^{1}

^{*}

^{2}

^{*}

We find that momentum portfolio returns are still unexplainable after addressing two major concerns in the “Investment Manifesto” of Lin and Zhang [1]: lack of economic basis in risk factor models and aggregate data measurement error. Our model represents a synthesis of the exchange economy model of Lucas and closed economy exogenous growth model of King and Rebelo. We mitigate data measurement error by utilizing firm-level financial data and production functions rather than macroeconomic data and utility functions. Although our results fail to completely explain momentum, they are consistent with the “Investment Manifesto” suggestion that firm-level market-to-book and productivity are important factors in describing returns.

The “Investment Manifesto” of Lin and Zhang [

One could argue that the market risk premium (MRP) of the capital asset pricing model (CAPM) is an identifiable and measurable proxy for aggregate market risk. However, the static CAPM is unable to explain many anomalies such as the size premium, distress premium, and momentum. This is largely due to necessarily time varying CAPM coefficients. Cochrane [

Lewellen and Nagel [

In this paper we make an attempt to address the lack of economic basis and measurement error concerns of Lin and Zhang [

Although our regression of momentum portfolio returns on our new productivity measure produces significant alphas, all betas in our model are significant. Furthermore, all HML betas are also significant in unconditional estimation of the momentum returns on the Fama-French three-factor model. The significant productivity and HML betas are consistent with the investment return equation of Lin and Zhang [

The exchange economy model of Lucas [

where

discounted

and defining

where

Marginal utility growth

The relationship between marginal utility growth and asset prices is represented by Equation (2). Given the unobservable nature of utility, obtaining a correct utility functional form is elusive. In this section, a discrete time general equilibrium macroeconomic growth model based on that of King and Rebelo [

We use a general equilibrium (central planner) approach for several reasons. First, general equilibrium models allow both supply (firm) and demand (shareholder) interactions while partial equilibrium models treat one side as exogenous. For instance, macroeconomic theory and intuition suggest an increase in productivity (supply side), even if only temporary, impacts consumption (demand side). Second, Abel and Blanchard ( [

“...very useful as it allows, when studying the effects of various shocks or policies, to use the equations of motion of the centralized economy with its unique shadow price rather than the equations of motion of the market economy with two shadow prices which themselves depend on market-determined interest rates.”

Therefore the complexities associated with the inclusion of market-determined interest rates, which are beyond the scope of this study, are bypassed in the central planner approach. Again, the goal here is to establish a connection between marginal utility growth and productivity.

The model developed here follows the exogenous growth model of King and Rebelo [

“...as Phelps showed, a necessary and sufficient condition for the existence of a steady state in an economy with exogenous technological progress is for this technological progress to be Harrod Neutral or Labor Aug- menting”

As such, we specify the production function in Cobb-Douglas form:

where

We chose the Cobb-Douglas production function for several reasons. First, by construction this production function exhibits constant returns to scale, consistent with the empirical findings of Jorgenson [

The infinitely-lived central planner maximizes discounted expected utility

where

In addition, capital stock evolves according to the “perpetual inventory method”:

where

A central planner with choice variables of consumption

where Equation (8) is obtained by combining Equations. (4), (5), and (6). The Lagrangian is:

The first order conditions are:

We note that

deterministic

Equations (10) and (12) are combined to reveal an alternative proxy for marginal utility growth:

Equation (13) provides a convenient and readily observable proxy for unobservable marginal utility growth. This function serves as a direct replacement for marginal utility growth in the Euler condition Equation (2). The first order approximation of Equation (13) provides a linear discount factor for use in linear regression:

where

Cochrane [

and we use this functional form later in subsequent empirical specifications.

In sum, the exchange economy of Lucas produces the Euler condition in Equation (2). The first order con- ditions of the macroeconomic growth model yield the productivity based marginal utility growth discount factor in Equation (13). Substitution of the linear approximation of this new discount factor into the Euler condition represents the synthesis of the exchange economy and macroeconomic growth models.

