Model X - Fuqua School of Business



Independent Study

on

Co-integration Trading Model

By

Paul Chong

Mike Cormier

Kristian Humer

David Jorgenson

Geoffrey Keegan

29 April 29, 2004

Executive Summary

The focus of the study was on improving on the co-integration trading model developed as part of the final project in Finance 453 – Global Asset Allocation - under the supervision of Professor Campbell Harvey.

The study has three distinct sections. The first section deals with using three screening levels to improve the performance of the model. While it worked extremely well in some cases, the strategy was frequently whipsawed around (trading based on recent convergence that did not continue to materialize) during long periods of divergence. Overall, the results conflicted with our intuition that the price ratio would converge to historical levels and that short-term momentum in the direction of the long-term ratio would accurately signal further convergence.

The second section covers a pairs trading strategy based on the area between curves. The intuition is that the area will give a consistent prediction of convergence, whether the ratios have diverged quickly, or slowly over a number of days. While this strategy deserves consideration as a pairs trading signal, it does not seem robust enough by itself to deserve our full confidence. There does not seem to be enough persistence in the behavior of our two simple variables to warrant trading on these signals alone. However, for certain stock pairs in certain situations, these signals can be quite strong, and seem to predict convergence with some accuracy.

Lastly, we use a probit model to predict the likelihood of a convergence or divergence of the price ratio to its moving average. The idea is to trade when the probability is either very high or very low. While the results are encouraging, the model needs to be more robust and stable as its out-of-sample performance is less than encouraging.

In conclusion, the three improvements that are considered exhibited hopeful improvements over the original model. However, the results are not as consistent, robust or stable as we envisioned.

Section 1 – Model improvement based on screening levels

1. Model Intuition

Stock prices of similar companies may be expected to converge over time to historical relationships. The greater the current disparity from the long-term stock price ratio between the two stocks, the more likely they are to converge. Finally, when there is short-term momentum in the direction of the long-term ratio, one can be more confident that there is a trend towards convergence.

The chart below shows the 5-day and 5-year moving averages for the GM/Ford stock price ratio. Ideally, the model would not trade at point A because, while the current price ratio is significantly higher than the long-term ratio, the ratio has been diverging, rather than converging, in the short-term. The model should trade at point B (short GM, long Ford), since now there is evidence of a convergence trend.

[pic]

2. Model Specifications

The model is convergence-based, so we constrained the trading to only shorting the expensive stock and going long the inexpensive stock based on the current (or short-term) price ratio. We also incorporated a momentum factor, which only yields a trade signal when the price ratio has been converging towards the long-term moving average. Lastly, we included a hurdle that the percentage difference between the short-term and long-term moving average price ratio had to be above in order to trade. This last model parameter attempts to eliminate some of the noise that exists when the short-term and long-term moving averages are close to each other.

In summary, the model has three screening levels:

(1) Current Valuation: Is the short-term moving average price ratio above or below the long-term moving average price ratio?

(2) Convergence Momentum: Is the price ratio moving towards the long-term moving average?

(3) Ratio Difference Threshold: Is the difference between the short- and long-term price ratios above a specific percentage in absolute terms?

We examined several different definitions of the (1) short-term price ratio (1-, 5-, 10-, and 15-day moving averages), (2) long-term price ratio (30-, 60- and 90-day, 1-, 2-, 3-, 5, and 10-year moving averages), and convergence momentum duration (5-, 10-, and 15-day trailing price ratio changes). We found that, in general, the trading strategy performed better with very short-term moving averages as a proxy for the current price ratio (we decided on a 5-day moving average), since 10- and 15-day averages limited the number of extreme observations and also resulted in the model being late to recognize a convergence trend. Also, longer-term moving averages worked better as a proxy for the long-term historical average ratio (we determined that 5 years was probably the minimum duration that should be used).

3. Results

Ford/GM:

Looking first at Ford and GM, the in-sample results were very interesting, if not consistently profitable. Based on an in-sample period of daily data from 11/6/1984 through 12/31/1998, the first screening factor alone (short expensive/long inexpensive based on the price ratio) consistently generated negative returns. This result indicates that convergence is not more common than divergence on a day-to-day basis. The addition of the second screen – requiring that the price ratio was converging towards the long-term average – boosted returns modestly, but still not significantly above 0% in many cases. Adding the third screen – requiring that the short- and long-term price ratio averages be different by a given amount in order to trade – generally resulted in higher returns, especially for thresholds in the 15% to 30% range.

