How Much Do Expected Stock Returns Vary Over Time WAYNE E. FERSON
Boston College and NBER
ANDREA HEUSON
University of Miami
TIE SU
University of Miami
first draft: January 30, 1998, this revision: December 11, 2002
This paper makes indirect inference about the time-variation in expected stock returns by comparing unconditional sample variances to estimates of expected conditional variances. The evidence reveals more predictability as more information is used, more reliable predictability in indexes than large common stocks, and no evidence that predictability has diminished over time. A "strong-form" analysis using options suggests that time-variation in market discount rates are economically important.
? 1998-2002 by Wayne E. Ferson, Andrea Heuson, and Tie Su. Ferson is the Collins Chair in Finance at Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA. 024678. ph (617) 552-6431, fax: 552-0431, email: wayne.ferson@https://www.doczj.com/doc/3717276069.html,, https://www.doczj.com/doc/3717276069.html,/~fersonwa. Heuson and Su are Associate Professors of Finance at the University of Miami, 5250 University Drive, Coral Gables, FL. 33124. Heuson may be reached at (305) 284-1866, fax 284-4800, aheuson@https://www.doczj.com/doc/3717276069.html,. Su may be reached at (305) 284-1885, fax 284-4800, tie@https://www.doczj.com/doc/3717276069.html,. We are grateful to Gurdip Bakshi, Hendrick Bessembinder, Charles Cao, John Cochrane, Pat Fishe, Bruce Grundy, Ravi Jagannathan, Herb Johnson, Avi Kamara, Terence Lim, Stewart Mayhew, Simon Pak, Mark Rubinstein, Robert Stambaugh, William Ziemba, and anonymous referees for helpful discussions, comments or other assistance. This paper was presented at the following universities: UC Riverside, Dartmouth, Emory, HEC School of Management, Houston, INSEAD, New South Wales, Oregon, the Stockholm School of Economics, Vanderbilt, Washington University in St. Louis, the University of Washington, the 1998 Maryland Symposium, the 2000 Midwestern Finance Association, the 1999 Utah Winter Finance Conference, the Spring, 1999 NBER Asset Pricing Meetings, the 1999 Western Finance Association, the 1999 European Finance Association, and the 2000 American Finance Association meetings. Parts of this work were competed while Ferson was the Pigott-PACCAR Professor of Finance at the University of Washington, a Visiting Scholar at the University of Miami, and a Visiting Scholar at Arizona State University. Su acknowledges financial support from the Research Council at the University of Miami.
I. Introduction
The empirical evidence for predictability in common stock returns remains ambiguous, even after many years of research. This paper makes indirect inference about the time-variation in expected stock returns by comparing unconditional, sample variances of return to estimates of expected conditional variances. The key to our approach is a sum-of-squares decomposition:
Var{R} = E{ Var(R|O) } + Var{ E(R|O) }, (1)
where R is the rate of return of a stock and O is the public information set. E(.|O) and Var(.|O) are the conditional mean and variance and Var{.} and E{.}, without the conditioning notation, are the unconditional moments. We are interested in the term Var{ E(R|O) }; that is, the amount of variation through time in expected stock returns. We infer this quantity by subtracting estimates of the expected conditional variance from estimates of the unconditional variance. We focus on the predictability in monthly stock returns. This is motivated by the empirical literature on asset pricing, which most commonly studies monthly returns.
We use three approaches to estimating the average conditional variances. These correspond to the classical description of increasing market information sets described by Fama (1970). Weak-form information considers only the information contained in past stock prices. This analysis, summarized in Table 1, builds on a comparison of daily and monthly sample variances, and is related to the variance ratios studied by Lo and MacKinlay (1988) and others. Semi-strong form information relates to lagged variables that are clearly publicly available. Our analysis uses regressions for individual stock returns, on lagged firm-specific characteristics. These results are reported in Table 2. Strong form refers to all relevant information that may be reflected in asset market prices. In this case, we use the implied volatilities from stock options to
proxy for the expected conditional variances. These results are reported in Table 3.
