The equity premium puzzle refers to the phenomenon that observed returns on stocks over the past century are a few percent higher than returns on government bonds. It is a term coined by Rajnish Mehra and Edward C. Prescott in 1985.[1][2] Economists expect arbitrage opportunities would reduce the difference in returns on these two investment opportunities to reflect the risk premium investors demand to invest in relatively more risky stocks.

The intuitive notion that stocks are much riskier than bonds is not a sufficient explanation as 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).[3] However, over any one decade, the premium shows great variability—from over 19% in the 1950s to 0.3% in the 1970s.

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.[4]

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

Investors are considered to be rational and optimize their utility. A person will maximize:

$E_0 \left[\sum_{t=0}^\infty \beta^t U(c_t)\right]$

where

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

where β and α are parameters.

Another utility function is:

$U_t = \left[c_t^{1-\rho}+\beta (E_t U_{t+1}^{1-\alpha})^{(1-\rho)/(1-\alpha)}\right]^{1/(1-\rho)}$

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

$p_t U'(c_t) = \beta E_t[(p_{t+1} + y_{t+1}) U'(c_{t+1})]$

as the fundamental equation.

For computing stock returns

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

where

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

gives the result.[5]

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:

• a contention that the equity premium does not exist: that the puzzle is a statistical illusion,
• modifications to the assumed preferences of investors, and
• imperfections in the model of risk aversion.

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.[6][7] Subsequent reviews of the literature have similarly found no agreed resolution.

The most basic 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.[citation needed]
• Low number of data points: the period 1900–2005 provides only 105 independent years which is not a large enough sample size to run statistical analyses with full confidence.
• 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 risk 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.

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

### 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 ("consumption growth").[8] 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.[9] 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,[10] based on a non-probabilistic treatment of uncertainty, which leads to the adoption of a robust satisficing approach to asset allocation.

### Equity characteristics

A second class of explanations focuses on characteristics of equity not captured by standard capital market models, but nonetheless consistent with rational optimization by investors in smoothly functioning markets. Writers including Bansal and Coleman (1996), Palomino (1996) and Holmstrom and Tirole (1998) focus on the demand for liquidity.

### Tax distortions

McGrattan and Prescott (2001) argue that the observed equity premium in the United States since 1945 may be explained by changes in the tax treatment of interest and dividend income. As Mehra (2003) notes, there are some difficulties in the calibration used in this analysis and the existence of a substantial equity premium before 1945 is left unexplained.

### 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.

### 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.[11] 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).

### Other explanations

Arguably more likely explanations are:

• Over the period, the observed outperformance of equities was substantially in excess of market expectations at the beginning of the relevant periods. This is not altogether surprising in view of the variability referred to above.
• Part of the reason for investment in fixed interest bonds was that many of the liabilities of insurance companies and pension funds requiring to be matched were expressed as guarantees of fixed currency amounts.

## Implications

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.

## 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.
2. ^ Handbook of the Equity Risk Premium, edited by Rajnish Mehra
3. ^ 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.
4. ^ Mankiw, N. Gregory; Zeldes, Stephen P. (1991). "The Consumption of Stockholders and Nonstockholders". Journal of Financial Economics 29 (1): 97–112. doi:10.1016/0304-405X(91)90015-C.
5. ^ The Equity Premium Puzzle: A Review
6. ^ Kocherlakota, Narayana R. (March 1996). "The Equity Premium: It's Still a Puzzle" (PDF). Journal of Economic Literature 34 (1): 42–71.
7. ^ Mehra, Rajnish; Edward C. Prescott (2003). "The Equity Premium Puzzle in Retrospect". In G.M. Constantinides, M. Harris and R. Stulz. Handbook of the Economics of Finance. Amsterdam: North Holland. pp. 889–938. ISBN 978-0-444-51363-2.
8. ^ Azeredo, F. (2014). "The Equity Premium: A Deeper Puzzle". Annals of Finance 10 (3): 347–373. doi:10.1007/s10436-014-0248-7.
9. ^ Benartzi, Shlomo; Richard H. Thaler (February 1995). "Myopic Loss Aversion and the Equity Premium Puzzle". Quarterly Journal of Economics (The MIT Press) 110 (1): 73–92. doi:10.2307/2118511. JSTOR 2118511.
10. ^ Yakov Ben-Haim, Info-Gap Decision Theory: Decisions Under Severe Uncertainty, Academic Press, 2nd edition, Sep. 2006. ISBN 0-12-373552-1.
11. ^ 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.