Beating the DAX, MDAX, and SDAX: investment strategies in Germany

Abstract

Motivated by two recent papers of Asness et al. (J Portf Manag Fall 40(5):75–92, 2014; J Portf Manag Fall 42(1):34–52, 2015), we investigate whether momentum and value strategies outperformed a buy-and-hold strategy in the three biggest German equity indices, DAX, MDAX, and SDAX from 1988 to 2015. Our findings show that a momentum premium was present only in the SDAX and that value strategies did not work in any of the three indices. Consequently, we conclude that at least the DAX and MDAX are efficient indices and that some supposedly abnormal returns could be illusionary, as limits to arbitrage obstruct any profitable exploitation in practice. Finally, we find a negative correlation between momentum and value in the DAX and show that mixing both strategies can substantially decrease a portfolio’s risk.

Appendices

Appendix 1: Details of sample construction

We pulled stock data from Thomson Reuters Datastream but were unable to find data for the firms “PWA” (MDAX), “Bonifatius Hospital und Senioren,” “Quante,” and “SG Holding” (all SDAX). Fibor and Euribor are used as a proxy for the risk-free rate and we pulled the data from the Deutsche Bundesbank. The yield of the 10-year German government bond is used as the long-term interest rate and was obtained from the Federal Reserve Bank of St. Louis. Table 12 reports the search codes used in Datastream.

Table 12 Datastream Data Types

Appendix 2: Details of strategy construction

1.1 Classic momentum

The Jegadeesh and Titman (1993) (JT 1993) momentum strategy, described in Sect. 3.1.1, is based on Jegadeesh and Titman (1993). The stock universe is split into five portfolios based on the previous return. The (6 1 6) strategy is based on the return of the previous 6  months and the (12 1 12) strategy on the previous twelve months. Both strategies skip one month between portfolio formation and the beginning of the holding period. The (6 1 6) strategy holds the stocks for 6 months, whereas the holding period for the (12 1 12) strategy is 12 months. An equally-weighted strategy (EW) weights all stocks in the portfolio equally, whereas a market-value-weighted strategy (MVW) weights the stocks in the portfolio according to their market capitalization in the previous month. After June 2002, the free floating market capitalization is used. The portfolio with the highest (lowest) returns is bought (sold) and the portfolios are rebalanced monthly to their original formation weights. The strategy creates overlapping portfolios, that is, each month a new portfolio is created and the return of the strategy for a month is the average return of every held portfolio for the month.

1.2 52-week high enhanced momentum

The 52-week high momentum strategy, described in Sect. 3.1.2, works similarly to the JT 1993 EW (6 1 6) momentum strategy and is based on George and Hwang (2004). But instead of sorting according to past returns, the stocks are sorted according to the sorting ratio of Eq. (6), where P is the price of stock i at month t and high is the 52-week high of stock i at month t. The portfolio with the highest (lowest) sorting ratio is bought (sold). The observation period and the holding period are 6  months and 1  month is skipped in between. The portfolios are EW and rebalanced monthly.

$$\begin{aligned} \mathrm{Sorting}~\mathrm{Ratio}_{i,t}=\frac{P_{i, (t-1)}}{\mathrm{High}_{i, (t-1)}} \end{aligned}$$

(6)

1.3 Volatility-scaled momentum

The strategy of volatility scaling of momentum, described in Sect. 3.1.3, is based on Barroso and Santa-Clara (2015). Momentum portfolios are created according to the EW (6 1 6) JT 1993 strategy, but the returns are scaled according to the forecasted variance. The variance of the strategy is forecast using daily returns of the previous 6  months. Since 21 trading days per month are assumed, this yields a look-back period to estimate the variance of 126 days. Equation (7) shows the formula for the variance forecast. WML$_t$ is the monthly momentum return, that is, the average return of all overlapping portfolios in month t, and $r_{\mathrm{WML},d}$ is the daily momentum return, that is, the average return of all overlapping portfolios per day in month t.

$$\begin{aligned} \hat{\sigma }^2_{\mathrm{WML},t}=21*\sum _{j=0}^{125} r^2_{\mathrm{WML},d_{(t-1)}-j}/126 \end{aligned}$$

