Investigación

Trabajos varios durante el posgrado

Resumen de la tesis de posgrado (inglés)

1. A pure exchange economy

Consider a three-period pure exchange economy where agents get utility by means of consumption of a single good. At \(t=0\) there is an open market for two financial assets: a risky one and a risk-free one. The agents, provided with an initial endowment \(\omega\) of the consumption good, must find an optimal investment portfolio to transfer its wealth between periods of time (as in an Arrow-Debreu economy). We introduce imperfect information by the following mechanism. There are two possible future states of the world: a positive state (\(P\)) and a negative state (\(N)\). The state of the world is revealed at \(t=1\). If the positive state is observed, then the agent can consume at every period. If the negative state is observed, then the agent can consume only up to \(t=1\). Therefore, a negative state can be regarded as a state where agents suffer a consumption shock, in the form of an illness, an emergency or other non-insured future event. The key insight here is that at time \(t=0\) the agents do not know the future state of the world, so they must make their investment decision facing a fundamental uncertainty about their future consumption path. Uncertainty in the model comes from two different sources:

  1. the intrinsic volatility of the risky asset (the returns of this asset are stochastic), and
  2. the risk to observe a negative state of the world and therefore to sell the whole portfolio at \(t=1\) to get maximal consumption utility.

We will assume that all the agents know the probability of a negative or positive state, namely \(\pi_1\) and \(\pi_2\) respectively. Therefore, the expected utility of the agent at time \(t=0\) can be expressed as

\[ E_0[u(c_0) + \beta u(c_1) + \pi_2 \beta^2(c_2)] \]

2. The equilibrium of this economy

The equilibrium of this economy is characterized as a Radner equilibrium where the following conditions must hold:

  1. Every agent maximises its expected utility subject to the budget constraint \(c_t \leq p_t \omega_t + z^k_t X^k_t\), where \(X_t\) is the payoff of the \(k\)-th asset a time \(t\): \[ \max_{z^k_0} \ E_0[u(c_0) + \beta u(c_1) + \pi_2 \beta^2(c_2)] \]
  2. Since there is no production, aggregate consumption equals the sum of the individual initial endowments: \[ \sum_i \sum_t c^i_t = \sum_i \sum_t \omega^i_t \] where \(i\) is the agent index.
  3. Holding assets provide no utility; therefore, every acquired asset must be sold to consume. This implies that the sum over time of the portfolios of every agent must be zero: \[ \sum_t z^k_t = 0 \] for every asset \(k\).

In equilibrium, the price \(q_k\) of an asset \(k\) at time \(t=0\) must satisfy: \[ q_k = E_0 \left[ \beta \frac{u’(c_1)}{u’(c_0)} X^k_1 \right] + \pi_2 E_0 \left[ \beta^2 \frac{u’(c_2)}{u’(c_0)} X^k_2 \right] \]

Since \(q_k\) is known at time zero, the pricing equation can be also written as \[ 1 = E_0 \left[ \beta \frac{u’(c_1)}{u’(c_0)} R^k_1 \right] + \pi_2 E_0 \left[ \beta^2 \frac{u’(c_2)}{u’(c_0)} R^k_2 \right] \] where \(R^k_t\) is the gross return of a unit invested at time zero on the asset \(k\), at time \(t\).

Since the latter pricing equation must hold for every asset (including the risk-free one), then the following expression must hold:

\[ E_0 \left[ \delta_1 R_1 \right] + \pi_2 E_0 \left[ \delta_2 R_2 \right] = E_0 \left[ \delta_1 \right] R^f_1 + \pi_2 E_0 \left[ \delta_2 \right] R^f_2 \]

where \(\delta_t = \beta^t u’(c_t) / u’(c_{t-1})\), \(R_t\) is the gross return of the risky asset and \(R^f_t\) is the gross return of the risk-free asset.

This equation shows the relationship that must hold in equilibrium between the risky asset and the risk-free asset. A risk premium can be derived after specifying a functional form for the utility function and making assumptions about the stochastic processes that governs the return of the risky asset and the aggregate consumption of the economy.

4. Results

After specifying the model, we found that it does not attain the observed risk premium from U.S. data for the period 1948-2020, given reasonable values of relative risk aversion. However, it performs better than a model with zero probability of a consumption shock. It does, indeed, perform increasingly better as the probability of a consumption shock (i.e \(\pi_2\), which are the odds of observing a negative state of the world) goes up. This suggests that this class of models, where an additional source of uncertainty is introduced as imperfect information about the future consumption paths of the agents, can potentially explain the equity premium puzzle under more complex specifications.

