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Ford-General Motors

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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 strategy involving Ford 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 GM because 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 periodic divergence 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 February 13, 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 % Maximum

    of Days Traded 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.

    PRICE RATIO0.10.20.30.40.50.60.70.80.91.01.52.02.50.123%27%33%33%21%24%21%14%12%12%15%72%154%0.230%37%46%49%36%43%41%33%29%29%24%83%154%DIV0.330%38%52%57%41%52%52%40%34%34%36%57%154%YIELD0.434%46%65%78%63%88%103%89%77%77%72%153%154%RATIO0.534%44%66%81%63%96%125%109%89%89%81%200%200%0.632%41%64%79%58%94%130%109%79%79%66%0.737%41%64%79%58%94%130%109%79%79%66%0.831%35%57%79%58%94%130%109%79%79%66%0.932%37%65%77%58%94%130%109%79%79%66%1.032%34%58%71%55%94%130%109%79%79%66%1.517%16%117%355%222%1229%2936%2.00%-23%197%5120%2.50%-23%197%5120%

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.

Model Results

    Number of Total Days3063

    Number of Days Traded429

    % of Days Traded14.01%

    Cumulative Return138.36%

    AACR - Total Days7.35%

    AACR - Days Traded65.90%

    Highest One-Day Return5.83%

    Lowest One-Day Return-4.52%

    Number of Positive Days237

    Number of Negative Days192

    Largest Drawdown-11.68%

    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.

Model Results

    Number of Total Days1257

    Number of Days Traded161

    % of Days Traded12.81%

    Cumulative Return39.09%

    AACR - Total Days6.78%

    AACR - Days Traded66.92%

    Highest One-Day Return10.86%

    Lowest One-Day Return-7.90%

    Number of Positive Days79

    Number of Negative Days82

    Largest Drawdown-15.32%

    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 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. Like with GM and Ford, our 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. These results told us

    that the interaction between stock price and dividend yield is critical in order to predict

    convergence. As a result of these findings, we again 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 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:

    Price Ratio0.80.91.01.11.21.31.41.51.551.60.146.31%51.63%53.08%67.7%59.5%60.6%74.0%133.1%143.9%140.9%0.246.03%52.38%53.97%68.9%59.5%60.6%74.0%133.1%143.9%140.9%0.361.82%68.06%58.78%75.9%67.2%60.6%74.0%133.1%143.9%140.9%0.470.76%72.45%65.26%86.1%80.8%75.7%74.0%133.1%143.9%140.9%

    0.558.69%61.06%61.07%78.0%70.8%64.2%63.9%133.1%143.9%140.9%

    0.659.89%53.54%52.34%74.6%72.3%65.8%65.7%132.8%143.9%140.9%0.782.63%78.91%54.84%71.7%70.5%63.7%63.3%130.4%133.4%140.9%0.862.55%61.52%37.00%52.2%69.4%59.6%58.7%123.7%125.9%132.3%0.955.23%51.67%33.72%46.8%69.3%59.3%58.7%123.7%125.9%132.3%1.090.54%90.20%65.66%77.8%97.5%84.9%87.8%153.8%157.4%135.1% Dividend Yield Ratio

Table 2:

    Price Ratio0.80.91.01.11.21.31.41.51.551.60.1-36.55%-32.30%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%0.2-36.55%-32.30%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%0.3-36.55%-32.30%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%0.4-36.55%-32.30%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%

    0.5-36.55%-32.30%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%

    0.6-36.55%-33.10%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%0.7-36.55%-32.30%-32.00%-24.5%-24.1%-17.8%-17.4%-17.4%-13.3%-13.3%0.8-36.55%-32.30%-32.00%-25.0%-22.8%-17.8%-17.4%-17.4%-13.3%-13.3%0.9-36.55%-32.76%-32.00%-26.3%-24.2%-17.8%-17.4%-17.4%-13.3%-13.3%1.0-36.55%-32.74%-32.00%-26.3%-24.2%-17.8%-17.4%-17.4%-13.3%-13.3% Dividend Yield Ratio

Table 3:

    Price Ratio0.80.91.01.11.21.31.41.51.551.60.143.13%36.83%31.31%26.2%21.4%18.0%14.5%12.1%10.7%9.4%0.242.51%36.70%31.18%26.0%21.4%18.0%14.5%12.1%10.7%9.4%0.339.63%34.80%30.33%25.2%20.5%18.0%14.5%12.1%10.7%9.4%0.436.66%32.26%28.21%23.6%19.0%16.6%14.5%12.1%10.7%9.4%

