# Betting against beta pdf

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Of course, when we regress our portfolios on standard risk factors, the realized factor loadings are not shrunk as above since only the ex-ante betas are subject to selection bias. Our results are robust to alternative beta estimation procedures as we report in the Appendix.

We compute betas with respect to a market portfolio, which is either specific to an asset class or the overall world market portfolio of all assets. While our results hold both ways, we focus on betas with respect to asset-class-specific market portfolios since these betas are less noisy for several reasons. First, this approach allows us to use daily data over a long time period for most asset classes, as opposed to using the most diversified market portfolio for which we only have monthly data over a limited time period.

Second, this approach is applicable even if markets are segmented. As a robustness test, Table B8 in the Appendix reports results when we compute betas with respect to a proxy for a world market portfolio comprised of many asset classes. We use the world market portfolio from Asness, Frazzini, and Pedersen To construct each BAB factor, all securities in an asset class are ranked in ascending order on the basis of their estimated beta.

The ranked securities are assigned to one of two portfolios: low- beta and high-beta. The low high beta portfolio is comprised of all stocks with a 15 See Asness, Frazzini, and Pedersen for a detailed description of this market portfolio. The market series is monthly and ranges from to Pedersen — Page 18 beta below above its asset-class median or country median for international equities. In each portfolio, securities are weighted by the ranked betas i.

The portfolios are rebalanced every calendar month. To construct the BAB factor, both portfolios are rescaled to have a beta of one at portfolio formation. The BAB is the self-financing zero-beta portfolio 8 that is long the low-beta portfolio and that short-sells the high-beta portfolio. For example, on average, the U. The holdings data run from March to March We focus our analysis on open-end, actively managed, domestic equity mutual funds. Our sample selection procedure follows that of Kacperzczyk, Sialm, and Zheng , and we refer to their Appendix for details about the screens that were used and summary statistics of the data.

This dataset has been used extensively in the existing literature on individual investors. For a detailed description of the brokerage data set, see Barber and Odean The data run from January to March The data run from March to March As an overview of these results, the alphas of all the beta-sorted portfolios considered in this paper are plotted in Figure 1.

We see that declining alphas across beta-sorted portfolios are general phenomena across asset classes. Figure B1 in the Appendix plots the Sharpe ratios of beta-sorted portfolios, which also shows a consistently declining pattern. We see all the BAB portfolios deliver positive returns, except for a small insignificantly negative return in Austrian stocks. The BAB portfolios based on large numbers of securities U. We discuss these results in detail below. We consider 10 beta-sorted portfolios and report their average returns, alphas, market betas, volatilities, and Sharpe ratios.

The average returns of the different beta portfolios are similar, which is the well-known relatively flat security market line. Hence, consistent with Proposition 1 and with Black , the alphas decline almost monotonically from the low-beta to high-beta portfolios. Indeed, the alphas decline when estimated relative to a 1-, 3-, 4-, and 5-factor model. Moreover, Sharpe ratios decline monotonically from low-beta to high-beta portfolios. Consistent with Proposition 2, the BAB factor delivers a high average return and a high alpha.

Last, we adjust returns using a 5-factor model by adding the traded liquidity factor by Pastor and Stambaugh , yielding an abnormal BAB return of 0. We note that while the alpha of the long-short portfolio is consistent across regressions, the choice of risk adjustment influences the relative alpha contribution of the long and short sides of the portfolio.

Pedersen — Page 21 Our results for U. Further, our results extend internationally. We consider beta-sorted portfolios for international equities and later turn to altogether different asset classes. The international portfolio is country neutral, i. Table V shows the performance of the BAB factor within each individual country. The BAB delivers positive Sharpe ratios in 18 of the 19 MSCI developed countries and positive 4-factor alphas in 13 out of 19, displaying a strikingly consistent pattern across equity markets.

The BAB returns are statistically significantly positive in 6 countries, while none of the negative alphas is significant. Of course, the small number of stocks in our sample in many of the countries makes it difficult to reject the null hypothesis of zero return in each individual country. Table B1 in the Appendix reports factor loadings. On average, the U. The larger long investment is meant to make the BAB factor market-neutral because the stocks that are held long have lower betas.