The in-sample results were disappointing in that they were highly sensitive to small changes in model specifications. For example, changing the short- versus long-term price ratio from 20% to 25% could potentially change the annual return by 20 percentage points. In the most severe example we uncovered, using 10-day and 5-year moving averages and 10-day momentum, changing the ratio difference threshold from 22% to 23% resulted in the annualized return leaping from negative 25.9% to positive 34.5%. This was due to several observations in 1998 where the difference in the 10-day and 5-year moving averages hovered between 22% and 24% and those observations above 23% performed extremely well. Similarly, using a 10-day rather than a 5-day momentum factor might lead to analogously diverse performance. These findings were discouraging, suggesting that the model would likely not be robust through time and across industries.

Overall, we concluded that the Ford-GM model that was most robust across time and for small changes in model parameters had the following specifications:

• Short-term moving average: 5 days

• Long-term moving average: 5 years

• Convergence momentum signal: Price ratio had converged towards long-term moving average over previous 5 days

• Short- and long-term price ratio difference threshold: 22% (at 25%, the model only traded on 3.7% of days and at 20% is was not nearly as profitable)

In-sample, this model generated a 29.9% annualized return. It traded on 7.9% of the days (179 out of 2,273). Annual results are summarized below:

|In-Sample |1990 |1991 |1992 |1993 |1994 |1995 |1996 |

|Total Return |9.5% |(2.8%) |25.6% |(15.6%) |(5.9%) |22.5% |(5.0%) |

|# Days Traded |422 |1 |56 |91 |113 |128 |34 |

The strategy generated an annualized return of 9.5% out-of-sample, which is fairly solid for a market-neutral strategy. It was also nice to see that the return was generated over a decent number of trading days – almost 35% of the potential days. From the beginning of 2000 through the rest of the out-of-sample period, the short-term GM/Ford price ratio remained significantly higher than the 5-year moving average, meeting the percentage difference threshold regularly and generating trades whenever the ratio started to converge. While the out-of-sample performance is fair, we were disappointed by the volatility of returns. Setting the percentage difference threshold to 25% would have generated a 17.3% annualized return out-of-sample, but (despite a 52.9% in-sample return) we concluded that these parameters did not result in enough trading frequency in-sample (3.7% of days).

4. Other Industry Pairs

In extending the model to other industry pairs, we took two approaches. First, we looked at how the best model for the entire Ford-GM history worked for other pairs, and the second was to determine the best in-sample model for each individual pair and then examine the out-of-sample results. The first approach could obviously not have been implemented in practice, as it uses data for the entire period. However, we thought it would be interesting to examine how the best model for one industry translates to another unrelated industry.

The best Ford-GM model for the entire period from 11/5/1984 to 4/19/2004 used 5-day and 5-year moving averages, 5-day momentum, and a 25% threshold for the percentage difference between the moving averages. These parameters yielded a 22.8% annual return and traded on 13.7% of days. This model also worked fairly well over time and for small changes in the parameters.

1) P&G/Colgate: The Ford-GM model worked fairly well in the case of P&G and Colgate, returning 65.7% annualized and trading on 5.9% of days. However, 198 of the 214 total trades took place in 2000 and 2001. During these two years, Colgate outperformed P&G fairly consistently, but when there was evidence of momentum in P&G’s favor, the model picked it up and performed very well.

During the in-sample period from 11/5/1984 to 12/31/1998, the best model for the P&G/Colgate pair used 5-day and 10-year moving averages, 5-day momentum, and a 10% difference in moving averages threshold. In-sample, this model generated a 23.9% annual return and traded on 12.5% of days. Out-of-sample the model was even better – a 36.8% annual return trading on 35.2% of days. Annual results are below:

|In-Sample |1995 |1996 |1997 |1998 |

|Total Return |(2.9%) |3.6% |5.0% |7.0% |

|# Days Traded |24 |34 |61 |13 |

|Out-of-Sample |1999 |2000 |2001 |2002 |2003 |2004 YTD |

|Total Return |19.2% |2.8% |20.3% |14.8% |(1.6%) |N/a |

|# Days Traded |31 |126 |117 |79 |77 |0 |

2) AIG/Progressive: The Ford-GM model failed in the case of AIG and Progressive, losing 17.5% per year while trading on 15.8% of days. A likely reason for this underperformance is that Progressive’s stock appreciated an incredible amount over the 20-year period examined – from $0.73 to $90.15 per share, split adjusted. AIG’s appreciation of 25 times was impressive, but paled in comparison. As a result, the stock price ratio regularly did not tend towards the moving average. Instead, the ratio frequently continued to move in Progressive’s favor, with a few days of momentum towards the long-term average initiating a trade that was not rewarded.

