Jagannathan and Wang 1996

1
Jagannathan and Wang (1996)

• Testing the Conditional CAPM

2
The Story

• When we test the static CAPM we find that
• We cant explain the cross-sectional variation in
  expected returns well at all.
• Other variables add explanatory power, when they
  should not
• Size
• Book-to-Market
• Fama-French (1992)
• relation between market beta and average return
  is flat
• Not good news for the theory or any MBA finance
  course.

3
Can We Understand This Result?

• Formal CAPM
• Equilibrium model providing a linear relation
  between expected returns and beta
• One period model
• Empirically, it is common to consider that
• Agents live many periods
• The parameters of the pricing model are constant
  over time
• Is this reasonable?

4
What If

• Suppose information about an assets next
  dividend came out only once a year, on January 5.
• Suppose this was true for every stock.
• What should the risk/return tradeoff look like
  over the course of a year?

5
What If cont

• Seems that time-varying expected returns are
  possible.
• What about time-varying risk premia?
• Other problems with an unconditional CAPM
• Leverage causes equity betas to rise during a
  recession (affects asset betas to a lesser
  extent)
• Firms with different types of assets will be
  affected by the business cycle in different ways
• Technology changes
• Consumers tastes change

6
The Plan

• Start by assuming the conditional CAPM.
• Then, rather than add conditioning information
  directly, the authors derive implications for the
  unconditional CAPM.
• This will nest the static (or unconditional)
  CAPM.
• Examine the performance of the enhanced
  unconditional CAPM.

7
The Results

• Unconditional model implied by the conditional
  CAPM explains 30 of the variance in the
  cross-section of expected returns using 100 stock
  portfolios similar to those used by Fama-French
  (1992).
• The rejection by the data and size effect are
  much weaker than when testing the static CAPM.

8
The Results cont

• The typical implementation (use of the VW proxy)
  may not be reasonable.
• When human capital is included in the proxy for
  the market (return on aggregate wealth), the
  unconditional model implied by the conditional
  CAPM explains over 50 of the cross-sectional
  variation in expected returns and the data fail
  to reject the model.
• Size and book to market have little ability to
  explain the unexplained cross-sectional variation.

9
Developing the Model

• Black CAPM
• where ?1 is the market risk premium.
• As stated above, this performs poorly.
• FF (1992) find ?1 close to zero.
• This is not necessarily evidence against the
  conditional CAPM.
• Assets on the conditional frontier need not be on
  the unconditional frontier.

10
Example from JW

• 2 stocks and 2 periods
• ?1t 0.5, 1.25 average 0.875
• ?2t 1.5, 0.75 average 1.125
• CAPM holds in each period but the risk premium
  differs across the periods.
• 10 at date 1
• 20 at date 2
• Expected risk premia on the stocks will be
• Premium on 1 0.5(.1) .05, 1.25(.2) .25
• Premium on 2 1.5(.1) .15, 0.75(.2) .15
• Both stocks have same expected return so the CAPM
  appears not to hold.

11
Example cont

• This example is a bit strained (contrived), but
  the point is well made.
• There are plenty of studies that show betas vary
  over time. BARRA takes this variation into
  account when producing BARRAs better betas.
• Next, they assume that the CAPM holds and derive
  implications for the unconditional model.

12
Conditional to Unconditional

• The conditional CAPM is a cheap trick, not an
  equilibrium model.
• Merton (1973) shows that the conditionally
  expected return on an asset should be jointly
  linear in its conditional market beta and hedge
  portfolio betas, where the hedge portfolios hedge
  against changes in the investment opportunity
  set.
• As Merton did, JW assume that the hedging motives
  are not important and the CAPM holds period by
  period.

13
Conditional CAPM

• The conditional CAPM they consider is
• where ?i,t-1 is the conditional market beta of
  asset i
• Now, take the unconditional expectation of the
  above equation to do the empirical analysis
• where ?1 is the unconditional expected market
  risk premium and is the unconditional
  expected beta.

14
Conditional CAPM cont

• If the covariance between the conditional beta of
  asset i and the conditional market risk premium
  is zero for all assets i, then this looks like
  the static CAPM we all know and love.
• However, in general, the conditional risk premium
  on the market portfolio and the asset betas are
  correlated. In bad times, the expected market
  risk premium may be relatively high and firms on
  the fringe and more levered firms may have
  higher conditional equity betas during such
  times.
• Something that should be testable.

15
Conditional CAPM cont

• If uncertainty about future growth opportunities
  is the cause for higher betas for fringe firms,
  then their conditional betas will be low,
  resulting in natural perverse market timing.
• This is because in bad times, uncertainty as well
  as the value of future growth opportunities is
  reduced, and this may offset increased leverage.
• Earlier studies have shown that the last term in
  (1) is not zero.
• Look at various papers by Ferson and Harvey.

