Ford-General Motors

Pairs Trading Analysis

Assignment #1

Global Asset Allocation

Spring 2004

The Hemline Theorists:

Fara Berkowitz

Paul Chong

Kristian Humer

Dave Jorgenson

Geoff Keegan

I. Introduction

In this document, we use the theory of cointegration to analyze various pairs trading models – going long one security while hedging market and industry-specific risk by shorting another similar security. We focus on a trading strategyinvolvingFord and General Motors (GM), but also test the model’s robustness by extending it to other industry pairs. Our initial objective was to determine whether a quantitative-based model could successfully generate excess returns through a pairs trading approach. We selected Ford and GMbecause both are relatively stable, mature companies with long trading histories. More importantly, both automakers should be similarly impacted by the same external factors, including interest rates, steel and energy prices, and unemployment. As a result, dissimilar stock price movements between the two companies should be largely due to idiosyncratic risk, rather than factors that influence the auto sector as a whole. Prior to conducting our research, we believed there might be an opportunity to capitalize on the periodicdivergence between the two stocks, generating positive excess returns while hedging market and industry exposure.

Our analysis is based on daily stock returns from January 1, 1987 to February13, 2004, covering a period of time after a more stable relationship between GM and Ford stock prices developed. See EXHIBIT I for a graph of the price ratio over time. Our model uses the historical stock price and dividend yield ratios of GM to Ford to identify periods where the current ratios are substantially different from historical norms and to employ a trading strategy centered around the belief that these ratios will converge to values more in line with history.

For the trading strategy we did not use economic or business cycle variables. The “industry pairs” are affected by the same external factors. For example Ford and GM are both affected by interest rates, fluctuations in the dollar, steel and energy prices and dividend yields. Dissimilar price movements between the two companies should be due to company specific risks such as for the automotive industry new product flow, cost cutting efforts, and pension liabilities. These variables will effect each company’s earnings. Company specific factors which would affect how Eli Lilly and Merck trade are pipeline, launch of a new drug class, loss of patent protection, and manufacturing problems.

II. Model I – Ford and GM

Model Explanation

Our first modeling approach examines the historical relationships between Ford and GM in terms of stock price and dividend yield. Based on the average historical ratio of GM stock price to Ford stock price, for example, the model would determine whether GM or Ford was more attractive based on the assumption that the future stock price ratio will converge toward the average historical ratio.

The model user can input the number of standard deviations away from the mean ratio that an observation must be for the model to issue a buy or short signal. For example, since 1987, the average ratio of GM’s stock price to Ford’s was 3.51 and the standard deviation was 0.76. If the user wants to issue a buy or short signal only if the current price ratio differs from the historical mean by at least one standard deviation, there will be no trade if the price ratio is between 2.76 and 4.27. However, if the ratio is less than 2.76, a buy signal will be issued for GM (and an offsetting short signal for Ford). Conversely, if the price ratio is greater than 4.27, the model will suggest shorting GM (and going long Ford).

The model user can also elect to “turn off” a variable and consider only stock price or dividend yield ratio in isolation. If the user wants to look at both the stock price and dividend yield ratios simultaneously, he has the option to trigger a trade when (1) both the stock price and dividend yield ratios give the same signal – buy or short, or (2) when one of the two variables signal a buy or short and the other variable is neutral or when both variables give the same signal. The second option is clearly less restrictive in terms of when trades are implemented, and results in a significantly higher number of trading days.

The trading strategy is very simple. An “overall buy signal” for each day is determined for GM, based upon the model user’s criteria for (1) which variables to include, (2) how many variable-level trade signals are required to trigger a trade, and (3) how far away each variable’s current ratio is from the historical average before a signal is issued. If GM receives an overall buy signal, the model goes long GM and takes an offsetting short position in Ford. If GM receives an overall short signal, it shorts GM and goes long Ford. The resulting trading strategy requires no net investment as the trader is long and short the same dollar value. (We assumed away transaction costs such as market impact and commissions and short-selling concerns such as the short rebate or margin requirements to simplify our analysis).

Model Objective

We discussed several alternative objectives that a trader implementing this strategy might be interested in. If a trader had only this single strategy to invest in, he would probably like to maximize cumulative return for the entire time period. However, we think it is more reasonable to assume that the Ford-GM trade would be one of many pairs trading strategies within a more diversified portfolio. Within this context, the trader would likely be more interested in the average return for this strategy on the days when a trade actually takes place, rather than over the entire time period, since he would have other trading strategies to choose from when the Ford-GM model did not offer a clear trade signal.

