Bubbles, Crashes, and Endogenous Expectations in Experimental Spot Markets

Vernon L. Smith, Gerry L. Suchanek, and Arlington W. Williams. “Bubbles, Crashes, and Endogenous Expectations in Experimental Spot Markets.” Econometrica 51 (1988): 1119-1151.

Paper Outline and Setting

Rational expectation and market efficiency predict that, in equilibrium, stock prices should change only when there is new information that changes investors' dividend expectations. The paper examines this prediction in an experimental spot market setting in which nine to twelve subjects, with a varying level of experience, make trades for a certain number of periods under the common knowledge on the probability distribution of dividends. 

Specific details on the setting of the experiments are as follows;

1. Price quotes progress so as to reduce the bid-ask spread

2. Each trader is given an asset endowment and cash endowment (or working capital)

3. Working capital amount reflects the accumulated capital gains (losses) and dividend earnings

4. Traders can buy assets as long as their working capital is sufficient to cover the purchase price

5. Everyone is informed that the dividend structure (distributed iid) and actual dividend draws are the same for everyone in the market.

Given this experimental setting, rational expectation theory predicts that possible temporary deviation of the price from risk-adjusted dividend value due to divergent individual expectations cannot persist because of uncertainty in arbitrage profit. In the end, it is expected that investor expectations become common and coincide with dividend value, which leads to a market price commensurate with risk-adjusted dividend value. The experimental setting of this paper enables us to control dividend structure and traders' knowledge of it so that we can explore whether common knowledge of a common dividend payout is sufficient to induce common expectations. By investigating the patterns of price evolution, we can analyze how long it takes until investors form common expectation, and whether, or in what form, such convergence process can be characterized.

More formally, the main objectives of this study are as below;

1. Will economic agents trade an asset whose dividend distribution is common knowledge?

2. If so, can we characterize the price adjustment process and interpret it as a pattern consistent with convergence to risk-adjusted dividend value?

3. Will we observe any price bubbles and crashes occurring as part of the adjustment process?

Observed Patterns from Experiments and Interpretation

The authors identify three general patterns that emerge from a series of experiments conducted. 

1. Stable Markets

: In this group of markets, prices appeared to follow dividend value, or were constant or followed approximately parallel with dividend value over most of the horizon.

2. Growing Markets

: This group consists of the experiments that do not belong to the stable markets or bubble-crash markets group.

3. Bubble-Crash Markets

: This group consists of markets that produced price bubbles that collapsed sometime before the final period.

A general summary of the observed trade and price patterns is as below;

1. Exchanges do occur even when identical probabilistic dividends are to be paid. Therefore, different private dividend values on different traders is not a necessary condition for trades to occur. This may indicate that there is sufficient homegrown diversity in agent price expectations and risk attitudes to induce subjective gains from exchange.

2. Fourteen out of twenty-two experiments exhibit price bubbles followed by crashes relative to the intrinsic value.

3. Observed bubbles can be interpreted as a form of temporary myopia from which agents learn that capital gains expectations are only temporarily sustainable. This realization by investors ultimately leads to the formation of common expectations or common "priors."

4. Sometimes, markets are sensitive to (or strongly dominated by) group endogenous expectational factors that are not reliably manipulated by controllable treatment variables such as experience, information, and horizon length.

5. Expectations and price adjustments are both adaptive, but the adaptation over time across experiments with increasing trader experience tends to a risk-adjusted, rational expectation equilibrium.

6. Common knowledge is not sufficient to induce initial common expectations due to agent uncertainty about the behavior of others. With experience and trial-and-error learning, expectations seem to tend to converge and yield an equilibrium consistent with rational expectations.

Overall, the predominating characteristics of the price patterns observed from experiments indicate the tendency for expectations and price adjustments to converge to intrinsic value across experiments with increasing subject experience.