Studies of predictability in stock index returns typically report regressions with small R-squares, as the fraction of the variance in returns that can be predicted with lagged variables is small. The R-squares are larger for longer-horizon returns, because expected returns are considered to be more persistent than returns themselves.1 However, because stock returns are very volatile, small R-squares can mask economically important variation in the expected return. To illustrate, consider the simple Gordon (1962) constant-growth model for a stock price: P = kE/(r-g), where P is the stock price, E is the earnings per share, k is the dividend payout ratio, g is the future growth rate of earnings and r is the discount rate. The discount rate is the required or expected return of the stock. Stocks are long "duration" assets, so a small change in the expected return can lead to a large fluctuation in the asset value. Consider an example where the price/earnings ratio, P/E = 15, the payout ratio, k = 0.6, and the expected growth rate, g = 3%. The expected return is 7%. Suppose there is a shock to the expected return, ceteris paribus. In this example a change of one percent in r leads to approximately a 20% change in the asset value.
Of course, it is unrealistic to hold everything else fixed, but the example suggests that small changes in expected returns can produce large and economically significant changes in asset values. Consistent with this argument, studies such as Kandel and Stambaugh (1996), Campbell and Viceira (2001), and Fleming, Kirby, and Ostdiek (2001) show that optimal portfolio decisions can be affected to an economically significant degree by predictability, even when the amount of predictability in returns,
1 Thus, the variance of the expected return accumulates with longer horizons faster than the variance of the return, and the R-squared increases (see, e.g. Fama and French, 1988, 1989).
as measured by R-squared, is small. Generalizing the Gordon model to allow for changes in growth rates, Campbell (1991) estimates that changes in expected returns through time may account for about half of the variance of equity index values.
Our estimates of the amount of time-variation in the expected returns of stocks are increasing with finer information. Weak-form tests find no reliable evidence of predictability in modern data. Semi-strong form tests find small but economically significant predictability. In contrast to recent studies that rely on aggregate predictor variables, we find no evidence that the predictability has diminished over time. Strong-form tests using option-implied expected conditional variances suggest that the variation in ex ante equity discount rates is highly economically significant for individual stocks, but the results for the index are ambiguous.
The rest of the paper is organized as follows. Section II discusses our three approaches to measuring the expected conditional variances of stock returns. Section III presents the empirical data and results. Conclusions are offered in Section IV. An appendix discusses data, estimation issues and technical details.
II. Measuring Average Conditional Variances
A. Weak Form Information
In order to use Equation (1) we need to estimate the average variance of the return around its conditional mean, E{Var(R|O)} = E{[R-E(R|O)]2}. The problem is that we don't know the conditional mean. Our approach in this section follows Merton (1980), who showed that while the mean of a stock return is hard to estimate, it is nearly irrelevant for estimating the conditional variance, when the time between observations is short. We use high frequency returns to estimate the conditional variance, subtract its average from the monthly unconditional variance, and the difference is the variance of
the monthly mean.
Nelson (1990, 1992) further develops Merton's idea. Suppose that the stock price can be approximated by a continuous process formed as a step function, with time intervals of length h between the steps. Take the interval [T-h,T), chop it into D pieces, and consider the average of the D squared price changes as an estimator for the conditional variance of the return over the interval. Nelson proves the estimator is consistent, in the sense that it approaches the conditional variance in the "continuous record" limit, as h approaches zero and D becomes infinite. The intuition is that for small h, the conditional mean is effectively constant, so the sample variance approaches the conditional variance as D grows. By similar logic, Nelson (1992) shows that misspecification of the conditional mean, which arises due to the inability to measure the information O, washes out as h gets small.
