9. Empirical Results
9.1 Results for Equally Weighted Portfolios
When one creates an investment strategy the most important question is often whether it produces a positive return. From table 9.1 it is evident that the winner portfolios for all the 16 distinct momentum strategies produce a positive return. Furthermore, all the of strategies’ winner portfolios have significantly positive returns at the 99% significance level. Furthermore, they all have an average monthly return above 1% with the average being 1.66%, and an average standard deviation of 5.78%. However, the loser portfolios of all the momentum strategies do not manage to create significant positive returns. Although they all produce positive average monthly returns, none of these results are significantly positive. The loser portfolios also exhibit substantially higher standard deviations than the winner portfolios, with the average standard deviation being 7.69%.
Furthermore, the strategies with longer formation periods perform better than those with shorter formation periods. A quick comparison between the different winner strategies reveals that on average, the average monthly return for the strategies with a 3-‐month formation period is 1.35%, while the strategies with a 12-‐month formation period on average have an
average monthly return of 1.89%. This is a noteworthy difference. However, these average returns and the fact that they are significantly different from zero doesn’t tell us whether the strategies are good or not, more on this soon.
In addition to the winner and loser strategies, which are based on holding portfolios of either previous winners or previous losers, this analysis also considers zero-‐cost strategies. These strategies consist of long positions in previous winners and short positions in previous losers. Just like the winner strategies, the zero-‐cost strategies manage to produce average monthly returns that are significantly greater than zero. The returns on the zero-‐cost strategies reveal that on average the winner portfolios outperform the loser portfolios by 1,45% per month, ranging from 1.14% to 1.81% across the different strategies. Furthermore, the best performing zero-‐cost strategies are also the ones with the most volatile return. Thus, there seems to be a positive correlation between average monthly return and standard deviation. Overall, the results provide the first indications that the momentum strategies work, and that the ‘momentum effect’ is real and observable on the Danish stock market.
To assess if the returns of the strategies are better (higher) than what an investor could have gotten by simply holding a standard index portfolio/fund, we compare the average monthly returns obtained in table 9.1 with those from our benchmark. In this analysis, the benchmark is the Danish OMXC index, which was described previously in the methodology section. To test if the momentum strategies based on winner portfolios and zero-‐cost portfolios produce abnormal returns compared to the benchmark, three different t-‐tests are applied. In short, each test has its own strengths and weaknesses that depend on the characteristics of the data being analyzed. In this study, it is assumed that the adjusted Welch t-‐test is the most fitting, given that the return on the different strategies and the return on the benchmark have unequal variances, but some degree of covariance. Nonetheless, the student’s-‐ and standard Welch t-‐test also provide information regarding the strategies relative performance, and are therefore also presented.
Table 9.1: Returns on winner, loser and zero-‐cost portfolios (Equally weighted)
The table shows the average monthly return for the winner, loser and zero-‐cost portfolios for all of the 16 distinct momentum strategies, while the numbers in parentheses are the standard deviations (decimal number). Furthermore, the table shows whether the results are significantly different from zero according to the student’s t-‐test (𝐻!: 𝑟=0).
* Significant at the 0.10 significance level.
** Significant at the 0.05 significance level.
*** Significant at the 0.01 significance level.
The test results in table 9.2 reveal that there are momentum strategies that struggle to produce returns high enough to significantly outperform the benchmark. This is particularly the case for the strategies with a 3-‐month formation period. Although these strategies do have a higher average monthly return compared to those of our benchmark, they are not significantly different from the benchmark according to both the student’s-‐ and the Welch t-‐test. However, when we take into consideration the covariance between the observations and apply the adjusted Welch t-‐test, the picture changes. The strategies based on winner portfolios with a short
3 6 9 12
3 Winner 1,39% *** 1,29% *** 1,31% *** 1,40% ***
(0,061) (0,056) (0,054) (0,054)
Loser 0,12% 0,15% 0,14% 0,23%
(0,077) (0,071) (0,068) (0,065)
Zero-‐C 1,26% *** 1,14% *** 1,18% *** 1,16% ***
(0,061) (0,05) (0,043) (0,039)
6 Winner 1,59% *** 1,67% *** 1,64% *** 1,56% ***
(0,06) (0,061) (0,058) (0,057)
Loser 0,13% 0,15% 0,15% 0,27%
(0,084) (0,08) (0,075) (0,071)
Zero-‐C 1,46% *** 1,52% *** 1,49% *** 1,29% ***
(0,071) (0,067) (0,058) (0,052)
9 Winner 1,86% *** 1,79% *** 1,74% *** 1,79% ***
(0,057) (0,057) (0,057) (0,057)
Loser 0,00% 0,08% 0,16% 0,46%
(0,086) (0,081) (0,077) (0,073)
Zero-‐C 1,85% *** 1,71% *** 1,58% *** 1,33% ***
(0,077) (0,071) (0,066) (0,061)
12 Winner 1,98% *** 1,83% *** 1,94% *** 1,81% ***
(0,06) (0,059) (0,059) (0,058)
Loser 0,17% 0,24% 0,46% 0,52%
(0,086) (0,082) (0,079) (0,077)
Zero-‐C 1,81% *** 1,59% *** 1,49% *** 1,29% ***
(0,078) (0,073) (0,07) (0,066)
Formation (J)
Holding (K)
formation period now manage to outperform the benchmark at the 90% significance level. In fact, the adjusted Welch t-‐test generally improves the significance of the results compared to the two other test statistics. Consequently, all winner strategies significantly outperform the benchmark at the 90% significance level. The winner strategies with a formation period of either 9 or 12 month produce returns that are above the benchmark and highly significant at the 99% level. Not surprisingly, it is also these strategies that show the most significant results in the student’s-‐ and Welch t-‐test.
