Equity premium puzzle

The economy has a single representative household whose preferences over stochastic consumption paths are given by:

${\displaystyle E_{0}\left[\sum _{t=0}^{\infty }\beta ^{t}U(c_{t})\right]}$

where ${\textstyle 0<\beta <1}$ is the subjective discount factor, ${\textstyle c_{t}}$ is the per capita consumption at time ${\textstyle t}$, U() is an increasing and concave utility function. In the Mehra and Prescott (1985) economy, the utility function belongs to the constant relative risk aversion class:

${\displaystyle U(c,\alpha )={\frac {c^{(1-\alpha )}}{1-\alpha }}}$

where ${\textstyle 0<\alpha <\infty }$ is the constant relative risk aversion parameter. When ${\displaystyle \alpha =1}$, the utility function is the natural logarithmic function. Weil (1989) replaced the constant relative risk aversion utility function with the Kreps-Porteus nonexpected utility preferences.

${\displaystyle U_{t}=\left[c_{t}^{1-\rho }+\beta (E_{t}U_{t+1}^{1-\alpha })^{(1-\rho )/(1-\alpha )}\right]^{1/(1-\rho )}}$

The Kreps-Porteus utility function has a constant intertemporal elasticity of substitution and a constant coefficient of relative risk aversion which are not required to be inversely related - a restriction imposed by the constant relative risk aversion utility function. Mehra and Prescott (1985) and Weil (1989) economies are a variations of Lucas (1978) pure exchange economy. In their economies the growth rate of the endowment process, ${\textstyle x_{t}}$, follows an ergodic Markov Process.

${\displaystyle P\left[x_{t+1}=\lambda _{j}|x_{t}=\lambda _{i}\right]=\phi _{i,j}}$

where ${\textstyle x_{t}\in \{\lambda _{1},...,\lambda _{n}\}}$. This assumption is the key difference between Mehra and Prescott's economy and Lucas' economy where the level of the endowment process follows a Markov Process.

There is a single firm producing the perishable consumption good. At any given time ${\textstyle t}$, the firm's output must be less than or equal to ${\textstyle y_{t}}$ which is stochastic and follows ${\textstyle y_{t+1}=x_{t+1}y_{t}}$. There is only one equity share held by the representative household.

We work out the intertemporal choice problem. This leads to:

${\displaystyle p_{t}U'(c_{t})=\beta E_{t}[(p_{t+1}+y_{t+1})U'(c_{t+1})]}$

as the fundamental equation.

For computing stock returns

${\displaystyle 1=\beta E_{t}\left[{\frac {U'(c_{t+1})}{U'(c_{t})}}R_{e,t+1}\right]}$

where

${\displaystyle R_{e,t+1}=(p_{t+1}+y_{t+1})/p_{t}}$

gives the result.[7]

They can compute the derivative with respect to the percentage of stocks, and this must be zero.

Much data exists that says that stocks have higher returns. For example, Jeremy Siegel says that stocks in the United States have returned 6.8% per year over a 130-year period.

Proponents of the capital asset pricing model say that this is due to the higher beta of stocks, and that higher-beta stocks should return even more.

Others have criticized that the period used in Siegel's data is not typical, or the country is not typical.

A large number of explanations for the puzzle have been proposed. These include:

• rejection of the Arrow-Debreu model in favor of different models,
• modifications to the assumed preferences of investors,
• imperfections in the model of risk aversion,
• the excess premium for the risky assets equation results when assuming exceedingly low consumption/income ratios,
• and a contention that the equity premium does not exist: that the puzzle is a statistical illusion.

Kocherlakota (1996), Mehra and Prescott (2003) present a detailed analysis of these explanations in financial markets and conclude that the puzzle is real and remains unexplained.[8][9] Subsequent reviews of the literature have similarly found no agreed resolution.

The magnitude of the equity premium has implications for resource allocation, social welfare, and economic policy. Grant and Quiggin (2005) derive the following implications of the existence of a large equity premium:

• Macroeconomic variability associated with recessions is expensive.
• Risk to corporate profits robs the stock market of most of its value.
• Corporate executives are under irresistible pressure to make short-sighted decisions.
• Policies—disinflation, costly reform that promises long-term gains at the expense of short-term pain, are much less attractive if their benefits are risky.
• Social insurance programs might well benefit from investing their resources in risky portfolios in order to mobilize additional risk-bearing capacity.
• There is a strong case for public investment in long-term projects and corporations, and for policies to reduce the cost of risky capital.
• Transaction taxes could be either for good or for ill.

• Ellsberg paradox
• Loss aversion
• Risk aversion
• List of cognitive biases
• Economic puzzle
• Forward premium anomaly
• Real exchange-rate puzzles

• Haug, Jørgen; Hens, Thorsten; Woehrmann, Peter (2013). "Risk Aversion in the Large and in the Small". Economics Letters. 118 (2): 310–313. doi:10.1016/j.econlet.2012.11.013. hdl:11250/164171.