Evidence from Nelson (1991) supports the idea that for monthly stock returns, chopping the month into days should work well. He finds that daily returns measured with versus without dividends, or with a simple adjustment for risk-free interest rates, produce virtually the same estimates of conditional variances. Similarly, Schwert (1990) finds that different dividend series have almost no affect on the daily variances of a long historical stock return series that we use in our analysis.
We estimate E{Var(R|O)} by the time series average of daily variances for each month. Using monthly returns data, we estimate the unconditional variance, Var(R). Then, we infer the variance of the conditional expected return by Equation (1). To fix things, let the return for month m be R m = ln(V m/V m-1) = S j e m?j, where V m is the value of the stock at time m and ?j is the daily log value change for day j. Assume that the conditional mean for month m is μm = E(S j?j|j e m), with E(?j|j e m) = μm/D, D being the number of days in the month. The unconditional mean monthly return is E(μm) = μ,
and we are interested in Var(μm), the variance of the monthly expected return. Define the average daily variance, ADV = E{E[(?j - μm/D)2|j e m]}, and the unconditional monthly variance, MV = E{(R m - μ)2}. Simple calculations show that Var(μm) = MV -
D(ADV).
The model outlined above uses the approximation that the means shift monthly, while daily returns fluctuate independently around the conditional means. However, there is weak serial dependence in daily stock returns, which would influence the sample estimates. The question is whether or not to attribute this serial dependence to changes in the conditional expected return.
On the one hand, much of the literature on predictability allows that serial dependence may reflect changing conditional means. Fama and French (1988) use rate-of-return autoregressions to study predictability. Studies such as Lo and MacKinlay (1988) and Conrad and Kaul (1988) model expected returns within the month as autoregressive processes. On the other hand, serial dependence in daily returns can arise from end-of-day price quotes that fluctuate between bid and ask (Roll, 1984) or from nonsynchronous trading of the stocks in an index. These effects should not be attributed to time-variation in the expected discount rate for stocks. We estimate
Var(μm) with and without adjustments for serial dependence. To illustrate the adjustment, let ? = E{ E[(?j - μm/D)(?j' - μm/D)|j,j'e m, |j-j'|=1] }. Assuming that the first order daily serial dependence reflects market microstructure effects unrelated to discount rates, we estimate Var(μm) = MV - D(ADV + 2?).
In the Appendix B we describe how the calculations are adjusted to obtain unbiased estimators in finite samples. Biases in the sample variances and autocovariances arise due to estimation error in the sample means. In addition, there is a "finite record" bias, which arises because h>0 and D<∞. To address these biases we
use Monte Carlo simulations.
B. Semi-strong Form Information: Using Individual Stock Regressions
Much of the empirical literature on asset-return predictability uses regressions
of stock-index returns on lagged, market-wide information variables. This approach raises two types of concerns. First, there are statistical problems associated with such regressions, especially when the data are heteroskedastic, the right-hand side variables are highly persistent or the left-hand side variables are overlapping in time.2 Second is the issue of data mining. If the lagged instruments result from many researchers sifting through the same data sets, there is a risk of spurious predictability (Lo and MacKinlay 1990; Foster, Smith, and Whaley 1997).
We use time-series regressions for individual stocks to estimate the sum-of-squares decomposition in Equation (1), focusing on the aggregate predictability. The individual-stock regressions use firm-specific variables. From basic portfolio theory, individual-stock expected returns teach us about index predictability, only to the extent that they covary. Consider the N x N covariance matrix of the conditional mean returns for N stocks, Cov{E(R|Z)}, where Z stands for the lagged, public information regressors. Letting 1 be an N-vector of ones, the variance of the conditional expected return on an equally-weighted portfolio, R p = (1/N)1'R, is Var{E(R p|Z)} = (1/N2)
1'Cov{E(R|Z)}1. Since there are N(N-1) covariance terms, but only N variances in the quadratic form, the expected return variance for the portfolio approaches the average of the firms' covariances, while the individual stock predictability vanishes as N gets large.