While seeking to maximize the average return on the days traded, we also need to consider how many days the model actually trades. For example, some model parameters may result in incredibly high daily returns on the days traded, but only trigger the model to trade on a handful of days per year. We wanted our model parameters to result in trades on at least 10% of days, to ensure more robust results and more frequently allow for a diversified portfolio with multiple pairs trades on simultaneously.

In determining the optimal model parameters for the in-sample period, we also wanted to limit the maximum drawdown from peak to trough to 15%. Our thinking was that more severe drawdowns would likely trigger capital redemptions and force us to close out some trades at unfavorable points in time. This is a long-term investment strategy, as convergence may take considerable periods of time, and the performance results must ensure that investors stick with the strategy when the relationships are diverging rather than converging.

Model Objectives

Variable to Maximize / Minimum %
of Days Traded / Maximum Drawdown
Average Annualized Return on Days Traded / 10% / 15%

Model Results

1. In-Sample Results

The in-sample analysis was based on the period from January 2, 1987 to February 12, 1999. We examined different model parameters (required standard deviations from the mean, stock price or dividend yield ratios alone and together, variable-level requirements for an overall trade signal) for the in-sample data. One important finding was that the model was rarely effective when the parameters forced the model to trade when one variable gave a neutral signal and the other a trade signal. Several models of this type actually resulted in negative average returns on days traded. We found the similarly poor results for models where one of the two variables was turned off. These results told us that the interaction between stock price and dividend yield is critical in order to predict convergence. In other words, GM’s stock price may be high relative to Ford’s, but it may also have increased its dividend yield relative to Ford’s, resulting in a higher total return even if the stock prices converge toward historical ratios. As a result of these findings, we focused on finding the optimal model parameters using both variables and requiring consistent trade signals for both variables for an overall trade signal.

The following table shows the average annualized return on days traded based on different standard deviation inputs for stock price and dividend yield.

We selected the highlighted point – 0.3 standard deviations for price ratio, 0.5 for yield ratio – based on our in-sample analysis. These parameters result in an annualized return of 65.9% on days traded, trades on 14.0% of days, and a maximum drawdown of 11.7%. The model generates 237 positive trading days to 192 negative trading days. While other standard deviation inputs result in higher average annualized returns in the above table, we feel that our chosen parameters have the most attractive characteristics in terms of high return, percentage of days traded, and low downside. For example, moving from our inputs to 0.6 standard deviations on price ratio and 0.5 on dividend yield results in a 30 percentage point increase in annual return and a maximum drawdown of 5.9%, but trades are only triggered 4.9% of total days. We did consider a 0.4 input for both variables, which resulted in a higher return and lower drawdown, but decided that the advantage our parameters offered in terms of higher trading frequency outweighed these benefits. Please see EXHIBIT II for more detailed sensitivity analysis on the model parameters.

In-sample results for our chosen model parameters are detailed below.

The implicit assumption is that the historical ratios will persist through the out-of-sample period, and our optimal parameters will maximize returns in those years as well. Below is a table of the resulting parameter values obtained through the in-sample optimization for use in the out-of-sample period.

In-Sample Optimization

Parameter / Value
Mean GM/Ford Price Ratio / 3.51
Standard Deviation of GM/Ford Price Ratio / 0.755
Sensitivity Band of Price Ratio (Std. Dev.’s around mean) / 0.3
Mean GM/Ford Dividend Yield Ratio / 1.63
Standard Deviation of GM/Ford Dividend Yield Ratio / .694
Sensitivity Band of DY Ratio (Std. Dev.’s around mean) / 0.5
Overall Trade Signal Value / 2

Both the mean and the standard deviation of the two ratios were simply computed from the in-sample dataset. There is certainly an opportunity to extend this model by attempting to forecast the movement of these ratios, but for our analysis we are assuming that the ratio is stable through time and mean reverting. Indeed, the ratios have been fairly consistent in the recent past and we are comfortable with our assumptions in this regard.

Returns are affected to a large degree by the sensitivity parameters, which set the band around the mean that is used by the model to make trading decisions. If we felt that the markets were extremely precise, and that the means for the ratios were exact, we would set these sensitivities to zero. This would in effect be saying that any movement above or below the mean ratio would be a trade signal. In reality, we do not make either of these assumptions, and the sensitivities act as a hurdle that is intended to avoid trading until prices diverge to a level where reversion is likely.

