approach, where two ad hoc fixes were considered, the updated weights approach also requires some creative thinking.

Our suggested way forward follows from considering the perspective of the investor. An investor asked what her portfolio is worth at the end of month 2 might reply: my investments in stocks A and B are currently worth $36.67 and $36, respectively, and although stock C did not trade this month, the last time it did trade, it was worth $16.67. Arguably, the investor's best guess as to the portfolio worth is $89.33. Accordingly, the month-2 portfolio return can be estimated (−1.11 per cent).10 Stock C trades in month 3, therefore allowing an update to it's true dollar investment ($26.67).

The suggested approach to handling missing returns under the updated weights method is ad hoc and clearly imperfect. In month 2, it uses current prices for stocks A and B and a stale price for stock C (the alternate treatment discussed above has the same implication). Further, the variance of portfolio returns may be overstated, because zero returns in nontrading months may be followed by a potentially large return when the stock next trades. For example, in Table 2, Stock C records a zero return in month 2 followed by a 60 per cent return in month 3. As such, the approach we advocate to handle missing returns has implications for the estimated portfolio variance. Of course, so too does the alternate treatment of turning a blind eye to missing returns and simply inserting zeros to fill the gaps.11 Nonetheless, the suggested approach has merit. In particular, it generates portfolio returns that cumulate to the correct holding-period return (0 per cent in this case). There is no look-ahead bias in the suggested ‘fix’.12 At the very least, it is no more ad hoc than the constant weights approaches that either ignore missing returns or insert a price relative of unity.

Finally, two important issues must be noted. First, while inserting price relatives of unity to fill gaps under the constant weights approach may be computationally convenient, doing so masks a potentially serious drawback. Specifically, the fact that Stock C moved from $5 in month 1 to $8 by the end of month 3 is totally missed. This ignorance drives the erroneous 4-month holding-period return (−6.06 per cent). In contrast, the updated weights approach captures this capital gain in month 3, thereby ensuring that the 4-month holding-period portfolio return is accurate.

Second, care must be taken with the treatment of missing returns when applying the analytic formula (1) of Liu and Strong (2008). If the researcher simply inserts price relatives of unity for Stock C in months 2 and 3, the resulting time series of monthly portfolio returns cumulate to an (incorrect) 4-month holding-period return of −12.50 per cent (not explicitly shown in Table 2). Instead, a price relative of unity must be inserted in month 2 (the nontrading month). Then, to capture the capital gain over the nontrading period, the month-3 price relative must be 1.60 ($8/$5).13 Specifically, in the presence of nontrading, Equation 1 must be augmented with (2) to capture capital gains/losses over nontrading periods:

(2)

2.3. Discussion

The stylised examples demonstrate several important flaws in the approach commonly used to estimate portfolio returns. As highlighted by Liu and Strong (2008), the constant weights approach implies that stock holdings are rebalanced to initial weights each month during the holding period. In addition to being practically infeasible, the implication is inconsistent with the buy-and-hold perspective intended in many empirical studies. As a consequence, monthly portfolio returns are mis-estimated and holding-period returns do not reflect the wealth effects to a buy-and-hold investor.

Further, the stylised examples extend the analysis of Liu and Strong (2008) to highlight the potentially important impact of nontrading. In the presence of nontrading, an ad hoc assumption is necessary to estimate monthly portfolio returns. Assumptions that are commonly employed in conjunction with the constant weights approach (e.g. replacing missing price relatives with unity) induce further error into the estimation of returns. The susceptibility of characteristic-sorted portfolios to these nontrading problems is a feature of the empirical analysis in the remainder of this study.

One caveat is necessary. Sections 2.1. and 2.2. demonstrate biases in estimated portfolio returns that are attributable to the method of averaging stock returns and the treatment of nontrading, respectively. Clearly, the stylised examples use somewhat extreme numbers for the purpose of illustrating the key points. Ultimately, the magnitude and direction of any bias is an empirical issue. The major contribution of this study is to empirically document the importance of each component of the bias in common research settings and demonstrate how to diagnose potential methodological problems with reference to stock characteristics.