2 Boudoukh and Richardson (1994) provide an overview of the statistical issues. Stambaugh (1999) and Ferson, Sarkissian, and Simin (2002) provide more recent analyses and references.
There is some correlation between our firm-specific variables and the instruments selected in studies of aggregate predictability, so we are not immune to data mining bias. However, such biases should be mitigated to some extent by our approach. Our measure does not rely on the direct index predictability that so many previous studies have explored. The number of studies that examine individual-stock return predictability with time-series regressions is still relatively small. We use Monte Carlo methods to handle the statistical issues, as described in Appendix B. Using only firm-specific instruments we probably understate the correlations among the expected returns, and therefore understate the aggregate predictability. If we use market-wide instruments for individual stocks, we probably overstate the correlations. Comparing the two cases we estimate a range of plausible values.
C. Strong Form: Using Implied Variances from the Options Markets
Empiricists cannot observe all of the information that economic agents might possess, and finer information is likely to reveal a greater degree of predictability. Our strong-form tests use the implied volatility at time t, s t, for an option that matures at time t+1, and assumes that the average implied variance is the average conditional variance of returns: E{s} = E{Var(R t+1|O t)}. We define the option-implied predictability using this assumption and equation (1):
Implied predictability = Var{E(R|O)}
= {Var(R)-E(s)}. (2)
Since the option-implied predictability derives directly from asset-market prices, the measure reflects the otherwise unobserved "market" information set, O.
The option-implied predictability uses insights from Lo and Wang (1995) and Grundy (1991), who emphasize that while standard option pricing formulae are invariant to the conditional mean of the stock, option prices should still reflect the predictability in asset returns. This works because option prices derive from stock returns "net" of their predictability. (For example, options may be priced using a "risk neutralized" distribution.) By comparing the implied volatility with the unconditional variance of the return, we draw inferences about the amount of predictability. The intuition is that, holding the unconditional variance constant, a situation with more predictability should imply a smaller conditional volatility, lower option prices; and thus, lower implied volatilities.
The key assumption of equation (2) is that the average implied variance from stock options is the average conditional variance of the stock return. This is consistent with empirical studies that find implied volatilities to be informative instruments for conditional variance (e.g. Day and Lewis 1992; Lamoureaux and Lastrapes 1993; Fleming 1998). If our assumption introduces less error than the error in explicit models for expected returns, we should obtain more reliable inferences about the predictability using our approach. Since the implied volatility from any particular option pricing model can be a biased proxy for the average conditional variance, we use several option pricing models in our analysis and we explore the potential biases.
D. Caveats about Implied Volatility
In the option pricing model of Black and Scholes (BS, 1973), the implied volatility is the fixed diffusion coefficient of the log-price process of the stock, representing the conditional variance of infinitesimal holding period returns. If the drift of the process is constant the implied variance is also the conditional variance per
period of discrete holding period returns, and the implied predictability is zero. Merton (1973) shows that if the diffusion coefficient in a lognormal diffusion varies deterministically over time, the BS model holds and the implied variance is the average of the time-varying conditional variance. Lo and Wang (1995) show that the relation between the diffusion coefficient in a continuous-time model and the conditional variances of holding period returns generally depends on the specification of the drift of the process. If we knew the correct continuous-time model we could compute the theoretical mapping between the option-implied variance and the conditional variance of return. In this case, we could also compute the implied predictability directly from the process.
It is easy to compute option-implied variances from several models, as we do, but it is not easy to agree on the "true" continuous time model. In fact, we doubt that such a model exists. We know that asset prices cannot literally follow continuous-time processes. Markets are not always open. Even when they are, assets tend to trade discretely and nonsynchronously.3 Approaching continuous time, we run into microstructural effects at high frequencies, such as bid-ask spreads and tick sizes. Options cannot be hedged continuously (Leland 1985; Figlewski 1989), although continuous-time models typically assume that they are. It is therefore unlikely that option-implied volatilities correspond exactly to the predictions of any continuous time model. Without the "true" stochastic process, we can't infer the errors in our empirical proxy from theory. We therefore conduct a series of experiments to evaluate the potential biases.