Lin and Zhang [

equating stock

where

Equation (16) does not explicitly relate any aggregate factors to firm returns. Lin and Zhang [

The implementation difficulty is in identifying and accurately measuring the relevant aggregate risk factors

Following Fama and French [

Item | Var | Frequency | Sample window |
---|---|---|---|

Output (SALEQ) | Quarterly | 1962 Q1-2012 Q4 | |

Capital (PPENTQ) | Quarterly | 1962 Q1-2012 Q4 | |

Excess market index returns | Daily | 19270101-20121231 | |

Small minus big size returns | Daily | 19270101-20121231 | |

High minus low book-to-market returns | Daily | 19270101-20121231 | |

10 prior-return portfolios | Daily | 19270101-20121231 | |

12 industry portfolios | Daily | 19270101-20121231 | |

One-month T-Bill rate | Daily | 19270101-20121231 | |

Long term (10-year) gov’t bond | Monthly | 196201-201212 | |

Daily | 19620102-20121231 | ||

High grade corporate bond | Monthly | 196201-201212 | |

Daily | 19830103-20121231 | ||

Low grade corporate bond | Monthly | 196201-201212 | |

Daily | 19860102-20121231 |

Fama-French Factors | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MRP | SMB | HML | ||||||||||

mean | 0.0207 | 0.0040 | 0.0118 | 0.0087 | ||||||||

stdev | 0.1123 | 0.0585 | 0.0673 | 0.0076 | ||||||||

N | 348 | 348 | 348 | 348 | ||||||||

Gross industry portfolio returns | ||||||||||||

INDxx | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 |

mean | 0.0307 | 0.0365 | 0.0331 | 0.0326 | 0.0329 | 0.0350 | 0.0266 | 0.0278 | 0.0314 | 0.0334 | 0.0322 | 0.0275 |

stdev | 0.0955 | 0.1764 | 0.1444 | 0.1071 | 0.1210 | 0.1532 | 0.0926 | 0.1141 | 0.1211 | 0.1118 | 0.1480 | 0.1456 |

N | 346 | 346 | 346 | 346 | 346 | 346 | 346 | 346 | 346 | 346 | 346 | 346 |

Gross momentum portfolio returns | ||||||||||||

MOMxx | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | ||

mean | 0.0195 | 0.0281 | 0.0286 | 0.0278 | 0.0277 | 0.0294 | 0.0302 | 0.0364 | 0.0362 | 0.0452 | ||

stdev | 0.2172 | 0.1914 | 0.1441 | 0.1305 | 0.1257 | 0.1148 | 0.1169 | 0.1106 | 0.1181 | 0.1351 | ||

