Financial Economics II Ch7- Lengwiler
Empirics and the Puzzles
by R. Ikeda & T. Kobayashi

［1］ Collecting the right data - The risk–free rate-ρ:

［1］ Collecting the right data - The risk–free rate-ρ The risk–free rate-ρ
An asset is free of risk if the cash flow it delivers in the future is independent of the state of the world.
So it rules out shares, options, and corporate bonds.
Government bonds of stable and developed economies are close to being risk free. But because of inflation, the return of a government bond is not risk free in terms of purchasing power of its cash flow. So inflation indexed government bonds is ideal risk-free asset.
But such bonds have become available only recently. Alternatively, we can consider bonds with a short time to maturity, and subtract from their return rate the realized inflation rate.

Collecting the right data -Stock indices-R:

Collecting the right data -Stock indices-R Stock indices-R
One specific asset that is at the center of much empirical research is a diversified portfolio of stocks, such as the stocks contained in the indices that are supposed to represent a significant part of the market.
Ex: S&P500, FTSE100, Nikkei, DAX, CAC40, SMI
Most of these are capital indices: weighted averages of the prices. So in order to compute rates of return, we have to add capital index to dividends. That is called “wealth index”.
Wealth index measures the wealth that an investor accumulates over time by holding the stocks contained in the index and reinvesting all dividends.

Collecting the right data -Consumption:

Collecting the right data -Consumption In our model consumption inputs utility. But what is “consumption” ?
It should be clear that expenses for non-durable consumption goods is part of consumption. But durable consumption goods also enter utility.
But it is not clear whether durables and non-durables are substitutes and empirical evidence on this is weak.
If they are substitutes, more durables reduce marginal utility of non-durables. Thus, the composite of durables and non-durables is less volatile than non-durables alone.
Maybe for that reason it has become accepted practice, to identify consumption with the expense for non-durable consumption goods.

[2] Mehra & Prescott (1985)(1) Stylized Fact:

[2] Mehra & Prescott (1985) (1) Stylized Fact The period and the universe
1889-1978, S&P500
The statistic data (per year)
Real per capita consumption growth
Average=1.83%
standard dev.= 3.57%
The real risk-free return rate
average =0.80%
standard dev.= 5%

The equity premium puzzle:

The equity premium puzzle The benchmark model of utility is the CRRA specification. We have seen in chapter 5 that this model implies a specific equation for the risk-free rate and for the risk premium,
with
Remember that these are all real per capita measures, so we need to correct nominal interest and growth rates for inflation and population growth.

Relative risk aversion by equation (5.38):

Relative risk aversion by equation (5.38) Needs very low g (very high risk aversion) to rationalize very low interest rate.

The problem:

The problem The problem lies with equation (5.39)
The historical return premium between an equity index and government bonds is substantially large.
The covariance between the return rate of equity and aggregate consumption is small simply because consumption is so smooth. This imply a huge γ*. large small

The problem (continued):

The problem (continued) The empirical estimation of (5.39) simply suggests that society is extremely risk averse (Kandel & Stambaugh, 1991).
The problem with this argument is that it leads directly to the empirical failure of (5.38).
If γ is very large, then not only do people dislike risk, but they also dislike intertemporal variations of consumption.
As a result, there is a very strong incentive to smooth intertemporally.
Yet, smoothing intertemporally is not possible in the aggregate.
Therefore if the economy grows and γis large, we should observe a very large risk-free rate.

Mehra & Prescott’s binominal formulation:

Mehra & Prescott’s binominal formulation Mehra and Prescott (1985) simplify the standard general equilibrium model of chapter 5 by assuming that there are only two states of the world, each with equal probability. Per capita consumption growth tree

Mehra & Prescott’s binominal formulation (continued) They calibrate the growth rates of these two states so that the mean and the variance of their binomial models fits the empirical mean and variance of U.S per capita consumption growth.
Mean per capita growth = +1.8%
Standard deviation =3.6%
g1:=+5.4%,g2:=-1.8%

Note that , and ,
thus risk-free rate is Mehra & Prescott’s binominal formulation (continued) In this two states model, we can easily compute the exact solutions.

Mehra & Prescott’s binominal formulation (continued) Besides the risk-free bond, Mehra & Prescott consider “Supershare” equity , whose cash flow equals state contingent per capita consumption.
Using the equilibrium SDF of the CRRA specification, the price of a risky asset is

Mehra & Prescott’s binominal formulation (continued) The expected return rate is thus, Subtracting (7.2) from (7.1) yields the equity premium,

Mehra and Prescott’s(1985) plot:

Mehra and Prescott’s(1985) plot Using (7.1) and (7.3), Mehra & Prescott plot the risk-free rate against the equity premium.

Mehra and Prescott (1985)’s conclusion:

Mehra and Prescott (1985)’s conclusion They conclude that they can justify a risk premium of at most 0.35% with the theoretical model if they constrain , and . Comparing these admissible regions with the historical data, the model cannot make this model fit the data.