3 Even in our sample of heavily traded stocks, CRSP records up to 15% of the daily returns as zeros, with more than 10% in six of 26 cases. See Lesmond, Ogden and Trzcinka (1999) for an analysis of transactions costs based on the frequency of zero returns.
The simple step function model behind our weak-form tests can motivate the assumption that option-implied variances are conditional variances. Nelson (1992) provides conditions under which the step function process becomes a diffusion process in the continuous-record limit. The diffusion coefficient of the limiting process is the conditional variance of return. Therefore, if the step function model is a reasonable approximation to our data, the option-implied variances should be a reasonable proxy for the conditional variances.
There are many reasons to think that option-implied variances contain measurement error. However, the implied predictability does not require a precise conditional variance at each date, because only the average value of s over time is used. To the extent that measurement errors average to zero, they do not bias the option-implied predictability. Errors in the implied variances will only create a problem if their sample means differ from zero.
The implied volatilities from any given model can be systematically biased for a variety of reasons. Some biases can produce implied predictability that appears too high, while other biases may work in the opposite direction. Still other factors have ambiguous effects on the implied volatility, or may bias both the implied and unconditional sample variances in the same direction, so the net effect on the implied predictability is an empirical issue. We conduct a series of experiments to assess the magnitudes of various sources of potential bias. The details are available by request to the authors, and Appendix C contains an abbreviated discussion.
III. Empirical Results
A. Results using Weak-form Information
Table 1 presents estimates of the variation in monthly conditional expected
returns based on the comparison of monthly and daily return variances. Appendix A describes the data. The first three columns give the name of the return series and the starting and ending dates. The rows present results for the Standard and Poors index over different subsamples, followed by a summary of the individual stocks. The fourth column is the unconditional standard deviation of the monthly returns, expressed as an annual percent (the monthly variance is multiplied by 12, then the square root of this result is multiplied by 100. All of the numbers in the tables are annualized this way.) Two estimators of predictability are shown. The estimator denoted by s(μm) ignores daily serial dependence, while the estimator denoted by s(μm)* adjusts for autocorrelation in daily returns, taking the view that daily serial correlation reflects microstructure issues unrelated to changes in discount rates. (The autocorrelation parameter, ?, is positive for the Standard and Poors 500 index, and negative for 16 of
the 26 firms.)
The estimates of predictability for the stock index are 6.2% and 4.8%, respectively, using Schwert's (1990) data for the 1885-1962 period; and 5.4% and 1.9% over the 1885-2001 period. However, over the 1962-2001 period where the CRSP daily data are used, there is little evidence of predictability. The estimated volatility of the expected returns is 2.0% if the serial dependence is allowed in the estimate, but -6.2% if serial dependence is assumed to be unrelated to the ex ante discount rate. Over the most recent 120 months of the sample the two estimates are -8.3% and -8.9%, respectively.4 For individual stocks the average point estimates of predictability are negative, and the individual estimates are negative in all but four cases, but none are
4 Our procedure does not constrain the estimated variance of the expected return to be positive. This could be done, following studies such as Boudoukh, Richardson, and Whitelaw (1993), and should produce more precise estimates.
significantly different from zero. Thus, no weak-form evidence for predictability is found at the firm level.
