Editing Equity premium puzzle (section)

== 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 (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.<ref>{{cite journal |last1=Kocherlakota |first1=Narayana R. |title=The Equity Premium: It's Still a Puzzle |journal=Journal of Economic Literature |date=1996 |volume=34 |issue=1 |pages=42–71 |jstor=2729409 }}</ref><ref>{{cite book |doi=10.1016/S1574-0102(03)01023-9 |title=Financial Markets and Asset Pricing |series=Handbook of the Economics of Finance |year=2003 |last1=Mehra |first1=Rajnish |last2=Prescott |first2=Edward C. |chapter=Chapter 14 the equity premium in retrospect |volume=1 |pages=889–938 |isbn=978-0-444-51363-2 }}</ref> 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.<ref>{{cite book |doi=10.1016/B978-044450899-7.50018-X |chapter=Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle |title=Handbook of the Equity Risk Premium |year=2008 |last1=Bakshi |first1=Gurdip |last2=Chen |first2=Zhiwu |pages=377–402 |isbn=978-0-444-50899-7 }}</ref>

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."<ref>McQuarrie, E. F. (2023). Stocks for the Long Run? Sometimes Yes, Sometimes No. Financial Analysts Journal, 80(1), 12–28. https://doi.org/10.1080/0015198X.2023.2268556</ref>

=== 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").<ref name=Azeredo2014/> 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.

=== Rare events hypothesis ===

One possible solution to the equity premium puzzle considered by Julliard and Ghosh (2008) is whether it can be explained by the rare events hypothesis, founded by Rietz (1988).<ref>{{cite journal |last1=Julliard |first1=Christian |last2=Ghosh |first2=Anisha |title=Can Rare Events Explain the Equity Premium Puzzle? |journal=The Review of Financial Studies |date=October 2012 |volume=25 |issue=10 |page=3037 |doi=10.1093/rfs/hhs078 |jstor=41678608 |url=https://www.jstor.org/stable/41678608 |access-date=30 April 2022}}</ref> 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.<ref>{{cite journal |last1=Gillard |first1=Christian |title=Can Rare Events Explain the Equity Premium Puzzle? |journal=The Review of Financial Studies |date=October 2012 |volume=25 |issue=10 |pages=3037–3076 |doi=10.1093/rfs/hhs078 |jstor=41678608 |url=https://www.jstor.org/stable/41678608 |access-date=30 April 2022}}</ref> As such, when these extreme economic events do not occur, equity holders are rewarded with higher returns.<ref name="jstor.org">{{cite journal |last1=Gillard |first1=Christian |last2=Ghosh |first2=Anisha |title=Can Rare Events Explain the Equity Premium Puzzle? |journal=The Review of Financial Studies |date=October 2012 |volume=25 |issue=10 |page=3037 |doi=10.1093/rfs/hhs078 |jstor=41678608 |url=https://www.jstor.org/stable/41678608 |access-date=30 April 2022}}</ref> 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.<ref name="jstor.org"/> Moreover, extreme economic events affect all assets (both equity and bonds) 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.<ref name="jstor.org"/>

=== Myopic loss aversion ===

Benartzi & Thaler (1995) 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.<ref name="Myopic Loss Aversion and the Equity">{{cite journal |last1=Benartzi |first1=Shlomo |last2=Thaler |first2=Richard H. |title=Myopic Loss Aversion and the Equity Premium Puzzle |journal=The Quarterly Journal of Economics |date=February 1995 |volume=110 |issue=1 |page=73 |doi=10.2307/2118511 |jstor=2118511 |s2cid=55030273 |url=https://www.jstor.org/stable/2118511 |access-date=1 May 2022}}</ref> They rely on two assumptions about decision-making to support theory; loss aversion and mental accounting.<ref name="Myopic Loss Aversion and the Equity"/> 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.<ref name="Myopic Loss Aversion and the Equity"/> 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.<ref name="Myopic Loss Aversion and the Equity"/> This makes investors more risk averse compared to if they were evaluating their stocks less frequently.<ref name="Myopic Loss Aversion and the Equity"/> Their study found that the difference between returns gained from stocks and returns gained from bonds decrease when stocks are evaluated less frequently.<ref name="Myopic Loss Aversion and the Equity"/> The two combined creates myopic loss aversion and Benartzi & Thaler concluded that the equity premium puzzle can be explained by this theory.<ref name="Myopic Loss Aversion and the Equity"/>

