Equity Premium Puzzle: What is puzzling?

Stock Prices Fluctuation

In 1982, Robert J. Shiller published an interesting working paper entitled “Consumption, Asset Markets and Macroeconomic Fluctuations”. The argument is quite simple: what causes the fluctuation in composite stock index. He also set up four objectives for this paper: (1)Whether consumption theory can be used to explain fluctuations in stock price (2)Whether the model can be evaluated if aggregate consumption data is not representative (3)Whether prices of long-term assets can be explained by the model (4)Whether the business cycle behavior of real short-term interest rate is explained by the model. First, let us go through this model quickly.

A fact is that, over the last century (1890~1980), the standard deviation of the real annual return on the Standard & Poor’s Stock Price Index was about 20 percentage points. In other work, if you bought and held a market portfolio from the beginning of a particular year to the end of it, you ended up with a 20% gain or loss. What was the key factor to cause such a fluctuation? Shiller showed, under an Arrow-Debreu framework, conventional efficient markets model was not able to reconcile with historical data. However, a time-varying real discount rate might help to explain volatility in stock index.

Let us see how the mechanism works here. If we assume individual consumer choose to invest in freely tradable assets in order to smooth his/her consumption, then we may use the following expected utility maximization problem to describe his/her behavior. Equation (1) is the explicit form of this maximization problem. From it, we can derive the first order condition as in (2). Here, we assume an additively separable utility function, frictionless world and perfectly competitive market (we refer these settings as an Arrow-Debreu framework). By iterative substitution, equation (3) gives us the relationship between current stock price and future dividends. Another key feature of this equation is the real discount factor. The real discount factor reflects the fact: when evaluating stock price, not only future dividends are taken into account but also the intertemporal marginal rate of substitution.

Moreover, if we assume individual preference takes the form as in equation (4), then we can explicitly formulate the relationship between three stochastic series: stock price (represented by Standard & Poor’s Stock Price Index ), consumption series (represented by aggregate consumption data) and future dividends.

Nevertheless, historical data over the last century (1890~1980) does not support this simple description. Let me elaborate that a little bit. From historical data, we can see the volatility of stock price but a smooth trend of future dividends and consumption is clearly there. We can interpret this point in two ways: (1) it shows the failure of efficient markets model, (2) if there is no measurement error in historical data and we accept that consumption data is representative, then models along this approach are also incapable to reconcile with historical data. First, by efficient markets model, current stock price is simply the present value of future dividends. However, historical data suggests a smooth trend in dividends, which implies stock price should not be too volatile. Second, by this expected utility maximization approach, we can relate the stock price series to consumption series and dividend series. If consumption series and dividend series are not varying too much, a relatively stable stock price series is expected. However, historical data shows the volatility of stock price is too much to be accounted for the fluctuation of both consumption series and dividend series.

Fortunately, we can do more to keep this model alive. First Shiller argued that a varying real discount factor is help to solve this annoying issue. He further showed that, with certain value of parameters (including the Constant Relative Risk Aversion coefficient and subjective real interest rate), we can generate higher predict power of this model , i.e., we can explain the high volatility of stock price. Second, using a varying real discount factor, not only stock price but also prices of other assets (including land, housing and long term Treasury bond) are also predictable.

Regardless of the performance of this modified model, some restrictions are in hand. First, the model is with high predict power during periods of large-scale economic fluctuation, for example, post world war two periods. Second, it is rejected by monthly data. Third, some assumptions are not compelling, for example, a frictionless world. Although the idea of a varying real discount factor is not yet complete, it is indeed a persuasive argument.

How about the four objectives, are they solved? First, with a varying real discount factor, we are able to explain the excess volatility of stock price. Second, although aggregate consumption data may not be representative, other works showed the validity of aggregate consumption data. Third, the idea varying real discount factor is not restricted only on stock price. This modified model is applicable on any other long-term assets, in principle. Finally, this modified model shows the relationship between business cycle and real discount factor, although not perfectly fitted. In other words, we explain the business cycle behavior of real short-term interest rate since real short-term interest rate is a component of real discount factor.

What is puzzling?

In fact, Shiller concluded this paper by posing a big question: rather than a varying real discount factor is there any other better explanation. Along with this, he also posed two further queries. First, what causes the real discount factor to shift? Indeed, we do not know the actual mechanism of a varying real discount factor nor does one mention the shift of real discount factor itself. Second, individual consumer seems to behave as if they have more information than that implied by this model. Namely, individual consumer adjusts his/her behavior due to an unidentified source in order to smooth his/her consumption. Moreover, the adjustment that is reflected is the movement of real discount factor. What is the possible source of the new information? Three questions remain unsettled.

Despite these three remaining questions, Shiller implicitly pointed out two puzzles: Equity Premium Puzzle and Risk-free Rate Puzzle. In his paper, CRRA coefficient is set to be four. Later works showed that a higher value is consistently estimated. As we all know, a high CRRA coefficient implies that individual consumer becomes more risk-averse. In other words, a higher equity premium is expected for those who buy stocks. Historical data also show that the average return on equity has far exceeded the average return on short-term debt (about 6 percentage points (Mehra & Prescott, 1985)). Does this high equity premium come from the conservative behavior of individual consumers? Or, we should ask: why people become more and more risk-averse? This issue is generally known as the “Equity Premium Puzzle”. Another puzzling issue is related to the low growth rate of consumption. Recall equation (4), if we assume that consumption is growing with a particular rate each period, it seems that the growth rate is always too low to account for the high variability in stock price. This issue is known as the “Risk-free Rate Puzzle.” Without lost of generality, we may consider these two puzzles as two interpretations of one historical fact that stock price is too volatile and the real return of it is much higher than other competitive assets (Weil, 1989, for a detailed explanation).