During the in-sample period from 11/5/1984 to 12/31/1998, the best model for the AIG/Progressive pair used 5-day and 10-year moving averages, 5-day momentum, and a 20% difference in moving averages threshold. In-sample, this model generated a 50.8% annual return and traded on 23.5% of days. Out-of-sample, however, the model was unsuccessful, losing 3% per year. Annual results are below:

|In-Sample |1995 |1996 |1997 |1998 |

|Total Return |(0.7%) |0.1% |(1.7%) |33.5% |

|# Days Traded |48 |20 |64 |113 |

|Out-of-Sample |1999 |2000 |2001 |2002 |2003 |2004 YTD |

|Total Return |(8.0%) |(20.9%) |17.9% |2.3% |13.0% |10.2% |

|# Days Traded |34 |131 |73 |76 |102 |43 |

3) Coke/Pepsi: The Ford-GM model failed in the case of Coke and Pepsi, losing 5.3% per year (trading on 18.1% of days). This result was surprising, given that the Coke/Pepsi stock price ratio traded within a pretty tight range between 1 and 1.5 for most of the period. However, graphical evidence demonstrates that there were long periods of divergence – the short-term ratio remained above the long-term ratio from 1993 to 1999 as Coke outperformed before a reversal early in 2000 leaving the short-term ratio below the long-term ratio to date. While the ratio ultimately converged in 1999-2000, the long time period of divergence created a lot of noise in the model and resulted in several unprofitable trading signals.

During the in-sample period from 11/5/1984 to 12/31/1998, the best model for the Coke/Pepsi pair used 5-day and 5-year moving averages, 5-day momentum, and a 35% difference in moving averages threshold. In-sample, this model generated a 69.0% annual return and traded on 6.3% of days. Out-of-sample, however, the model was unsuccessful, losing 23.2% per year. Again, we believe the model failed because of the lack of a tendency for the two stocks to converge towards a long-term average. The strong return in 1998 demonstrates the potential of the model to generate outsized returns during convergence periods. We would expect there to be another period of convergence soon in which Coke outperforms Pepsi, but until such a trend develops the out-of-sample results are extremely poor. Annual results are below:

|In-Sample |1990 |1991 |1992 |1993 |1994 |1995 |

|Total Return |N/a |N/a |(5.5%) |(1.8%) |N/a |N/a |

|# Days Traded |0 |0 |53 |20 |0 |0 |

4) Merck/Pfizer: The Ford-GM model performed well with annual returns of 9.5% and trading on 12.7% of days in our total sample period. The price ratio of Merck to Pfizer has been dropping precipitously over our data sample period. The ratio currently stands around 1.25 but totaled approximately 5.0 in the mid-1980s. The continual and steady slide in the price ratio means that the current price ratio almost never crosses the historical moving average. Unfortunately this makes it difficult to predict returns based on our model.

During the in-sample period from 11/5/1984 to 12/31/1998, the best model for the Merck/Pfizer pair used 5-day and 10-year moving averages, 5-day momentum, and a 40% difference in moving averages threshold. In-sample, this model generated a 25.2% annual return and traded on 15.8% of days. The model was not as successful out-of-sample, as annual returns only equaled 1.9%, with trading activity on 19.7% of days. Annual results are below:

|In-Sample |1995 |1996 |1997 |1998 |

|Total Return |N/A |N/A |0.6% |12.1% |

|# Days Traded |0 |0 |27 |93 |

|Out-of-Sample |1999 |2000 |2001 |2002 |2003 |2004 YTD |

|Total Return |10.8% |(2.5%) |25% |7.6% |1.4% |(13.2%) |

|# Days Traded |84 |72 |30 |49 |19 |36 |

5) ChevronTexaco/ExxonMobil: The Ford-GM model essentially does not work for this pair. In our entire sample period, a 25% difference in moving averages threshold only produces 1 trading day. Interestingly, that 1 day produces a hefty 1.2% return. Reviewing the data, there is a very tight band, especially within our in-sample periods, and although there are several days where the difference is in the low 20% range, only once does it actually cross our threshold. The price ratio between the two stocks does change through time, although quite gradually. In 1984 the price ratio was 3.0, 2.0 in late 1987, back to 3.0 in 1990, and does not cross 2.0 again until 2000. The ratio currently stands at 2.1. This slow change in price ratio allows the moving average to act as a reliable proxy for the current price ratio, making it unlikely that the 25% deviation would occur.