=== 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]].<ref>{{cite journal |last1=Benartzi |first1=S. |last2=Thaler |first2=R. H. |title=Myopic Loss Aversion and the Equity Premium Puzzle |journal=The Quarterly Journal of Economics |date=1 February 1995 |volume=110 |issue=1 |pages=73–92 |doi=10.2307/2118511 |jstor=2118511 |s2cid=55030273 |url=http://www.nber.org/papers/w4369.pdf }}</ref> 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]],<ref>Yakov Ben-Haim, ''Info-Gap Decision Theory: Decisions Under Severe Uncertainty,'' Academic Press, 2nd edition, Sep. 2006. {{ISBN|0-12-373552-1}}.</ref> 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 principal-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.<ref name="ageconsearch.umn.edu">{{cite journal |last1=Grant |first1=Simon |last2=Quiggin |first2=John |title=The Risk Premium for Equity: Implications for Resource Allocation, Welfare and Policy |journal=Australian Economic Papers |year=2006 |volume=45 |issue=3 |pages=253–268 |doi=10.1111/j.1467-8454.2006.00291.x|s2cid=15693644 |url=http://ageconsearch.umn.edu/record/151167 }}</ref>

Another explanation related to the characteristics of equity was explored by a variety of studies including Holmstrom and Tirole (1998),<ref name="Private and Public Supply of Liquid">{{cite journal |last1=Holmstrom |first1=Bengt |last2=Tirole |first2=Jean |title=Private and Public Supply of Liquidity |journal=Journal of Political Economy |date=February 1998 |volume=106 |issue=1 |pages=1–40 |doi=10.1086/250001|hdl=1721.1/64064 |s2cid=158080077 |hdl-access=free }}</ref> Bansal and Coleman (1996)<ref>{{cite journal |last1=Bansal |first1=Ravi |last2=Coleman |first2=Wilbur |title=A Monetary Explanation of the Equity Premium, Term Premium, and Risk-Free Rate Puzzles |journal=Journal of Political Economy |date=December 1996 |volume=104 |issue=6 |pages=1135–1171 |doi=10.1086/262056|s2cid=54871134 }}</ref> and Palomino(1996)and was in relation to liquidity.<ref name="Noise Trading in Small Markets">{{cite journal |last1=Palomino |first1=Frederic |title=Noise Trading in Small Markets |journal=The Journal of Finance |date=September 1996 |volume=51 |issue=4 |pages=1537–1550 |doi=10.2307/2329404|jstor=2329404 |hdl=1814/498 |hdl-access=free }}</ref> 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.<ref name="Noise Trading in Small Markets"/> 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.<ref name="Private and Public Supply of Liquid"/>

=== Tax distortions ===

Another explanation related to the observed growing equity premium was argued by McGrattan and Prescott (2001)<ref>{{cite report |last1=McGrattan |first1=Ellen |last2=Prescott |first2=Edward |title=Taxes, Regulations, and Asset Prices |date=December 2001 |doi=10.3386/w8623 |doi-access=free |website=National Bureau of Economic Research}}</ref> 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.<ref>{{cite report |last1=Mehra |first1=Rajnish |title=The Equity Premium: Why is it a Puzzle? |date=February 2003 |doi=10.3386/w9512|doi-access=free |website=National Bureau of Economic Research}}</ref> 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.<ref>{{Cite book |last1=Dimson |first1=Elroy |last2=Staunton |first2=Mike |last3=Marsh |first3=Paul |title=Handbook of the Equity Risk Premium |date=9 November 2007 |publisher=Elsevier Science |isbn=9780444508997 |url=https://www.sciencedirect.com/book/9780444508997/handbook-of-the-equity-risk-premium }}{{page needed|date=July 2021}}</ref>

=== 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.<ref>{{cite journal |title=The Equity Risk Premium in January 2007: Evidence from the Global CFO Outlook Survey |first1=John R. |last1=Graham |first2=Campbell R. |last2=Harvey |year=2007 |journal=Working Paper |ssrn=959703 }}</ref> 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 (finance)|Volatility]] Index).