The other factor loadings indicate that, relative to high-beta stocks, low-beta stocks are likely to be larger, have higher book-to- 17 We keep the international portfolio country neutral because we report the result of betting against beta across equity indices BAB separately in Table VIII. Pedersen — Page 22 market ratios, and have higher return over the prior 12 months, although none of the loadings can explain the large and significant abnormal returns.

The Appendix reports further tests and additional robustness checks. In Table B2, we report results using different window lengths to estimate betas and different benchmarks local, global. We split the sample by size Table B3 and time periods Table B4 , we control for idiosyncratic volatility Table B5 and report results for alternative definition of the risk-free rate B6.

Finally, in Table B7 and Figure B2 we report an out-of-sample test. Treasury bonds. As before, we report average excess returns of bond portfolios formed by sorting on beta in the previous month. In the cross section of Treasury bonds, ranking on betas with respect to an aggregate Treasury bond index is empirically equivalent to ranking on duration or maturity. The rightmost column reports returns of the BAB factor.

Abnormal returns are computed with respect to a one-factor model where alpha is the intercept in a regression of monthly excess return on an equally weighted Treasury bond excess market return. The results show that the phenomenon of a flatter security market line than predicted by the standard CAPM is not limited to the cross section of stock returns. Indeed, consistent with Proposition 1, the alphas decline monotonically with beta.

Likewise, Sharpe ratios decline monotonically from 0. Furthermore, the bond 18 DataStream international pricing data start in while Xpressfeed Global coverage starts in Pedersen — Page 23 BAB portfolio delivers abnormal returns of 0.

Since the idea that funding constraints have a significant effect on the term structure of interest may be surprising, let us illustrate the economic mechanism that may be at work. Suppose an agent, e. This higher leverage is needed because the long-term Treasures are 11 times more volatile than the short-term Treasuries.

If the agent has leverage limits or prefers lower leverage , then he would strictly prefer the year Treasuries in this case. According to our theory, the 1-year Treasuries therefore must offer higher returns and higher Sharpe ratios, flattening the security market line for bonds. While a constrained investor may still prefer an un-leveraged investment in year bonds, unconstrained investors now prefer the leveraged low-beta bonds, and the market can clear.

While the severity of leverage constraints varies across market participants, it appears plausible that a 5-to-1 leverage on this part of the portfolio makes a difference for some large investors such as pension funds. In Panel A, columns 1 to 5 , the test assets are monthly excess returns of corporate bond indexes by maturity.

We see that the credit BAB portfolio delivers abnormal returns of 0. Furthermore, alphas and Sharpe ratios decline monotonically. Pedersen — Page 24 In columns 6 to 10 , we attempt to isolate the credit component by hedging away the interest rate risk.

We compute market returns by taking the equally weighted average of these hedged returns, and we compute betas and BAB portfolios as before. Abnormal returns are computed with respect to a two-factor model where alpha is the intercept in a regression of monthly excess return on the equally weighted average pseudo-CDS excess return and the monthly return on the Treasury BAB factor.

Consistent with all our previous results, we find large abnormal returns of the BAB portfolios 0. The BAB portfolio delivers positive returns in each of the four asset classes, with an annualized Sharpe ratio ranging from 0. Pedersen — Page 25 are only able to reject the null hypothesis of zero average return for equity indexes, but we can reject the null hypothesis of zero returns for combination portfolios that include all or some combination of the four asset classes, taking advantage of diversification.

We construct a simple equally weighted BAB portfolio. We report results for an All Futures combo including all four asset classes and a Country Selection combo including only Equity indices, Country Bonds and Foreign Exchange. Figure 1 illustrates the remarkably consistent pattern of declining alphas in each asset class, and Figure 2 shows the consistent return to the BAB factors.

Clearly, the relatively flat security market line, documented by Black, Jensen, Scholes for U. Pedersen — Page 26 We take this prediction to the data using the TED spread as a proxy of funding conditions. The first column simply regresses the U. Hence, the coefficient for change is consistent with the model, but the coefficient for the lagged level is not, under this interpretation of the TED spread. Hence, our test relies on an assumption that such variation of other variables does not lead to an omitted variables bias.

To partially address this issue, column 2 provides a similar result when controlling for a number of other variables. The control variables are the market return to account for possible noise in the ex ante betas used for making the BAB portfolio market neutral , the 1-month lagged BAB return to account for possible momentum in BAB , the ex-ante Beta Spread, the Short Volatility Returns, and the Lagged Inflation.