N | 345 | 345 | 345 | 345 | 345 | 345 | 345 | 345 | 345 | 345 | ||

Federal Reserve Board H.15 data | ||||||||||||

mean | 0.0164 | 0.0187 | 0.0213 | |||||||||

stdev | 0.0068 | 0.0063 | 0.0069 | |||||||||

N | 210 | 210 | 210 |

I | II | III | IV | V | VI | |
---|---|---|---|---|---|---|

Begin | 19610131 | 19700101 | 19800101 | 19900101 | 20000101 | 19610131 |

End | 19691231 | 19791231 | 19891231 | 19991231 | 20121231 | 20121231 |

M01 | 0.0122 | −0.0148 | 0.0202 | 0.0246 | 0.0020 | 0.0084 |

0.5757 | 0.6685 | 0.4523 | 0.3906 | 0.9666 | 0.5964 | |

M02 | 0.0110 | 0.0089 | 0.0594 | 0.0422 | 0.0299 | 0.0307 |

0.5493 | 0.7576 | 0.0086 | 0.0721 | 0.3878 | 0.0121 | |

M03 | 0.0281 | 0.0212 | 0.0715 | 0.0564 | 0.0257 | 0.0400 |

0.1082 | 0.3909 | 0.0006 | 0.0047 | 0.3677 | 0.0001 | |

M04 | 0.0195 | 0.0297 | 0.0662 | 0.0538 | 0.0327 | 0.0404 |

0.2320 | 0.2171 | 0.0014 | 0.0033 | 0.1852 | 0.0000 | |

M05 | 0.0317 | 0.0196 | 0.0609 | 0.0502 | 0.0312 | 0.0384 |

0.0461 | 0.3872 | 0.0032 | 0.0038 | 0.1652 | 0.0000 | |

M06 | 0.0305 | 0.0347 | 0.0588 | 0.0588 | 0.0321 | 0.0427 |

0.0683 | 0.1167 | 0.0049 | 0.0007 | 0.1200 | 0.0000 | |

M07 | 0.0317 | 0.0243 | 0.0681 | 0.0657 | 0.0205 | 0.0411 |

0.0641 | 0.2663 | 0.0017 | 0.0001 | 0.3137 | 0.0000 | |

M08 | 0.0454 | 0.0376 | 0.0787 | 0.0712 | 0.0363 | 0.0530 |

0.0131 | 0.0774 | 0.0003 | 0.0000 | 0.0793 | 0.0000 | |

M09 | 0.0474 | 0.0423 | 0.0701 | 0.0701 | 0.0199 | 0.0483 |

0.0185 | 0.0650 | 0.0032 | 0.0003 | 0.3684 | 0.0000 | |

M10 | 0.0843 | 0.0599 | 0.0761 | 0.1057 | 0.0319 | 0.0690 |

0.0013 | 0.0227 | 0.0064 | 0.0000 | 0.2796 | 0.0000 | |

UMD | 0.0721 | 0.0747 | 0.0559 | 0.0811 | 0.0299 | 0.0606 |

0.0004 | 0.0025 | 0.0133 | 0.0008 | 0.4749 | 0.0000 |

The portfolio long on the best performing stocks (M10) and short the worst performing stocks (M01), UMD, is positive and significant for the entire sample (January 31, 1961 to December 31, 2012) and all subsamples except the January 1, 2000 to December 31, 2012 subsample (Column V). Looking to Column VI, mean daily returns increase approximately monotonically with prior 12-month performance. These univariate results are qualitatively similar to those of Chordia and Shivakumar [

Aggregate firm-level financial data from Compustat over the 1962 Q1 to 2012 Q4 time period. Quarterly aggregate output Y is the sum of quarterly firm-level sales. Quarterly aggregate capital K is the sum of quarterly firm-level net property plant and equipment.

We approach the macroeconomic data measurement error in a new way. Rather than use the traditional BEA macroeconomic data, we construct our own macroeconomic variables by aggregating firm-level data from Compustat. The Compustat dataset has 628,144 firm-quarter observations of NYSE, AMEX, and NASDAQ listed stocks over the 1962 Q1 to 2012 Q4 time period. The firm-quarter observations are aggregated by calendar quarter by summation with 3049 firms per quarter on average. The resulting aggregate sales and net property, plant, and equipment (net PP&E) are treated as aggregate output

We note a possible relation between productivity and momentum profitability when looking at

where

i | Symbol | Description | p-value | |
---|---|---|---|---|

1 | IND01 | Non-durables | 0.0849 | |

2 | IND02 | Durables | 0.0879 | 0.1386 |

3 | IND03 | Manufacturing | 0.4668 | |

4 | IND04 | Energy | 0.0618 | 0.3148 |

5 | IND05 | Chemicals | 0.0747 | 0.3827 |

6 | IND06 | Business equipment | 0.0714 | 0.4739 |

7 | IND07 | Telecommunications | 0.9978 | |

8 | IND08 | Utilities | 0.7550 | |

9 | IND09 | Shops (retail) | 0.0954 | |

10 | IND10 | Healthcare | 0.0271 | |

11 | IND11 | Financials | 0.0346 | 0.7755 |

12 | IND12 | Other | 0.0790 | |

13 | Long term government | 0.0001 | ||

14 | High grade corporate | 0.0050 | ||

15 | Low grade corporate | 5.1212 | 0.1366 | |

16 | CRSP value-weighted portfolio | 0.1520 |

and

Compustat company financial data is lower frequency (quarterly) than momentum return data (daily). Given greater statistical power associated with more observations (higher frequency data) we convert the quarterly Compustat data to daily by employing the factor-mimicking-portfolio approach of Breeden et al. [

where

The results in

The estimated coefficients from Equation (18) are combined with daily returns for the 16 assets listed in