(7)

The variance forecast is used to scale the momentum returns of each month according to Eq. (8). The target volatility used is 12 % per year or 3.464 % per month. WML$^*_t$ is the return of the strategy in month t.

$$\begin{aligned} \mathrm{WML}^*_t=\frac{\sigma _{\mathrm{target}}}{\hat{\sigma }_{\mathrm{WML},t}}*\mathrm{WML}_t \end{aligned}$$

(8)

1.4 Information discreteness enhanced momentum

The information discreteness strategy, described in Sect. 3.1.4, is built on Da et al. (2014). First, a measure for information discreteness (ID) is created for stock i in month t as in Eq. (9). PRET is the cumulative return over the previous year. % neg and % pos are the percentage of daily negative and positive returns over the past year, respectively.

$$\begin{aligned} \mathrm{ID}_{i,t}=\mathrm{sgn}[\mathrm{PRET}_{i,(t-1)}]*[\%\mathrm{neg}_{i,(t-1)}-\%\mathrm{pos}_{i,(t-1)}] \end{aligned}$$

(9)

The stock universe is then split into conditionally double-sorted portfolios, first into three portfolios based on PRET and second into three portfolios based on ID. Hence, there are $3*3=9$ portfolios. The portfolio with the lowest ID within the portfolio of the highest return is then bought and the portfolio with the highest ID within the portfolio with the lowest return is sold. As in the JT 1993 momentum strategy, EW overlapping portfolios are created and the portfolios are rebalanced monthly. The holding period is 6 months with a 1-month lag between portfolio formation and the beginning of the holding period.

1.5 Skewness enhanced momentum

The skewness enhanced momentum strategy, described in Sect. 3.1.5, is based on Jacobs et al. (2015). The proxy for expected skewness is the maximum daily return of the previous month, that is, the past 21 days. Then, EW $3*3=9$ conditionally sorted portfolios are created; first, on expected skewness and, second, on the cumulative past return over the previous 12 months. The portfolio with the highest (lowest) past cumulative return within the portfolio of the lowest (highest) expected skewness is bought (sold). The portfolios are held for 1 month.

1.6 Value enhanced momentum

Value is measured according to the B/M ratio as described in Asness and Frazzini (2013) and shown in Eq. (10) (see Sect. 3.1.6). Market value is the market capitalization of stock i at month t and book value is the book value of stock i at the end of the last fiscal year z. $3*3=9$ conditionally sorted portfolios are created. First, on the B/M–devil ratio and then on the cumulative past return over the previous 6  months. The portfolio with the highest (lowest) cumulative past return within the portfolio with the highest (lowest) B/M–devil ratio is bought (sold). The holding period is 6  months and 1  month is skipped between portfolio formation and the beginning of the holding period. Overlapping EW portfolios are created and the portfolios are rebalanced monthly.

$$\begin{aligned} \mathrm{B/M-Devil}_{i,t}=\frac{\mathrm{Book}~\mathrm{Value}_{i,z}}{\mathrm{Market}~\mathrm{Value}_{i,(t-1)}} \end{aligned}$$

(10)

1.7 Pure value B/M

The B/M measure at time t, described in Sect. 3.2.1, is based on Fama and French (1992) and is calculated as in Eq. (11). Book value and market value are the book value of equity and the market capitalization at the end of the last fiscal year z, respectively. Five portfolios are created based on the B/M ratio and the portfolio with the highest (lowest) B/M ratio is bought (sold). Overlapping EW portfolios are created with a holding period of 6  months. The portfolios are rebalanced monthly.

$$\begin{aligned} B/M_{i,t}=\frac{\mathrm{Book}~\mathrm{Value}_{i,z}}{\mathrm{Market}~\mathrm{Value}_{i,z}} \end{aligned}$$

(11)

1.8 Pure value devil

The B/M measure, described in Sect. 3.2.2, is based on Asness and Frazzini (2013) and is calculated as in Eq. (10). Market value is the market capitalization of stock i at month t and book value is the book value of stock i at the end of the last fiscal year z. Five portfolios are created based on the B/M–devil ratio and the portfolio with the highest (lowest) B/M–devil ratio is bought (sold). Overlapping EW portfolios are created with a holding period of 6  months and the portfolios are rebalanced monthly.

