The previous discussion provides a general framework in which to describe empirically the long volatility exposure embedded within CTA trend- following strategies. To fit the kinked regression demonstrated in Figures 9.1 through 9.4, we use a piecewise linear capital asset pricing model (CAPM)–type model. The model can be described as:
Rtf−Rf=(1−D)[alow+blow(ROEX−Rf)]+
D[ahigh+bhigh(ROEX−Rf)] (9.1) where
Rtf =return to the trend-following strategy Rf =risk-free rate
ROEX =return to the S&P 100
alow,blow=regression coefficients to the left-hand side of the kink ahigh,bhigh=regression coefficients to the right-hand side of the kink
D =1 if ROEX−Rf> the threshold
D =0 if ROEX−Rf< or equal to the threshold.
In essence we plot two regression lines that have different alpha and beta coefficients depending on which side of the kink the market returns fall. The trick is to maintain continuity at the kink in the fitted regression line. To insure this, we impose this following condition:
alow+blow(Threshold)=ahigh+bhigh(Threshold) (9.2) Our regression equation then becomes:
Rtf−Rf=(1−D)[alow+blow(ROEX−Rf)]+
D[alow+(blow−bhigh)(Threshold)+bhigh(ROEX−Rf)] (9.3) We express our regression equation in this fashion to demonstrate how the threshold value is explicitly incorporated into the solution. Table 9.1 pres- ents the results for our fitted regression lines.
For the Barclay Commodity Trading Index, the threshold value (the kink) is −5.2 percent.2Several observations can be made from the regresion Measuring the Long Volatility Strategies of Managed Futures 189
2We found the threshold value through a recursive method that minimizes the residual sum of squares in equation 9.3.
TABLE 9.1Two-Step Regression Coefficients Commodity TradingDiversified TradingSystematic TradingMLM Index Coefficientt-statisticCoefficientt-statisticCoefficientt-statisticCoefficientt-statistic Threshold−0.0526−0.0868−0.0485−0.0926 Alpha_low−0.0158−1.2699−0.0793−1.7150−0.0175−1.2127−0.0437−1.7743 Beta_low−0.3962−2.1083−1.1018−2.1911−0.4923−2.1703−0.5893−2.3223 Alpha_high0.00140.00430.00290.0022 Beta_high−0.0676−1.1759−0.1384−2.0820−0.0717−0.9365−0.0929−3.2138 S.E. 0.02640.03530.03430.0155 Regression Rsquare0.05550.07450.05200.1203 AdjRsquare0.04370.06290.04020.1094
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coefficients. First, the value of blowis negative and significant at the 5 per- cent level, with a t-statistic of −2.11. This demonstrates that when the returns to the S&P 100 are negative, the commodity trading strategies earn positive excess returns. In particular, the value of blowis−0.396, indicating that CTAs earn, on average, about a 0.4 percent excess return for every 1 percent decline in the S&P 100 below the threshold value.
This is similar to a put option being exercised by the CTA manager when the returns to the stock market are negative, but created synthetically as a consequence of the trend-following strategy. As long as stock market returns remain positive, CTAs earn a constant return equal to a cash (treas- ury bill) rate. However, when the stock market suffers a negative volatility event that drives market returns into negative territory, the synthetic put option is exercised, leading to large positive returns.
The coefficient for bhighis close to zero (−0.067). It is neither econom- ically nor statistically significant.3Trend-following CTAs do not earn excess returns when the returns to the stock market are positive. When the returns to the S&P 100 are positive, there is no need to exercise the put option. In addition,ahighis also close to zero, indicating a lack of excess returns over this part of the graph. Managed futures earn a treasury bill rate of return when the returns to the stock market are positive. The lack of any excess return over this part of the graph can be considered the payment for the put option premium. That is, trend-following CTAs forgo excess returns when the returns to the stock market are positive in return for a long put option exposure to be exercised when the returns to the stock market are negative.
Similar results are presented in Table 9.1 for diversified trading man- aged futures, systematic trading, and the passive MLMI index. In each case, blow is economically and statistically significant. In addition, blowalways has a negative sign, indicating positive returns to managed futures when the stock market earns negative returns. Also, ahighis close to zero for each cat- egory of managed futures. Once again, this indicates that managed futures do not generate any excess returns when the returns to the stock market are positive. All that is received is a cash return equal to treasury bills.
bhighis statistically significant in two categories: diversified trading and the MLMI. The sign of the bhighis negative, indicating a downward slop- ing curve. However, the coefficient is small and lacks economic significance.
Still, this indicates that managed futures can be countercyclical when the stock market has positive returns.
Measuring the Long Volatility Strategies of Managed Futures 191
3There is no t-statistic for ahighbecause this coefficient is a linear combination of the other regression coefficients (see equation 9.2).