Given the numerous flaws in the constant weights approach, we can only conjecture why its use remains prevalent. One possible explanation is careless adherence to the textbook definition that the time-t return on a portfolio of N stocks is . This formula is correct, if and only if the vector of weights are the correctly updated weights at the beginning of period t as shown in (1). An alternate motivation is programming simplicity – updating stock weights each month as per (1) and (2) is considerably more difficult than simply inserting unity and averaging.

3.1. Size effects

The size effect is one of the most-heavily researched anomalies. Brown et al. (1983), Beedles et al. (1988), Gaunt et al. (2000), Durand et al. (2007), O'Brien (2008) and Gharghori et al. (2009) have all studied the size effect in Australian equities. At face value, the weight of evidence suggests that there is a distinct small firm premium. Notably, each of these studies reports incredulously high average returns to the smallest size portfolio (ranging from 5 to 10 per cent per month).

In a recent study, Gaunt (2011) explores the legitimacy of such large apparent returns to small-stock portfolios by adopting a methodology that strives to estimate the returns that could realistically be achieved. To mitigate market-microstructure return bias and illiquidity concerns, his portfolio-formation procedure excludes stocks with price below 20 cents. In brief, Gaunt (2011) finds no evidence of a size premium given these methodological choices. The disturbingly high average returns to small stocks reported in prior work are absent, and there is no statistically significant difference in average returns across size deciles.

Our study of the size effect proceeds as follows. Starting in December 1973, all stocks are ranked by market capitalisation and sorted into decile portfolios. Buy-and-hold positions in the constituent stocks are maintained for 12 months, after which the ranking and portfolio-formation procedure is repeated. Over the 37-year sample period (444 months), 970 stocks on average per annum are assigned to decile size portfolios.

Table 3 reports the empirical findings. When returns to size portfolios are estimated under the constant weights approach, there is evidence of a substantial size effect. Irrespective of whether stocks are equal-weighted (Panel A) or value-weighted (Panel B) into portfolios, the magnitude of the size premium (as measured by the spread portfolio) exceeds 4 per cent per month and is highly statistically significant.15 These findings under the constant weights approach are highly consistent with prior research. However, a material proportion of the apparent size premium reflects bias induced by this methodology. Table 3 quantifies the total bias in portfolio returns and facilitates its decomposition into stock weighting and nontrading components.

Table 3. Average monthly returns to size portfolios

	Small	2	3	4	5	6	7	8	9	Big	Spread
Panel A: Equally weighted portfolios
Constant weights	5.72	3.35	2.32	1.56	1.38	0.88	1.05	0.96	1.07	1.14	4.58	(p < 0.001)
Updated weights	4.34	2.48	1.59	1.26	1.30	0.94	1.12	1.16	1.20	1.20	3.14	(p < 0.001)
Total bias	1.38	0.87	0.73	0.30	0.08	−0.06	−0.07	−0.20	−0.13	−0.06	1.44
Weighting bias	1.54	0.87	0.65	0.29	0.05	−0.09	−0.09	−0.20	−0.15	−0.07	1.61
Nontrading bias	−0.16	0.00	0.08	0.01	0.03	0.03	0.02	0.00	0.02	0.01	−0.17
Panel B: Value-weighted portfolios
Constant weights	5.11	3.21	2.21	1.48	1.32	0.85	0.99	0.95	1.06	1.09	4.02	(p < 0.001)
Updated weights	3.90	2.42	1.54	1.24	1.27	0.93	1.11	1.16	1.19	1.13	2.77	(p < 0.001)
Total bias	1.21	0.79	0.67	0.24	0.05	−0.08	−0.12	−0.21	−0.13	−0.04	1.25
Weighting bias	1.31	0.78	0.59	0.23	0.01	−0.11	−0.14	−0.21	−0.15	−0.04	1.35
Nontrading bias	−0.10	0.01	0.08	0.01	0.04	0.03	0.02	0.00	0.02	0.00	−0.10
Panel C: Summary statistics
Mkt Cap ($m): mean	1	3	5	7	11	19	33	67	181	2600
Mkt Cap ($m): median	1	3	5	7	11	18	32	65	164	1036
Price ($): mean	0.23	0.31	0.44	0.55	0.71	0.93	1.22	1.81	2.57	6.18
Price ($): median	0.12	0.19	0.27	0.37	0.48	0.65	0.90	1.36	1.90	3.88
Missing prels (%)	20.45	12.97	9.84	8.74	7.75	6.81	6.41	5.51	4.64	3.58

