## Monday, April 20, 2015

### Consumption-based model and value premium

The consumption based model is not as bad as you think. (This is a problem set for my online PhD class, and I thought the result would be interesting to blog readers.)

I use 4th quarter to 4th quarter nondurable + services consumption, and corresponding annual returns on 10 portfolios sorted on book to market and the three Fama-French factors. (Ken French's website)
The graph is average excess returns plotted against the covariance of excess returns with consumption growth. (The graph is a distillation of Jagannathan and Wang's paper, who get any credit for this observation.  The lines are OLS cross-sectional regressions with and without a free intercept.)

By comparison, the CAPM is the usual disaster. If we plot average returns against the covariance of returns with the market (rmrf) or against market betas, there is very little pattern. In particular, the hml portfolio, which by itself captures almost all the pricing information in the ten b/m portfolios (that's the point of the Fama-French model) has a 5% average return and a slightly negative market beta. The fact that the hml portfolio is right on the line in the previous graph is the main point of that graph.
There is an essentially correct story in the consumption-based model: value stocks and small stocks have higher average returns. And they have correspondingly higher covariance with consumption growth. Value and small stocks tend to do poorly in years of bad consumption growth, though they have little systematic correlation with 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 growth and $$\gamma$$ is the risk aversion coefficient. The mean returns are so large -- and the volatility of consumption growth so small -- that the slope coefficient = risk aversion coefficient is 80, a bit hard for most people to swallow.

Also, this is the linearized model. The true nonlinear model is $$E(R^e) = -cov(R^e_{t+1}, (c_{t+1}/c_t)^{-\gamma})$$, and raising things to the 80th power is a lot different than multiplying by 80. On the other hand, perhaps this is the key to good performance. If you think the underlying correct model works in continuous time,  which is linear, $$E_t(dR^e) = -E_t(dR^e, dc) \gamma$$, then perhaps the linearized model is a better approximation to annual time-averaged data than is the discrete-time model that pretends all consumption happens in one big lump every December 31. Furthermore, if you raise consumption growth to the 80th power, all the covariance of returns with marginal utility comes in one or two big spikes. The model becomes a model of rare disasters in marginal utility, not one of repeated events. Perhaps, but life would be so much easier if markets were about repeated risks not once per century disaster covariances.

The larger point: Very few researchers have really given the consumption model a good go to see just how full the glass might be. Hansen and Singleton famously rejected the model, but they used monthly seasonally adjusted consumption data, a bunch of low-power instruments, and no treatment of time aggregation (consumption is sum for the month, returns are 30th to 30th), or the durability of most "nondurable" goods. (Shirts are "nondurable." I get all mine at Christmas, hence 4th quarter to 4th quarter works pretty well for me!) Their point was mostly an illustrative example of GMM methodology not a serious Fama-French style empirical investigation of just how far a model can go. (The Fama-French model is also rejected!) It took 25 years before Jagannathan and Wang produced this simple graph. Can we do even better?

Sure, the consumption-based model won't work at a 5 minute interval. But is there some essence of truth in it, that stocks which fall more in business cycles, as measured by consumption, must pay a higher rate of return.  Just how far does that truth go? I think one could do far better by thinking hard about time aggregation,  data construction, durability, seasonal adjustment, and the appropriate frequency to evaluate such a model. And by trying to see just how far the model can go, rather than statistically rejecting its perfection.

In the end  "why are people afraid of value stocks and leave attractive returns on the table?" must come down to 1) they're morons, they haven't figured it out 2) the value premium isn't really there or 3) value stocks do badly in bad times, so make a portfolio riskier. That consumption is also low in these bad times seems pretty natural.

Update

From "Cross-Sectional Consumption-Based Asset Pricing: A Reappraisal" by Tom Engsted and Stig Vinther Møller at University of Aarhus. Thanks to Stig for the link. BOP and EOP are beginning of period and end of period consumption. In a discrete time model, do you treat the sum of consumption over the year as happening at the beginning of the year, or the end of the year? Treating it at the beginning produces the dramatic graph on the left.

This is a small instance of the many explorations one can do to see if there is some power to the consumption-based model, rather than just take it literally and reject it.

A bigger point. Means are pretty insensitive to timing. But covariances and correlations of white noise series are exquisitely sensitive to timing, measurement error, and so forth. $$cov(a_t,b_t)$$ may be large, and $$cov(a_{t-1} b_t)=0$$. Another approach is to create time averaged returns. I did this a long time ago here. Average january-january, feburary-february, march-march, etc. returns and compare them to the growth of annual macro data. The right thing to do is to explicitly model time aggregation -- the fact that consumption is reported as an annual average -- along with seasonal adjustment.

1. Your online class is awesome :)

2. I didn't have as much luck verifying this until I realized you used annual data. I was also using the 5x5 Fama-French portfolios (so 25 total instead of 10), which adds a little variability to the results.

I had noticed that Nominal GDP was doing better than some of the real consumption variables I had downloaded (also much better than real GDP), so I tried switching from real consumption to nominal consumption. When I was analyzing returns vs. real consumption, I had broken out PCE by durables, non-durables, services, and then tried to get some estimate of PCE ex durables, so I did that again with nominal versions of them all. In this case, nominal PCE durables actually had the highest R squared (headline nominal PCE's R^2 was below nominal GDP's R^2). So I'm not quite sure why you would want to exclude durables.

1. Theory says nondurables or the service flow from durables. Including durables properly is another important item on the see-if-it-works agenda.

2. BTW, my data should be available at the "problem set" link above.

3. book to market and high minus low?

4. John, I feel like you're overlooking someone:

5. Thanks for this really interesting post.

The model actually also works surprisingly well with annual consumption data from the NIPA tables over the period from 1929 to 2014 if Campbell’s beginning-of-period timing convention is used. Mean excess returns on the ten value portfolios and consumption growth covariances:

BM1 7.0 0.25
BM2 8.2 0.24
BM3 7.9 0.22
BM4 8.4 0.28
BM5 9.2 0.27
BM6 9.7 0.32
BM7 9.8 0.31
BM8 12.1 0.34
BM9 12.6 0.31
BM10 14.0 0.39

With this sample period, estimates of the risk aversion coefficient are about 20 (much less than 80). See: http://ssrn.com/abstract=1754019

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