16
Conditional CAPM cont

• Notice that the last term in (1) depends only on
  the part of the conditional beta that is in the
  span of the market risk premium.
• Thus decompose the conditional beta of any asset
  into 2 orthogonal components by projecting the
  conditional beta on the market risk premium.
• For each asset i, define the beta-premium
  sensitivity as

17
Conditional CAPM cont

• ?i measures the sensitivity of the conditional
  beta to the market risk premium. We can show
  that
• The way to show this is to regress
• Then, the regression coefficient and error are as
  shown above, and the fact that the error is mean
  zero and unrelated to the regressor you get for
  free.

18
Conditional CAPM cont

• So far, we have the conditional beta can be
  written in three parts
• The expected (unconditional) beta.
• A random variable perfectly correlated with the
  conditional market risk premium.
• Something mean zero and uncorrelated with the
  conditional market risk premium.

19
Implications for Unconditional Expected Returns

• Substituting (2) into (1) yields
• (3) says that the unconditional expected return
  on any asset i is a linear function of
• Expected beta
• Beta-prem sensitivity
• The larger the sensitivity, the larger the
  variability of the second part of the
  conditional beta.
• Hence, the beta-prem sensitivity measures
  instability of beta over the business cycle.
  Stocks with betas that vary more over the cycle
  have higher expected returns.

20
How to Estimate

• We need both estimates of expected beta and
  estimates of beta-prem sensitivity.
• How do we do this?
• We need assumptions about the nature of the
  stochastic process governing the joint temporal
  evolution of conditional market betas and the
  conditional risk premium.
• From (3) we can see that ? does not affect
  expected return. Therefore we can concentrate on
  the first two parts of the conditional beta.

21
How to Estimate cont

• They look directly at how stock returns respond
  to the market risk premium on average and how
  they respond t changes in the risk premium
• The first unconditional beta is the market beta
  and the second unconditional beta is the premium
  beta. They measure average market risk and beta
  instability risk.

22
How to Estimate cont

• In appendix A of the paper they show that, under
  some assumptions, the unconditional expected
  return is a linear function of these two betas
• This two beta model is not a special case of the
  general equilibrium multi-beta CAPM from Merton
  (1973).
• In those models, expected return is linear in
  several conditional betas, one of which is the
  market beta.
• Here, its linear only in the conditional market
  beta and this implies that the unconditional
  expected return is linear in the unconditional
  market beta and the premium beta.

23
On to the Estimation?

• No, while equation (4) forms the basis for
  empirical work, we still need further
  assumptions.
• Need observations on the conditional risk
  premium, ?1t-1, so that we can compute the
  prem-beta, ?i?.
• Actually will have to settle for some estimate of
  the conditional premium.
• We also need observations on the market
  portfolio.
• A constant problem that this study also must find
  a way to deal with.

24
Conditional Risk Premium

• The risk premium varies over the business cycle.
• How can we predict the business cycle? They go
  to the relevant literature and pick the single
  variable that best predicts the cycle.
• Stock Watson (1989) find that spread between
  different bond yields helps predict.
• Bernanke (1990) finds that the best single
  variable is the spread between commercial paper
  and t-bill rates.

25
Conditional Risk Premium cont

• Here, they choose the spread between BAA and AAA
  rated bonds (denote it Rpremt-1) and further
  assume
• Assumption 1 (A fairly heroic assumption) The
  conditional risk premium is linear in the spread
  between BAA and AAA bonds
• and then
• Under assumption 1, the expected return is linear
  in its prem-beta and its market beta. To see
  this, substitute (using the papers numbers) (14)
  into (12) and make use of (15) and theorem 1.

26
Conditional Risk Premium cont

• The resulting relation is
• Suppose that ?i? is not linear in ?i and that
  assumption 1 holds. Then
• The linearity is preserved because covariance is
  a linear operation and the actual conditional
  market risk premium is assumed to be linear in
  the proxy.
• This is an important result, because now all the
  returns necessary to calculate the ?iprem are
  observable.

27
Market Portfolio

• Usually a value weighted stock index portfolio is
  used.
• The implicit assumption is that the return on the
  market portfolio (return on aggregate wealth) is
  linear in the value weighted index return.
• and then
• This is the standard CAPM regression (FM style).
• Of course, the market proxy could matter a lot
  (Roll (1977) and Mayers (1972)).

28
Market Portfolio cont

• Mayers (1972) points out that human capital forms
  a large part of the total capital in the economy.
• Note that monthly per-capita income from
  dividends i the US for 1959-1992 was less than 3
  of the monthly personal income from all sources.
  
• Salaries and wages were 63.
• Common view is that human capital is not tradable
  and must be treated differently.
• But note that mortgage loans are based, in part,
  on labor income.
• There is an important difference between human
  capital and other kinds of capital.
• All cash from corporations is promised through
  securities.
• Only some cash from human capital is promised
  through mortgage payments.

29
How to Measure Human Capital?

• Assume the return on human capital is an exact
  linear function of the growth rate in per-capita
  labor income.
• Suppose to a first order approximation that the
  expected rate of return on human capital is a
  constant, r, and that date-t per-capita labor
  income, Lt, follows an AR process
• Then the realized capital gain part of the rate
  of return on human capital will be the same as
  the realized growth rate in per-capita labor
  income

30
How to Measure Human Capital?

• This follows because wealth due to human capital
  is
• The rate of change in this wealth is

31
Market Portfolio cont

• They note that even though stocks are only a
  small fraction of wealth, the index return could
  be an excellent proxy for the return on aggregate
  wealth. Why?
• Nevertheless, they allow for their measure of
  human capital to augment the standard market
  proxy.
• Let Rtlabor be the growth rate in per-capita
  labor income which proxies for the return on
  human capital.
• Then let the true market return be linear in Rtvw
  and Rtlabor.