• The initial sample comprises all stocks listed in the AGSM CRIF database from 1973 to 2010. Stocks that have nonordinary shares and/or are classified as real estate, property trusts, investment and financial services are excluded. Starting in December 1973, stocks are ranked annually by market capitalisation and sorted into decile portfolios. The table reports average monthly portfolio returns calculated under the constant and updated weights approaches. Panel A (Panel B) reports average monthly portfolio returns when stocks are equal (value) weighted. The statistical significance of return bias is shown, with ***, ** and * indicating significance at 1, 5 and 10 per cent levels, respectively. Panel C summary statistics are the time series average of annual cross-sectional means and medians.

Holding the treatment of nontrading constant, the stock weighting bias isolates the error induced by applying constant-initial weights to monthly stock returns. Liu and Strong (2008) provide an analytic representation of the bias induced by the constant weights method and suggest that it will over (under) state returns to portfolios comprising low (high)-price stocks. The empirical results in Table 3 are highly consistent with this conjecture. Summary statistics in Panel C demonstrate that share prices are lowest in the smaller deciles, and it is these deciles where the stock weighting bias is greatest (Method 1 – Method 2 ≈ 130–155 basis points per month for the smallest portfolio). Similarly, the constant weights approach results in a modest understatement of average returns for the larger size portfolios, where share prices are highest. In terms of the size premium (far right column labelled ‘spread’), using constant stock weights overstates the average spread return by 161 (135) basis points per month for equal (value) weighted portfolios.

A second pertinent aspect of Table 3 is the influence of nontrading bias. Holding constant the method of weighting stocks into portfolios, the nontrading bias quantifies the return differential resulting from using approach (2), compared to simply inserting a price relative of unity to cover nontrading. Panel C documents a high incidence of nontrading clustered in the small-stock portfolio. Over the 12 months following portfolio formation, stocks in the smallest size portfolio do not trade in 20.45 per cent of these months on average (this is nearly six times higher than the frequency of nontrading for large stocks). A priori, estimated returns for this portfolio are expected to be most sensitive to methodological choices regarding the treatment of nontrading. Somewhat surprisingly, average returns to the smallest size portfolio exhibit a very modest bias (Method 2 – Method 3 ≈ negative 10–16 basis points per month), which is statistically insignificant. This preliminary finding suggests that the approach taken to handle missing returns is less important than the approach to weighting stock returns into the portfolio.

After aggregating the two sources of bias, the nontrading bias partially offsets the stock weighting bias, yet a significant upward bias to the size premium remains. Specifically, when genuine buy-and-hold portfolio returns are calculated under the updated weights approach (1), and after applying (2) to accommodate nontrading, the size premium decreases by 144 (125) basis points per month under equal (value) weighting. This total bias in the size premium is almost entirely attributable to the stock weighting bias in the small-stock portfolio. To summarise, while there is still evidence of a size effect in Australian equity returns, Table 3 suggests that its magnitude is grossly overstated by popular methodologies.

Throughout this study, we repeatedly stress the importance of examining the share price and nontrading characteristics of portfolios. As demonstrated in Table 3 in the context of size-sorted portfolios, the low-price characteristics of stocks in small size portfolios makes them acutely vulnerable to the bias induced by the constant weights approach to estimating portfolio returns. Similarly, the prevalence of nontrading amongst small stocks signals that assumptions made over missing returns may be influential. In such circumstances, a careful study of portfolio characteristics may guide key methodological choices.