(2) Hansen-Jagannathan bounds:

(2) Hansen-Jagannathan bounds Hansen & Jagannathan(1991) use the CCAPM to put a bound on the volatility of the SDF.
From CCAPM,
Because , Sharpe ratio is bounded above as
To a first-order approximation, thus

Hansen-Jagannathan bounds (continued):

Hansen-Jagannathan bounds (continued) According to the Mehra-Prescott data, and , but the Sharpe ratio of the risky asset they study is 6.18%/16.54%=0.37.
Thus no less than is required to make compatible with data. And it is only a lower bound.
The smoothness of consumption growth together with the assumption of moderate risk aversion imply a small volatility of the SDF. Yet, the high Sharpe ratios observed in the data suggest a very volatile SDF.
The incompatibility of these statements is an alternative way of stating the equity premium.

Mean reversion on H-J bounds:

Mean reversion on H-J bounds Fama & French(1988), Poterba &Summers (1988)
Empirically, equity returns in U.S. data from 1926 to 1985 are positively autocorrelated over a short horizon, but negatively autocorrelated over a long horizon.
That implies that stocks are actually less risky for long-term investors than what the one-period standard deviation of the return rate suggests.
Also, if negative serial correlation exists, the longer-period Sharpe ratio becomes larger than the shorter-period one
So the Hansen-Jagannathan bound becomes more difficult to fulfill as the horizon lengthens, because even more risk aversion is required to satisfy the bound.

(3) Production:

(3) Production Introducing production into the model frees up the equality between aggregate endowment and aggregate consumption.
There can be aggregate saving, which is invested into productive capital, and consumption becomes an endogenous variable.
But Rouwenhorst(1995) has noted the endogeneity of consumption can make the equity premium puzzle harder to solve, because any increase in the effective coefficient of risk aversion will also make consumption smoother.
So in order to explain the large equity premium, the introduction of a production technology that allows for easy intertemporal substitution, but not for easy transformation across states is needed.

[3] Alternative interpretations of the data:

[3] Alternative interpretations of the data Siegel (1992)
He argued that Mehra & Prescott’s sample from 1889 to 1979 covered an exceptional period.
He extended the sample to almost 200 years from 1802 to 1998.
Average real return
stocks =7.0%
(≒ 6.98% from Mehra & Prescott’s sample)
risk-free bills =2.9%
(>0.80% from Mehra & Prescott’s sample)
equity premium=7.0-2.9=4.1%
(<6.18 from Mehra & Prescott’s sample)

Unobserved disaster state :

Unobserved disaster state Rietz(1988)
He entertains the idea that there is a disaster state in which endowment falls dramatically, but the state has a very small probability and was not observed.
Adding such a third state to Mehra-Prescott’s formulation clearly increases the risk and the equity premium without increasing the price of risk.
But his third “disaster” state is always very extreme. His representative agent model has coefficient of relative risk aversion > 5, and δ<0.9(rather impatient). Also, the disaster state will come every year with 0.3% probability and destroy 50% of output.
Mehra & Prescott(1988)
Rietz’s model is very difficult to test empirically and clearly does not convince everyone.

Is the U.S. market really representative?:

Is the U.S. market really representative? If the success of the American market was not fully expected, the expected rate of return on stocks was smaller than it turned out be.
In other words, the historic experience overstates the expected performance of American stocks, and underestimates the expected performance of less lucky markets.
Survival bias (Brown et.al(1995))
If the market that turned out to be successful is only studied, an important source of loss is missed.
Siegel(1998)
Comparing British, German and Japanese stock returns with the performance of American stocks for the period of 1926-1997.
All 3 countries’ stock returns were worse than the performance of American stocks ,which clearly justifies the hypothesis of survival bias.

Are stocks riskier than “risk-free” bonds?:

Are stocks riskier than “risk-free” bonds? It is true that realized returns of American stocks exceeded expected returns because of the survival bias. But all economies whose financial markets experienced serious disruptions experiences very high inflation.
So historically, stocks have provided much better insurance against inflation than bonds and “risk-free” bonds with long times to maturity are actually riskier than risky stocks in this respect.
Correcting for the risk embodied in bonds would make the true equity premium even larger.

Treasury Inflation Protected Securities (TIPS):

Treasury Inflation Protected Securities (TIPS) Recently inflation-indexed bonds (TIPS) issued by the U.S. Treasury have become available.
Siegel (1999) reports that the yield is about 4%, which is significantly more than historic real ex post yields of non-indexed bonds.
Indexed bonds are not subject to inflation risk, so we can take this 4% as an accurate measurement of the real risk-free interest rate.
From Siegel’s estimate, the long-run return of American stocks is 7%.So the equity premium of the U.S. market is estimated at 7%-4%=3%.

Forward-looking return rates:

Forward-looking return rates Our asset pricing relationships concern expected values. One reason for the puzzle may therefore be that the historical average equity premium is an imprecise proxy for the expected premium.
One way of measuring expectations is by asking professionals.
Welch(2000) reports that popular views about the equity premium are extremely optimistic and that academic financial economists are far more conservative than the broad public.
Welch(2001) updated his survey after the stock market had started to declined. This time the forecasts were more pessimistic.