Firm characteristics, consumption risk, and firm-level risk exposures

Robert F. Dittmar, Christian T. Lundblad. "Firm characteristics, consumption risk, and firm-level risk exposures." Journal of Financial Economics 125 (2017): 326-343.

Using a consumption-based asset pricing model, the authors measure the relation between exposures to consumption risk and portfolio-level characteristics. This model is then used to calculate risk exposures at the firm level.

Consumption Risk Premia in Average Returns

The authors choose the following model to represent an asset's price as determined by it's conditional covariance with respect to an inter-temporal marginal rate of substitution,

To estimate risk exposures and cross sectional risk premium, the authors regress returns on anomaly portfolios that are sorted on six firm characteristics: asset growth, book to market ratio, market capital, past 12 month returns, stock issues, and total accruals. This is done for different windows, K, where R is the portfolio return and Eta is the innovation (the difference in log consumption growth in quarter t minus j and its unconditional mean).

Consumption is measured as per capita real personal consumption expenditures on non-durable goods and services.

Risk exposures for a window of K=4 (this window produces risk exposures with the best cross sectional fit) show the following results,

The results suggest a positive relation between average returns and consumption risk exposures. Equities are positively exposed to consumption risk, and this suggests a positive premium for consumption risk.

The authors continue to investigate whether risk exposures are related to average returns. This is done in a two-step process:

1) Returns are regressed on sources of risk

2) Average returns are then regressed on the resulting risk exposure estimates

They use the following cross sectional regression with previous risk exposure estimates,

Following are results of the cross sectional regression,

The best results are found with K=4 windows. This can also be seen graphically with average returns over predicted returns,

Using Characteristics to Estimate Risk Exposures

The authors assume that the relation between an asset’s consumption risk exposure and a set of relevant characteristics can be captured by projecting the risk exposure onto a set of characteristics,

They hypothesize the reason portfolio characteristics are related to average returns is through the link between the characteristics and their exposure to risk in the stochastic discount factor. It is assumed that a linear relation exists between risk exposures and characteristics,

The following are results of the relation between portfolio betas and characteristics,

The authors hypothesize the relation that holds at the portfolio level also holds at the firm level. The relation between firm level exposures and portfolio level exposures and characteristics is given by,

They use firm level characteristics to form portfolios that are characterized by differences in ex-ante consumption risk exposure. For each month, t, they calculate a consumption growth innovation exposure for every firm using the panel regression coefficients (deltas) and firm characteristics at time t. They rank firms into quintiles by the exposure and form equal-weighted and value-weighted portfolios

The authors propose three questions:

1) Does sorting firms into portfolios by ex-ante predicted betas produce a positive risk premium?

2) Do resulting portfolios exhibit ex-post risk exposures that are consistent with the ex-ante ranking?

3) Can the time series variation in the ex-ante risk exposure of these portfolios be characterized in an economically meaningful way?

Following are results for mean returns, ex ante beta, ex post beta for 5 value weighted and equally weighted portfolios,

The model replicates increasing expected returns and ex post risk exposures in value weighted portfolios. These results indicate that with equally weighted portfolios, the ex-ante procedure produces consistent ex post beta rankings.

The authors were also interested in using their model for standard market betas, however, results are less than ideal. It is suggested that this is due to a weak relationship between market betas and firm characteristics.

Industry Costs of Capital

The authors investigate cost of capital for firms grouped by primary industry classification.

They create 24 Industry portfolios based on S&P GICS industry groups. Ex-ante risk exposure are found for each firm using the portfolio-level coefficients from the 55 portfolios sorted by characteristics. Equal-weighted portfolios are formed by industry groups and ex-ante betas are examined. Average industry betas are reported with the results of Fama-Macbeth regressions of industry portfolios at times t on portfolio betas.