A final point to note is that the average returns to smaller size deciles in Table 3 are notably larger than reported in Gaunt (2011). The summary statistics in Table 3 allow us to reconcile the findings. Panel C suggests that the difference is primarily attributable to Gaunt's (2011) exclusion of stocks with sub-20c share price as at portfolio-formation date. Examining the mean and median share prices, many of the stocks in our smallest 3–4 size portfolios would be excluded by a sub-20c filter. In fact, when we repeat our analysis with a similar exclusion, our results almost precisely match Gaunt (2011). While Gaunt's motivation for a sub-20c filter is to mitigate the market-microstructure issues with low-price stocks, inadvertently, it also alleviates problems raised in the current study arising from stock weighting bias in portfolio returns.

3.2. Momentum profits

In the jargon of Jegadeesh and Titman (1993), our momentum study is a 6/6 strategy. Starting in June 1974, prior 6-month momentum is calculated for all stocks with valid prices in December 1973 and June 1974. The momentum metric includes both capital gains/losses and dividends. Stocks are ranked by their prior momentum and sorted into decile portfolios. Buy-and-hold positions in the constituent stocks are maintained for 6 months, after which the ranking and portfolio-formation procedure is repeated. Over the sample period, this generates 72 nonoverlapping 6-month test periods (hence 432 months), encompassing 899 stocks on average.16

As noted above, one of the aims of this study is to demonstrate the importance of carefully considering the characteristics of stocks assigned to various portfolios, with an eye towards how this may combine with the chosen research methodology to influence estimated portfolio returns. Accordingly, before examining the empirical evidence on momentum profitability, we take a brief look at the summary statistics in Table 4 Panel C.

Table 4. Average monthly returns to momentum portfolios

	Losers	2	3	4	5	6	7	8	9	Winners	Spread
Panel A: Equally weighted portfolios
Constant weights	3.17	1.88	1.51	1.66	1.50	1.69	1.58	1.98	2.07	2.16	−1.01	(p = 0.135)
Updated weights	2.75	1.58	1.35	1.46	1.40	1.66	1.51	1.96	2.06	2.02	−0.73	(p = 0.294)
Total bias	0.42	0.30	0.16	0.20	0.10	0.03	0.07	0.02	0.01	0.14	−0.28
Weighting bias	0.56	0.28	0.16	0.20	0.09	0.02	0.06	0.05	0.05	0.14	−0.42
Nontrading bias	−0.14	0.02	0.00	0.00	0.01	0.01	0.01	−0.03	−0.04	0.00	0.14
Panel B: Value-weighted portfolios
Constant weights	0.40	−0.15	0.79	0.81	1.11	1.34	0.95	1.47	1.43	1.90	1.50	(p = 0.019)
Updated weights	0.24	−0.16	0.87	0.83	1.14	1.35	1.00	1.50	1.46	2.01	1.77	(p = 0.005)
Total bias	0.16	0.01	−0.08	−0.02	−0.03	−0.01	−0.05	−0.03	−0.03	−0.11	−0.27
Weighting bias	0.16	−0.02	−0.09	−0.01	−0.04	−0.02	−0.06	−0.02	−0.03	−0.12	−0.28
Nontrading bias	0.00	0.03	0.01	−0.01	0.01	0.01	0.01	−0.01	0.00	0.01	0.01
Panel C: Summary statistics
Mkt Cap ($m): mean	34	115	212	294	464	526	518	497	383	164
Mkt Cap ($m): median	5	9	14	25	29	32	32	32	29	18
Price ($): mean	0.38	0.78	1.18	1.50	1.88	2.02	2.13	2.15	2.05	1.46
Price ($): median	0.19	0.37	0.54	0.71	0.86	0.98	1.04	1.06	0.99	0.60
Missing prels (%)	8.34	5.87	5.49	5.54	4.92	5.16	5.39	5.33	6.33	7.48

• The sample comprises all stocks listed in the AGSM CRIF database over the period 1973 to 2010. Stocks that have nonordinary shares and/or are classified as real estate, property trusts, investment and financial services are excluded. Starting in June 1974, stocks are ranked by their prior 6-month momentum and sorted into decile portfolios. The composition of the portfolios remain fixed for 6 months, after which the ranking and portfolio-formation procedure is repeated. The table reports average monthly portfolio returns calculated under the constant and updated weights approaches. Panel A (Panel B) reports average monthly portfolio returns when stocks are equal (value) weighted. The statistical significance of return bias is shown, with ***, ** and * indicating significance at 1, 5 and 10 per cent levels, respectively. Summary statistics are the time series average of annual cross-sectional means and medians.

