Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles

Bansal, Ravi, and Amir Yaron, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles,” The Journal of Finance Vol. 59 (2004), 1481-1509.

Like Barro (2006), Bansal & Yaron try to resolve the Mehra-Prescott (1985) equity premium puzzle by adding more risk.  Whereas Barro’s risk is the small probability of a major economic disaster, Bansal and Yaron’s risk is a permanent component in the consumption growth process.

With a permanent component, a negative shock at any point in time to consumption should have effects reaching far into the future.  The shocks compound over time, in a sense.  This means consumers/investors should be very sensitive to news about consumption growth.

The authors decline to give economic rationale for why consumption growth should have a permanent component.  They say in the same breath that it is econometrically impossible to distinguish between iid consumption growth and a growth process with a permanent component, and that their model is consistent with observed data.  It follows that the opposite story (iid growth) is also consistent with the data!

Bansal and Yaron also equate consumption to expenditures, thus ignoring both savings and durable goods, both of which I believe have important implications for consumer behavior and asset pricing.

Rare Disasters and Asset Prices in the Twentieth Century

Barro, Robart J, “Rare Disasters and Asset Prices in the Twentieth Century,” The Quarterly Journal of Economics Vol. 121 No. 3 (2006), 823-866.

Mehra and Prescott (1985) find an equity premium puzzle, which is that either

  • the observed equity premium and volatility imply very high consumer risk aversion, or
  • a more reasonable level of risk aversion would imply a risk free rate that is higher and more variable than the one we see in the data.

At a very high level, Barro and others attempt to solve the puzzle by introducing more risk into their models.  Rather than suggesting that consumers are indeed extremely risk averse, they allow risk aversion to be low but suggest Mehra and Prescott simply didn’t account for enough risk.  This explanation is extremely intuitive and flexible enough, I think to encompass a wide variety of methods for adding risk.  So, I find the first few pages of this article instructive.  The probability of a rare disaster could certainly be something consumers have in mind.  The logic is solid.  The rest of the paper discusses a calibration that I find unconvincing at best.

The problems with the calibration are as follows.

  • Barro relies on a very small sample of rare disasters.  Although he cites 60 instances, they can mostly be reduced to four “events”: WWI, WWII, the Great Depression, and two decades of revolutions in South America.  It is difficult to believe that anything can be inferred from the timing of these occurrences.
  • In order to make inferences, Barro relies on three very strong assumptions, which are that (1) events of major social unrest are uncorrelated with one another, that (2) they are randomly and uniformly distributed across countries, and that (3) they are randomly and uniformly distributed across time.  This third assumption means that the probability of disaster is constant across time.
  • Finally, Barro conflates consumption and expenditures.  In other words, there are no durable goods and the representative consumer has zero savings.  When GDP falls, Barro assumes that consumption also falls and so expected stock returns must be high to offset this risk.  Durable goods and savings are two methods that consumers can use to smooth actual consumption.

Barro’s base model is meant to be simple, and he does acknowledge in closing that stochastic default probability would be an obvious way to extend the model.  I can allow for the theoretical limitations.  The calibration, however, I find too incredible to be very useful.  If anything, he shows that a 1.7% chance of a bad event is about the right amount of extra risk to solve the equity premium puzzle.  He is not convincing that 1.7% is anywhere close to the actual probability of another world war or communist revolution.

Endogenous Disasters and Asset Prices

Petrosky-Nadeau, Nicolas, Lu Zhang, Lars-Alexander Kuehn, “Endogenous Disasters and Asset Prices,” Charles A. Dice Center Working Paper No. 2012-1 (October 1, 2013).

Purpose: This model produces a realistic equity premium and stock return variance, and endogenously leads to rare economic disasters, at the confluence of small corporate profits, large job flows, and frictions in the matching process that connects unemployed workers with job vacancies.

Model:

  1. A representative household, with both employed and unemployed members, chooses its optimal consumption and asset allocation (holdings of shares in a representative firm and of a risk-free bond).
  2. A representative firm posts job vacancies, and unemployed workers apply for them.
    1. vacancies are costly for the firm.
  3. The labor market is a matching function that produces jobs using vacancies and unemployed workers as inputs.
    1. Matching frictions are composed of fixed and variable hiring costs.
    2. The wage rate is determined by a Nash bargaining process.

