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# Momentum Study

Submitted By Neelaksh
Words 1965
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MxN strategy

Here, we specifically use a strategy that selects stocks on the basis of returns over the past M months (i.e. formation period) and holds them for N months (holding period). This is called as the M x N strategy. At the beginning of each month t, the candidate stocks are ranked in descending order on the basis of their returns in the past M months. The top decile portfolio is called the “winners” portfolio and the bottom decile is called the “losers” portfolio. This strategy involves, simultaneously buying the winner portfolio and selling the loser portfolio and then holding this position for N months (total number of months).

The strategies we have considered in our project involve selecting stocks based on their 3-month and 6month returns (for formation period). We then consider holding periods of 3, 6 and 12 months for stocks selected on the basis of returns in 3-month formation period--thus generating trading strategies: 3x3, 3x6 and 3x12. On the basis of returns in 6-month formation period, however, we consider a holding period of only 12 months, thus generating the 6x12 trading strategy. However in the project report analysis of only 6x12 strategy is laid down due to non-completion of other strategies.

Methodology

In this paper, we explain the methodology for 6 x 12 strategy for illustration purpose, although similar methodology has been used in other strategies as well. The analysis is performed using first six months data for portfolio formation and next twelve months for portfolio testing period i.e. 6x12 strategy. As the study uses 104 months data (from January 2003 to August 2011), there are 86 winner and loser portfolios each for the testing period.

Measurements of returns
Stock returns are measured monthly on adjusted monthly price of the companies by using the formula ln(Pn/Pn-1) where Pn and Pn-1 are the adjusted closing prices of the last trading day of the two relevant months.

Formation Period
In case of a 6x12 strategy, if the portfolio has to be formed for the month of August 2003, then the cumulative returns of the selected stocks for last 6 months are calculated and then arranged in a descending order. For every stock i in the sample (CNX 100), the cumulative returns (CR) for the prior 6 months will be calculated as:

CR1= ln(P1/P0) + ln(P2/P1) + ln(P3/P2) + ln(P4/P3) + LN(P5/P4) + ln(P6/P5)

Where, P0, P1, P2, P3, P4, P5, P6 are adjusted monthly closing prices for January, February, March, April, May ,June and July respectively. The stocks are arranged in descending order. Based on these rankings, ten decile portfolios are formed. Each decile portfolio consists of stocks weighed equally in that decile. Top decile portfolio forms winner portfolio and bottom decile portfolio forms loser portfolio.
It is to be noted that a portfolio is constructed by going long on the winner portfolio and short on the loser portfolio. This is done on a monthly basis and the step is repeated 86 times for the period starting August 2003 and ending on August, 2011 as mentioned above.
Holding Period
After the winner and loser portfolios are identified in a given month for a given formation period, the following calculations are to be done for the holding period.

Step 1: Calculating month-on-month stock returns for all stocks selected in winner and loser portfolio
The first step involves calculating month-on-month returns for all candidate stocks in the winner and loser portfolio for each of the 12 months holding period as shown in the fig. below.

Where:
R(1,1) is the month-on-month return of stock 1 for the month of August 2003;
R(12,10) is the month-on-month return of stock 10 for July 2004 and so on.

Step 2: Calculating Average Return of Stocks
After Step 1, average of the stock returns are calculated for each month of the 12-month holding period for each winner and loser portfolio. Likewise, average returns are calculated for the winner and loser portfolios separately for each other 86 iterations as shown in the fig. below.

Step 3: Calculating Cumulative Average Returns
The monthly ARs are used to calculate the Cumulative Average Returns (CARs) in each month (t), where t=1…..12 during holding period, this step is repeated 86 times each i.e. for 86 winner and 86 loser portfolios for 6x12 strategy.

Where:
CARW,1 denotes the Cumulative average returns for the winner portfolio for the one month and
CARw,12 denotes the cumulative average returns for the winner portfolio for two months and so on.
Similarly, cumulative average returns are calculated for loser portfolio.

Step 4: Mean Cumulative Average Returns
Then average the CARs for these 86 portfolios are used to get Mean Cumulative Average
Returns (MCARs).

Step 5: Mean Average Returns
The mean average returns (MARs) are calculated by averaging the ARs for the 86 portfolios for the 86 months.

Where,
AR(1,1) and AR(1,86) are the Average Returns for the 1 month of the holding period for the 1st and 86th portfolios respectively.

AR (12,1) and AR(12,86) are the Average Returns for the 12stmonth of the holding period for the 1st and 86th portfolios respectively.

Test of Significance
MCARW (MCARL) indicates how much cumulated returns stocks in the winner (loser) portfolio earn on an average during 12 months in test period.