Dennis, Mayhew & Stivers (2006) <ref>{{cite journal |title=Stock Returns, Implied Volatility Innovations, and the Asymmetric Volatility Phenomenon |first1=Patrick|last1=Dennis |first2=Stewart|last2=Mayhew |first3=Chris|last3=Stivers |journal=Journal of Financial Economics |date=2006 |volume=41 |issue=2 |pages=381–406 }}</ref> find that changes in implied volatility have an asymmetric effect on stock returns. They found that negative changes in implied volatility have a stronger impact on stock returns than positive changes in implied volatility. The authors argue that such an asymmetric volatility effect can be explained by the fact that investors are more concerned with downside risk than upside potential. That is, investors are more likely to react to negative news and expect negative changes in implied volatility to have a stronger impact on stock returns. The authors also find that changes in implied volatility can predict future stock returns. Stocks that experience negative changes in implied volatility have higher expected returns in the future. The authors state that this relationship is caused by the association between negative changes in implied volatility and market downturns.

Yan (2011) <ref>{{cite journal |title=Jump risk, stock returns, and slope of implied volatility smile |first=Shu |last=Yan |journal=Journal of Financial Economics |date=2011 |volume=99 |issue=1 |pages=216–233 |doi=10.1016/j.jfineco.2010.08.011 }}</ref> presents an explanation for the equity premium puzzle using the slope of the implied volatility smile. The implied volatility smile refers to the pattern of implied volatilities for options contracts with the same expiration date but different strike prices. The slope of the implied volatility smile reflects the market's expectations for future changes in the stock price, with a steeper slope indicating higher expected volatility.
The author shows that the slope of the implied volatility smile is a significant predictor of stock returns, even after controlling for traditional risk factors. Specifically, stocks with steeper implied volatility smiles (i.e., higher jump risk) have higher expected returns, consistent with the equity premium puzzle. The author argues that this relationship between the slope of the implied volatility smile and stock returns can be explained by investors' preference for jump risk. Jump risk refers to the risk of sudden, large movements in the stock price, which are not fully captured by traditional measures of volatility. Yan argues that investors are willing to accept lower average returns on stocks that have higher jump risk, because they expect to be compensated with higher returns during times of market stress.

=== Information derivatives ===

The simplest scientific interpretation of the puzzle suggests that consumption optimization is not responsible for the equity premium. More precisely, the timeseries of aggregate consumption is not a leading explanatory factor of the equity premium.<ref name="Soklakov2020S">{{cite encyclopedia 
|last1=Soklakov |first1=Andrei N.
|year=2020 
|title=One trade at a time -- unraveling the equity premium puzzle
|url=https://www.mdpi.com/1099-4300/22/8/860/s1 
|encyclopedia=Supplementary Materials for "Economics of Disagreement", Entropy, 22(8):860 }}</ref>

The human brain is (simultaneously) engaged in many strategies. Each of these strategies has a goal. While individually rational, the strategies are in constant competition for limited resources. Even within a single person this competition produces a highly complex behavior which does not fit any simple model.<ref name="Soklakov2020">{{cite journal 
|last1=Soklakov |first1=Andrei N.
|year=2020 
|title=Economics of Disagreement -- Financial Intuition for the Rényi Divergence 
|journal=Entropy 
|volume=22
|issue=8 
|page=860 
|doi=10.3390/e22080860 |pmid=33286632
|pmc=7517462
|arxiv=1811.08308
|bibcode=2020Entrp..22..860S
|doi-access=free
}}</ref>

Nevertheless, the individual strategies can be understood. In finance this is equivalent to understanding different financial products as information derivatives i.e. as products which are derived from all the relevant information available to the customer. If the numerical values for the equity premium are unknown, the rational examination of the equity product would have accurately predicted the observed ballpark values.<ref name="Soklakov2020S" /><ref name="Soklakov2020" />

From the information derivatives viewpoint consumption optimization is just one possible goal (which never really comes up in practice in its pure academic form). To a classically trained economist this may feel like a loss of a fundamental principle. But it may also be a much needed connection to reality (capturing the real behavior of live investors). Viewing equities as a stand-alone product (information derivative) does not isolate them from the wider economic picture. Equity investments survive in competition with other strategies. The popularity of equities as an investment strategy demands an explanation. In terms of data this means that the information derivatives approach needs to explain not just the realized equities performance but also the investor-expected equity premia. The data suggest the long-term equity investments have been very good at delivering on the theoretical expectations.<ref name="Soklakov2020S" /> This explains the viability of the strategy in addition to its performance (i.e. in addition to the equity premium).<ref name="Soklakov2020S" /><ref name="Soklakov2020" />