Hence, the TED spread at the end of the return period is a measure of the credit conditions at that time even if the TED spread is a difference in interest rates that would be earned over the following time period. Pedersen — Page 27 difference between the long and short side of the BAB portfolios, which should positively predict the BAB return as seen in Proposition 2. Consistent with the model, Table IX shows that the estimated coefficient for the Beta Spread is positive in all specifications, but not statistically significant.

Lagged Inflation is equal to the 1-year U. CPI inflation rate, lagged 1 month, which is included to account for potential effects of money illusion as studied by Cohen, Polk, and Vuolteenaho , although we do not find evidence of this effect. These regressions include fixed effects and standard errors are clustered by date.

Beta Compression We next test Proposition 4 that betas are compressed toward 1 when funding liquidity risk is high. Table X presents tests of this prediction. We use the volatility of the TED spread to proxy for the volatility of margin requirements. Since we are computing conditional moments, we use the monthly volatility as of the prior calendar month, which ensures that the conditioning variable is known as the beginning of the measurement period.

The sample runs from December to March Each calendar month, we compute cross-sectional standard deviation, mean absolute deviation, and inter-quintile range of the betas for all assets in the universe. Pedersen — Page 28 cross-sectional dispersion measure on the full set of dummies without intercept.

In Panel C, we compute the monthly dispersion measure in each asset class and average across assets. All standard errors are adjusted for heteroskedasticity and autocorrelation up to 60 months. Table X shows that, consistent with Proposition 4, the cross-sectional dispersion in betas is lower when credit constraints are more volatile. The average cross-sectional standard deviation of U. The tests based on the other dispersion measures, the international equities, and the other assets all confirm that the cross-sectional dispersion in beta shrinks at times where credit constraints are more volatile.

The Appendix contains an additional robustness check. Since we are looking at the cross-sectional dispersion of estimated betas, one could worry that our results was driven by higher beta estimation errors, rather than a higher variance of the true betas. To investigate this possibility, we run simulations under the null hypothesis of a constant standard deviation of true betas and tests whether the measurement error in betas can account for the compression observed in the data.

Figure B3 shows that the compression observed in the data is much larger than what could be generated by estimation error variance alone. Naturally, while this bootstrap analysis does not indicate that the beta compression observed in Table X is likely due to measurement error, we cannot rule out all types of measurement error.

Panels D, E, and F report conditional market betas of the BAB portfolio returns based on the volatility of the credit environment for U. The dependent variable is the monthly return of the BAB portfolio. The explanatory variables are the monthly returns of the market portfolio, Fama and French mimicking portfolios, and Carhart momentum factor. Market betas are allowed to vary across TED volatility regimes low, neutral and high using the full set of TED dummies.

To understand this test, recall first that the BAB factor is market neutral conditional on the information set used in the estimation of ex ante betas which determine the ex ante relative position sizes of the long and short sides of the portfolio. Hence, if the TED spread volatility was used in the ex-ante beta estimation, then the BAB factor would be market neutral conditional on this information. However, the BAB factor was constructed using historical betas that do not take into account the effect of the TED spread and, therefore, a high TED spread volatility means that the realized betas will be compressed relative to the ex-ante estimated betas used in portfolio construction.

Therefore, a high TED spread volatility should increase the conditional market sensitivity of the BAB factor because the long-side of the portfolio is leveraged too much and the short side is deleveraged too much. Indeed, Table X shows that when credit constraints are more volatile, the market beta of the BAB factor rises.

The rightmost column shows that the difference between low- and high- credit-volatility environments is statistically significant t-statistics 3. Controlling for three or four factors yields similar results. The results for our sample of international equities Panel E and for the average BAB across all assets Panel F are similar, but are weaker both in terms of magnitude and statistical significance.

Importantly, the alpha of the BAB factor remains large and statistically significant even when we control for the time-varying market exposure. This means that, if we hedge the BAB factor to be market neutral conditional on the TED spread volatility environment, then this conditionally market-neutral BAB factor continues to earn positive excess returns.

