The equity premium puzzle refers to the inability of an important class of economic models to explain the average equity risk premium (ERP) provided by a diversified portfolio of U.S. equities over that of U.S. Treasury Bills, which has been observed for more than 100 years.

Description

The term was coined by Rajnish Mehra and Edward C. Prescott in a study published in 1985 titled The Equity Premium: A Puzzle,.[1][2] 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.[3] Azeredo (2014) 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.[4]

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 (ERP), 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 (2006).[6] 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 [7]

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.[8]

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:

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

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.[9]

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

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:

• 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.[10][11] 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. [12]

The equity premium: a deeper puzzle

Azeredo (2014) 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 ("consumption growth").[13] 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 (1985) generates negative equity premium for reasonable risk-aversion levels, thus further exacerbating the equity premium puzzle.

Individual characteristics

Some explanations rely on assumptions about individual behavior and preferences different from those made by Mehra and Prescott. Examples include the prospect theory model of Benartzi and Thaler (1995) based on loss aversion.[14] A problem for this model is the lack of a general model of portfolio choice and asset valuation for prospect theory.

A second class of explanations is based on relaxation of the optimization assumptions of the standard model. The standard model represents consumers as continuously-optimizing dynamically-consistent expected-utility maximizers. These assumptions provide a tight link between attitudes to risk and attitudes to variations in intertemporal consumption which is crucial in deriving the equity premium puzzle. Solutions of this kind work by weakening the assumption of continuous optimization, for example by supposing that consumers adopt satisficing rules rather than optimizing. An example is info-gap decision theory,[15] based on a non-probabilistic treatment of uncertainty, which leads to the adoption of a robust satisficing approach to asset allocation.

Equity characteristics

Another explanation of the equity premium puzzle focuses on the characteristics of equity that cannot be captured by typical models but are still consistent with optimisation by investors.

The most significant characteristic that is not typically considered is the requirement for equity holders to monitor their activity and have a manager to assist them. Therefore, the principle-agent relationship is very prevalent between corporation managers and equity holders. If an investor was to choose to not have a manager, it is likely costly for them to monitor the activity of the corporations that they invest in and often rely heavily on auditors or they look to the market hypothesis in which information about asset values in the equity markets are exposed. This hypothesis is based on the theory that an investor who is inexperienced and uninformed can bank on the fact that they will get average market returns in an identifiable market portfolio, which is questionable as to whether or not this can be done by an uninformed investor. Although, as per the characteristics of equity in explaining the premium, it is only necessary to hypothesise that people looking to invest do not think they can reach the same level of performance of the market.[16]

Another explanation related to the characteristics of equity was explored by a variety of studies including Holmstrom and Tirole (1998),[17] Bansal and Coleman (1996)[18] and Palomino(1996)and was in relation to liquidity.[19] Palomino described the noise trader model that was thin and had imperfect competition is the market for equities and the lower its equilibrium price dropped the higher the premium over risk-free bonds would rise.[20] Holmstrom and Tirole in their studies developed another role for liquidity in the equity market that involved firms willing to pay a premium for bonds over private claims when they would be facing uncertainty over liquidity needs. [21]

Tax distortions

Another explanation related to the observed growing equity premium was argued by McGrattan and Prescott (2001)[22] to be a result of variations over time of taxes and particularly its effect on interest and dividend income. It is difficult however to give credibility to this analysis due to the difficulties in calibration utilised as well as ambiguity surrounding the existence of any noticeable equity premium before 1945.[23] Even given this, it is evident that the observation that equity premium changes arising from the distortion of taxes over time should be taken into account and give more validity to the equity premium itself.

Related data is mentioned in the Handbook of the Equity Risk Premium. Beginning in 1919, it captured the post-World War I recovery, while omitting wartime losses and low pre-war returns. After adding these earlier years, the arithmetic average of the British stock premium for the entire 20th century is 6.6%, which is about 21/4% lower than the incorrect data inferred from 1919-1999. [24]

Implied volatility

Graham and Harvey have estimated that, for the United States, the expected average premium during the period June 2000 to November 2006 ranged between 4.65 and 2.50.[25] They found a modest correlation of 0.62 between the 10-year equity premium and a measure of implied volatility (in this case VIX, the Chicago Board Options Exchange Volatility Index).

