The distance method identifies candidate pairs by historical co-movement; the Engle-Granger and Johansen tests certify that the spread is currently stationary. But neither says why two stocks should be cointegrated — they are purely backward-looking. For a pairs desk, this matters: a pair identified on historical grounds may lose its cointegrating relationship as fundamentals shift, while a pair with aligned factor exposures should remain cointegrated as long as those exposures persist.
Arbitrage pricing theory (APT) provides the economic theory that connects factor exposure alignment to cointegration. If two stocks have identical risk-factor exposure profiles (up to a scalar), their common-factor return components are identical up to the same scalar — which is precisely the condition for cointegration in the common-trends model. This note derives that connection, shows how to measure factor-exposure alignment, and explains how tracking baskets exploit the same logic for multi-stock portfolios.
Learning objectives. By the end of this note you should be able to:
- write the APT return decomposition and identify the systematic and idiosyncratic components;
- express the variance of a stock return in matrix form using the factor covariance matrix;
- state the two necessary conditions for cointegration in terms of APT factor exposures;
- explain why the angle between two stocks' factor-exposure vectors determines their cointegration likelihood;
- construct a tracking basket and explain how it generalises pairs trading to the multi-asset setting.
In this lesson
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- The APT return decomposition
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- From APT to cointegration: two necessary conditions
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- The distance measure in APT space
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- Tracking baskets: pairs trading for multiple securities
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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