Consistent with this prediction, Table XI presents evidence that mutual funds and individual investors hold high-beta stocks while LBO firms and Berkshire Hathaway buy low-beta stocks. Pedersen — Page 30 Before we delve into the details, let us highlight a challenge in testing Proposition 5. For example, while a hedge fund may be able to apply some leverage, its leverage constraint could nevertheless be binding if its desired volatility is high especially if its portfolio is very diversified and hedged.

Given that binding constraints are difficult to observe directly, we seek to identify groups of investors that are plausibly constrained and unconstrained, respectively. One example of an investor who may be constrained is a mutual fund. A second class of investors that may face borrowing constraints is individual retail investors.

Although we do not have direct evidence of their inability to employ leverage and some individuals certainly do , we think that at least in aggregate it is plausible that they are likely to face borrowing restrictions. The flipside of this portfolio test is identifying relatively unconstrained investors. Thus, one needs investors that may be allowed to use leverage and are operating below their leverage cap so that their leverage constraints are not binding.

Admittedly, we do not have direct evidence of the maximum leverage available to these LBO firms relative to the leverage they apply, but anecdotal evidence suggests that they achieve a substantial Betting Against Beta - Andrea Frazzini and Lasse H. Pedersen — Page 31 amount of leverage. Second, we examine the holdings of Berkshire Hathaway, a publicly traded corporation run by Warren Buffett that holds a diversified portfolio of equities and employs leverage by issuing debt, via insurance float, and other means.

The advantage of using the holdings of a public corporation that holds equities like Berkshire is that we can directly observe its leverage. It is therefore plausible to assume that Berkshire at the margin could issue more debt but choose not to, making it a likely candidate for an investor whose combination of risk aversion and borrowing constraints made it relatively unconstrained during our sample period.

Table XI reports the results of our portfolio test. We first aggregate all holdings for each investor group, compute their ex-ante betas equal and value-weighted, respectively , and take the time series average. To compute the realized betas, we compute monthly returns of an aggregate portfolio mimicking the holdings, under the assumption of constant weight between reporting dates. The realized betas are the regression coefficients in a time series regression of these excess returns on the excess returns of the CRSP value- weighted index.

Panel A shows evidence consistent with the hypothesis that constrained investors stretch for return by increasing their betas. Panel A. These findings are consistent with those of Karceski , but our sample is much larger, including all funds over year period. Pedersen — Page 32 Panel B. For each target stock in our database, we focus on its ex-ante beta as of the month end prior to the initial announcements date.

This focus is to avoid confounding effects that result from changes in betas related to the actual delisting event. The first two lines report results of all delisting events. The last two lines in Panel B. The results are consistent with Proposition 5 in that investors executing leverage buyouts tend to acquire or attempt to acquire in case of a non-successful bid firms with low betas, and we are able to reject the null hypothesis of a unit beta.

The results for Berkshire Hathaway Panel B. We find empirically that portfolios of high-beta assets have lower alphas and Sharpe ratios than portfolios of low-beta assets. We show how this deviation from the standard CAPM can be captured using betting- against-beta factors, which may also be useful as control variables in future research Proposition 2. The return of the BAB factor rivals those of all the standard asset pricing factors e.

Pedersen — Page 33 statistical significance, and robustness across time periods, sub-samples of stocks, and global asset classes. Extending the Black model, we consider the implications of funding constraints for cross-sectional and time-series asset returns. We show that worsening funding liquidity should lead to losses for the BAB factor in the time series Proposition 3 and that increased funding liquidity risk compresses betas in the cross section of securities toward 1 Proposition 4 , and we find consistent evidence empirically.

To test this, we identify investors that are likely to be relatively constrained and unconstrained. Conversely, we show that leveraged buyout funds and Berkshire Hathaway, all of which have access to leverage, buy stocks with betas below 1 on average, another prediction of the model. Hence, these investors may be taking advantage of the BAB effect by applying leverage to safe assets and being compensated by investors facing borrowing constraints who take the other side.

Buffett bets against beta as Fisher Black believed one should. Pedersen — Page 34 References Acharya, V. Ang, A. Hodrick, Y. Xing, X. Ashcraft, A. Garleanu, and L. Asness, C. Frazzini and L. Baker, M. Bradley, and J. Barber, B, and T. Black, F. Jensen, and M. Jensen ed. Brennan, M. Brunnermeier, M. Calvet, L. Campbell, and P. Carhart, M. Cohen, R. Polk, and T. Cuoco, D. De Santis , G.