During the in-sample period from 11/5/1984 to 12/31/1998, the best model for the ChevronTexaco/ExxonMobil pair used 5-day and 5-year moving averages, 5-day momentum, and a 15% difference in moving averages threshold. In-sample, this model generated a 72.1% annual return but traded on only 4.6% of days. The model was also successful out-of-sample, as annual returns totaled 13.0%, and 9.3% of trading days. Annual results are below:

|In-Sample |1995 |1996 |1997 |1998 |

|Total Return |N/A |0.3% |N/A |8.0% |

|# Days Traded |0 |2 |0 |21 |

|Out-of-Sample |1999 |2000 |2001 |2002 |2003 |2004 YTD |

|Total Return |2.8% |7.2% |1.1% |N/A |(1.8%) |N/A |

|# Days Traded |7 |80 |22 |0 |10 |0 |

Model limitation

We wanted to examine the model results using a stock and a relevant industry index rather than two stocks within the same industry. Our intuition was that a stock might be more likely to have a consistent long-term relationship and a stronger trend towards convergence with a diversified industry index rather than with a single company from that index. However, upon further investigation, such analysis proved problematic. There are a limited number of industries with more than a handful of companies that have the following necessary characteristics: (1) 20-year histories as public companies; (2) relatively consistent business profiles (products, exposure to specific macro risks, etc.); and (3) limited merger activity altering the nature of the business (consolidation was a significant barrier to creating a diversified index in many industries). The indices we were able to construct were often dominated by a single firm or two firms. As a result, we felt it made sense to create trading pairs with the two most appropriate companies (most stable, largest, most homogenous companies) in selected industries.

5. Conclusion

We were not very satisfied with the results of this model. While it worked extremely well in some cases, the strategy was frequently whipsawed around (trading based on recent convergence that did not continue to materialize) during long periods of divergence. Therefore, it is imperative to pick company pairs where the short-term and long-term moving averages cross frequently. Another discouraging finding was the model’s high sensitivity to small changes in the sample period examined and model parameters. Overall, the results conflicted with our intuition that the price ratio would converge to historical levels and that short-term momentum in the direction of the long-term ratio would accurately signal further convergence.

Section II: Pairs Trading Strategy Based on the Area Between Curves

We attempted a simple strategy based entirely on quantifying the area between two moving averages as they diverge. Our intuition is that the area will give a consistent prediction of convergence, whether the ratios have diverged quickly, or slowly over a number of days.

1. Variables

• Price Ratio – As in the other models, we are basing our trading strategy upon the simple ratio of the closing prices of two similar stocks. In this model, we chose to use a three day moving average as the short term variable, in an attempt to smooth some of the daily noise in this variable.

• Cumulative Area – Area per measurement period (in this case one day) is defined as simply the absolute value of the distance between the short term moving average and the long term moving average. The cumulative area is the sum of all of the areas since the curves have intersected. So, as two curves diverge, the area will necessarily increase.

• Steepness – Steepness is our attempt to quantify the rate at which curves are diverging (or the steepness of the short term spike). It is simply the long term moving average divided by the number of days since a crossover.

2. Model

Our model is arranged with three possible trading evaluations for each pair of stocks under consideration. We compare the short term moving average (STMA) with three different long term moving averages (LTMA). Our first LTMA is 1 Year, then 5 Year, and finally 10 Year. Using three different possible curves should allow us to find the optimum curve for each pair. This model depends greatly upon the crossover of the two moving average curves, so finding an LTMA curve that intersects the STMA curve greatly increases performance.

The model makes daily trading decisions based upon the closing prices of the stock pairs, or more precisely the level of the LTMA and STMA at the end of the day. It compares the cumulative area and the steepness of the STMA to threshold levels set by the user, and if conditions are met indicates a long/short or short/long strategy for tomorrows trading (which stock to short depends on whether the STMA is above or below the LTMA). Within each moving average pair, the user can turn on or off each of the two variables used to evaluate trades. If both are turned on, trades will only be made if the cumulative area and steepness of the divergence exceed their thresholds.

Our theory is that, for each pair, the cumulative area will max out at a predictable level, and that once this level is reached convergence is likely. Similarly, it might be that very rapidly diverging curves (high steepness) are more likely to converge. One question that presented itself before the model was built was whether optimal threshold levels found for one pair would work for other stock pairs.

Our in sample period for all pairs is 11/84 to 12/98, and our out of sample period 1/99 to 4/04.

3. GM/Ford

As with our other models, we first analyzed Ford and General Motors. Our first step was to try and find consistent behavior in the cumulative area and steepness variables. Below are charts of this data, along with charts of the STMA and LTMA for each of the three curve comparisons.

3 Day STMA – 1 Year LTMA

[pic]

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3 Day STMA – 5 Year LTMA

[pic]

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3 Day STMA – 10 Year LTMA

[pic]

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As these graphs show, our variables tend to climb as expected then plummet at the crossover days to zero. The cumulative area variable climbs more steadily, while the steepness variable is more humped in shape. To the eye, this suggests that the steepness variable would be less effective, because we need our variables to climb past a trigger point then quickly plummet.