1.9 Pure value P/E

The pure value P/E strategy, described in Sect. 3.2.3, sorts the portfolios according to the price–earnings (P/E) ratio of the last month, a measure previously employed by Fama and French (1992). Five portfolios are created based on the P/E ratio and the portfolio with the highest (lowest) P/E ratio is bought (sold). Overlapping EW portfolios are created with a holding period of 6  months and the portfolios are rebalanced monthly.

1.10 Stock’s average B/M

The stock’s average B/M strategy, described in Sect. 3.2.4, sorts the stocks according to the sorting ratio as shown in Eq. (12). The average-B/M–devil is the average of the B/M–devil ratio of stock i at month t over the past 5  years. Five portfolios are created based on the sorting ratio and the portfolio with the highest (lowest) sorting ratio is bought (sold). Overlapping EW portfolios are created with a holding period of 6  months and the portfolios are rebalanced monthly.

$$\begin{aligned} \mathrm{Sorting}~\mathrm{Ratio}_{i,t}=\frac{\mathrm{B/M}-\mathrm{Devil}_{i,(t-1)}}{\mathrm{Average}-\mathrm{B/M}-\mathrm{Devil}_{i,(t-1)}} \end{aligned}$$

(12)

1.11 Momentum enhanced value

The momentum enhanced value strategy, described in Sect. 3.2.5, measures value according to the B/M–devil ratio described in Asness and Frazzini (2013) and shown in Eq. (10). Market value is the market capitalization of stock i at month t and book value is the book value of stock i at the end of the last fiscal year z. $3*3=9$ conditionally sorted portfolios are created; first, on the cumulative past return over the previous six months and then on the B/M–devil ratio. The portfolio with the highest (lowest) B/M–devil ratio within the portfolio with the highest (lowest) cumulative past return is bought (sold). The holding period is 6 months and 1 month is skipped between portfolio formation and the beginning of the holding period. Overlapping EW portfolios are created and the portfolios are rebalanced monthly.

1.12 Average value and momentum rank

The average value and momentum rank strategy, described in Sect. 3.2.6, is based on Fisher et al. (2015). Value is measured by the B/M–devil ratio as shown in Eq. (10) and described in the previous section. Momentum is the cumulative past return of the previous 6 months. All stocks are ranked separately every month on value and momentum and the average rank is created as in Eq. (13), where value is the rank of stock i at month t based on the B/M–devil ratio and momentum is the rank of stock i at month t based on the cumulative return over the previous 6 months. Five EW portfolios are then created based on the average rank. The portfolio with the highest (lowest) average rank is bought (sold). The strategy uses overlapping portfolios that are rebalanced monthly. The holding period is 6 months.

$$\begin{aligned} \mathrm{Average}~\mathrm{Rank}_{i,t}=\frac{\mathrm{Value}_{i,(t-1)}+\mathrm{Momentum}_{i,(t-1)}}{2} \end{aligned}$$

(13)

1.13 Short-term reversal

The short-term reversal strategy, described in Sect. 4.2.2, is based on De Groot et al. (2012). Stocks are sorted into five EW portfolios according to their return in the previous month. Portfolios are created weekly and are held for 1 week and rebalanced daily. The portfolio with the highest (lowest) return in the previous month is sold (bought).

1.14 Long-term reversal

The long-term reversal strategy, described in Sect. 4.2.2, is based on De Bondt and Thaler (1985). Stocks are sorted into five EW portfolios according to their return in the previous 3 years. Because our sample size is much smaller than that of De Bondt and Thaler (1985), we create overlapping portfolios. Every month, portfolios are created based on the return of the last 3 years and are held for 3 years and rebalanced monthly. The portfolio with the highest (lowest) return in the previous three years is sold (bought).