We estimate the finite sample bias in the predictability using Monte Carlo simulations.5 The adjusted estimates, after subtracting the expected bias, are shown in columns 7 and 8 of Table 1. They tell a similar story. The estimates for the index are
8.9% and 7.5% for the 1885-1962 period. However, using CRSP data for 1962-2001, one estimate is 4.2% while the other is -4.1%. Over the last 120 months the adjusted estimates are -6.6% and -7.3%, respectively. The values for the average individual stocks are typically negative, and insignificantly different from zero.6 Thus, our weak-form tests find only weak evidence of time-varying expected returns on the market index, concentrated in the pre-1962, pre CRSP data, but no evidence for predictability at the firm level.7
B. Semi-Strong form Tests Using Individual Stock Regressions
Table 2 presents our estimates of predictability based on the covariances of
5 We resample from the actual data for a given stock or index, randomly with replacement. For each simulation trial we generate an artificial time series with the same number of daily observations as the original data series. The artificial data satisfy the null hypothesis that the expected return is constant. We compute the estimators on the artificial sample in exactly the same way as on the original samples. We repeat this for 1,000 trials. The average across the trials is the expected finite sample bias. We use the distribution of the simulated estimates to generate empirical p-values. These are the fraction of the simulations where the variance estimates are larger than the sample values.
A small p-value means that the sample estimate is unlikely to occur if expected returns are constant.
6 We also run simulations for the estimators without adjustment for finite sample biases, relying on the simulations to control the biases. The results are similar.
7 Nelson and Startz (1993) also find that weak-form evidence for stock index predictability is thin in post World War II data.
individual stock regressions on the lagged variables. The regressions use monthly data from 1969 through 2001 and twenty large common stocks. We estimate predictability
for an equally weighted portfolio. The first column shows the results when each regression uses both firm-specific and index characteristics as the lagged regressors, in which case the covariances among the expected returns are probably overestimated. The estimated standard deviations of monthly expected returns are between 6% and 7% annualized, depending on whether or not we exclude the diagonals from the calculation. (Excluding the diagonals provides some information on what would be expected to happen as the number of similar stocks used in the calculation is increased.) Using only firm-specific variables in the regressions reduces the estimated predictability to the 2.4% to 3.3% range, as shown in the second column.
The estimates of predictability in Table 2 are similar whether we use the full sample or concentrate on the last 120 months. This is interesting in view of recent empirical finance studies that find predictability, measured using lagged variables like aggregate dividend yields and bond yield spreads, has weakened in recent samples. It may be that the predictability was "real" when first publicized, but diminished as traders attempted to exploit it.8 It may also be that the predictability was spurious in the first place. If the predictability is spurious we would expect instruments to appear
in the empirical literature, then fail to work with fresh data (e.g., Ferson, Sarkissian, and Simin, 2002). In this case, we should be suspicious of any "stylized facts" based on the aggregate instruments. The aggregate dividend yield rose to prominence in the 1980s, but fails to work in post 1990 data. The book-to-market ratio seems also to have
8 Lo and MacKinlay (1999) present weak form tests with less evidence for predictability in more recent data, and suggest this may be related to "statistical arbitrage" trading by Wall Street firms.
weakened in recent data. (See e.g. Bossaerts and Hillion (1999), Goyal and Welch, (1999), and Schwert, 2002.) But Table 2 presents no evidence that predictability is weaker in the recent ten-year period. As the measures in Table 2 do not rely on index predictability using aggregate predictor variables, the table provides interesting evidence that the underlying predictability has not diminished.
The regressions behind Table 2 are subject to statistical biases, which we control via simulation as discussed in Appendix B. The bias-adjusted estimates are reported in the fourth column. Using only firm-specific lagged variables the estimates are 1.6% to 2.3% annualized, which is statistically significant according to the empirical p-values. Again, the values are similar for the last 120 months of the sample.
Our semi-strong form estimates of predictability provide more reliable evidence of time-variation in monthly stock returns than our weak-form tests, which is expected if returns are more easily predicted using more information. But is 2% on an annual basis an economically significant effect? The simple Gordon model example from the introduction provides an illustration. Consider a month in which the required expected return jumps by two standard deviations, from 7% to 11%. Other things held fixed, the stock price would fall to half of its former value in response.