Of course, cumulative area will increase while the curves are converging as well as diverging. Our model will be most effective if we can find the point on the cumulative area upslope that corresponds to the beginning of the convergence of the ratios. Not only that, we need to find a consistent area value above which the curves are likely to be converging rather than diverging. After a quick scan of these first charts it seems like the 3Day/5Year strategy offers the most consistent cumulative area and steepness performance. Our hope is that this behavior persists through the out of sample period. Based on the chart, it looks like setting a threshold level of approximately 100 for cumulative area and near .2 for steepness might provide good returns. In fact, some manual optimization provides the following results:

|On/Off |Value | | | | | | |

|1 |105 |Area Threshold | | | | |

|1 |0.2 |Steepness Threshold | | | |

| | | | | | | | |

|In-Sample Results | |Out of Sample Results |

|2314 |Total Days | |1331 |Total Days | |

|603 |Days Traded | |841 |Days Traded | |

|26.1% |% Days Traded |63.2% |% Days Traded | |

|55.9% |Total Return | |-36.4% |Total Return | |

|20.8% |AACR Days Traded |-12.9% |AACR Days Traded |

In sample performance is positive, reflecting the consistent performance of the data. However, the data obviously does not maintain this performance through the out of sample period. In fact, in 2000 the ratios entered a long period of divergence as shown in the following chart:

[pic]

We would have been killed had we relied on these two metrics to trade these stock pairs, and we must question the persistence of these variables. How much confidence can we have that the in sample results will provide effective trading signals out of sample? Hopefully another trading pair will provide more consistent results.

4. Colgate / Proctor & Gamble

[pic]

As was the case with Ford and GM, this pair seems to have the most consistent behavior with the 3Day/5Year curves, and the chart above suggests that a cumulative area value near 23 and a steepness value near .06 might be best. Results below were positive for the out of sample period, and the strategy resulted in a high number of trading days.

|On/Off |Value | | | | | | |

|1 |22 |Area Threshold | | | | |

|1 |0.05 |Steepness Threshold | | | |

| | | | | | | | |

|In-Sample Results | |Out of Sample Results |

|2314 |Total Days | |1331 |Total Days | |

|520 |Days Traded | |569 |Days Traded | |

|22.5% |% Days Traded |42.7% |% Days Traded | |

|52.6% |Total Return | |52.7% |Total Return | |

|23.2% |AACR Days Traded |21.1% |AACR Days Traded |

5. AIG and Progressive

[pic]

The 3Day/1Year strategy appeared most promising for this pair, but we were unable to obtain a positive return when including cumulative area. When using steepness alone, returns in the in sample period were positive, but as shown below returns out of sample were a disaster:

|On/Off |Value | | | | | | |

|0 |6 |Area Threshold | | | | |

|1 |0.31 |Steepness Threshold | | | |

| | | | | | | | |

|In-Sample Results | |Out of Sample Results |

|3326 |Total Days | |1331 |Total Days | | |

|489 |Days Traded | |723 |Days Traded | |

|14.7% |% Days Traded | |54.3% |% Days Traded | |

|36.0% |Total Return | |-65.6% |Total Return | |

|17.5% |AACR Days Traded |-31.6% |AACR Days Traded | |

Clearly, this model does not inspire confidence when trading this pair of stocks.

6. Coke and Pepsi

When using the longer LTMA’s, the moving averages never cross, so our variables just increase throughout the sample period. The 3Day/1Year chart is below, and indicates that there might be some repeatability in the cumulative area variable near the 20-30 range and in the steepness variable near .15.

[pic]

In fact, the best model in sample does not use the steepness variable, and out of sample results are positive.

|On/Off |Value | | | | | | |

|1 |15 |Area Threshold | | | | |

|0 |0.2 |Steepness Threshold | | | |

| | | | | | | | |

|In-Sample Results | |Out of Sample Results |

|3326 |Total Days | |1331 |Total Days | | |

|695 |Days Traded | |468 |Days Traded | |

|20.9% |% Days Traded | |35.2% |% Days Traded | |

|16.8% |Total Return | |38.0% |Total Return | |

|5.9% |AACR Days Traded |19.3% |AACR Days Traded | |

7. Merck and Pfizer

[pic]

Similar to Coke and Pepsi, the 5 and 10 year LTMA strategies do not provide enough crossover to provide easy optimization of our threshold variables. The 3Day/1Year chart above suggests that a cumulative area setting near 25 and a steepness setting near .2 might work. In fact, as the results below demonstrate, a positive return in sample was only available when using the steepness variable alone, and this resulted in zero out of sample trading days.