As pre-empted in footnote,¹⁵ portfolios sorted according to momentum are more heterogeneous in terms of market capitalisation and share price than size-sorted portfolios. With greater dispersion of market caps within a portfolio, we expect to see a larger difference between equal- and value-weighted portfolio returns. Further, low-price stocks are more evenly distributed across decile portfolios (not necessarily evenly, but more so than with size-sorted portfolios). As such, the stock weighting bias in portfolio returns is likely to be less extreme than observed with size-sorted portfolios. With respect to nontrading, Panel C shows that the occurrence of nontrading is both less clustered in specific portfolios and less prevalent than observed for size portfolios.17 These characteristics should to some extent mitigate the nontrading bias. Each of these conjectures is confirmed in Table 4.

In Panel A, average returns to equal-weighted momentum-sorted portfolios show no evidence of momentum profits. In fact, recent losers outperform recent winners, although the spread portfolio is statistically insignificant. As was the case with the size effect, the stock weighting bias is greatest for the portfolio comprising the lowest-priced stocks (Losers in this case).

In Panel B, average portfolio returns are notably lower when stocks are value-weighted into portfolios (compared with Panel A equal-weighted portfolios). This occurs when small stocks have higher returns, and there is heterogeneity in market caps within each portfolio. The nontrading bias is virtually nonexistent, suggesting that nontrading within portfolios is confined to small-cap stocks that carry relatively little weight.18 Importantly, the stock weighting bias shows that the constant weights approach overstates (understates) returns to the Loser (Winner) portfolio. As such, there is a double whammy effect that increases momentum profits when genuine buy-and-hold returns are calculated using correctly updated stock weights. Providing further support for the notion that the constant weights approach understates the momentum effect, average returns to the momentum spread strategy are approximately 30–40 basis points per month higher after correcting the stock weighting bias.19

With reference to the extant Australian momentum literature, the results in Table 4 warrant two final comments. First, the current findings shed light on prior Australian momentum findings, which have been inconclusive. Using all ASX-listed stocks, Durand et al. (2006) find no evidence of momentum profits. While they do not articulate how stocks are weighted into their portfolios, the findings in Table 4 Panel A suggest that equal weighting is likely. Second, 4 replicates the dramatic reversal in momentum findings under equal- and value-weighted momentum portfolios previously reported in Brailsford and O'Brien (2008). In Section 4., we show that momentum effects are unduly affected by small stocks under equal-weighted portfolios. When the analysis is restricted to the Top 500 stocks, a clear momentum effect is apparent regardless of how stocks are weighted into portfolios.

3.3. Book-to-market effects

Early Australian research into the book-to-market (BM) effect was severely hampered by the limited availability of financial-statement data required to calculate book value. Utilising modest subsets of the population of stocks, early studies by Halliwell et al. (1999) and Gaunt (2004) provide cautious evidence of a significant value premium. With the benefit of an extended time series of accounting data (1981–2005) and near-complete cross-sectional coverage, O'Brien (2008) finds evidence of a value premium only for equal-weighted BM-sorted decile portfolios. Gharghori et al. (2009) report a significant value premium to six value-weighted BM portfolios over the period 1993–2005. Gharghori et al. (2013) document a positive relationship between BM and average returns to BM-sorted decile portfolios spanning 1993–2009. Recently, Dou et al. (2012) show that the value premium in the population is driven by micro-cap stocks, with little evidence amongst large caps.