The following are mean returns, betas, and standard deviations of betas for industry portfolios,

Firm Costs of Capital

To examine implied firm-level costs of capital, the authors run Fama-Macbeth firm-level regressions. They compute consumption beta exposures, then compute risk exposures for the five risk factors (Fama, French 2014) using rolling regressions of the past 60 months with respect to when return was measured. Month-by-month Fama-Macbeth regressions are then run.

Following are resulting average prices of risk and their respective t-statistics,

From the results, substantial variation in risk premia occur over time and differs significantly across different firms. The point estimates of the periodic regression coefficients are more stable for the consumption beta than the return factor betas. The authors suggest this is likely due to the difficulty in estimating risk exposures at the firm-level.

A skeptical appraisal of asset pricing tests

Lewellen, Jonathan, Stefan Nagel, and Jay Shanken. “A skeptical appraisal of asset pricing tests” Journal of Financial Economics Vol. 96 (2010) 175-194.

Summary:

The authors bring to light some problems with empirical asset pricing tests which are common in the literature and then offer up several (partial) solutions as a way to have more confidence in the implications of asset pricing tests moving forward.

Many proposed asset pricing models offer explanations for the size and book-to-market effects. They then test their models, at least in part, using the Fama and French (1993) 25 size and book-to-market portfolios, concluding that the models do a good job of explaining these effects because they generate a high cross-sectional R-squared. The authors, however, suggest that this is a low hurdle to meet and that, while the models may give important economic insights, they may not perform as well as originally thought.

The problem is the strong factor structure of these test portfolios. The Fama and French (1993) factors explain “more than 90% of the time variation in the portfolios’ realized returns and more than 80% of the cross-sectional variation in their average returns.” Thus, a high R-squared for a proposed factor is achieved even when having a weak correlation with SMB or HML but not with the idiosyncratic residuals.

Illustrating the Problem:

The authors randomly generate artificial factors which are correlated with the three factors of Fama and French (1993), but not with the idiosyncratic residuals. They then obtain R-squared from regressions of returns from the 25 size and book-to-market portfolios on 1 to 5 of these randomly generated factors. Their figure 1 gives the results.

As we see, it is easy to obtain a high cross-sectional R-squared in each case when the estimation uses more than one factor. This is true even in the case of Panel C where only randomly drawn factors with an expected return close to zero are kept in the analysis (it should be zero).

Proposed (Partial) Solutions:

The authors suggest four:

1. Add additional test portfolios sorted on some other characteristic (the authors use the Fama and French 30 industry portfolio returns).

2. Impose restrictions on risk premia when guided by theory rather than just allowing it to find the best fit.

3. Report GLS cross-sectional R-squared rather than, or at least in addition to, those for OLS.

4. Report confidence intervals for test statistics. In particular, this will illustrate the (sometimes wide) range of parameters which are consistent with the data.

Empirical Results:

The authors test their suggestions on a total of eight models, which they report in Table 1 and the effectiveness of each of the models is clearly weaker than initial tests suggested.

In summary, the authors have pointed to some problems with how we evaluate the empirical successes of proposed asset pricing models and then offered some (partial) solutions to these problems. In all, they seem to have raised the hurdle which must be cleared in order to declare a proposed model a success in explaining cross-sectional variation in asset returns.

Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time-Varying

Martin Lettau, Sydney Ludvigson. “Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time-Varying.” Journal of Political Economy Vol. 109, No. 6 (2001): 1238-1287. (link)

In this paper, Lettau and Ludvigson provide an asset pricing model that combines both the consumption CAPM and conditional CAPM in order to develop a factor model with reference to macroeconomic factors that drive risk premia and returns.

The authors review the effectiveness of the traditional CAPM, Fama French three factor model (FF3), and consumption CAPM (CCAPM) at explaining the cross-section of size and book-to-market related returns. Pricing errors for each model are shown in Figure 1 with a) CAPM, b) Fama French, c) CCAPM, d) the scaled consumption CAPM, ©CAPM, developed in this paper. For each two digit number, the first digit represents portfolios sorted on size (with 1 the smallest) while the second number represents portfolios sorted by book-to-market (with 1 the lowest book-to-market ratio).