|On/Off |Value | | | | | | |

|0 |30 |Area Threshold | | | | |

|1 |0.4 |Steepness Threshold | | | |

| | | | | | | | |

|In-Sample Results | |Out of Sample Results |

|3326 |Total Days | |1331 |Total Days | | |

|297 |Days Traded | |0 |Days Traded | | |

|8.9% |% Days Traded | |0.0% |% Days Traded | |

|11.5% |Total Return | |0.0% |TotalReturn | | |

|9.9% |AACR Days Traded |N/A |AACR Days Traded | |

8. Chevron and Exxon

[pic]

The 3Day/5Year chart above shows some repetitive behavior in the cumulative area variable, but not much consistency in the steepness variable. In fact, the best in sample returns were obtained with a cumulative area threshold of 30 and the steepness variable turned off. The results below, however, demonstrate a high number of trading days in the out of sample period with a relatively low return.

|On/Off |Value | | | | | | |

|1 |30 |Area Threshold | | | | |

|0 |0.03 |Steepness Threshold | | | |

| | | | | | | | |

|In-Sample Results | |Out of Sample Results |

|2314 |Total Days | |1331 |Total Days | | |

|1135 |Days Traded | |996 |Days Traded | |

|49.0% |% Days Traded |74.8% |% Days Traded | |

|52.3% |Total Return | |9.3% |Total Return | |

|10.0% |AACR Days Traded |2.3% |AACR Days Traded | |

9. Conclusion

Below is a chart of our best model performance for all the pairs:

|  |Strategy |  |Settings |  |In Sample |  |Out of Sample |

Pair |STMA |LTMA | |Area |Steepness | |% Days |AACR | |% Days |AACR | |GM/F |3 Day |5 Year | |105 |0.2 | |26.1 |20.8 | |63.2 |(12.90) | |CL/PG |3 Day |5 Year |  |22 |0.05 |  |22.5 |23.2 |  |42.7 |21.10 | |AIG/PGR |3 Day |1 Year | |N/A |0.31 | |14.7 |17.5 | |54.3 |(31.60) | |KO/PEP |3 Day |1 Year |  |15 |N/A |  |20.9 |5.9 |  |35.2 |19.30 | |MRK/PFE |3 Day |1 Year | |N/A |0.4 | |8.9 |9.9 | |0 |0.00 | |CVX/XOM |3 Day |5 Year |  |30 |N/A |  |49.0 |10.0 |  |74.8 |2.30 | |

We believe that this strategy deserves consideration as a pairs trading signal, but it does not seem robust enough by itself to deserve our full confidence. There does not seem to be enough persistence in the behavior of our two simple variables to warrant trading on these signals alone. However, for certain stock pairs in certain situations, these signals can be quite strong, and seem to predict convergence with some accuracy.

Section III: Probit Analysis

1. Introduction

The aim of this section is to construct a co-integration model that will predict the future returns of two comparable companies. We analysed the price ratio of the General Motors Company (“GM”) and the Ford Motors Company (“F”) from November 1984 until March 2004. The 3-day rolling average price ratio was then compared to the 90-day rolling average price ratio. We separated the data into two parts: the 3-day rolling average price ratio is above the 90-day rolling average price ratio (part 1); the 3-day rolling average price ratio is below the 90-day rolling average price ratio (part 2). In both parts one and two of the data we assigned a 1 to every point that is mean reverting and a 0 to every point that was not. By using a Probit analysis, we estimated the probabilities of the next point mean reverting. The idea is to trade when the probability is either very high or very low. The results of this analysis are encouraging with significant positive returns; however the stability of the model needs to be improved.

2. Methodology

We constructed a co-integration model analysing the price ratios of two comparable companies over the sample period from November 1984 to March 2004. Comparable companies are companies that have operated in the same industry, have been subject to the same external macroeconomic factors and have had similar market capitalisations over the sample period in question. The two companies used in our analysis are “GM” and “F”.

The key assumption we are making is that stock prices of comparable companies are expected to converge over time to historical relationships. Illustration 1 shows that the stock price conversion assumption applies to our two sample companies “GM” and “F”.

[pic]

The greater the current disparity from the long-term stock price ratio between the two stocks, the more likely they are to converge. Finally, when there is short-term momentum in the direction of the long-term ratio, one can be more confident that there is a trend towards convergence.

Having observed that the “GM”/”F” price ratio trades around an average, we compared the 3-day rolling average price ratio to the 90-day rolling average price ratio. The data was separated into two parts: 3-day rolling average price ratio is above the 90-day rolling average price ratio (part 1); 3-day rolling average price ratio is below the 90-day rolling average price ratio (part 2). The reasoning behind separating the data into two parts and running two separate Probit regressions is that the behavior of the 3-day rolling average price ratio above the 90-day rolling average price ratio is asymmetrical to the behavior of the 3-day rolling average price ratio below the 90-day rolling average price ratio. This is proven in the section “Results” of this section where it can be observed that the probability cut-off points for equation 1 (above the 90-day rolling average) are different to the cut-off points for equation 2 (below the 90-day rolling average) when determining maximum returns.