    0.534.12%30.33%26.67%22.5%17.9%15.5%13.8%12.1%10.7%9.4%

    0.632.45%29.48%26.12%22.1%17.6%15.2%13.5%12.0%10.7%9.4%0.728.63%26.71%24.94%21.9%17.4%15.0%13.3%11.9%10.5%9.4%0.826.18%24.45%23.15%20.8%17.1%14.9%13.2%11.8%10.4%9.3%0.923.96%22.36%21.32%19.8%17.0%14.8%13.2%11.8%10.4%9.3%1.020.21%18.71%17.83%17.2%15.7%13.6%12.0%10.9%9.9%9.2% Dividend Yield Ratio

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:

    Price Ratio0.10.20.30.40.50.60.70.81.014.64%14.64%14.64%14.64%14.64%13.76%11.63%12.65%1.115.49%15.49%15.49%15.49%15.49%15.32%12.32%9.78%1.220.31%20.31%20.31%20.31%20.31%20.31%20.16%22.79%1.330.88%30.88%30.88%30.88%30.88%30.88%30.88%34.96%1.4106.25%106.25%106.25%106.25%106.25%106.25%106.25%117.47%

    1.552.92%52.92%52.92%52.92%52.92%52.92%52.92%52.92%

    1.659.03%59.03%59.03%59.03%59.03%59.03%59.03%59.03%1.7159.89%159.89%159.89%159.89%159.89%159.89%159.89%159.89%1.8231.47%231.47%231.47%231.47%231.47%231.47%231.47%231.47%1.967.01%67.01%67.01%67.01%67.01%67.01%67.01%67.01%2.097.88%97.88%97.88%97.88%97.88%97.88%97.88%97.88% Dividend Ratio

Table 2:

    Price Ratio0.10.20.30.40.50.60.70.81.0-51.9%-51.9%-51.9%-51.9%-51.9%-51.9%-51.9%-51.9%1.1-52.3%-52.3%-52.3%-52.3%-52.3%-52.3%-52.3%-52.3%1.2-51.3%-51.3%-51.3%-51.3%-51.3%-51.3%-51.3%-51.3%1.3-41.6%-41.6%-41.6%-41.6%-41.6%-41.6%-41.6%-41.6%1.4-31.4%-31.4%-31.4%-31.4%-31.4%-31.4%-31.4%-31.4%

    1.5-28.0%-28.0%-28.0%-28.0%-28.0%-28.0%-28.0%-28.0%

    1.6-27.2%-27.2%-27.2%-27.2%-27.2%-27.2%-27.2%-27.2%1.7-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%1.8-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%1.9-14.6%-14.6%-14.6%-14.6%-14.6%-14.6%-14.6%-14.6%2.0-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%-12.9%-12.9% Dividend Ratio

Table 3:

    Price Ratio0.10.20.30.40.50.60.70.81.038.9%38.9%38.9%38.9%38.9%38.4%37.9%35.7%1.134.6%34.6%34.6%34.6%34.6%34.5%34.1%32.4%1.228.2%28.2%28.2%28.2%28.2%28.2%28.0%27.6%1.321.1%21.1%21.1%21.1%21.1%21.1%21.1%20.6%1.415.6%15.6%15.6%15.6%15.6%15.6%15.6%15.3%

    1.510.2%10.2%10.2%10.2%10.2%10.2%10.2%10.2%

    1.66.7%6.7%6.7%6.7%6.7%6.7%6.7%6.7%1.75.1%5.1%5.1%5.1%5.1%5.1%5.1%5.1%1.83.7%3.7%3.7%3.7%3.7%3.7%3.7%3.7%1.92.6%2.6%2.6%2.6%2.6%2.6%2.6%2.6%2.02.2%2.2%2.2%2.2%2.2%2.2%2.2%2.2% Dividend Ratio

    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:

Model Results

    Number of Total Days1257

    Number of Days Traded67

    % of Days Traded5.33%

    Cumulative Return42.32%

    AACR - Total Days7.27%

    AACR - Days Traded273.18%

    Highest One-Day Return53.11%

    Lowest One-Day Return-5.01%

    Number of Positive Days29

    Number of Negative Days38

    Largest Drawdown-14.25%

Coca-Cola & Pepsi:

Model Results

    Number of Total Days1257

    Number of Days Traded80

    % of Days Traded6.36%

    Cumulative Return26.66%

    AACR - Total Days4.81%

    AACR - Days Traded109.27%

    Highest One-Day Return12.13%

    Lowest One-Day Return-6.52%

    Number of Positive Days38

    Number of Negative Days42

    Largest Drawdown-17.49%

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