The issue of the CAPM for explaining the cross-sectional returns of size and book-to-market are clear in the figure (1a) and have been empirically show in Fama and French 1992, 1993, among others. The consumption CAPM has not held up empirically but has theoretical weight which the authors utilize in their model. FF3 is successful at capturing much of the cross-sectional returns but Lettau and Ludvigson are interested in understanding the underlying economic, non-deversifiable risk which is proxied for by size and book-to-market.

Lettau and Ludvigson argue that the empirical evidence points to time-variation of risk premia which implies that investors’ expectations of future returns will affect the stochastic discount factor of the model. This approach contrasts with the traditional and consumption CAPM which assume constant risk premia.

To measure the expectations of future returns, the authors use the cointegration of c consumption, a log asset wealth, and y log labor income, referred to as cay. This measure is observable and Lettau and Ludvigson 2001 show that it serves as a reasonable proxy for the unobservable ratio of consumption to aggregate wealth, human and non-human. The factors in their model are scaled by the estimated value, $\widehat{cay}$, with stars representing measured values.

\[\widehat{cay_{t}}=c_{t}^*-0.31a_{t}^*-0.59y_{t}^*-0.60\]

Table 1 gives the results of Fama-MacBeth regressions on the 25 Fama French portfolios. The first row is the traditional CAPM and, as expected, the t-statistic for beta is insignificant and the $R^2$ is only 1%. The human capital CAPM, row 2, and FF3, row 3, explain increasing amounts of the cross-sectional variation in returns.

\[E[R_{i,t+1}]=E[R_{0,t}]+\beta_{zi}\lambda{z}+\beta_{vwi}\lambda_{vw}+\beta_{vwzi}\lambda_{vwz}\>\>\>\>\>\>\>(13)\]

\[E[R_{i,t+1}]=E[R_{0,t}]+\beta_{zi}\lambda_{z}+\beta_{vwi}\lambda_{vw}+\beta_{vwzi}\lambda_{vwz}+\beta_{\Delta yi}\lambda_{\Delta y}+\beta_{\Delta yzi}\lambda_{\Delta yz}\>\>\>\>\>(14)\]

Rows 4 and 5 of table 1 gives results for (13), the scaled CAPM, where $z_{t}=\widehat{cay}$, is the scaled factor and $f_{t+1}=R_{vw,t+1}$ is the fundamental factor.  Row 4 shows the time-varying intercept term, $\lambda_{z}$, is not significant and so row 5 gives the results without the intercept, showing it has little effect.   Rows 6 presents the regression (14), the scaled human capital CAPM, which succeeds in explaining 71% of the cross-sectional returns after correcting the t-statistic. Finally, row 7 presents regression (14) but without the time-varying intercept, $\lambda_{z}$, which has little effect on the results.

\[E[R_{i,t+1}]=E[R_{0,t}]+\beta_{zi}\lambda_{z}+\beta_{\Delta ci}\lambda_{\Delta c}+\beta_{\Delta czi}\lambda_{\Delta cz}\>\>\>\>\>\>(15)\]

Table 3 presents the results of the scaled multi factor consumption CAPM (15) developed in this paper.  Where $\Delta c$ represents the log difference of consumption, measured in Lettau and Ludvigson 2001. Row 1 presents the unconditional consumption CAPM for comparison and row 2 presents the results of (15). Row 3 is (15) but without the time-varying intercept as it is statistically insignificant. Note that $\lambda_{\Delta cz}$, the conditional scaled coefficient, is very significant and the $R^2$ statistic is nearly 70%, almost as high as the 80% cross-sectional returns explained by FF3. Furthermore, there is a dramatic increase in $R^2$ from the unconditional CCAPM in row 1 and the conditional CCAPM in row 2.

