The consumption based model is non every bit bad every bit you lot think. (This is a problem set for my online PhD class, in addition to I idea the lawsuit would live on interesting to weblog readers.)
I purpose fourth quarter to fourth quarter nondurable + services consumption, in addition to corresponding annual returns on 10 portfolios sorted on mass to marketplace in addition to the 3 Fama-French factors. (Ken French's website)
The graph is average excess returns plotted against the covariance of excess returns amongst consumption growth. (The graph is a distillation of Jagannathan in addition to Wang's paper, who teach whatsoever credit for this observation. The lines are OLS cross-sectional regressions amongst in addition to without a costless intercept.)
By comparison, the CAPM is the commons disaster. If nosotros plot average returns against the covariance of returns amongst the marketplace (rmrf) or against marketplace betas, at that spot is rattling picayune pattern. In particular, the hml portfolio, which yesteryear itself captures almost all the pricing information inward the x b/m portfolios (that's the indicate of the Fama-French model) has a 5% average supply in addition to a slightly negative marketplace beta. The fact that the hml portfolio is right on the describe inward the previous graph is the primary indicate of that graph.
There is an essentially right flush inward the consumption-based model: value stocks in addition to pocket-sized stocks stimulate got higher average returns. And they stimulate got correspondingly higher covariance amongst consumption growth. Value in addition to pocket-sized stocks tend to do poorly inward years of bad consumption growth, though they stimulate got picayune systematic correlation amongst the market.
Is this perfect? No. The model is \(E(R^e) = cov(R^e, \Delta c)) \times \gamma\) where \(R^e\) = excess return, \(\Delta c\) is consumption increase in addition to \( \gamma\) is the guide a opportunity aversion coefficient. The hateful returns are so large -- in addition to the volatility of consumption increase so pocket-sized -- that the gradient coefficient = guide a opportunity aversion coefficient is 80, a chip difficult for most people to swallow.
Also, this is the linearized model. The truthful nonlinear model is \(E(R^e) = -cov(R^e_{t+1}, (c_{t+1}/c_t)^{-\gamma})\), in addition to raising things to the 80th powerfulness is a lot dissimilar than multiplying yesteryear 80. On the other hand, mayhap this is the cardinal to practiced performance. If you lot intend the underlying right model industrial plant inward continuous time, which is linear, \( E_t(dR^e) = -E_t(dR^e, dc) \gamma \), so mayhap the linearized model is a ameliorate approximation to annual time-averaged information than is the discrete-time model that pretends all consumption happens inward 1 large lump every Dec 31. Furthermore, if you lot enhance consumption increase to the 80th power, all the covariance of returns amongst marginal utility comes inward 1 or ii large spikes. The model becomes a model of rare disasters inward marginal utility, non 1 of repeated events. Perhaps, but life would live on so much easier if markets were virtually repeated risks non 1 time per century disaster covariances.
The larger point: Very few researchers stimulate got actually given the consumption model a practiced snuff it to run across merely how total the drinking glass mightiness be. Hansen in addition to Singleton famously rejected the model, but they used monthly seasonally adjusted consumption data, a bunch of low-power instruments, in addition to no handling of fourth dimension aggregation (consumption is total for the month, returns are 30th to 30th), or the durability of most "nondurable" goods. (Shirts are "nondurable." I teach all mine at Christmas, thence fourth quarter to fourth quarter industrial plant pretty good for me!) Their indicate was by in addition to large an illustrative representative of GMM methodology non a serious Fama-French trend empirical investigation of merely how far a model tin go. (The Fama-French model is also rejected!) It took 25 years earlier Jagannathan in addition to Wang produced this elementary graph. Can nosotros do fifty-fifty better?
Sure, the consumption-based model won't run at a five infinitesimal interval. But is at that spot roughly essence of truth inward it, that stocks which autumn to a greater extent than inward work concern cycles, every bit measured yesteryear consumption, must pay a higher charge per unit of measurement of return. Just how far does that truth go? I intend 1 could do far ameliorate yesteryear thinking difficult virtually fourth dimension aggregation, data construction, durability, seasonal adjustment, in addition to the appropriate frequency to evaluate such a model. And yesteryear trying to run across merely how far the model tin go, rather than statistically rejecting its perfection.
In the destination "why are people afraid of value stocks in addition to move out attractive returns on the table?" must come upwardly downward to 1) they're morons, they haven't figured it out 2) the value premium isn't actually at that spot or 3) value stocks do badly inward bad times, so brand a portfolio riskier. That consumption is also depression inward these bad times seems pretty natural.
Update
From "Cross-Sectional Consumption-Based Asset Pricing: Influenza A virus subtype H5N1 Reappraisal" yesteryear Tom Engsted in addition to Stig Vinther Møller at University of Aarhus. Thanks to Stig for the link. BOP in addition to EOP are commencement of menstruum in addition to destination of menstruum consumption. In a discrete fourth dimension model, do you lot process the total of consumption over the twelvemonth every bit happening at the commencement of the year, or the destination of the year? Treating it at the commencement produces the dramatic graph on the left.
This is a pocket-sized lawsuit of the many explorations 1 tin do to run across if at that spot is roughly powerfulness to the consumption-based model, rather than merely accept it literally in addition to turn down it.