C. Strong-form Answers from the Options Markets
We estimate the option implied predictability using the Generalized Method of Moments (GMM, Hansen 1982). The GMM estimates are obtained by replacing the population moments in (2) by their sample analogues. The standard errors and test statistics then follow from standard results in Hansen (1982).9 The key input is the
9 Given monthly returns R t and implied variances s, the system of moment conditions is:
implied volatility, s t, which comes from an option pricing model. We use the BS model and two more general option pricing models. The general models, which are described in Appendix D, allow for stochastic volatility and jump risk, following Bakshi, Cao, and Chen (1997). All of the implied volatilities are calculated using one month-to-maturity, at-the-money options as described in Appendix A. At-the-money options typically have the highest trading volumes and the smallest bid-ask spreads. At-the-money option prices are the most sensitive to the volatility of the underlying stock. Conversely, the implied volatilities are relatively insensitive to errors in the option price data.
Our sample includes 26 large, highly-traded common stocks and the SP500 index option (SPX). These large stocks may not be representative of the larger universe of individual stocks, and our results should be interpreted accordingly. Larger and more heavily traded firms are likely to be of relatively low volatility. Fleming (1998) argues that lower volatilities may be more accurately estimated by the BS model.
The results are summarized in Table 3. For the S&P 500 the average implied variance is larger than the unconditional sample variance, so the implied predictability over the May, 1986 - December, 1997 period is negative. In contrast, for the individual stocks negative value is rare, and the average estimates are between 10.5% and 16%, depending on the option pricing model. The average predictability is between three
u1t = R t - μ
u2t = v - (R t - μ)2
u3t = IP - v + s,
where the parameters are {IP,v,μ} and IP is the implied predictability.
and eight standard errors from zero, as indicated in the last row of the table.10 It is a puzzle that the implied predictability based on the index options is so different. The subperiod for the index options is short, but when we run the analysis
for the individual stocks over the matching subperiod, we get predominantly positive values.
The estimates suggest that the extent of the predictability in the individual stocks, as assessed by the options markets, is larger than the predictability delivered by the weak form tests. This makes sense, as a finer information set should reveal more predictability.
A comparison across the alternative option pricing models reveals that the implied volatilities are the largest for the SVJ model, followed by the SV and the BS models. The differences in the implied predictabilities, however, tend to be relatively small compared with the standard errors. Statistically small differences for the alternative models should not be surprising given the empirical findings of Bakshi, et al. (1997) for the S&P 500. Chernov and Ghysels (2000) also find that option pricing errors are not highly sensitive to the choice of the asset pricing kernel, which suggests that implied volatilities should not be highly sensitive to the pricing model. Oveall, the average implied predictability is 14.7% for the SV model and 10.5% for the SVJ model, the latter brought down by two outlying, negative values.
The average monthly return for the individual stocks, over the 1975-1997
10 The variance of the estimate of the average implied predictability is computed as Var( S i IP i2 /n) = (1/n) s'?s, where s is the n-vector of the individual IP's estimated by the GMM, and ? is the n x n matrix of sample correlations between the estimators of the n IPs. Consistent estimates of the elements of ? are obtained as the sample correlations between the error terms corresponding to the IPs, the u3t's of the system in footnote 4, using the common months where all the stock returns are available.
period, is about 5.4% on an annual basis. With a standard deviation of expected return between 10% and 15%, the conditional expected return is frequently negative. Negative expected returns are troublesome for some rational models. For example, negative expected returns can only occur in the Capital Asset Pricing Model (Sharpe, 1964) if the betas are negative, which is not the case for these stocks. In intertemporal asset pricing models, the expected return can be negative if the stock provides a hedge against consumption risk. Stock returns can also be predictable if investors irrationally under or overreact to information. Alternatively, the average expected return could be higher than the 4.5% sample mean over this period, but it would have to exceed 20-30% per year to avoid negative expected returns, which seems unlikely. Accounting for taxes and transactions costs it may be easier to justify negative ex ante returns over short monthly periods. This remains an interesting topic for future research.