If markets are efficient but weak then MCARW minus MCAR = 0
The momentum hypothesis implies that MCARw minus MCAR >0
The two tests -- MCAR and MAR are used to test the hypothesis. The test of MCAR verifies significance of momentum returns and show if the returns grow stronger or are reversed at some stage during holding period on a cumulative basis. The test of MAR helps one to identify on a monthly basis whether the momentum returns are getting built up or reversed.

Results and Analysis
We have analysed the results of the 6 x 12 strategy using MCAR and MAR tests

Mean Cumulative Average Returns Test
As shown below, the winner portfolio delivers a 1.64 percent return in the first month of testing period that goes on increasing to 12.14 percent in the 12th month of the testing period.
Similarly, the MCAR of loser portfolio in the 1st month of the testing period is 1.53 percent, which increases to 11.03 percent at the end of the 12th month of the testing period. Column 3 of table gives the mean cumulated average return of the long winner and short loser portfolio. The return on the momentum portfolio i.e. the difference between MCARw and MCARL becomes 3.88 percent in the 6th month, which is the highest in the 12 months testing period and reduces to 1.11 percent at the end of the 12 month testing period. Interestingly in this 12 month testing period, the MCAR of loser portfolio remains positive. This can be contributed to the fact that in the test period of January 2003 to August 2011, the CNX 100 saw a huge rally during which it climbed to a peak of 6204 on 4th January 2008, fell to a low of 2456 on October 24, 2008 and rose again to close at 4921 on 30th August 2011.

It may be observed from Chart, that although the returns of both the loser and the winner portfolio rise throughout the 12-month period, the pace of increase slows down for both portfolios after the 6th month. It may also be observed that the MCAR of winner portfolio to diverge from the loser portfolio till the 6th month and after remaining widely divergent till the 8th month, the gap narrows subsequently. Thus, from MCAR test, it becomes clear that irrespective of the market direction, the difference remains positive indicating that the strategy is market neutral (i.e. non-directional market strategy)

Mean Average Test (MAR)
The Mar of a portfolio denotes the mean average return of the portfolio in the particular month of the testing period. From the table below, of winner portfolio, it is clear that for the initial 5 months there is significantly high MAR. MAR test helps us to identify whether it is the winner or loser portfolio that runs out of the momentum and returns of which portfolio are reversed in the first instance.

Thus we can see a strong reversal in the momentum returns for the winner portfolio in the sixth month of the test period.
Interestingly MAR of the loser portfolio shows an increase in the 2 month to 1.78 percent from 1.53 percent in the 1st month. In the third month there is a strong fall in the MAR which becomes 0.54 percent and thereafter a small upsurge to become 1.07 percent and 0.89 percent in the 4th and 5th month respectively before dropping to -0.07 percent in the sixth month of the testing period. After the sixth month there is a reversal and loser portfolio starts to rally. Sixth month onwards the MAR of the portfolio increases to 1.18 percent in the twelfth month.

Thus, overall we can see from the MAR figures that the winner portfolio shows a reversal in momentum in the fifth month while the loser portfolio shows a reversal in momentum in the fourth and seventh month of the holding period.
To answer the question as to whether the superior returns derived by investing in loser portfolio are just the compensation of higher risk or there is presence of genuine momentum profits, we calculate the average betas of the winner and loser portfolios during the test period. The average beta of 86 loser portfolios is 1.0364 and that of the winner portfolio is 0.9502 and the two are not significantly different from each other. Hence, the superior returns observed in momentum portfolio cannot be attributed only to compensation for higher risk.

To test whether the momentum profits do exist in shorter formation and holding period, the study is required to be done for shorter duration formation and holding periods. It is evident from the discussion of 6x12 strategy that though MCAR remains positive for the entire holding period, from seventh month onwards loser portfolio shows superior MAR to the winner portfolio, implying that the trend of winner portfolio outperforming loser portfolio is completely getting reversed. It shows that going by the results of this study, for a momentum portfolio formed using six months returns, it is better to have a holding period for six months rather than of 12 months. Because one can see, that momentum loses steam from seventh month onwards. Having said that, it is worth mentioning here that this conclusion is based on the result of current study only.

Conclusion There is strong evidence of momentum profit for the short-term formation-test period. For the trading strategy and 6x12, we found presence of momentum profits. After a period of 6 to 8 months, reversal in momentum takes place, as the winner and loser portfolio returns start to converge. Further, it was found that the average risk of the winner portfolios was not significantly different to loser portfolio, thus proving evidence that the superior momentum returns are not only due to compensation for higher risk. In other words, there is empirical evidence against weak form of market efficiency in the Indian market. These results are consistent with those of the seminal studies by Jegadeesh and Titman (1993, 2001) and De Bondt & Thaler (1985, 1987 and 1990) in the US markets. To conclude, the study provides a strong evidence of short-term profits through the use of momentum strategy.

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