5. Some key references

  1. About the premium puzzle itself:
    • Abel, Andrew B (1990). “Asset Prices under Habit Formation and Catching up with the Joneses”. The American Economic Review, 80, pp. 38–42. ISSN 00028282. http://www.jstor.org/stable/2006539.
    • Brav, Alon; Constantinides, George and Geczy, Christopher (1999). Asset Pricing with Heterogeneous Consumers and Limited Participation: Empirical Evidence. doi: 10.3386/w7406.
    • Campbell, John Y. (2000). Asset Pricing at the Millennium. The Journal of Finance, 55. ISSN 00221082. doi: 10.1111/0022-1082.00260.
    • Campbell, John Y. and Cochrane, John H. (1999). By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior. Journal of Political Economy, 107. ISSN 0022-3808. doi: 10.1086/250059.
    • Constantinides, G. M.; Donaldson, J. B. and Mehra, R. (2002). Junior Can’t Borrow: A New Perspective on the Equity Premium Puzzle. The Quarterly Journal of Economics, 117. ISSN 0033-5533. doi: 10.1162/003355302753399508.
    • Constantinides, George M. (1990). Habit Formation: A Resolution of the Equity Premium Puzzle. Journal of Political Economy, 98. ISSN 0022-3808. doi: 10.1086/261693.
    • Constantinides, George M and Duffie, Darrell (1996). Asset Pricing with Heterogeneous Consumers. Journal of Political Economy, 104, pp. 219–240. ISSN 00223808, 1537534X. http://www.jstor.org/stable/2138925.
    • DaSilva, Amadeu; Farka, Mira and Giannikos, Christos (2019). Age-Dependent Increasing Risk Aversion and the Equity Premium Puzzle. Financial Review, 54. ISSN 0732-8516. doi: 10.1111/fire.12191.
    • Epstein, Larry G and Zin, Stanley E (1991). Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis. Journal of Political Economy, 99. doi: 10.1086/261750.
    • Mehra, Rajnish and Prescott, Edward C (1985). The Equity Premium, a Puzzle. Journal of Monetary Economics, 15, pp. 145–161.
    • Mehra, Rajnish and Prescott, Edward C (2003). The Equity Premium in Retrospect. Handbook of the Economics of Finance, pp. 889–938. doi: 10.1016/S1574-0102(03)01023-9.
    • Rietz, Thomas A. (1988). The equity risk premium a solution. Journal of Monetary Economics, 22. ISSN 03043932. doi: 10.1016/0304-3932(88)90172-9.
    • Shirvani, Abootaleb; Stoyanov, Stoyan Veselinov; Fabozzi, Frank J. and Rachev, Svetlozar T. (2020). Equity Premium Puzzle or Faulty Economic Modelling? SSRN Electronic Journal. ISSN 1556-5068. doi: 10.2139/ssrn.3530635.
    • Wang, Yuanping and Mu, Congming (2019). Can ambiguity about rare disasters explain equity premium puzzle? Economics Letters, 183. ISSN 01651765. doi: 10.1016/j.econlet.2019.108555.
    • Wilson, Matthew S. (2020). Disaggregation and the equity premium puzzle. Journal of Empirical Finance, 58. ISSN 09275398. doi: 10.1016/j.jempfin.2020.05.002.
  2. About the relevance of consumption shocks:
    • Bunn, Philip; Roux, Jeanne Le; Reinold, Kate and Surico, Paolo (2018). The consumption response to positive and negative income shocks. Journal of Monetary Economics, 96. ISSN 03043932. doi: 10.1016/j.jmoneco.2017.11.007.
    • Christelis, Dimitris; Georgarakos, Dimitris; Jappelli, Tullio; Pistaferri, Luigi and van Rooij, Maarten (2019). Asymmetric Consumption Effects of Transitory Income Shocks. The Economic Journal, 129. ISSN 0013-0133. doi: 10.1093/ej/uez013.
    • Karahan, Fatih and Ozkan, Serdar (2013). On the persistence of income shocks over the life cycle: Evidence, theory, and implications. Review of Economic Dynamics, 16. ISSN 10942025. doi: 10.1016/j.red.2012.08.001.
  3. About the estimation of the model parameters:
    • Ahmed, Waqas; Haider, Adnan y Iqbal, Javed (2012). Estimation of discount factor (beta) and coefficient of relative risk aversion (gamma) in selected countries. MPRA. https://EconPapers.repec.org/RePEc:pra:mprapa:39736.
    • Booij, Adam S. and van Praag, Bernard M.S. (2009). A simultaneous approach to the estimation of risk aversion and the subjective time discount rate. Journal of Economic Behavior and Organization, 70, pp. 374–388. ISSN 01672681. doi: 10.1016/j.jebo.2009.01.005.