Our study of the BM effect proceeds as follows. Using financial-statement data sourced from Aspect Huntley, book value of equity is calculated as described in Section 3.. Starting in December 1989, stocks are ranked by BM and sorted into decile portfolios. Buy-and-hold positions in the constituent stocks are maintained for 12 months, after which the ranking and portfolio-formation procedure is repeated each year through to December 2009. Over the 21-year period (252 months), 946 stocks per annum are assigned to decile BM portfolios on average.

The results in Table 5 cast further doubt over the existence of a BM effect in Australian equity returns. Under the constant weights approach, there is apparent evidence of a statistically significant value premium regardless of how the stocks are weighted into portfolios (2.07 and 1.41 per cent under equal- and value-weighting, respectively). These findings are highly consistent with prior results. O'Brien (2008) reports an average value premium of 1.87 per cent per month for equal-weighted portfolios. Gharghori et al. (2009) report an average value premium of 1.28 per cent per month for value-weighted portfolios.

Table 5. Average monthly returns to book-to-market portfolios

	Growth	2	3	4	5	6	7	8	9	Value	Spread
Panel A: Equally weighted portfolios
Constant weights	0.80	1.06	1.41	1.36	1.54	1.71	1.91	2.05	2.47	2.87	2.07	(p = 0.003)
Updated weights	0.78	1.00	1.51	1.32	1.65	1.73	1.64	2.00	2.11	2.38	1.60	(p = 0.024)
Total bias	0.02	0.06	−0.10	0.04	−0.11	−0.02	0.27	0.05	0.36	0.49	0.47
Weighting bias	0.16	0.01	0.01	−0.02	−0.03	0.10	0.19	0.18	0.50	0.62	0.46
Nontrading bias	−0.14	0.05	−0.11	0.06	−0.08	−0.12	0.08	−0.13	−0.14	−0.13	0.01
Panel B: Value-weighted portfolios
Constant weights	0.14	0.45	0.90	0.89	0.98	0.79	1.18	1.26	1.01	1.55	1.41	(p = 0.019)
Updated weights	0.37	0.52	0.98	0.91	0.98	0.83	1.20	1.36	1.11	1.33	0.96	(p = 0.096)
Total bias	−0.23	−0.07	−0.08	−0.02	0.00	−0.02	−0.02	−0.10	−0.10	−0.22	0.45
Weighting bias	−0.23	−0.07	−0.09	−0.02	0.00	−0.04	−0.02	−0.11	−0.10	0.17	0.40
Nontrading bias	0.00	0.00	0.01	0.00	0.00	0.02	0.00	0.01	0.00	0.05	0.05
Panel C: Summary statistics
Mkt Cap ($m): mean	415	711	920	932	754	623	365	221	95	36
Mkt Cap ($m): median	36	40	43	38	31	27	23	14	10	6
Price ($): mean	2.10	2.35	2.59	2.29	1.84	1.71	1.31	0.98	0.66	0.38
Price ($): median	0.48	0.63	0.75	0.63	0.57	0.53	0.44	0.33	0.24	0.13
BM ($): mean	0.12	0.28	0.40	0.52	0.65	0.80	0.99	1.26	1.70	3.26
BM ($): median	0.13	0.28	0.40	0.52	0.65	0.80	0.99	1.25	1.68	2.77
Missing prels (%)	7.39	5.35	5.25	5.38	6.77	6.43	7.36	8.38	9.93	13.56

• The initial sample comprises all stocks listed in the Aspect Huntley database from 1989 to 2009. Stocks that have nonordinary shares and/or are classified as real estate, property trusts, investment and financial services are excluded. The book value of stocks is calculated as the total value of equity, less outside equity interests, less preferences shares, less future tax benefits. Market value is drawn from AGSM CRIF as at December of the portfolio-formation year. Starting December 1989, stocks are ranked annually by BM and sorted into decile portfolios. The table reports average monthly portfolio returns calculated under the constant and updated weights approaches. Panel A (Panel B) reports average monthly portfolio returns when stocks are equal (value) weighted. The statistical significance of return bias is shown, with ***, ** and * indicating significance at 1, 5 and 10 per cent levels respectively. Summary statistics are the time series average of annual cross-sectional means and medians.