Influenza A virus subtype H5N1 bigger point. Means are pretty insensitive to timing. But covariances in addition to correlations of white vibrations serial are exquisitely sensitive to timing, touchstone error, in addition to so forth. \(cov(a_t,b_t)\) may live on large, in addition to \( cov(a_{t-1} b_t)=0\). Another approach is to do fourth dimension averaged returns. I did this a long fourth dimension agone here. Average january-january, feburary-february, march-march, etc. returns in addition to compare them to the increase of annual macro data. The right matter to do is to explicitly model fourth dimension aggregation -- the fact that consumption is reported every bit an annual average -- along amongst seasonal adjustment.
I purpose fourth quarter to fourth quarter nondurable + services consumption, in addition to corresponding annual returns on 10 portfolios sorted on mass to marketplace in addition to the 3 Fama-French factors. (Ken French's website)
By comparison, the CAPM is the commons disaster. If nosotros plot average returns against the covariance of returns amongst the marketplace (rmrf) or against marketplace betas, at that spot is rattling picayune pattern. In particular, the hml portfolio, which yesteryear itself captures almost all the pricing information inward the x b/m portfolios (that's the indicate of the Fama-French model) has a 5% average supply in addition to a slightly negative marketplace beta. The fact that the hml portfolio is right on the describe inward the previous graph is the primary indicate of that graph.
Is this perfect? No. The model is \(E(R^e) = cov(R^e, \Delta c)) \times \gamma\) where \(R^e\) = excess return, \(\Delta c\) is consumption increase in addition to \( \gamma\) is the guide a opportunity aversion coefficient. The hateful returns are so large -- in addition to the volatility of consumption increase so pocket-sized -- that the gradient coefficient = guide a opportunity aversion coefficient is 80, a chip difficult for most people to swallow.
Also, this is the linearized model. The truthful nonlinear model is \(E(R^e) = -cov(R^e_{t+1}, (c_{t+1}/c_t)^{-\gamma})\), in addition to raising things to the 80th powerfulness is a lot dissimilar than multiplying yesteryear 80. On the other hand, mayhap this is the cardinal to practiced performance. If you lot intend the underlying right model industrial plant inward continuous time, which is linear, \( E_t(dR^e) = -E_t(dR^e, dc) \gamma \), so mayhap the linearized model is a ameliorate approximation to annual time-averaged information than is the discrete-time model that pretends all consumption happens inward 1 large lump every Dec 31. Furthermore, if you lot enhance consumption increase to the 80th power, all the covariance of returns amongst marginal utility comes inward 1 or ii large spikes. The model becomes a model of rare disasters inward marginal utility, non 1 of repeated events. Perhaps, but life would live on so much easier if markets were virtually repeated risks non 1 time per century disaster covariances.
The larger point: Very few researchers stimulate got actually given the consumption model a practiced snuff it to run across merely how total the drinking glass mightiness be. Hansen in addition to Singleton famously rejected the model, but they used monthly seasonally adjusted consumption data, a bunch of low-power instruments, in addition to no handling of fourth dimension aggregation (consumption is total for the month, returns are 30th to 30th), or the durability of most "nondurable" goods. (Shirts are "nondurable." I teach all mine at Christmas, thence fourth quarter to fourth quarter industrial plant pretty good for me!) Their indicate was by in addition to large an illustrative representative of GMM methodology non a serious Fama-French trend empirical investigation of merely how far a model tin go. (The Fama-French model is also rejected!) It took 25 years earlier Jagannathan in addition to Wang produced this elementary graph. Can nosotros do fifty-fifty better?
Sure, the consumption-based model won't run at a five infinitesimal interval. But is at that spot roughly essence of truth inward it, that stocks which autumn to a greater extent than inward work concern cycles, every bit measured yesteryear consumption, must pay a higher charge per unit of measurement of return. Just how far does that truth go? I intend 1 could do far ameliorate yesteryear thinking difficult virtually fourth dimension aggregation, data construction, durability, seasonal adjustment, in addition to the appropriate frequency to evaluate such a model. And yesteryear trying to run across merely how far the model tin go, rather than statistically rejecting its perfection.
In the destination "why are people afraid of value stocks in addition to move out attractive returns on the table?" must come upwardly downward to 1) they're morons, they haven't figured it out 2) the value premium isn't actually at that spot or 3) value stocks do badly inward bad times, so brand a portfolio riskier. That consumption is also depression inward these bad times seems pretty natural.
Update
From "Cross-Sectional Consumption-Based Asset Pricing: Influenza A virus subtype H5N1 Reappraisal" yesteryear Tom Engsted in addition to Stig Vinther Møller at University of Aarhus. Thanks to Stig for the link. BOP in addition to EOP are commencement of menstruum in addition to destination of menstruum consumption. In a discrete fourth dimension model, do you lot process the total of consumption over the twelvemonth every bit happening at the commencement of the year, or the destination of the year? Treating it at the commencement produces the dramatic graph on the left.
This is a pocket-sized lawsuit of the many explorations 1 tin do to run across if at that spot is roughly powerfulness to the consumption-based model, rather than merely accept it literally in addition to turn down it.
Influenza A virus subtype H5N1 bigger point. Means are pretty insensitive to timing. But covariances in addition to correlations of white vibrations serial are exquisitely sensitive to timing, touchstone error, in addition to so forth. \(cov(a_t,b_t)\) may live on large, in addition to \( cov(a_{t-1} b_t)=0\). Another approach is to do fourth dimension averaged returns. I did this a long fourth dimension agone here. Average january-january, feburary-february, march-march, etc. returns in addition to compare them to the increase of annual macro data. The right matter to do is to explicitly model fourth dimension aggregation -- the fact that consumption is reported every bit an annual average -- along amongst seasonal adjustment.