Market failure explanations

Two broad classes of market failure have been considered as explanations of the equity premium. First, problems of adverse selection and moral hazard may result in the absence of markets in which individuals can insure themselves against systematic risk in labor income and noncorporate profits. Second, transaction costs or liquidity constraints may prevent individuals from smoothing consumption over time.

A final possible explanation is that there is no puzzle to explain: that there is no equity premium.[citation needed] This can be argued from a number of ways, all of them being different forms of the argument that we don't have enough statistical power to distinguish the equity premium from zero:

• Selection bias of the US market in studies. The US market was the most successful stock market in the 20th century. Other countries' markets displayed lower long-run returns (but still with positive equity premiums). Picking the best observation (US) from a sample leads to upwardly biased estimates of the premium.
• Survivorship bias of exchanges: exchanges often go bust (just as governments default; for example, Shanghai stock exchange during 1949 communist takeover), and this risk needs to be included – using only exchanges which have survived for the long-term overstates returns. Exchanges close often enough for this effect to matter.[26]
• Low number of data points: the period 1900–2005 provides only 105 years which is not a large enough sample size to run statistical analyses with full confidence, especially in view of the black swan effect.
• Windowing: returns of equities (and relative returns) vary greatly depending on which points are included. Using data starting from the top of the market in 1929 or starting from the bottom of the market in 1932 (leading to estimates of equity premium of 1% lower per year), or ending at the top in 2000 (vs. bottom in 2002) or top in 2007 (vs. bottom in 2009 or beyond) completely change the overall conclusion. However, in all windows considered, the equity premium is always greater than zero.

A related criticism is that the apparent equity premium is an artifact of observing stock market bubbles in progress.

• David Blitz, head of Quant Research at Robeco, suggested that the size of the equity premium is not as large as widely believed. It is usually calculated, he said, on the presumption that the true risk-free asset is the one month T bill. If one recalculates, taking the five-year T-bond as the risk free asset, the equity premium is less impressive. Furthermore, he gives his reasons for believing one should recalculate in that way.

Note however that most mainstream economists agree that the evidence shows substantial statistical power.

Implications

The magnitude of the equity premium brings about substantial implications for policy, welfare and also resource allocation.

Policy and Welfare Implications

Campbell and Cochrane (1995) have found in a study of a model that simulates equity premium value's consistent with asset prices, welfare costs are similar in magnitude to welfare benefits.[27] Therefore essentially, a large risk premium in society where asset prices are a reflection of consumer preferences, implies that the cost of welfare is also large. It also means that in recessions, welfare costs are excessive regardless of aggregate consumption. As the equity premium rises, recession-state income marginal values steadily increase also thus further increasing the welfare costs of recessions. This also brings about questions regarding the need for microeconomic policies that operate by way of higher productivity in the long run by trading off short-term pain in the form of adjustment costs. Given the impact on welfare through recessions and the large equity premium, it is evident that these short-term trade offs as a result of economic policy are likely not ideal, and would be preferred to take place in times of normal economic activity.[28]

