Trading strategies based on past returns: evidence from Germany

Abstract

Among the various strategies studied in this paper, only momentum investing appears to earn persistently nonzero returns: From 1965 to 2014, the classical momentum strategy based on performance over the previous 2–12 months earned an average return of 1.57% per month (excluding microcap stocks and value-weighted returns). In the most recent 10-year period, this return was even larger—2.27%—which is much larger than in the USA. However, profitability net of transaction costs is weak because the strategy involves trading in disproportionately small stocks with high transaction costs, something that is particularly true for the loser portfolio. A strategy that concentrates only on the winner portfolio and thus avoids potential problems associated with (short) selling the costly loser portfolio appears to earn strong and persistently abnormal profits, even after transaction costs.

Appendices

Appendix 1: Fama–MacBeth regressions with single and/or compounded returns over certain contiguous and non-contiguous past horizons

To further explore whether certain lags contribute to a given strategy, I run multivariate Fama and MacBeth (1973) cross-sectional regressions by including (1) single lagged returns and/or (2) compounded returns over certain contiguous and non-contiguous past horizons as explanatory variables. The compounded and single returns over certain past horizons are selected on the basis of the results reported in Table 2. Specifically, the following are included:

• (i) single lagged returns (e.g., lag 1),

• (ii) compounded (contiguous) lagged returns (e.g., lags 5–12),

• (iii) multiple cumulative (non-contiguous) lagged returns (e.g., lags 13–23 plus 25–35 plus 37–47 plus 49–59, which, in fact, are lags 13–60, excluding the annual lags 24, 36, 48, 60), and

• (iv) annual cumulative (non-contiguous) lagged returns (e.g., lag 12 plus 24 plus 36 plus 48 plus 60).

Table 7 shows the results for multiple setups. In the first setup (A.I), the insignificant average coefficient for the second lag indicates that a short-term contrarian strategy is not improved by including this lag. Thus, the results reported in Table 2, which suggest the opposite, are not robust in this respect. For a momentum strategy, it seems useful to include not only lags 5 to 12 but also the third lag because of its positive and significant coefficient. Excluding the second lag does not appear to significantly improve the strategy.

Setups A.IV and A.V of Table 7 support the results of Table 2: a long-run contrarian strategy seems to improve after excluding the multiple annual lags. The average coefficient for the multiple compounded lagged return (lags 14–22, 26–34, 38–46, 50–58) is negative (–0.30) and highly statistically significant (t-statistic –4.05), while the average coefficient for the annual compounded lagged return (lag 12, 24, 36, 48, 60) is positive (1.11) and highly statistically significant (t-statistic 5.54). Table 7 also supports the results of Table 2 with respect to the improvements that could be made to the portfolio strategy proposed by Heston and Sadka (2008). The coefficient in Column 7 is positive (0.31) and statistically significant (t-statistic 2.08) for the previous and next lag to the multiple annual lags (lag 11, 13, 23, 25, 35, 37, 47, 49, 59).

Table 7 Multivariate Fama–MacBeth regression results

Appendix 2: Additional descriptive statistics

Table 8 Descriptive statistics of momentum, contrarian, and seasonality strategies, 1965–2014

Appendix 3: Bid–ask spread estimation procedure and comparison of estimates

1.1 Quoted spread procedure

Based on Stoll and Whaley (1983) and following Lesmond et al. (2004), I obtain bid–ask spread estimates based on quoted bid and ask prices. Estimates are calculated individually for each stock based on daily closing bid and ask quotes obtained from Datastream. The spread is estimated over Frankfurt Stock Exchange trading day $d-15$ to $d-6$ ($=$two weeks) relative to portfolio formation to mitigate any influence of turn-of-the-month effects in quotes (Lesmond et al. 2004). The quoted spread measure for stock i at the beginning of month t is

$$\begin{aligned} {\text {Quoted Spread}}_{i,t} =\frac{1}{10}\mathop \sum \limits _{\tau =-15}^{-6} \frac{a_{i,d+\tau } -b_{i,d+\tau }}{\frac{1}{2}\left( {a_{i,d+\tau }+b_{i,d+\tau }} \right) } \end{aligned}$$

(3)

where $a_{i,d+\tau }$ is the quoted ask price and $b_{i,d+\tau }$ is the quoted bid price on day d for stock i.