=== 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 cost]]s or liquidity constraints may prevent individuals from [[consumption smoothing|smoothing consumption over time]]. In relation to transaction costs, there are significantly greater costs associated with trading stocks than trading bonds.<ref>{{cite journal |last1=Kocherlakota |first1=Narayana R. |title=The Equity Premium: It's Still a Puzzle |journal=Journal of Economic Literature |volume=34 |issue=1 |pages=42 and 71|citeseerx=10.1.1.334.8403 }}</ref> These include costs to acquire information, broker fees, taxes, load fees and the bid-ask spread.<ref name="The Equity Premium: It's Still a Pu">{{cite journal |last1=Kocherlakota |first1=Narayana R. |title=The Equity Premium: It's Still a Puzzle |journal=Journal of Economic Literature |volume=34 |issue=1 |pages=42 and 71 |citeseerx=10.1.1.334.8403 }}</ref> As such, when shareholders attempt to capitalise on the equity premium by  adjusting their asset allocation and purchasing more stocks, they incur significant trading costs which eliminate the gains from the equity premium.<ref name="The Equity Premium: It's Still a Pu"/> However, Kocherlakota (1996) contends that there is insufficient evidence to support this proposition and further data about the size and sources of trading costs need to be collected before this proposition could be validated.<ref name="The Equity Premium: It's Still a Pu"/>

=== Denial of equity premium ===

{{more citations needed section|date=February 2010}}

A final possible explanation is that there is no puzzle to explain: that there is no equity premium.{{citation needed|date=May 2013}} 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: This refers to the equity holder's fear about an economic crash such as the 1929 stock market crash eventuating, even when the probability of that event occurring is minute.<ref name="Anomalies: The Equity Premium Puzzl">{{cite journal |last1=Siegel |first1=Jeremy J. |last2=Thaler |first2=Richard H. |title=Anomalies: The Equity Premium Puzzle |journal=Journal of Economic Perspectives |date=Winter 1997 |volume=11 |issue=1 |page=191 |doi=10.1257/jep.11.1.191 |jstor=2138259 |url=https://www.jstor.org/stable/2138259 |access-date=1 May 2022}}</ref> The justification here is that over half of the stock exchanges that were operating in early 1900s were discontinued, and the equity risk premium calculated does not account for this.<ref name="Anomalies: The Equity Premium Puzzl"/> As such, the equity risk premium is "calculated for a survivor" such that if returns from these stock exchanges were included in the calculations, there may not have been such a great disparity between returns gleaned from bonds compared to stocks.<ref name="Anomalies: The Equity Premium Puzzl"/> However, this hypothesis cannot be easily proven and Mehra and Prescott (1985) in their studies, included the effect on the equity returns following the Great Depression.<ref name="Anomalies: The Equity Premium Puzzl"/> Although shares lost 80% of their value, comparisons of returns from stocks against bonds showed that even in those periods, significantly higher returns were gained from investing in stocks.<ref name="Anomalies: The Equity Premium Puzzl"/>
*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 theory|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 bubble]]s in progress.
*[[David C. Blitz|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 smaller and the risk-return relation becomes more positive.<ref>{{Cite SSRN |last=Blitz|first=David|date=2019|title=The Risk-Free Asset Implied by the Market: Medium-Term Bonds instead of Short-Term Bills |ssrn=3529110}}</ref>

Note however that most mainstream economists agree that the evidence shows substantial statistical power. Benartzi & Thaler analyzed the equity returns over a 200-year period, between 1802 and 1990 and found that whilst equity returns were remained stable between 5.5% and 6.5%, return on government bonds fell significantly from around 5% to 0.5%.<ref>{{cite journal |last1=Benartzi |first1=Shlomo |last2=H. Thaler |first2=Richard |title=Myopic Loss Aversion and The Equity Premium Puzzle |journal=The Quarterly Journal of Economics |date=February 1995 |volume=110 |issue=1 |page=73 |doi=10.2307/2118511 |jstor=2118511 |s2cid=55030273 |url=https://www.jstor.org/stable/2118511 |access-date=1 May 2022}}</ref> Moreover, analysis of how faculty members funded their retirement showed that people who had invested in stocks received much higher returns than people who had invested in government bonds.<ref name="Myopic Loss Aversion and the Equity"/>