The equity size puzzle:

For example, if the potential GDP growth , the leakage out of the stock market annually (as estimated by Welch) and the real risk-free interest rate , maximum long-run equity premium So the population average return of any asset is bounded above by the growth rate of the economy plus leakage out of this asset . The equity size puzzle Welch(1999)’s argument is that aggregate returns on stocks need to be bounded above because if they are not, the total value of stocks would exceed GDP. Unless the coefficient of risk aversion is substantially different from 1, the net effect of potential growth on the long run equity premium is small. He calls this the equity size puzzle. However, long-run GDP growth and the interest rate are related by

The recent decline of the equity premium:

The recent decline of the equity premium Fama & French (2002) reported that for 1951-2000, equity price increased much faster than firms’ earnings.
There are three possible explanations for this.
A bubble on equity prices
Much greater earnings expectations than they were earlier (“New economy” explanation)
Decrease of expected equity premium
Fama & French don’t consider the bubble explanation. They also dismiss the new economy interpretation because they find no evidence that earnings can be forecast.
As a result, they conclude that the expected equity premium has decreased.
Other researchers have reached the same conclusion (Blanchard(1993), Jagannathan, McGrattan & Scherbina(2000)). And the diagnosis has recently surfaced also in the investment community (Arnott & Ryan(2001), Best & Byrne(2001)).
But some authors are still quite upbeat about the prospects of investing into equity (Ibobotson & Chen(2003))

[4] Variance bounds tests:

[4] Variance bounds tests LeRoy & Porter (1981) and Shiller (1979, 1981)
Empirically asset prices are more volatile than what is compatible with the theory.
The model
The current price q of an asset is the discounted value of its dividends, given today’s information:
where r denots the realized dividend sequence.
Notice that, the price q change only if there is new information so that expectation changes.
Suppose for a moment that investors had perfect foresight. We call the resulting asset price the “ex post rational price”,

Variance bounds tests (continued):

The model (continued)
If agents had perfect information, then the two price is the same. But there is uncertainty and thus q will be a noisy estimate of q*:
Rational expectations dictates that the expected dividends are an unbiased but possibly noisy estimate of the true dividends, and expectations errors are orthogonal. Formally,
This implies that Variance bounds tests (continued)

Estimate of q*:

q* is not observable since it is an infinite sum.
Shiller (1981) constructs an estimate of q* by applying
recursively for t=1,….,T-1.
Imposing the terminal condition:
which says that the observed price q is equal on average to the ex post rational price q* . Estimate of q*

Excessive volatility:

Excessive volatility Visual inspection of q and q* suggests that q* is conspicuously smoother than q, so that variance bounds relation seems flagrantly violated.

A Critique :

A Critique One of the objections to the apparent violation of the present value relation is based on the time series properties of dividends.
According to the present value equation, it is only the information about future dividends that drives today’s price.
From this it follows that the stochastic process one assumes for the discount process is of utmost importance for relating current dividend to expected future dividends, and thus for the relationship between the relative variances of dividends and stock prices.

Another critique:

Another critique Agents may be risk averse, which would invalidate the constant discount factor assumption. If the discount factor is stochastic, prices may become more volatile.
LeRoy & LaCivita show that the coefficient of dispersion of the stock price is smaller than the coefficient of dispersion of dividends if and only if the coefficient of relative risk aversion is smaller than one.

Another critique (continued):

Another critique (continued) The intuitive reason for LeRoy & LaCIvita (1981)’s result is that risk-averse agents try to smooth consumption, but aggregate risk cannot be smoothed in equilibrium.
Thus, asset prices have to behave in such a way that the representative agent willingly abstains from consumption smoothing and instead consumes his endowment.
For this to happen, stocks in order to save, and they must be cheap in a recession, to prevent agents from selling stocks in order to dissave.
The problem is that , in order to induce enough volatility of the SDF to make relative stock price and dividend volatility compatible with each other, we need to assume a very large risk aversion parameter because consumption growth is so smooth.

[5] Other Anomalies:

[5] Other Anomalies The weekend effect (French(1980))
S&P 500 returns are significantly negative between Fridays and Mondays.
The size effect (Banz(1981) and Reinganum(1983))
Shares of firms with small capitalization earn abnormal returns.
The January effect (Keim (1983) and Reinganum(1983))
Many of these abnormal returns occur in January. Researcher have uncovered an impressive collection of even stranger patterns in financial market data.

Does the anomalies offer risk-free arbitrage opportunities? –Two opposing answers:

Does the anomalies offer risk-free arbitrage opportunities? –Two opposing answers Barberis & Thaler (2003)
Even if price offer risk-free arbitrage opportunities, we can’t be sure that prices return to their arbitrage-free equilibrium position (Noise trader risk). So arbitrage is limited.
Schwert (2003)
Many researchers could find anomalous statistics in samples of data, but none of the anomalies has proved to be reliably present out of sample. In other words, the anomalies have never in fact really been there, or have been arbitraged away since their discovery.

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