The authors demonstrate that their conditional version of the consumption CAPM performs nearly as well as the Fama French 3 factor model at explaining cross-sectional returns of portfolios sorted on size and book-to-market. Their intuition is that the risk associated with a value portfolio comes from the covariance with consumption growth that is time-variant rather than unconditional. During times of high risk aversion (high $\widehat{cay}$) this correlation of value with consumption growth will be high where as during times of low risk aversion it will be low. Figure 2 charts the conditional consumption betas for four pairs of size and book-to-market portfolios during good states ( $\widehat{cay}$ at least one standard deviation above its mean) and bad states (at least one standard deviation below). As expected, the high value portfolios have larger consumption betas during bad times than good times and higher consumption betas during bad times, than value stocks during bad times.  Since this high correlation with consumption growth comes when investors are most averse to it (times of high risk aversion) the value portfolios have greater time-variant risk and hence, greater return.

The scaled multi-factor model presented by Lettau and Ludvigson attempts to explain the cross-section of expected stock returns of portfolios sorted on size and book-to-market. The failure of the traditional CAPM to explain these returns has lead to research into the source of risk. The static consumption CAPM provides strong economic theory but ultimately falls short after empirical tests. The Fama French 3 factor model provides strong empirical results but does not offer an explanation of the fundamental macro economic sources of risk. The (C)CAPM model derived by the authors provides an economic framework in which to explain the cross-section of size and book-to-market while empirically explaining a large amount of the expected returns.

Understanding Risk and Return

Campbell, John Y. “Understanding Risk and Return.” Journal of Political Economy Vol. 104, No. 2 (Apr. 1996), pp. 298-345

The author seeks to address some of the failings and critiques of prominent asset pricing models by deriving a model of his own. He begins by asking two fundamental questions:

1.       How should we measure risk?

2.      What determines the how much an investor should be compensated for bearing that risk?

Previous models:

CAPM:

This model measures the risk of an asset by its covariance with the market return.

1.       Merton (1973) says it should be measured by its covariance with the marginal utility of investors.

2.       Roll (1977) questions the validity of using stock market return as a proxy for the “true” market return. (i.e. we can’t truly test the CAPM because we don’t know the market return.

In the CAPM, the price of risk is determined by the risk aversion of the investors.

Multi-Factor Models:

These models measure the risk of an asset by its covariance with common factors which have broad explanatory power in asset returns.

1.       These models are very weak on theory and, thus, give little guidance as to what factors should be used. Therefore, factors could just come from spurious correlations as an artefact of the sample being used.

Multi-Factor models have nothing to say about the question of what determines the price of risk.

CCAPM:

This model measures risk by an asset’s covariance with consumption.

1.       In particular, the model performs poorly when being tested empirically.

The price of risk is determined by the risk aversion of a representative investor.

To attempt to better address the critiques of the CAPM than the other models have, the author derives a “multi-factor model” of his own which brings in both changing investment and human capital. He then tests the model empirically. He assumes Epstein and Zin (1989, 1991) preferences and begins with a budget constraint:

The preferences contain both a coefficient of relative risk aversion, γ, and elasticity of intertemporal substitution, σ. Through process of derivation and substitution he eliminates consumption and, therefore, σ from the final model. The main equation from his derivation is:

The left-hand side is the expected excess return on an asset with an adjustment for Jensen’s Inequality. The right-hand side is a weighted average of covariances of the asset with the stock market, good news about current and future labor income, and good news about future expected returns on the market, respectively. As we see, only the coefficient of relative risk aversion, γ, enters the final equation.

This derivation motivates the choice of variables which have some ability to predict market returns and labor income growth as well as to explain the cross-section of asset returns. He tests for this by estimating Vector Autoregressions to show both time series and cross-sectional explanatory power. Having done that the author comes to the main takeaway for asset pricing in Table 8.