Resource Allocation

References

1. ^ Mehra, Rajnish; Edward C. Prescott (1985). "The Equity Premium: A Puzzle" (PDF). Journal of Monetary Economics. 15 (2): 145–161. doi:10.1016/0304-3932(85)90061-3. Archived (PDF) from the original on 2009-03-27. Retrieved 2007-05-01.
2. ^ Handbook of the Equity Risk Premium Archived 2020-12-29 at the Wayback Machine, edited by Rajnish Mehra
3. ^ "Consumption, Asset Markets, and Macroeconomic Fluctuations," Carnegie Rochester Conference Series on Public Policy 17 203-238
4. ^ Azeredo, F. (2014). "The equity premium: a deeper puzzle" (PDF). Annals of Finance. 10 (3): 347–373. doi:10.1007/s10436-014-0248-7. S2CID 7108921. Archived (PDF) from the original on 2020-12-29. Retrieved 2019-09-01.
5. ^ Larson, Francis (2016). "Can Myopic Loss Aversion Explain the Equity Premium Puzzle? Evidence from a Natural Field Experiment with Professional Traders". WORKING PAPER 22605. doi:10.3386/w22605. Retrieved 2016-09-23. Cite journal requires |journal= (help)
6. ^ Dimson, Elroy; Marsh, Paul; Staunton, Mike (2008). "The Worldwide Equity Premium: A Smaller Puzzle". Handbook of the Equity Risk Premium. Amsterdam: Elsevier. ISBN 978-0-08-055585-0. SSRN 891620.
7. ^ Siegel, Jeremy; Thaler, Richard. "Download Limit Exceeded". citeseerx.ist.psu.edu. Retrieved 2021-04-27.
8. ^ Mankiw, N. Gregory; Zeldes, Stephen P. (1991). "The Consumption of Stockholders and Nonstockholders". Journal of Financial Economics. 29 (1): 97–112. CiteSeerX 10.1.1.364.2730. doi:10.1016/0304-405X(91)90015-C. S2CID 3084416.
9. ^ "The Equity Premium Puzzle: A Review" (PDF). Archived (PDF) from the original on 2020-12-29. Retrieved 2014-07-04.
10. ^ Kocherlakota, Narayana R. (March 1996). "The Equity Premium: It's Still a Puzzle" (PDF). Journal of Economic Literature. 34 (1): 42–71. Archived (PDF) from the original on 2020-12-29. Retrieved 2006-10-21.
11. ^ Mehra, Rajnish; Edward C. Prescott (2003). "The Equity Premium Puzzle in Retrospect" (PDF). In G.M. Constantinides, M. Harris and R. Stulz (ed.). Handbook of the Economics of Finance. Amsterdam: North Holland. pp. 889–938. ISBN 978-0-444-51363-2. Archived (PDF) from the original on 2020-12-29. Retrieved 2007-05-01.
12. ^ Bakshi, Gurdip; Chen, Zhiwu (2008-01-01). "Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle". Handbook of the Equity Risk Premium: 377–402. doi:10.1016/B978-044450899-7.50018-X. ISBN 9780444508997.
13. ^ Azeredo, F. (2014). "The equity premium: a deeper puzzle" (PDF). Annals of Finance. 10 (3): 347–373. doi:10.1007/s10436-014-0248-7. S2CID 7108921. Archived (PDF) from the original on 2020-12-29. Retrieved 2019-09-01.
14. ^ Benartzi, Shlomo; Richard H. Thaler (February 1995). "Myopic Loss Aversion and the Equity Premium Puzzle" (PDF). Quarterly Journal of Economics. 110 (1): 73–92. doi:10.2307/2118511. JSTOR 2118511. S2CID 55030273. Archived (PDF) from the original on 2020-12-29. Retrieved 2019-09-23.
15. ^ Yakov Ben-Haim, Info-Gap Decision Theory: Decisions Under Severe Uncertainty, Academic Press, 2nd edition, Sep. 2006. ISBN 0-12-373552-1.
16. ^ Grant, Simon; Quiggin, John (2006). "The Risk Premium for Equity: Implications for Resource Allocation, Welfare and Policy". Australian Economic Papers. 45 (3): 253–268. doi:10.1111/j.1467-8454.2006.00291.x. S2CID 15693644.