We continued by assigning the number 1 to any point which is mean reverting (in both part 1 and part 2 of the data) and the number 0 to all others. A Probit regression was created to determine the probability of mean-reversion occurring in period t. The dependent variable was defined as a 1 or a 0 – 1 equals mean-reversion and 0 equals a move away from the mean. We created three independent variables: distance from the 90-day rolling average price ratio in period t-1, the number of days since crossing the moving average in period t-1 and the area above/below the 90-day rolling average price ratio curve in period t-1.

Running the two Probit regressions results in probability estimates that period t is mean-reverting. Two probability cut-off points are chosen in part one and part two of the data. In part one of the data – 3-day rolling average price ratio is above the 90-day rolling average price ratio – a high probability of mean reversion indicates that we should go short “GM” and long “F” and a low probability of mean reversion indicates that we should go long “GM” and short “F”. In part two of the data – 3-day rolling average price ratio is below the 90-day rolling average price ratio – a high probability of mean reversion indicates that we should go long “GM” and short “F” and vice versa when the probability of mean reversion is low. Illustration 2 shows that by pursuing this trading strategy we potentially can trade at all points of the curve.

The current model is coded in such a way that for the probit equation above the MV, ‘1’ occurs when price ratio is lower today as compared to yesterday. Hence, the circles shaded in yellow exhibits high probability that the price ratio will be lower today as compared to yesterday’s price ratio. Thus, the strategy would be to short GM and long Ford (price ratio is calculated as GM / Ford). On the other hand, the red circle represents the price ratio diverging from the moving average and the appropriate strategy would be to long GM and short Ford. Consequently, the probability that the dependent variable is ‘1’ is rather low.

For the probit equation below the MV, ‘1’ occurs when price ratio is higher today as compared to yesterday. Hence the green circle represents high probability of ‘1’ occurring and the blue circles representing low probability of ‘1’ occurring. Effectively, the 4 different colored circles represent the 4 quadrants that will be captured by the 2 probit equations

3. Data

We determined the price ratio of “GM” and “F” for the sample period of November 1984 to March 2004. The length of the sample period was determined by the availability of individual stock prices and stock returns. Our database went back only as far as November 1984. The 3-day rolling average price ratio was compared to the 90-day rolling average price ratio. The 3-day rolling average price ratio performs better than the daily price ratio as it is less noisy and it incorporates momentum when comparing it to a short term or long term rolling average price ratio. A further analysis to be done would be to look at whether a 5-day, 10-day or 15-day rolling average would outperform the 3-day rolling average price ratio results.

We continued by comparing the 3 day-rolling average price ratio to the 30-day, 60-day, 90-day, 250-day, 1-year, 2-year, 3-year and 5-year rolling average price ratios to determine whether a short-term or long-term rolling average price ratio delivers better results when predicting mean-reversion. The 90-day average performed best with R2s of 27.960% and 20.710% for part one of the data and part two of the data respectively.

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The source for the daily stock prices and stock returns for “GM” and “F” for the sample period is Fact Set. We used the Statsgraphics 5.1 software to perform a Probit analysis with which to determine the probability of mean-reversion.

4. Results

Our in-sample period is defined as March 12, 1984 to December 31, 1998 and our out-of-sample period is defined as January 1, 1999 to March 19, 2004.

We first defined our probability cut-off points for part one of the data – 3-day rolling average price ratio is above the 90-day rolling average price ratio. Exhibit A in the appendix shows the average annualized returns for different probability cut-off point combinations. We constrained trading to when the probability of mean-reversion is greater than 70% in period t and to when the probability of mean-reversion is smaller than 10% in period t and the 3-day rolling average price ratio is above the 90-day rolling average price ratio in period t-1. These probability cut-off points were chosen on the basis that we trade on a significant amount of days. The results are summarized in the following table.

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The results from imposing these constraints are a cumulative return of 31.61% or an annual average return of 87.74%. The percentage of days traded in our out-of-sample period is 13.80%. The annualized variance is 19.6% and the largest drawdown over the out-of-sample period is 15.63%.

The limitation of the results is that it is not clear how to choose the probability cut-off points in the in-sample period. From exhibit A in the appendix it is not obvious why the 70-10 cut-off points were chosen. The probability cut-off points 70-15 for example resulted in historically higher returns. However, had we chosen the 70-15 cut-off points our returns in the out-of-sample period would have been negative (Exhibit B in appendix). Comparing exhibit A and exhibit B in the appendix, we observe a general shift from probability cut-off points that have resulted in negative annualized returns in the in-sample period to the same cut-off points resulting in positive annualized returns in the out-of-sample period. This suggests that improvements could be made in determining the variables for the Probit regression and that results might improve if different in- and out-of-sample periods were chosen.