2. Out-of-Sample Results

We held out the past five years of data to test our model out-of-sample, beginning February 16, 1999. The out-of-sample performance was impressive, with a 66.9% annualized return on days traded and trades on 12.8% of days. On the negative side, there was a 15.3% drawdown between February and March of 2003. However, the strategy did rebound to produce and overall return of 15.5% for the year. We also would have liked to have seen a higher number of positive days than negative days, but the greater magnitude of returns on the positive days more than outweighed three more negative days during the sample period.

Out-of-sample results for our chosen model parameters are detailed below.

II. Model I – Extending the Model to Other Pairs

We extended the model to two other pairs – Coca-Cola Corp & Pepsi Corp and Eli Lilly Corp. & Merck Corp. We selected these pairs because they include relatively stable, mature companies with long trading histories. More importantly, the chosen companies are similarly impacted by the same industry and macroeconomic factors. As a result, dissimilar stock price movements between the two compared companies should be largely due to idiosyncratic risk. The model objectives remained the same – maximizing Average Annual Return on Days Traded and constraining the minimum % of Days Traded and the maximum Drawdown to 10% and 15% respectively.

1. In-Sample Results:

The following tables show

(i)Table 1: the average annualized return on days traded based on different standard deviation inputs for stock price and dividend yield.

(ii)Table 2: the largest drawdown based on different standard deviation inputs for stock price and dividend yield.

(iii)Table 3: the % of days traded

Eli Lilly & Merck:

Table 1:

Table 2:

Table 3:

We selected the boxed point – 1.55 standard deviations for price ratio, 0.5 for yield ratio – based on our in-sample analysis. These parameters result in an annualized return of 143.9% on days traded, trades on 10.7% of days, and a maximum drawdown of 13.3%. The model generates 186 positive trading days to 142 negative trading days.

Coca-Cola & Pepsi:

Table 1:

Table 2:

Table 3:

We selected the boxed point – 0.5 standard deviations for price ratio, 1.5 for yield ratio – based on our in-sample analysis. These parameters result in an annualized return of 52.92% on days traded, trades on 10.2% of days, and a maximum drawdown of 28.0%. As there was no combination of trading on at least 10% of days and a maximum drawdown of 15%, we extended the drawdown constraint to 30%. The model generates 163 positive trading days to 148 negative trading days.

2. Out-of-Sample Results:

As before, we held out the past five years of data to test our model out-of-sample, beginning February 16, 1999.

The out-of-sample performance for Eli Lilly & Merck was extraordinary, with a 273% annualized return on days traded and a drawdown of 14%. On the negative side, we traded on only 6% of all days. We also would have liked to have seen a higher number of positive days than negative days.

The out-of-sample performance for Coca-Cola & Pepsi was also impressive, with a 110% annualized return on days traded and a drawdown of 17.5%. On the negative side, we traded on only 6% of all days. We also would have liked to have seen a higher number of positive days than negative days.

Eli Lilly & Merck:

Coca-Cola & Pepsi:

III. Model II

While our primary focus is on Model I, we were curious whether a regression-based approach based on the same GM to Ford ratios could be used to find attractive trading opportunities.

Model Explanation

Our second model is regression-based but uses the same valuation convergence concept as the first. It attempts to forecast the excess return of GM over Ford over some future time period based on the current ratio of the two company’s stock prices and dividend yields (independent variables). Similar to the first model, if model II predicts a positive excess return (GM – Ford), we would go long on GM and simultaneously short on Ford. Likewise, a negative excess return would signal the opposite. A key difference between the two models is the number of days traded. The first model has constraints that result in a minimum number of trading days. On the other hand, without any trading constraints, model II would require us to be in the market every day.

Model Results

As expected, we found that the stock price and dividend yield ratios had limited predictive power (adjusted R2=0.01%) for predicting the next day’s excess return of GM over Ford even though the t-statistic is significant for the GM/Ford Price Ratio. This is borne out by the in-sample cumulative return of 1.69% over the period and the model is only correct 50% of the time. On an out-of-sample test, the model performed even more poorly with a negative 3.54% cumulative return and getting positive returns on only half the number of days. This is in line with our intuition that the convergence strategy takes several weeks, months, or even years to completely play itself out. The current relationships often diverge from historical averages before they converge.