However, when the procedure of (1) and (2) is applied to estimate genuine buy-and-hold returns, the evidence in favour of a book-to-market effect is less compelling. The magnitude of the value premium is reduced by 47 (45) basis points per month under equal (value)-weighted portfolios. Statistical support for a value premium exists for equally weighted portfolios (p = 0.024), but is weak for value-weighted portfolios (p = 0.096).

Again, the share price and nontrading statistics are the key to understanding how bias is induced. Focusing on value-weighted portfolios in Panel B, the Value portfolio (high BM) comprises low-price stocks, for which the constant weights approach overstates average returns (17 basis points). Similarly, the Growth portfolio comprises higher-priced stocks, for which the constant weights approach understates average returns (23 basis points). This results in the value premium being overstated by 40 basis points per month under the constant weights approach. Note also that the constituents of the value portfolio are small-cap stocks, with high nontrading frequencies thereby inducing a modest bias (5 basis points).

3.4. Simulation evidence

Several points emerge from our empirical exploration of size, momentum and book-to-market anomalies. When a buy-and-hold investment strategy is intended, estimating portfolio returns using the constant weights approach can induce a significant bias. Depending on the characteristics of stocks involved, this bias can approach 150 basis points per month. While the treatment of nontrading induces an additional bias, its magnitude is modest and virtually negligible for value-weighted portfolios. Nonetheless, taken together, the stock weighting and nontrading biases combine for individual portfolios and flow through to hedge portfolio, potentially influencing empirical findings. As such, examining the size, price and nontrading characteristics of a portfolio's constituent stocks is a useful starting point in understanding the direction and magnitude of the bias.

Arguably, these insights are limited in the sense that they derive from our empirical analysis in very specific settings (the study of size, momentum and book-to-market effects). With an eye towards providing a more general analysis, we conduct a simple simulation experiment. A random month is chosen between December 1973 and December 2009. The population of stocks at that point is filtered using either share price or market capitalisation (as indicated below). From the resulting subset of stocks, 100 stocks are randomly selected and formed into a buy-and-hold portfolio. Monthly portfolio returns are estimated over the subsequent 12 months using the approach of Method 1, Method 2 and Method 3 described above. The total bias in average monthly portfolio returns is decomposed into stock weighting and nontrading bias. This simulation procedure is repeated 5000 times, with average bias reported.

Table 6 provides an insight into the complexity of the bias induced by methodological choices. In each case, the reported bias is basis points per month. A positive (negative) bias indicates that an ‘inferior’ methodological combination over (under) states average portfolio returns relative to a ‘superior’ approach. Consistent with Liu and Strong's (2008) analytic representation, Table 6 documents positive (negative) bias for low (high) priced stocks. Similarly, as there is a strong correlation between share price and market capitalisation, the constant weights approach under (over) estimates monthly returns to big (small) firms. Curiously, however, the bias is not a monotonic function of price or firm-size. For example, as we move from Top 200 stocks to those ranked 201 to 500, the understatement of returns increases. However, for partitions outside the Top 500, the constant weights approach overstates monthly returns.20

In terms of the relative importance of each bias component, Table 6 documents that the nontrading bias is less problematic than the stock weighting bias. With the exception of stocks outside the top 1000, the nontrading bias is negligible. The stock weighting bias, however, has the potential to be empirically significant. For stocks with share price below 20 cents or outside the top 1000, using the constant weights approach generates an overstatement of average monthly returns in the vicinity of 100 basis points per month. To some extent, value-weighting stocks into the portfolio reduces the severity of the stock weighting bias in extreme cases. Nonetheless, the total remaining bias can be material. To grasp the potential influence of this bias, one simply needs to note that there were 2045 stocks in the AGSM database as at December 2010. Hence, at least half the population lies outside the Top 1000. Further, just over 900 of these stocks have share price below 20 cents. Clearly, there is enormous potential for empirical studies to be affected when portfolios are formed under the constant weights approach.