Second, we defined the probability cut-off points for part two of the data – 3-day rolling average price ratio is below the 90-day rolling average price ratio. Exhibit C in the appendix shows the average annualized returns for different probability cut-off point combinations. We constrained trading to when the probability of mean-reversion is greater than 80% in period t and to when the probability of mean-reversion is smaller than 20% in period t and the 3-day rolling average price ratio is below the 90-day rolling average price ratio in period t-1. These probability cut-off points were chosen on the basis that we trade on a significant amount of days. The results are summarized in the following table.

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The results from imposing these constraints are a cumulative return of 4.55% or an annual average return of 13.81%. The percentage of days traded in our out-of-sample period is 15.09%. The annualized variance is 10.8% and the largest drawdown over the out-of-sample period is 19.26%.

The results for part two of the data contain the same problems as observed in part one of the data. It is not clear-cut which probability cut-off points need to be chosen and again, we observe a general shift from probability cut-off points that have resulted in positive annualized returns in the in-sample period to the same cut-off points resulting in negative annualized returns in the out-of-sample period.

Combining the results of when the 3-day rolling average price ratio is above the 90-day rolling average price ratio and below yields the following results:

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The results from imposing the 70/10 and 80/20 probability constraints when the 3-day rolling average price ratio is above and below the 90-day rolling average price ratio are a cumulative return of 36.72% or an annual average return of 49.32%. The percentage of days traded in our out-of-sample period is 14.33%. The annualized variance is 15.6% and the largest drawdown over the out-of-sample period is 19.42%.

We would like to stress that these are results prior to further stabilizing the model and do not include our best results. Please refer to exhibit E for a yearly breakdown of returns in the in-sample and out-of sample periods.

5. Model Limitations and Improvements

As discussed above, the key limitation of the model is that the probability cutoffs are determined arbitrarily. Also, while using the 3-day rolling moving average price ratio is less noisy than using the daily price ratio, there is no precise way to determine the best way to capture the dependent variable. Additionally, the number of days traded are rather low when we uses more extreme probability cutoffs. Conversely, the performance of the model suffers when we try to increase the number of days traded. A key reason for this that most of the probabilities calculated are clustered together within the 0.3 to 0.6 band.

Improvements that we can focus on include finding the optimal rolling moving average price ratio to use instead of the current 3-day moving average. Also, when performing backtesting on the model, a more robust way is to use 5 years of data to forecast the probability for the following year. This is repeated through time so as to verify the stability of the probability cutoffs that are chosen. Another area that could be improved on is the explanatory power of the probit analysis. While the adjusted R2 is encouraging, we believe that a higher R2 will lead to better predictability and hence yield better probability for use in our trading strategy. More independent variables such as volume, and ratio of call/put options traded, could be use to improve on the predictability of the model. Moreover, as the R2 is not constant through time, an improvement would be to determine the appropriate out-of-sample period to use to forecast the probabilities for different periods, i.e. using longer out-of-sample periods during periods of stability and a shorter period when the price ratio exhibits higher volatility. Lastly, while the performance of the model for this particular pair of stocks is encouraging, its performance would need to be replicated across a portfolio of stocks so as to provide diversification as well as to increase the total number of days traded so that the trading strategy becomes commercially viable.

6. Conclusion

The premise of the model is that price ratios are mean reverting, i.e. asset prices are tied together in the long term by a common stochastic trend. Thus, a trading strategy can be established to generate positive alpha to take advantage of the short-term deviations from equilibrium before they are corrected.

The model we developed uses probit analysis to help us in determining the likelihood of mean reversion in the price ratio for a given period. While the results are certainly encouraging, we felt that there are many areas where we can improve to make the model more robust and stable through time.

Exhibit A – In-Sample Probability Cut-Off Combination for Part 1 of the Data:

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Exhibit B – Out-of-Sample Probability Cut-Off Combination for Part 1 of the Data:

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Exhibit C – In-Sample Probability Cut-Off Combination for Part 2 of the Data:

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Exhibit D – Out-of-Sample Probability Cut-Off Combination for Part 2 of the Data:

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Exhibit E – Yearly Breakdown of Returns for In-Sample and Out-of Sample Periods:

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Exhibit E – Yearly Breakdown of Returns for In-Sample and Out-of Sample Periods (Continued):

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3-day price ratio

Below 90-days Moving Average

Above 90-days Moving Average

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