As shown in Cochrane ( [

Regressions are performed using generalized methods of moments (GMM) with standard errors corrected for heteroskedasticity and autocorrelation [

We perform the full sample unconditional regressions on the CAPM, Fama-French three-factor model (FF3FM), and the single factor productivity-based CAPM model (PCAPM) using our newly constructed factor mimicking portfolio time series

We assess model fit by performing the J-test on a restricted regression

As mentioned in the Introduction, coefficients for the CAPM and similar linear factor models are necessarily time-varying. Identifying specific macroeconomic variables that cause this time variance is difficult. Following Lewellen and Nagel [

We perform the short-window regressions on the CAPM, the FF3FM, and the PCAPM:

Mean values of the alpha and beta time series are computed and significance is tested via p-values based on Andrews [

Results for the unconditional GMM estimation of Equations (20)-(22) are presented in

Estimations are performed using daily excess returns of the CRSP market value weighted portfolio

CAPM | FF3FM | PCAPM | ||||||
---|---|---|---|---|---|---|---|---|

M01 | −0.0399 | 1.3136 | −0.0353 | 1.4220 | 0.4861 | 0.5000 | -0.0213 | 0.1546 |

0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.1918 | 0.0003 | |

M02 | −0.0139 | 1.1433 | −0.0131 | 1.2130 | 0.1494 | 0.4059 | 0.0015 | 0.1480 |

0.0315 | 0.0000 | 0.0370 | 0.0000 | 0.0000 | 0.0000 | 0.9063 | 0.0002 | |

M03 | −0.0019 | 1.0203 | −0.0025 | 1.0705 | 0.0187 | 0.3382 | 0.0112 | 0.1424 |

0.7099 | 0.0000 | 0.5988 | 0.0000 | 0.5514 | 0.0000 | 0.3100 | 0.0000 | |

M04 | −0.0007 | 0.9842 | −0.0019 | 1.0224 | −0.0353 | 0.2822 | 0.0105 | 0.1594 |

0.8668 | 0.0000 | 0.6268 | 0.0000 | 0.0862 | 0.0000 | 0.3039 | 0.0000 | |

M05 | −0.0019 | 0.9469 | −0.0032 | 0.9789 | −0.0540 | 0.2489 | 0.0083 | 0.1624 |

0.5972 | 0.0000 | 0.3400 | 0.0000 | 0.0025 | 0.0000 | 0.3931 | 0.0000 | |

M06 | 0.0029 | 0.9245 | 0.0015 | 0.9487 | −0.0677 | 0.2022 | 0.0114 | 0.1794 |

0.3673 | 0.0000 | 0.6268 | 0.0000 | 0.0009 | 0.0000 | 0.2273 | 0.0000 | |

M07 | 0.0014 | 0.9172 | 0.0002 | 0.9325 | −0.0698 | 0.1422 | 0.0092 | 0.1894 |

0.6421 | 0.0000 | 0.9446 | 0.0000 | 0.0000 | 0.0000 | 0.3289 | 0.0000 | |

M08 | 0.0132 | 0.9268 | 0.0120 | 0.9381 | −0.0682 | 0.1129 | 0.0201 | 0.2050 |

0.0002 | 0.0000 | 0.0005 | 0.0000 | 0.0004 | 0.0001 | 0.0339 | 0.0000 | |

M09 | 0.0073 | 0.9775 | 0.0078 | 0.9808 | 0.0379 | 0.0039 | 0.0150 | 0.2101 |

0.0605 | 0.0000 | 0.0452 | 0.0000 | 0.0105 | 0.8827 | 0.1393 | 0.0000 | |

M10 | 0.0244 | 1.1449 | 0.0294 | 1.1234 | 0.3415 | −0.3240 | 0.0328 | 0.2553 |

0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0101 | 0.0000 | |

52.38 | 56.49 | 41.53 | ||||||

0.0000 | 0.0000 | 0.0000 |

CAPM | FF3FM | PCAPM | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

