The risk return tradeoff in the long run: 1836–2003

Previous studies typically find a statistically insignificant relation between the market risk premium and its expected volatility. Further, several of these studies estimate a negative risk return tradeoff, contrary to the predictions of mainstream theory. Using simulations, I demonstrate that even 100 years of data constitute a small sample that may easily lead to this finding even though the true risk return tradeoff is positive. Small-sample inference is plagued by the fact that conditional volatility has almost no explanatory power for realized returns. Using the nearly two century history of U.S. equity market returns from Schwert [1990. Indexes of United States stock prices from 1802 to 1987. Journal of Business 63, 399–426], I estimate a positive and statistically significant risk return tradeoff. Finally, exploratory analysis suggests a role for a time-varying relation linked to the changing nature of the U.S. economy.     

The intertemporal relation between expected returns and risk

This paper explores the time-series relation between expected returns and risk for a large cross section of industry and size/book-to-market portfolios. I use a bivariate generalized autoregressive conditional heteroskedasticity (GARCH) model to estimate a portfolio's conditional covariance with the market and then test whether the conditional covariance predicts time–variation in the portfolio's expected return. Restricting the slope to be the same across assets, the risk-return coefficient is highly significant with a risk–aversion coefficient (slope) between one and five. The results are robust to different portfolio formations, alternative GARCH specifications, additional state variables, and small sample biases. When conditional covariances are replaced by conditional betas, the risk premium on beta is estimated to be in the range of 3% to 5% per annum and is statistically significant.     

The CAPM and value at risk at different time-scales

By resorting to wavelet analysis, we estimate the capital asset pricing model (CAPM) at different time-scales for the Chilean stock market. Our sample comprises 24 stocks that were actively traded on the Santiago Stock Exchange over 1997-2002. We find evidence in support of the CAPM at a medium-term horizon. We extend the literature in this area to analyze the impact of time scaling on the computation of value at risk. We conclude that risk is concentrated at higher frequencies of the data.     

The return impact of realized and expected idiosyncratic volatility

We show that the negative relation between realized idiosyncratic volatility, measured over the prior month, and returns is robust in non-January months. Controlling for realized idiosyncratic volatility, we show that the relation between returns and expected idiosyncratic volatility is positive and robust. Realized and expected idiosyncratic volatility are separate and important effects describing the cross-section of returns. We find the negative return on a zero-investment portfolio that is long high realized idiosyncratic volatility stocks and short low realized idiosyncratic volatility stocks is dependent on aggregate investor sentiment. In cross-sectional tests, we find the negative relation is weaker for stocks with a large analyst following and stronger for stocks with high dispersion of analyst forecasts. The positive relation between expected idiosyncratic volatility and returns is not due to mispricing.     

Efficient tests of stock return predictability

Conventional tests of the predictability of stock returns could be invalid, that is reject the null too frequently, when the predictor variable is persistent and its innovations are highly correlated with returns. We develop a pretest to determine whether the conventional t-test leads to invalid inference and an efficient test of predictability that corrects this problem. Although the conventional t-test is invalid for the dividend–price and smoothed earnings–price ratios, our test finds evidence for predictability. We also find evidence for predictability with the short rate and the long-short yield spread, for which the conventional t-test leads to valid inference.     

Omitted debt risk, financial distress and the cross-section of expected equity returns

The study of Ferguson and Shockley (2003) shows that, if the Merton (1974) model can reflect reality, the omission of debt claims from the market portfolio proxy may explain the poor pricing ability of the CAPM in empirical tests. We critically re-assess this argument by first reviewing existing, but also new avenues through which the Merton (1974) model can point to a systematic bias in market beta estimates. However, we also show that some avenues are diversifiable, and that they all rely on excessive economy-wide default risk to create a non-negligible bias. We then use the Merton (1974) model to proxy for the total debt portfolio, but find that its application in empirical tests cannot improve pricing performance. We conclude that there are (so far) no valid theoretical reasons to believe that omitted debt claims undermine CAPM tests.     

Risk and return: Long-run relations,fractional cointegration,and return predictability

Univariate dependencies in market volatility, both objective and risk neutral, are best described by long-memory fractionally integrated processes. Meanwhile, the ex post difference, or the variance swap payoff reflecting the reward for bearing volatility risk, displays far less persistent dynamics. Using intraday data for the Standard & Poor's 500 and the volatility index (VIX), coupled with frequency domain methods, we separate the series into various components. We find that the coherence between volatility and the volatility-risk reward is the strongest at long-run frequencies. Our results are consistent with generalized long-run risk models and help explain why classical efforts of establishing a naïve return-volatility relation fail. We also estimate a fractionally cointegrated vector autoregression (CFVAR). The model-implied long-run equilibrium relation between the two variance variables results in nontrivial return predictability over interdaily and monthly horizons, supporting the idea that the cointegrating relation between the two variance measures proxies for the economic uncertainty rewarded by the market.     