1.2 Corwin and Schultz procedure

Bid–ask spread estimates based on Corwin and Schultz (2012) are also estimated over Frankfurt Stock Exchange trading day $d-15$ to $d-6$ ($=$two weeks) relative to portfolio formation. Based on daily high and low prices, the Corwin and Schultz measure for stock i at the beginning of month t is

$$\begin{aligned} \text {CS Spread}_{i,t} =\frac{1}{10}\mathop \sum \limits _{\tau =-15}^{-6} \text {max } \left( {\frac{2\left( {e^{\alpha _{i,d+\tau } }-1} \right) }{1+e^{\alpha _{i,d+\tau } }},0} \right) \end{aligned}$$

(4)

with

$$\begin{aligned} \alpha _{i,d} =\frac{\sqrt{2\beta _{i,d} }-\sqrt{\beta _{i,d} }}{3-2\sqrt{2}}-\sqrt{\frac{\gamma _{i,d} }{3-2\sqrt{2}}} \end{aligned}$$

(5)

where

$$\begin{aligned} \beta _{i,d} =\left[ {\mathrm{ln}\left( {\frac{P_{i,d}^{H,\mathrm{adj}} }{P_{i,d}^{L,\mathrm{adj}} }} \right) } \right] ^{2}+\left[ {\mathrm{ln}\left( {\frac{P_{i,d-1}^H }{P_{i,d-1}^L }} \right) } \right] ^{2} \end{aligned}$$

(6)

and

$$\begin{aligned} \gamma _{i,d} =\left[ {\mathrm{ln}\left( {\frac{{\text {max}} \left( {P_{i,d-1}^H ,P_{i,d}^{H,\mathrm{adj}} } \right) }{\text {min} ({P_{i,d-1}^L ,P_{i,d}^{L,\mathrm{adj}} })}} \right) } \right] ^{2}. \end{aligned}$$

(7)

$P_{i,d}^H$ and $P_{i,d}^L$ are the high and low prices, respectively, $P_{i,d}^{H,\mathrm{adj}} =P_{i,d}^H -\Delta \mathrm{PON}$ and $P_{i,d}^{L,\mathrm{adj}} =P_{i,d}^L -\Delta \mathrm{PON}$ are the high and low adjusted prices, respectively, with $\Delta \mathrm{PON}= \max ( {P_{i,d}^L -P_{i,d-1}^C ,0} )+ \max ( {P_{i,d-1}^C -P_{i,d}^H ,0} )$, the estimated overnight price change with $P_{i,d}^C$ the closing price of stock i on Frankfurt Stock Exchange trading day d. Thus, I make use of the overnight return adjustment described by Corwin and Schultz (2012).

1.3 Spread estimates based on actual turnover

When incorporating transaction costs, it should be noted that typically not all stocks in the winner and/or loser portfolios are replaced in every rebalancing interval. Hence, the stock positions that are kept do not incur transaction costs.

To incorporate the fact that the weight of a stock in a portfolio at the beginning of a month typically differs from the weight at the end,³⁸ I first calculate for each stock i in a given portfolio the end of month t portfolio weight:

$$\begin{aligned} w_{i,t}^{\mathrm{End}} =\frac{w_{i,t}^{\mathrm{Start}} \left( {1+r_{i,t} } \right) }{\mathop \sum \nolimits _{i=1}^N w_{i,t}^{\mathrm{Start}} \left( {1+r_{i,t}} \right) } \end{aligned}$$

(8)

with $w_{i,t}^{\mathrm{Start}}$ the beginning of month portfolio weight of stock i, which is the weight (equal- or value-weighted) obtained based on the portfolio formation described in Sect. 4.2.

The difference between $w_{i,t}^{\mathrm{Start}}$ and the end of month weight of $t-1$, $w_{i,t-1}^{\mathrm{End}}$, is the change in a stock’s portfolio weight due to the new weights obtained from the formation ($w_{i,0}^{\mathrm{End}} =0)$. I assume that all transaction costs accrue when a stock position is opened, which, in turn, implies that only additional stock positions are subjected to trading costs. Thus, the proportion of stock holdings of stock i in a given portfolio that is associated with additional transaction costs is

$$\begin{aligned} \Delta p_{i,t} =\left\{ {{\begin{array}{c@{\quad }l} 0&{} \quad {\textit{if } w_{i,t}^{\mathrm{Start}} =0} \\ \text {max} \left( {\frac{w_{i,t}^{{\text {Start}}} -w_{i,t-1}^{\mathrm{End}} }{w_{i,t}^{{\text {Start}}} },0} \right) &{} \quad {\mathrm{else}} \\ \end{array} }} \right. \end{aligned}$$