M01 | −0.0339 | 1.1647 | −0.0227 | 1.1900 | 0.4060 | 0.0815 | −0.0213 | −0.0204 | ||

0.0000 | 0.0000 | 0.0009 | 0.0000 | 0.0000 | 0.3773 | 0.115 | 0.795 | |||

M02 | −0.0114 | 1.0287 | −0.0028 | 1.0547 | 0.1174 | 0.1017 | −0.0001 | 0.0189 | ||

0.0336 | 0.0000 | 0.6435 | 0.0000 | 0.0073 | 0.0623 | 0.990 | 0.794 | |||

M03 | −0.0004 | 0.9533 | 0.0006 | 0.9840 | 0.0030 | 0.0951 | 0.0070 | 0.0484 | ||

0.8317 | 0.0000 | 0.8248 | 0.0000 | 0.9036 | 0.0170 | 0.415 | 0.493 | |||

M04 | 0.0000 | 0.9280 | 0.0036 | 0.9624 | −0.0263 | 0.0963 | 0.0074 | 0.0661 | ||

0.9847 | 0.0000 | 0.2880 | 0.0000 | 0.1719 | 0.0067 | 0.297 | 0.349 | |||

M05 | −0.0015 | 0.9093 | −0.0010 | 0.9387 | −0.0316 | 0.0774 | 0.0041 | 0.0690 | ||

0.5717 | 0.0000 | 0.7123 | 0.0000 | 0.1226 | 0.0112 | 0.538 | 0.354 | |||

M06 | 0.0021 | 0.9145 | 0.0016 | 0.9466 | −0.0167 | 0.0971 | 0.0065 | 0.0864 | ||

0.3123 | 0.0000 | 0.5835 | 0.0000 | 0.3415 | 0.0035 | 0.345 | 0.253 | |||

M07 | 0.0007 | 0.9431 | −0.0021 | 0.9685 | −0.0121 | 0.0884 | 0.0047 | 0.0965 | ||

0.7575 | 0.0000 | 0.4677 | 0.0000 | 0.5555 | 0.0074 | 0.564 | 0.216 | |||

M08 | 0.0105 | 0.9884 | 0.0090 | 1.0048 | −0.0133 | 0.0852 | 0.0156 | 0.1009 | ||

0.0000 | 0.0000 | 0.0019 | 0.0000 | 0.6551 | 0.0194 | 0.053 | 0.214 | |||

M00 | 0.0054 | 1.0626 | 0.0011 | 1.0655 | 0.0708 | 0.0244 | 0.0105 | 0.0903 | ||

0.1160 | 0.0000 | 0.7715 | 0.0000 | 0.0570 | 0.6171 | 0.272 | 0.281 | |||

M10 | 0.0247 | 1.2695 | 0.0210 | 1.2226 | 0.3450 | −0.1465 | 0.0334 | 0.0193 | ||

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.1247 | 0.006 | 0.850 | |||

89.4 | 73.1 | 129 | ||||||||

0.0000 | 0.0000 | 0.0000 |

Another interesting observation is that all PCAPM betas are insignificant in contrast to significant CAPM betas. Lewellen and Nagel [

Rolling alphas

We find that momentum portfolio returns are still unexplainable after addressing two major concerns in the “Investment Manifesto” of Lin and Zhang [

Specifically, we find evidence in support of the significance of productivity and market-to-book in pricing momentum portfolios. We observe significant productivity betas in unconditional estimations and

In sum, we document productivity and market-to-book ratios are related to momentum portfolio returns at the aggregate level. Future research that incorporates firm-level market-to-book and productivity may move the literature closer to a more complete explanation of momentum portfolio returns and anomalies in general.

Earlier versions of this paper were presented at the 2010 Oxford Business and Economics Conference, Pep- perdine University, The University of Memphis, and The University of Tennessee. The authors would like to thank participants of those presentations for their constructive comments. We would also like to thank Wayne Ferson for helpful comments and suggested readings relevant to this paper and James Kuhle for editorial sug- gestions.