17. ^ Holmstrom, Bengt; Tirole, Jean (February 1998). "Private and Public Supply of Liquidity". Journal of Political Economy. 106 (1): 1–40. doi:10.1086/250001. hdl:1721.1/64064. S2CID 158080077.
18. ^ Bansal, Ravi; Coleman, Wilbur (December 1996). "A Monetary Explanation of the Equity Premium, Term Premium, and Risk-Free Rate Puzzles". Journal of Political Economy. 104 (6): 1135–1171. doi:10.1086/262056. S2CID 54871134.
19. ^ Palomino, Frederic (September 1996). "Noise Trading in Small Markets". The Journal of Finance. 51 (4): 1537–1550. doi:10.2307/2329404. hdl:1814/498. JSTOR 2329404.
20. ^ Palomino, Frederic (September 1996). "Noise Trading in Small Markets". The Journal of Finance. 51 (4): 1537–1550. doi:10.2307/2329404. hdl:1814/498. JSTOR 2329404.
21. ^ Holmstrom, Bengt; Tirole, Jean (February 1998). "Private and Public Supply of Liquidity". Journal of Political Economy. 106 (1): 1–40. doi:10.1086/250001. hdl:1721.1/64064. S2CID 158080077.
22. ^ McGrattan, Ellen; Prescott, Edward (December 2001). "axes, Regulations, and Asset Prices". National Bureau of Economic Research. Working Paper 8623. doi:10.3386/w8623.
23. ^ Mehra, Rajnish (February 2003). "The Equity Premium: Why is it a Puzzle?". NBER Working Papers. 9512. doi:10.3386/w9512.
24. ^ Dimson, Elroy; Staunton, Mike; Marsh, Paul. "Handbook of the Equity Risk Premium | ScienceDirect". www.sciencedirect.com. Retrieved 2021-04-27.
25. ^ Graham, John R.; Harvey, Campbell R. (2007). "The Equity Risk Premium in January 2007: Evidence from the Global CFO Outlook Survey". Working Paper. SSRN 959703.
26. ^ Performance Persistence - Stephen J. Brown and William N. Goetzman (1995) he Journal of Finance Vol. 50, No. 2 (Jun., 1995), pp. 679-698 (20 pages) https://www.jstor.org/stable/2329424?seq=1#metadata_info_tab_contents Archived 2020-12-29 at the Wayback Machine
27. ^ Campbell, John; Cochrane, John (April 1999). "By Force of Habit: A Consumption‐Based Explanation of Aggregate Stock Market Behavior". Journal of Political Economy. 107 (2): 205–251. doi:10.1086/250059. S2CID 18750189.
28. ^ Grant, Simon; Quiggin, John (2006). "The Risk Premium for Equity: Implications for Resource Allocation, Welfare and Policy". Australian Economic Papers. 45 (3): 253–268. doi:10.1111/j.1467-8454.2006.00291.x. S2CID 15693644.
29. ^ Grant, Simon; Quiggin, John (2006). "The Risk Premium for Equity: Implications for Resource Allocation, Welfare and Policy". Australian Economic Papers. 45 (3): 253–268. doi:10.1111/j.1467-8454.2006.00291.x. S2CID 15693644.
30. ^ Shiller, Robert (16 August 2016). Irrational Exuberance (3 ed.). Princeton University Press.
31. ^ Jagannathan, Ravi; McGrattan, Ellen; Scherbina, Anna (2001). "The Declining U.S. Equity Premium". Quarterly Review. Federal Reserve Bank of Mineneapolis. 24 (4): 3–19. doi:10.21034/qr.2441.
32. ^ McGrattan, Ellen; Prescott, Edward (December 2001). "axes, Regulations, and Asset Prices". National Bureau of Economic Research. Working Paper 8623. doi:10.3386/w8623.
33. ^ Grant, Simon; Quiggin, John (2006). "The Risk Premium for Equity: Implications for Resource Allocation, Welfare and Policy". Australian Economic Papers. 45 (3): 253–268. doi:10.1111/j.1467-8454.2006.00291.x. S2CID 15693644.
34. ^ Grant, Simon; Quiggin, John (2006). "The Risk Premium for Equity: Implications for Resource Allocation, Welfare and Policy". Australian Economic Papers. 45 (3): 253–268. doi:10.1111/j.1467-8454.2006.00291.x. S2CID 15693644.