Political regimes,business cycles,seasonalities, and returns

This paper provides a method for testing for regime differences when regimes are long-lasting. Standard testing procedures are generally inappropriate because regime persistence causes a spurious regression problem – a problem that has led to incorrect inference in a broad range of studies involving regimes representing political, business, and seasonal cycles. The paper outlines analytically how standard estimators can be adjusted for regime dummy variable persistence. While the adjustments are helpful asymptotically, spurious regression remains a problem in small samples and must be addressed using simulation or bootstrap procedures. We provide a simulation procedure for testing hypotheses in situations where an independent variable in a time-series regression is a persistent regime dummy variable. We also develop a procedure for testing hypotheses in situations where the dependent variable has similar properties.     

Exchange-rate return predictability and the adaptive markets hypothesis: Evidence from major foreign exchange rates

This study examines return predictability of major foreign exchange rates by testing for martingale difference hypothesis (MDH) using daily and weekly nominal exchange rates from 1975 to 2009. We use three alternative tests for the MDH, which include the wild bootstrap automatic variance ratio test, generalized spectral test, and Dominguez–Lobato consistent tests. We evaluate time-varying return predictability by applying these tests with fixed-length moving sub-sample windows. While exchange rate returns are found to be unpredictable most of times, we do observe a number of episodes of statistically significant return predictability. They are mostly associated with the major events such as coordinated central bank interventions and financial crises. This finding suggests that return predictability of foreign exchange rates occurs from time to time depending on changing market conditions, consistent with the implications of the adaptive markets hypothesis.     

Correlation and return dispersion dynamics in Chinese markets

This paper studies the relation between firm-level return dispersions and correlations among Chinese stocks during periods of unusually large upward and downward swings. We analyze individual stock returns across 18 sectors and test if return dispersions and stock correlations show asymmetric patterns for extreme up and down markets. Evidence from studies on U.S. stocks suggests that equity return correlations tend to be much greater on the downside than on the upside and that the degree of comovement gets even stronger during extreme market states. However, in the case of Chinese stock market, we find that higher downside correlations apply to only stocks within the Financial sector. With the exception of Financial stocks, we find that stock correlations are significantly higher during up markets, rather than down markets. Regarding firm-level return dispersions, our findings are consistent with rational asset pricing model predictions. We find that equity return dispersions are significantly higher during periods of large price changes.     

Estimation of expected return: CAPM vs. Fama and French

Most practitioners favour a one-factor model (CAPM) when estimating expected return for an individual stock. For estimation of portfolio returns, academics recommend the Fama and French three-factor model. The main objective of this paper is to compare the performance of these two models for individual stocks. First, estimates for individual stock returns based on CAPM are obtained using different time frames, data frequencies, and indexes. It is found that 5 years of monthly data and an equal-weighted index, as opposed to the commonly recommended value-weighted index, provide the best estimate. However, performance of the model is very poor; it explains on average 3% of differences in returns. Then, estimates for individual stock returns are obtained based on the Fama and French model using 5 years of monthly data. This model, however, does not do much better; independent of the index used, it explains on average 5% of differences in returns. These results therefore bring into question the use of either model for estimation of individual expected stock returns.     

Liquidity risk and expected corporate bond returns

This paper studies the pricing of liquidity risk in the cross section of corporate bonds for the period from January 1994 to March 2009. The average return on bonds with high sensitivities to aggregate liquidity exceeds that for bonds with low sensitivities by about 4% annually. The positive relation between expected corporate bond returns and liquidity beta is robust to the effects of default and term betas, liquidity level, and other bond characteristics, as well as to different model specifications, test methodologies, and a variety of liquidity measures. The results suggest that liquidity risk is an important determinant of expected corporate bond returns.     

CAN STOCK MARKET LIBERALIZATION IN EMERGING ECONOMIES MITIGATE LEGAL SYSTEMS DEFICIENCIES?

We document positive valuation effects around the time of stock market liberalization. We find that the valuation effects are larger for countries with civil law traditions compared with countries with common law origins. Similarly, we find that countries with weaker shareholder protections exhibit greater market valuation increases compared with countries with stronger shareholder protections. The results reinforce the importance of country legal systems and of shareholder protections in determining the quality of corporate governance systems and financial market outcomes. The findings suggest that stock market liberalization may mitigate deficiencies in the existing institutional environments not supportive of effective corporate governance systems.     