1.15 Idiosyncratic volatility

The idiosyncratic volatility strategy, described in Sect. 4.2.2, is based on Ang et al. (2006). First, the idiosyncratic risk is measured as the residual of the regression in Eq. (14). R is the daily return of stock i at day t. Fama and French factors for Germany are provided by Brückner et al. (2015). Note that the daily factors are provided only from 1990 onward and because the betas in Eq. (14) are estimated over the past two years, this strategy begins in 1992.

$$\begin{aligned} r_{i,t}-\mathrm{rf}_t=\alpha +b_1*\mathrm{rm}_t+b_2*\mathrm{smb}_t+b_3*\mathrm{hml}_t+\epsilon _{i,t} \end{aligned}$$

(14)

Then, the standard deviation of the residual of each stock is calculated over the past 21  days and used as a proxy for the expected idiosyncratic volatility. Five EW portfolios are created based on the idiosyncratic volatility in the previous month. The portfolio with the highest (lowest) idiosyncratic volatility is bought (sold) and held for 1  month.

1.16 Maxing out

The maxing out strategy, described in Sect. 4.2.2, is based on Bali et al. (2011). First, the maximum daily return over the previous month for each stock is identified. Second, five EW portfolios are created based on that maximum daily return. The portfolio containing the stocks that had the lowest (highest) maximum daily return in the previous month is bought (sold). The portfolios are held for 1  month.

1.17 Betting against beta

The betting against beta (BAB) strategy, described in Sect. 4.2.2, is based on Frazzini and Pedersen (2014). First, the betas for the daily return of stock i at day t are estimated according to the regression in Eq. (15). rm is the return of the respective index j, that is, either the return of the DAX, MDAX, or SDAX. rf is the return of the risk-free rate in month t. To adjust for outliers the betas are compressed to 1 according to Eq. (16). Because the daily factors are provided only from 1990 onward and the betas are estimated over the past 2 years, this strategy begins in 1992.

$$\begin{aligned} r_{i,t}-\mathrm{rf}_t= & {} \alpha +b_1*(\mathrm{rm}_{j,t}-\mathrm{rf}_t)+\epsilon _{i,t} \end{aligned}$$

(15)

$$\begin{aligned} \hat{\beta }_{i,t}= & {} 0.6*\beta _{i,t}+0.4 \end{aligned}$$

(16)

The stock universe is split in half into high and low beta stocks. Next, the rank within each group is calculated according to the beta of the stock. The weight of a stock in the final portfolio is then given by Eq. (17). z is the rank of stock i at month t and $\bar{x}$ is the average rank of the high or low beta stock group m at month t. k is a normalizing constant and is given by two divided by the sum of the absolute deviation from the average rank in each group, high or low beta. However, only the positive weights are taken from each group. More informally, the stock universe is split into four portfolios based on the beta, the top and bottom portfolio are kept, and within these portfolios the stocks are weighted according to their rank of beta in their portfolio. The higher the beta, the higher the rank in the top beta group and vice versa for the low beta group.

$$\begin{aligned} w_{\mathrm{H}_{i,t}}= & {} k(z_{i,(t-1)}-\bar{x}_{m,(t-1)}) \nonumber \\ w_{\mathrm{L}_{i,t}}= & {} k(z_{i,(t-1)}-\bar{x}_{m,(t-1)}) \end{aligned}$$

(17)

Finally, using the risk-free rate as leverage, the low beta portfolio is scaled to have a beta of one at portfolio formation. Similarly, the high beta portfolio is scaled down to a beta of one. Hence, the beta of the strategy is constructed as zero at portfolio formation. Equation (18) shows the final equation for the return of the BAB strategy. $\beta ^\mathrm{L}$ and $\beta ^\mathrm{H}$ is the beta of the low and high beta portfolio at month t, respectively. $r^\mathrm{L}$ and $r^\mathrm{H}$ is the return of the low and high beta portfolio at month t. The low (high) beta portfolio is bought (sold) and the portfolios are held for 1 month.

$$\begin{aligned} r^{\mathrm{BAB}}_{t+1}=\frac{1}{\beta ^\mathrm{L}_t}*(r^\mathrm{L}_{(t+1)}-\mathrm{rf}_t)-\frac{1}{\beta ^\mathrm{H}_t}*(r^\mathrm{H}_{(t+1)}-\mathrm{rf}_t) \end{aligned}$$

(18)