Equity premium puzzle

The equity premium puzzle refers to the inability of an important class of economic models to explain the average equity risk premium provided by a diversified portfolio of equities over that of government bonds, which has been observed for more than 100 years. There is a significant disparity between returns produced by stocks compared to returns produced by government treasury bills. The equity premium puzzle addresses the difficulty in understanding and explaining this disparity. This disparity is calculated using the equity risk premium:
The equity risk premium is equal to the difference between equity returns and returns from government bonds. It is equal to around 5% to 8% in the United States.
The risk premium represents the compensation awarded to the equity holder for taking on a higher risk by investing in equities rather than government bonds. However, the 5% to 8% premium is considered to be an implausibly high difference and the equity premium puzzle refers to the unexplained reasons driving this disparity.

Description

The term was coined by Rajnish Mehra and Edward C. Prescott in a study published in 1985 titled "The Equity Premium: A Puzzle". An earlier version of the paper was published in 1982 under the title "A test of the intertemporal asset pricing model". The authors found that a standard general equilibrium model, calibrated to display key U.S. business cycle fluctuations, generated an equity premium of less than 1% for reasonable risk aversion levels. This result stood in sharp contrast with the average equity premium of 6% observed during the historical period.
In 1982, Robert J. Shiller published the first calculation that showed that either a large risk aversion coefficient or counterfactually large consumption variability was required to explain the means and variances of asset returns. Azeredo shows, however, that increasing the risk aversion level may produce a negative equity premium in an Arrow-Debreu economy constructed to mimic the persistence in U.S. consumption growth observed in the data since 1929.
The intuitive notion that stocks are much riskier than bonds is not a sufficient explanation of the observation that the magnitude of the disparity between the two returns, the equity risk premium, is so great that it implies an implausibly high level of investor risk aversion that is fundamentally incompatible with other branches of economics, particularly macroeconomics and financial economics.
The process of calculating the equity risk premium, and selection of the data used, is highly subjective to the study in question, but is generally accepted to be in the range of 3–7% in the long-run. Dimson et al. calculated a premium of "around 3–3.5% on a geometric mean basis" for global equity markets during 1900–2005. However, over any one decade, the premium shows great variability—from over 19% in the 1950s to 0.3% in the 1970s.
In 1997, Siegel found that the actual standard deviation of the 20-year rate of return was only 2.76%. This means that for long-term investors, the risk of holding the stock of a smaller than expected can be derived only by looking at the standard deviation of annual earnings. For long-term investors, the actual risks of fixed-income securities are higher. Through a series of reasoning, the equity premium should be negative.
To quantify the level of risk aversion implied if these figures represented the expected outperformance of equities over bonds, investors would prefer a certain payoff of $51,300 to a 50/50 bet paying either $50,000 or $100,000.
The puzzle has led to an extensive research effort in both macroeconomics and finance. So far a range of useful theoretical tools and numerically plausible explanations have been presented, but no one solution is generally accepted by economists.

Theory

The economy has a single representative household whose preferences over stochastic consumption paths are given by:
where is the subjective discount factor, is the per capita consumption at time, U is an increasing and concave utility function. In the Mehra and Prescott economy, the utility function belongs to the constant relative risk aversion class:
where is the constant relative risk aversion parameter. Note that. Weil replaced the constant relative risk aversion utility function with the Kreps-Porteus nonexpected utility preferences.
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 and Weil economies are a variations of Lucas pure exchange economy. In their economies the growth rate of the endowment process,, follows an ergodic Markov Process.
where. 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, the firm's output must be less than or equal to which is stochastic and follows. There is only one equity share held by the representative household.
We work out the intertemporal choice problem. This leads to:
as the fundamental equation.
For computing stock returns
where
gives the result.
The derivative of the Lagrangian with respect to the percentage of stock held must equal zero to satisfy necessary conditions for optimality under the assumptions of no arbitrage and the law of one price.

Data

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.

Possible explanations

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

Rare events hypothesis,
Myopic loss aversion,
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, Mehra and Prescott present a detailed analysis of these explanations in financial markets and conclude that the puzzle is real and remains unexplained. Subsequent reviews of the literature have similarly found no agreed resolution.
The mystery of stock premiums occupies a special place in financial and economic theories, and more progress is needed to understand the spread of stocks on bonds. Over time, as well as to determine the factors driving equity premium in various countries / regions may still be active research agenda.
A 2023 paper by Edward McQuarrie argues the equity risk premium may not exist, at least not as is commonly understood, and is furthermore based on data from a too narrow a time period in the late 20th century. He argues a more detailed examination of historical data finds "over multi-decade periods, sometimes stocks outperformed bonds, sometimes bonds outperformed stocks and sometimes they performed about the same. New international data confirm this pattern. Asset returns in the US in the 20th century do not generalize."

The equity premium: a deeper puzzle

Azeredo showed that traditional pre-1930 consumption measures understate the extent of serial correlation in the U.S. annual real growth rate of per capita consumption of non-durables and services. Under alternative measures proposed in the study, the serial correlation of consumption growth is found to be positive. This new evidence implies that an important subclass of dynamic general equilibrium models studied by Mehra and Prescott generates negative equity premium for reasonable risk-aversion levels, thus further exacerbating the equity premium puzzle.

Rare events hypothesis

One possible solution to the equity premium puzzle considered by Julliard and Ghosh is whether it can be explained by the rare events hypothesis, founded by Rietz. They hypothesized that extreme economic events such as the Great Depression, the World Wars and the Great Financial Crisis resulted in equity holders demanding high equity premiums to account for the possibility of the significant loss they could suffer if these events were to materialise. As such, when these extreme economic events do not occur, equity holders are rewarded with higher returns. However, Julliard and Ghosh concluded that rare events are unlikely to explain the equity premium puzzle because the Consumption Capital Asset Pricing Model was rejected by their data and much greater risk aversion levels were required to explain the equity premium puzzle. Moreover, extreme economic events affect all assets and they all yield low returns. For example, the equity premium persisted during the Great Depression, and this suggests that an even greater catastrophic economic event is required, and it must be one which only affect stocks, not bonds.

Myopic loss aversion

Benartzi & Thaler contend that the equity premium puzzle can be explained by myopic loss aversion and their explanation is based on Kahneman and Tversky's prospect theory. They rely on two assumptions about decision-making to support theory; loss aversion and mental accounting. Loss aversion refers to the assumption that investors are more sensitive to losses than gains, and in fact, research calculates utility of losses felt by investors to be twice that of the utility of a gain. The second assumption is that investors frequently evaluate their stocks even when the purpose of the investment is to fund retirement or other long-term goals. This makes investors more risk averse compared to if they were evaluating their stocks less frequently. Their study found that the difference between returns gained from stocks and returns gained from bonds decrease when stocks are evaluated less frequently. The two combined creates myopic loss aversion and Benartzi & Thaler concluded that the equity premium puzzle can be explained by this theory.