Results:

  1. The model generates an equity premium of 5.70%, versus 5.07% in the data (adjusted for financial leverage).
  2. Annual stock market volatility in the model is 10.83%, versus 12.94% in the data.
  3. The model’s interest rate volatility is 1.34%, versus the observed 1.87%.
  4. The equity premium is countercyclical, both in the model and in the data.
  5. The ratio of vacancies to unemployed workers forecasts (with a negative slope) excess returns; this is confirmed in the data.
  6. Rare disasters are endogenous.
    1. The average peak-to-trough magnitude of a disaster is roughly 20%, both modeled and observed.
    2. The probability of a consumption disaster is 3.08% in the model and 3.63% in the data.
    3. The probability of a GDP disaster is 4.66% in the model and 3.69% in the data.
  7. Comparative statics
    1. The value of workers’ activities in unemployment are assumed to have a high value, which makes wages inelastic. When output falls in hard times, wages fall less, and so the cyclical nature of profits and dividends is magnified. This raises the equity premium and makes the stock market more volatile compared to other models.
    2. Job flows are assumed to be about 5%, consistent with previous literature (5% of the workforce quits each month), so frictions in the matching process contribute to macroeconomic risk.
    3. Matching frictions (especially fixed hiring costs) cause marginal hiring costs to fall slowly in a recession and to rise quickly in an expansion.
      1. In a recession, there are many unemployed workers and few vacancies. An additional vacancy has only a slight impact on the likelihood of an existing vacancy being filled, so hiring costs fall slowly.  As workers continue to attrite  at a 5% rate, hiring may not keep up and the economy may fall off a cliff.
      2. In an expansion, there are few unemployed workers and many vacancies.  An additional vacancy in an expansion has a large (negative) impact on the likelihood of a vacancy being filled, so marginal hiring costs rise quickly, hampering the expansion.

Asset Prices and Business Cycles with Financial Shocks

Nezefat, Mahdi, and Ctirad Slavik, “Asset Prices and Business Cycles with Financial Shocks,” American Finance Association, 75th Annual Meeting, Boston (2015).

Purpose:  This paper introduces a DSGE asset pricing model in which shocks to firm productivity and to firm financial constraints lead to asset price volatility.

Model:

  • Setup
    • two consumers (entrepreneurs and laborers)
    • two goods (a consumption good and a capital good)
    • infinite and discrete time, with two subperiods in each period
      • in subperiod 1, all entrepreneurs hire labor and produce using the same technology.
      • in subperiod 2, a fraction of entrepreneurs are randomly presented with investment opportunities (i.e. the ability to transform the consumption good one-to-one into capital, without adjustment costs).
        • Firms not investing in new projects can purchase equity in other firms.
    • Equity is the only asset traded in the market (incomplete markets).
    • Firms’ financial constraint (the financial friction) is that there is a limit on how much of each new project can be sold as equity.
      • This limit changes over time, which is a theoretical contribution of this model.
    • Entrepreneurs and laborers maximize the present value of their consumption subject to a budget constraint, and wages and return on equity are determined competitively, and markets clear.
  • Productivity and financial shocks:
    • Productivity shocks affect the wealth of all firms, changing how much they can spend on equity.
    • Financial shocks affect the funding of firms with investment projects, and determine how much equity is available to the market.
    • These two shocks directly influence the amount of equity traded and investors’ budget constraints, and so directly contribute to fluctuations in asset prices.

Results:

  • After calibrating the model to the U.S. economy, productivity shocks alone explain little of the observed volatility.
  • With both types of shocks, modeled asset price volatility is about 80% of the observed volatility of the stock market.
  • The model explains 70% of the observed equity premium.
  • This model also generates the volatility in investment that is observed in the data.
  • Unlike previous models, the equilibrium here is not Pareto optimal.  The government could increase all agents’ welfare by relaxing the financial constraints of entrepreneurs with investment projects by extending loans to them.