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### Jagannathan and Wang (1996)

Testing the Conditional CAPM

The Story

- When we test the static CAPM we find that
- We can’t 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.

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?

What If

- Suppose information about an asset’s 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?

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

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.

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.

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.

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.

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.

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 “BARRA’s better betas.”
- Next, they assume that the CAPM holds and derive implications for the unconditional model.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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, it’s 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.

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.

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.

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 paper’s numbers) (14) into (12) and make use of (15) and theorem 1.

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.

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)).

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.

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

How to Measure Human Capital?

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

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.

Market Portfolio cont…

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

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? (Won’t 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.

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.

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.

Data

- NYSE and AMEX firms covered by CRSP 1962-90 (don’t 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.

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 MV’s. 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.

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.

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.

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.

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.

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 R2’s 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.

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.

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