Equity Premium: A Puzzle

It was in 1985, Mehra & Prescott first recognized this as a puzzle. In an important paper entitled “The equity premium: a puzzle”, they pointed out that models with Arrow-Debreu framework may not be able to reconcile with historical data (Mehra & Prescott, 1985). They claimed:

“… most likely, an equilibrium model which is not an Arrow-Debreu economy will be the one that simultaneously rationalizes both historically observed large average equity return and the small average risk-free return”.

Moreover, a challenge was set in this paper. With a CRRA coefficient less than ten, the results were essentially the same. This implies that any later researches with CRRA coefficient less than ten will be considered as eligible for valuation. Nevertheless, this criteria was set according to previous studies which may be traced back as early as 1971 in Arrow’s paper (Arrow, 1971). The validity of it demands further profound researches.

Solutions

An emerging industry was born as soon as Mehra and Prescott recognized it as a puzzle (Cochrane & Hansen, 1992). Tons of papers were released to try to solve this puzzle. In effect, these papers were conducted along three main streams. Next, we will discuss the three main approaches set to tackle the Equity Premium Puzzle.

First, as Mehra & Prescott pointed out, any Arrow-Debreu economy considering assumption that is more general may help to solve it. Heaton & Lucas (1996) and Luttmer (1999) showed that transaction cost might be a possible solution to the puzzle. Heaton & Lucas incorporated transaction cost into their model with incomplete market settings. With a high transaction cost level, they were able to generate the high equity premium but it seems implausible that we have high transaction cost, especially in recent decades. Luttmer followed a slightly different way. He concentrated on measuring the lower bound of possibly cost incurred during transaction. As in Heaton & Lucas (1996), the fixed cost level remains too high. Regardless of the high level of fixed cost, Luttmer actually generated the lower bound, which is a major contribution to the issue.

Second, some studies stick to the Arrow-Debreu framework with more restrictions on individual consumer’s behavior, such as Campbell & Cochrane (1999), Cecchetti, Lam and Mark (2000). Campbell & Cochrane (1999) argued that consistent consumption habit formation is a cure. With certain level of habit formation, they generated a virtually identical stock price series. Cochrane himself was satisfied with this achievement and published another paper to explain the poor performance in consumption based assets pricing model. Although his model successfully fitted the historical data, this result was expected. If you go back to equation (4) and take a closer look of it, you can find the trade off between the variability of consumption series and CRRA coefficient. Thus, Cochrane could have generated a lower parameter with a more variable consumption series that he conducted with consumption formation in his paper. Hence, Shiller did a major contribution back in 1982. As a saying goes “there may be superior techniques, however, there may not be superior insight.”

Finally, some other papers work also argued regarding the validity of aggregate consumption data, such as Mankiw & Zeldes (1991). We will not elaborate at this point, since Shiller (1982) already examined a similar argument. We want to point out that aggregate consumption data may not be as representative as we required but, with historical evidence, it is hard to neglect that aggregate consumption is somehow lower than we expected.

Concluding Remarks

Up to this moment, researchers are still working on the solution of this puzzle. In fact, if you believe what Shiller said in 1982, then it seems the puzzle is no longer puzzling. Instead, we need more general researches on human behavior to find out how people can react so fast or how people can acquire more information than we expect. In addition, this will help to explain the shift in real discount factor and solve the puzzle in principle. However, what he said may not be valid and we need some more works to be done on this issue. There are three approaches to tackle the puzzle if you try to solve the puzzle. The first one is a debate over reasonable assumptions. Can one find a set of assumptions to describe the real world and can the model be solved with these assumptions? It is a matter of belief over the abstract explaining ability of models. Second seems to become a deeper question of human behavior. Individual consumer does not behave as we assumed in the model. There may be some other elements affecting his/her decision. As for the third, it depends on the way you interpret available data. Is equity premium puzzle still puzzling? I believe it is.

Reference:

J. Campbell and J. Cochrane (1999), “By Force of Habit: A consumption-Based Expectation of Aggregate Stock Market Behavior,” JPE 107, 205-251

J. Heaton and D. Lucas (1996), “Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Pricing,” JPE 104, 668-712

E. Luttmer (1999), “What Level of Fixed Costs Can Reconcile Consumption and Stock Returns,” JPE 107, 969-997

N. Mankiw and S. Zeldes (1991), “The Consumption of Stockholders and Non-Stockholder,” JFE 29, 97-112

R. Mehra and E. Prescott (1985), “The Equity Premium: A Puzzle,” JME 15, 145-161

R. Shiller (1982), “Consumption, Asset Markets, and Macroeconomic Fluctuations,” CR 17, 203-238

Other Readings:

J. Campbell and J. Cochrane (1999), “Explaining the Poor Performance of Consumption Based Asset Pricing Model,” NBER Working Papers #7237

J. Cochrane and L Hansen (1992), “Asset Pricing Explorations for Macroeconomics,” NBER Working Paper #4088

R. Shiller (1990), “Market Volatility and Investor Behavior,” AER, 80, 2 , 58-62