Relative Trends - Cumulative Abnormal Return Trends - CARTs


Below we introduce our approach to relative trends. They are used in many places at to obsereve trends of single instruments and groups of instruments in relation to the broad market.

CARTs and Capital Asset Pricing Model

According to the classic Capital Asset Pricing Model (CAPM), an expected rate of return of the instrument i (Ri) can be computed as follows:

Ri = Rf + βi (Rm – Rf)


  • βi is the beta (risk measure) of the instrument;
  • Rm is the expected broad market rate of return, which has β = 1 (Rm = Rf + Rm – Rf = Rm);
  • Rf is the risk-free rate of return, which has β = 0 (Rf = Rf + 0 = Rf).

In our model Rm is the rate of return from the SPDR S&P 500 ETF Trust (SPY), which corresponds to the price and yield of the S&P 500 Index. SPY is very often identified as the broad market index and given β = 1, e.g. by the Yahoo Finance.

The risk-free rate is more and more often assumed to equal zero. For instance, the return of 3-month Treasury Bills (which are often identified as risk-free securities) has not reached 0.2% since 2008 and for a year it has not exceeded 0.05%. Furthermore, even 3-month Treasury are not entirely free of risk, so the hypothetical risk-free rate is even lower. With Rf = 0 the expected return can be computed as follows:

Ri = βi * Rm

In fact, in the market one can observe a lot of instruments with returns above the levels based on their betas, lasting for a short or sometimes for a long period of time. These excess returns are called abnormal returns. Abnormal log-returns are approximated as follows:

Ra = Ri – βi * Rm


An ongoing systematic change of compounded excess returns forms the Cumulative Abnormal Returns Trend (CART).

CARTs are very useful for short-term analysis as well as for early detection of medium to long-term events. Furthermore, they can be used anytime, even in case of a market turmoil or panic. During such periods it is hardly possible to observe normal trends. Most financial instruments move in the same direction and the volatility reaches high level. However, CARTs are still able to detect systematic relative changes.

If beta is close to 1, CARTs can be approximated by the price ratio series. If additionally price changes are moderate, CART Trend move can be approximated by the difference: instrument final change – index final change.

CART Strategy

CART Investment Strategy is based on observing ex-post CARTs and selecting the most promising ones in terms of continuation potential. CARTs are generally assumed to continue but in case of economic, political or corporate events that makes CART trend probable to break the portfolio positions are closed. Closing positions may be also a result of finding more promising CARTs.

For more see the presentation.



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