Extreme downside risk and expected stock returns

We propose a measure for extreme downside risk (EDR) to investigate whether bearing such a risk is rewarded by higher expected stock returns. By constructing an EDR proxy with the left tail index in the classical generalized extreme value distribution, we document a significantly positive EDR premium in cross-section of stock returns even after controlling for market, size, value, momentum, and liquidity effects. The EDR premium is more prominent among glamor stocks and when high market returns are expected. High-EDR stocks are generally characterized by high idiosyncratic risk, large downside beta, lower coskewness and cokurtosis, and high bankruptcy risk. The EDR premium persists after these characteristics are controlled for. Although Value at Risk (VaR) plays a significant role in explaining the EDR premium, it cannot completely subsume the EDR effect.     

The impact of risk and uncertainty on expected returns

We study asset pricing in economies featuring both risk and uncertainty. In our empirical analysis, we measure risk via return volatility and uncertainty via the degree of disagreement of professional forecasters, attributing different weights to each forecaster. We empirically model the typical risk-return trade-off and augment these models with our measure of uncertainty. We find stronger empirical evidence for an uncertainty-return trade-off than for the traditional risk-return trade-off. Finally, we investigate the performance of a two-factor model with risk and uncertainty in the cross section.     

Lévy jump risk: Evidence from options and returns

Using index options and returns from 1996 to 2009, I estimate discrete-time models where asset returns follow a Brownian increment and a Lévy jump. Time variations in these models are generated with an affine GARCH, which facilitates the empirical implementation. I find that the risk premium implied by infinite-activity jumps contributes to more than half of the total equity premium and dominates that of the Brownian increments suggesting that it is more representative of the risks present in the economy. Overall, my findings suggest that infinite-activity jumps, instead of the Brownian increments, should be the default modeling choice in asset pricing models.     

The risk-return tradeoff: A COGARCH analysis of Merton's hypothesis

We analysed daily returns of the CRSP value weighted and equally weighted indices over 1953-2007 in order to test for Merton's theorised relationship between risk and return. Like some previous studies we used a GARCH stochastic volatility approach, employing not only traditional discrete time GARCH models but also using a COGARCH — a newly developed continuous-time GARCH model which allows for a rigorous analysis of unequally spaced data. When a risk-return relationship symmetric to positive or negative returns is postulated, a significant risk premium of the order of 7-8% p.a., consistent with previously published estimates, is obtained. When the model includes an asymmetry effect, the estimated risk premium, still around 7% p.a., becomes insignificant. These results are robust to the use of a value weighted or equally weighted index.The COGARCH model properly allows for unequally spaced time series data. As a sidelight, the model estimates that, during the period from 1953 to 2007, the weekend is equivalent, in volatility terms, to about 0.3-0.5 regular trading days.     

The empirical risk–return relation: A factor analysis approach

Existing empirical literature on the risk–return relation uses relatively small amount of conditioning information to model the conditional mean and conditional volatility of excess stock market returns. We use dynamic factor analysis for large data sets, to summarize a large amount of economic information by few estimated factors, and find that three new factors—termed “volatility,” “risk premium,” and “real” factors—contain important information about one-quarter-ahead excess returns and volatility not contained in commonly used predictor variables. Our specifications predict 16–20% of the one-quarter-ahead variation in excess stock market returns, and exhibit stable and statistically significant out-of-sample forecasting power. We also find a positive conditional risk–return correlation.     

Time-varying rare disaster risk and stock returns

This study provides empirical support for theoretical models that allow for time-varying rare disaster risk. Using a database of 447 international political crises during the period 1918-2006, we create a crisis index that shows substantial variation over time. Changes in this crisis index, our proxy for changes in perceived disaster probability, have a large impact on both the mean and volatility of world stock market returns. Crisis risk is positively correlated with the earnings-price ratio and the dividend yield. Cross-sectional tests also show that crisis risk is priced: Industries that are more crisis risk sensitive yield higher returns.     

Habit formation,surplus consumption and return predictability: International evidence

On an international post World War II dataset, we use an iterated GMM procedure to estimate and test the Campbell and Cochrane (1999, By force of habit: a consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107, 205–251.) habit formation model with a time-varying risk-free rate. In addition, we analyze the predictive power of the surplus consumption ratio for future stock and bond returns. We find that, although there are important cross-country differences and economically significant pricing errors, for the majority of countries in our sample the model gets empirical support in a variety of different dimensions, including reasonable estimates of risk-free rates. Further, for the majority of countries the surplus consumption ratio captures time-variation in expected returns. Together with the price-dividend ratio, the surplus consumption ratio contains significant information about future stock returns, also during the 1990s. In addition, in most countries the surplus consumption ratio is also a powerful predictor of future bond returns. Thus, the surplus consumption ratio captures time-varying expected returns in both stock and bond markets.