The preceding calculations are sensitive to the assumed holding period for the return. Since the expected return increases in proportion to the interval, while the standard deviation grows approximately as the square root of the interval, negative expected returns become less likely over longer holding periods.
IV. Conclusions
Small changes in expected returns can produce large and economically significant changes in asset values. Our estimates of the time-variation in the expected returns of stocks are increasing with finer information. Weak-form tests find no reliable evidence of predictability in modern data. Semi-strong form tests find small but economically significant predictability. In contrast to recent studies that rely on aggregate predictor variables, we find no evidence that the predictability has diminished over time. Strong-form tests using option-implied expected conditional
variances suggest that the variation in ex ante equity discount rates is highly economically significant for individual stocks, but the results for the index are ambiguous.
Appendix A: Data
Our analysis includes several data sets. We initially collect individual stock price, option price and Treasury bill yields from the Wall Street Journal. These data cover all option-expiration Fridays during the June, 1975-December, 1997 period. By mid-1975 there were 67 firms with listed options. We select a sample of the fifteen largest, most heavily-traded firms in 1975, to minimize problems associated with infrequent trading. We do not impose a survival criterion on this sample; data for one firm ends in 1995. Subsequently, we purchased daily options price data from the Chicago Board Options Exchange (1985-1997, and 1998-2001). These data include the SPX and 20 large, highly-traded firms that exist in 2001, some of which overlap with our original sample. The daily data allow us to precisely match returns and option maturity dates to one month. When a firm appears in both data sets, we use the daily data, which we believe to be more reliable. However, there may be a mild survival bias in the daily sample. Of the twenty firms in that sample, fifteen exist in July of 1962. A comparison with the hand-collected sample reveals an average monthly return standard deviation of 29.7% in our full sample, versus 32.0% in the hand collected.
We also obtain from CRSP, the actual cash dividends paid between the date that we observe the prices and the expiration date of the option. We take the ask yield of the Treasury bill that matches the maturity of the option to represent the risk-free rate.
Our semi-strong form tests use data on lagged, firm-specific instruments and macro instruments for the index. The firm-specific instruments are obtained from the
CRSP and the COMPUSTAT. They include for each stock month: (1) current month stock return; (2) one-month lagged 12-month holding period return; (3) book to market ratio, defined as the most recently-available book value per share (book-value of equity divided by the number of shares outstanding, quarterly COMPUSTAT data items Data60 and Data61, respectively) divided by then-current stock price; (4) annualized dividend yield, defined as trailing four-quarter dividends (COMPUSTAT Data16) divide by stock price, and (5) earnings-to-price ratio, defined to be trailing four-quarter earnings (COMPUSTAT Data11) divided by stock price; (6) all stock prices and returns are directly obtained from daily and monthly CRSP files.
The aggregate instruments include: (1) the three-month Treasury Bill secondary market yield computed on a discount basis; (2) the one-month holding period return on a three-month Treasury Bill; (3) the spread between this Treasury Bill yield and the
ten-year constant maturity Treasury Bond yield to maturity, a proxy for the slope of the term structure; and (4) the spread between Moody's Seasoned AAA and BAA corporate bond yields, a proxy for default risk in the economy. All of these series are measured as the lagged monthly average of daily values, from the Federal Reserve Bank of St. Louis. In addition, we use the one-month lagged holding period return on the Standard and Poor's Index, excluding dividends, and the lagged 12-month holding period return on the S&P 500 Index, from CRSP. We also use the dividend yield on the SPX index. The dividend yield is computed as the trailing 12-month dividends divided by the SPX index level. Monthly dividends are obtained from Bloomberg.
Our approach to implied volatility is similar to the construction of the VIX volatility series of the Chicago Board Options Exchange (see Fleming, Ostdiek, and Whaley 1995). Four options are used in the computation each month: two calls and two puts, with exercise prices that bracket the current value of the stock. The implied