32
Market Portfolio cont

• We can then have a labor beta
• And let
• Which leads to
• This is the premium labor model.

33
Econometric Tests

• In light of the existing Fama-French results we
  have natural tests
• Is the size anomaly explained?
• Is the market to book anomaly explained? (Wont
  consider.)
• Let size be log(MVE), where MVE is a time-series
  average. Then, the alternative model is
• and csize should be zero under the null.
• The methodology could be FM (1973) or BJS (1972).
• But the regressors are measured with error.
  Shanken (1992) gives a correction procedure.

34
Econometric Tests cont

• Can also use GMM technique.
• Substitute into the PL model for the betas and
  massage into a stochastic discount factor form
• where ?0, ?vw, ?prem, and ?labor are
• Note that the stochastic discount factor has 4
  parameters.

35
Econometric Tests cont

• Now we have N assets in our econometric tests.
  Let 1N be an N dimensional vector of ones. Then
• and
• Now dt Yt?.
• The pricing errors are wt(?) ? Rtdt 1N and we
  pick the ? vector (4 parameters) using GMM.
• The optimal weighting matrix is not used.

36
Data

• NYSE and AMEX firms covered by CRSP 1962-90
  (dont need compustat why?) Slightly different
  from FF (1992).
• Create 100 portfolios of NYSE/AMEX stocks as in
  FF.
• For every year, starting in 1964, sort into size
  deciles based on MV at end of June.
• For each size decile, estimate beta of each firm,
  using 24 60 months of past data to do so.
• This is the pre-ranking beta estimate.
• Then, sort each size decile into beta deciles
  based on estimates of pre-beta.
• This yields 100 portfolios. Compute the return
  for the next 12 months equally weighting the
  stocks in the portfolio.
• Repeat, yielding a time series of monthly
  observations for the 100 pfs.

37
Data Table I

• Rates of return vary from a low of .51 per month
  to 1.71 per month, panel A.
• The ?ivw range from 0.57 to 1.70.
• The size of the portfolio is the EW average of
  the log of the MVs. This time series is in
  panel C of table 1. This is all similar to FF
  (1992).
• The numbers in panels D and E are the parts of
  ?iprem and ?ilabor that are orthogonal to ?ivw
  (for the first) and also to ?iprem for the labor
  beta.
• Note the funky construction for growth in labor
  income on page 21 to deal with reporting
  convention.

38
Main Results Table II

• Traditional CAPM Panel A
• R2 1.35 and the t on cvw is 0.28, consistent
  with FF.
• When size is added to the model, the t for csize
  is 2.30 and the R2 is 57.36.
• For the GMM test, the pricing error is
  significantly different from zero and the p-value
  of 27.59 on ?vw suggests that Rvw does not play
  a significant role in determining the SDF.

39
(No Transcript)
40
Figure 1 Static Model
41
Main Results cont

• Now let the betas vary over time, but measure the
  market using only the VW portfolio Panel B
• The coefficient cprem is significantly different
  from zero.
• Size still adds explanatory power.
• The pricing errors are still significantly
  different from zero, but Rprem, the spread
  between high and low risk corporate bonds that is
  used to capture the variation of the betas across
  the business cycle enters the SDF.

42
(No Transcript)
43
Figure 2 Time Varying Betas
44
Main Results cont

• The main model in the paper Panel C.
• Much better explanatory power.
• Size no longer adds explanatory power when it is
  included.
• GMM estimation can not reject model.
• Rlabor and Rprem are included in the SDF.
• However
• cvw is negative (but insignificant).
• The zero beta rate is higher than average
  t-bill rates.

45
(No Transcript)
46
Figure 3 Main Model
47
Figure 4 Main Model Plus Size
48
Comparisons with Chen, Roll and Ross

• Are lagged prem factor and labor income growth
  factor proxies for the macro factors of CCR?
• They consider (essentially)
• Spread between long bond and t-bill rates.
• Spread between long corporate and long government
  bonds.
• Growth rate in industrial production.
• Expected inflation.
• Unexpected inflation.
• The CRR model does not fit the data as well.

49
(No Transcript)
50
Comparison with FF (1993)

• The FF model
• Nest their model in the one here a 5 factor
  model.
• Combined model has R2 of 64. Individual models
  have R2s of 55. The HJ distance for the FF
  model is larger.
• The results suggest that the FF factors may be
  proxies for the return on human capital and for
  beta instability.

51
(No Transcript)
52
Conclusions

• Advocate caution in interpreting their results as
  strong support for the conditional CAPM.
• Simple modeling of the time variation in betas.
• Impact of events that occur at deterministic
  frequencies and failure to model these events.
• Ability of this model to fit other choices of
  portfolios is in question.