Table 9.2: Excess return on winner and zero-‐cost portfolios
The table shows the average monthly excess return for the winner and zero-‐cost portfolios for all of the 16 distinct momentum strategies, while the numbers in parentheses are the standard deviations. Furthermore, the table includes indicators from 3 different tests, all testing whether the average monthly return on the portfolios are significantly larger than the average monthly return on the benchmark. Test 1 (T1): Student’s t-‐test; Test 2 (T2): Welch’s t-‐test, Test 3 (T3): Adjusted Welch’s t-‐test.
* Significant at the 0.10 significance level.
** Significant at the 0.05 significance level.
*** Significant at the 0.01 significance level.
3 (T1) (T2) (T3) 6 (T1) (T2) (T3) 9 (T1) (T2) (T3) 12 (T1) (T2) (T3)
3 Winner 0,53% * 0,43% * 0,37% * 0,43% **
(0,061) (0,056) (0,054) (0,054)
Zero-‐C 0,41% 0,28% 0,23% 0,20%
(0,061) (0,05) (0,043) (0,039)
6 Winner 0,79% ** * ** 0,75% ** * ** 0,66% * ** 0,61% * **
(0,06) (0,061) (0,058) (0,057)
Zero-‐C 0,66% * 0,60% 0,51% 0,33%
(0,071) (0,067) (0,058) (0,052)
9 Winner 1,01% *** ** *** 0,84% ** * *** 0,76% ** * *** 0,71% * ***
(0,057) (0,057) (0,057) (0,057)
Zero-‐C 1,00% ** * * 0,76% 0,61% 0,24%
(0,077) (0,071) (0,066) (0,061)
12 Winner 1,10% *** ** *** 0,89% ** * *** 0,84% ** * *** 0,75% ** ***
(0,06) (0,059) (0,059) (0,058)
Zero-‐C 0,92% * * 0,64% 0,38% 0,22%
(0,078) (0,073) (0,07) (0,066)
Formation (J)
Holding (K)
While all winner strategies are producing good results that are significantly positive and above the benchmark, they don’t perform equally well. In this analysis, two strategies stand out slightly in comparison to the rest. The 9/3-‐ and the 12/3-‐strategies manage to produce some of the highest average monthly returns while also creating the most significant results in all tests.
The average monthly return of the 9/3 and the 12/3-‐strategies are respectively 1.86% and 1.98%, which is more than double the return on the benchmark in the same period. In section 9.5 these strategies among a few others will be analyzed in greater detail.
Focusing on the zero-‐cost strategies in table 9.2, all of them manage to create average monthly returns larger than those of the benchmark. On average, these 16 zero-‐cost strategies have an average monthly return of 1,45%, with an average standard deviation of 6.25%, and similar to the strategies based solely on previous winners, the strategies with a longer
formation period also exhibit higher returns compared to those with shorter formation periods.
Furthermore, the zero-‐cost strategies with a shorter holding period perform relatively better than those with a longer holding period, and this is particularly noticeable for the strategies with a longer formation period. But the similarities between the zero-‐cost-‐ and winner strategies stop here. Because in contrast to the winner strategies, most of the zero-‐cost strategies are not able to produce returns that are statistically significantly different from the benchmark. For the adjusted Welch’s t-‐test, only one zero-‐cost strategy is able beat the benchmark at a 90% significance level.
The 9/3-‐zero-‐cost strategy generates an impressive average monthly return of 1.85%, more than double the return on the benchmark of 0.85%.
To illustrate the performance of the strategies, figure 9.1 presents the results from table 9.1 in a visual manner. Doing so highlights some of the key points, particularly the patterns emerging from the different formation-‐ and holding periods.
Figure 9.1: Return over time on zero-‐cost portfolios (Index: 1 = Strategy start)
The figure shows the average monthly return on the winner, loser and zero-‐cost portfolios for all of the 16 distinct momentum strategies. The x-‐axis represents the formation period and portfolio type, while the color of the individual bars represents the holding period (see the legend).