(9)

with $0\le p\le 1$. The quoted spread measure adjusted for actual turnover is, in turn,

$$\begin{aligned} \text {Quoted Spread}_{i,t}^{\mathrm{act}} =QS_{i,t} \Delta p_{i,t} , \end{aligned}$$

(10)

and for the Corwin and Schultz measure

$$\begin{aligned} \text {CS Spread}_{i,t}^{\mathrm{act}} =CS_{i,t} \Delta p_{i,t} . \end{aligned}$$

(11)

Fig. 5

Average bid–ask spread estimates for size quintiles. At the beginning of each month between 01/1990 and 06/2014, for each stock in the sample (no exclusions), a bid–ask spread estimate is obtained based on Corwin and Schultz (2012) and quoted bid and ask prices; see the formulas in “Appendix 3.” The figure shows the average spread estimates for size quintiles over time

Fig. 6

Percentage of zero bid–ask spread estimates for size-quintiles based on Corwin and Schultz. At the beginning of each month between 01/1990 and 06/2014, for each stock in the sample, a bid–ask spread estimate based on Corwin and Schultz (2012) is obtained; see the formulas in “Appendix 3.” The figure shows the average percentage of zero estimates for three size quintiles (the smallest, Q1; the largest, Q5; and medium-sized stocks, Q3) over time

1.4 Comparison of bid–ask spread estimates

Datastream offers bid and ask quotes for the German stock market from the end of 1996 onward, which enables me to estimate the quoted spread measure from January 1997 onward. Daily high, low, and closing prices are available for most of the stocks in the sample from September 1988 onward. Because Brückner (2013) does not recommend German stock market data from Datastream before 1990, I calculate the Corwin and Schultz measure only from January 1990 onward. Thus, I am able to assess the profitability of momentum strategies net of bid–ask spread over the period from 1990 to 2014 by using the Corwin and Schultz measure and over the period from 1997 to 2014 based also on the quoted spread measure.

Figure 5 shows the average estimates of bid–ask spreads for size quintiles based on all stocks in the sample.³⁹ The five plots altogether are based on 126,384 individual estimates (stock months) for the quoted bid–ask spread beginning in January 1997 and 157,976 estimates for the Corwin and Schultz measure beginning in January 1990. The first plot shows the average bid–ask spread estimates for the largest stocks in the sample (Q5). The lines are close to each other and indicate that the two measures produce similar results, although the average estimates obtained from the quoted bid and ask data are generally higher. Within the size quintiles containing medium-sized stocks (Q4, Q3, Q2), the bid–ask spread estimates are not as close. While the quoted bid–ask spread estimates increase with decreasing stock size, the estimates based on Corwin and Schultz change only marginally. Among the stocks that are smallest in size, the difference between the lines is remarkable: the spreads based on the Corwin and Schultz measure are flat and close to zero, while the bid–ask measure based on quoted prices reaches average estimates of 10% around the turn of the millennium and up to 30–50% after 2008 (note the bandwidth of the y-axis).

For the quintile containing the smallest stocks (Q1), the estimates based on quoted bid and ask prices appear odd, especially for the most recent time period. This again highlights the need to carefully select an adequate sample of stocks and/or use alternative weighting schemes, such as value weighting. For the other quintiles, the time series and cross-sectional variation of the average estimates appear reasonable.

On the other hand, the very limited time series and cross-sectional variation in the average estimates based on Corwin and Schultz is less satisfying. A thorough analysis reveals that many of the individual estimates are zero [actually negative and forced to zero; see Eq. (4)]. Between 1990 and 1996, approximately 47.5% of the individual estimates are zero. Between 1997 and 2014, approximately 19.4% are zero. As Corwin and Schultz note (2012, p. 727), the individual spread estimates on single days can be zero. In fact, in a number of cases, all 10 individual daily estimates are zero, and, thus, so is the average. Figure 6 shows that the number of zero estimates largely diverges among the size quintiles: with decreasing stock size, the number of zero estimates increases. For this reason, the Corwin and Schultz estimates largely diverge from the quoted spread estimates in the mid- and small-sized quintiles. In sum, based on the available data from Datastream and in the context of this paper, I find the Corwin and Schultz measure to be inappropriate for estimating bid–ask spreads for German smallcap stocks.