Here we see that RVW (the stock market factor) is the most important determinant of stock market portfolio returns by far. For bond portfolio returns, columns 3, 6, and 7 are the most important where columns 6 and 7 are the short-term rate and long-term spread, respectively. Despite the theoretical derivation of a multi-factor model, this table ultimately shows that the CAPM does still explain most of the variation in stock returns, at least for those portfolios here. As for the intertemporal model then, the author states, “its main contribution is to explain why investors use covariance with an aggregate stock index to determine expected returns on assets.”

Bad Beta, Good Beta

Bad Beta, Good Beta

By: Campbell and Vuolteenaho

            Campbell and Vuolteenaho (C and V) break down the CAPM model into separate betas to hopefully find a new approach to asset pricing than the single beta CAPM.  They break down betas into two categories, one using news of the market’s future cash flows (CF) and the other reflecting news about the market’s discount rate (DR).  Based on ICAPM theory, CF should be the beta with the higher price of risk.  As a metaphor, C and V compare the good and bad beta to good and bad types of cholesterol, and say that not all beta is bad just like not all cholesterol is bad.  After breaking down the betas into these categories, C and V find that small and value stocks have much higher CF beta than large and growth stocks and explains why they have higher returns.

            Robert Merton (1973) suggests that the price of risk for DR beta should be equal to the variance of the market return, while the price of risk for CF beta should be multiplied by gamma to compensate an investor for their risk aversion.  C and V build on this notion using a model that follows Merton’s suggestion.  Using the two-beta model, they find that the two-beta model outperforms the standard CAPM in the modern period from July 1963 to December 2001.  This is due to the fact that growth stocks, with low returns, have high betas, but they are mostly the good beta that comes with low risk prices.  On the contrary, value stocks with higher average returns, have higher bad beta than growth stocks. The single-beta CAPM adequately explained the data from the early period (1929-1963) because during this time, they find that the ratio of good to bad beta remains fairly constant.

In the lower left section of table 3 shows the correlation f shocks to individual state variables with the news terms for CF and DR.  r^e_M is log excess return, TY is the term yield spread, PE is the Price-earnings ratio, and VS is the small-stock value spread.

            Figure 1 shows market cash flows and market discount rate lined up with the occurrence of recession (indicated by the dotted vertical line).  In some cases, a recession occurs when CF decreases (valuation recession).  In other cases, a recession occurs when DR increases (profitability recession).  There are also some that occur as a result of both types (mixed recession).  Combining this with table 3, you can see what changes in state variables could do to CF and DR, and can in turn, do to the market.  For example, a valuation recession is characterized by a declining PE ration, a steepening yield curve, and larger declines in growth stocks than in value stocks.

            Table 5 shows the level of CF beta and DR beta based on size of the company and their measure from growth to value stocks.  The highest level of CF beta is recognized as you move to smaller value stocks.  The highest level of DR beta is when you move toward small growth stocks.

 

This is the equation used to find the results in table 7.  Expected returns are on the left side of the equation. Variance of the beta for Cash Flows and Discount rate are shown on the right, with gamma in front of the Cash flow beta term.  This is due to Merton’s suggestion mentioned above.

Table 7 shows the differences in beta premium between cash flows and discount-rate and follows the overall results talked about above.

            Based on these results, C and V conclude that value stocks and small stocks have higher cash-flow betas than growth stocks and large stocks, and this can explain their higher average returns.  The post-1963 negative CAPM alphas of growth stocks are explained by the fact that their betas are mostly of the good variety.  This model also explains why CAPM betas induced very little spread in the post-1963 period.  It is because the CAPM beta sort induces a post-ranking spread only in the good discount-rate beta, which carries the low premium (because it is the good beta).  For investors, C and V say these results show that risk-averse long-term investors who hold only equities should view the high average returns on value and small stocks as appropriate compensation for risk.  Lower risk-averse investors should overweight these stocks while investors with higher risk aversion should underweight them.