Dimson, E. Duffee, G. Elton, E. Gruber, S. Brown and W. Falkenstein, E. Fama, E. Frazzini, A. Kabiller, and L. Fu, F. Pedersen — Page 36 Journal of Financial Economics, vol. Garleanu, N. Gibbons, M. Grinblatt, M. Keloharju, and J. Gromb, D. Hindy, A. Kacperczyk, M. Sialm and L.

Kandel, S. Karceski, J. Lewellen, J. Markowitz, H. Mehrling, P. Merton R. Moskowitz, T. Ooi, and L. Pastor, L , and R. Polk, C. Thompson, and T. Pedersen — Page 37 Scholes, M. Shanken, J. Tobin, J. Vasicek, O. The sample includes all commons stocks on the CRSP daily stock files "shrcd" equal to 10 or 11 and Xpressfeed Global security files "tcpi" equal to 0. Means are pooled averages as of June of each year.

Returns, - This table shows calendar-time portfolio returns. Column 1 to 10 report returns of beta-sorted portfolios: at the beginning of each calendar month stocks are ranked in ascending order on the basis of their estimated beta at the end of the previous month. The ranked stocks are assigned to one of ten deciles portfolios based on NYSE breakpoints.

All stocks are equally weighted within a given portfolio, and the portfolios are rebalanced every month to maintain equal weights. The rightmost column reports returns of the zero-beta BAB factor. To construct BAB factor, all stocks are assigned to one of two portfolios: low beta and high beta. In this instead of adding the market itself as Black, Jensen, and Scholes case, higher funding liquidity risk means that betas are compressed do. This table shows summary statistics as of June of each year.

The sample includes all commons stocks on the Center for Research in Security Prices daily stock files shrcd equal to 10 or 11 and Xpressfeed Global security files tcpi equal to zero. Mean ME is the average market value of equity, in billions of US dollars. Means are pooled averages as of June of each year. Excess returns are above the US Treasury bill rate.

We have the common stocks on the Xpressfeed Global daily security following natural result for the agents' positions. Uncon- universe between January and March We strained agents hold a portfolio of risky securities that has a assign each stock to its corresponding market based on beta less than one; constrained agents hold portfolios of risky the location of the primary exchange. Betas are computed securities with higher betas.

If securities s and k are identical with respect to the corresponding MSCI local market except that s has a larger market exposure than k, bs 4 bk, index. We compute alphas with The reverse is true for any agent with jt o t. Beyond matching the data qualitatively, Appendix C available liquidity risk. Table 2 match the magnitude of the estimated BAB returns. We obtain US Treasury bond data 3. Data and methodology from the CRSP US Treasury Database, using monthly returns in excess of the one-month Treasury bill on the The data in this study are collected from several sources.

We report these tests in Appendix B. S counterparts and follow Fama and database. Our US equity data include all available common French , , This table reports the securities included in our data sets and the corresponding date range.

Each weight across commodities. Only non-callable, non-flower notes and bonds binding [as in Garleanu and Pedersen and others]. Treasuries rate. Hub database. Estimating ex ante betas month Treasury bill of four aggregate US credit indices with maturity ranging from one to ten years and nine We estimate pre-ranking betas from rolling regressions investment-grade and high-yield corporate bond portfo- of excess returns on market excess returns.

Whenever lios with credit risk ranging from AAA to Ca-D and possible, we use daily data, rather than monthly data, as Distressed. The data are collected from a variety of sources and First, we use a one-year rolling standard deviation for contain daily return on futures, forwards, or swap con- volatilities and a five-year horizon for the correlation to tracts in excess of the relevant financing rate.

The type of account for the fact that correlations appear to move more contract for each asset depends on availability or the slowly than volatilities. Prior to expira- returns to estimate volatilities and overlapping three-day i;t k 0 ln1 r t k , for correlation to con- log returns, r 3d 2 i tion, positions are rolled over into the next most-liquid contract. The rolling date's convention differs across con- trol for nonsynchronous trading which affects only corre- tracts and depends on the relative liquidity of different lations.

We require at least six months trading days maturities. The data cover the period between January of non-missing data to estimate volatilities and at least and March , with varying data availability three years trading days of non-missing return data depending on the asset class. For more details on the for correlations. If we have access only to monthly data, we computation of returns and data sources, see Moskowitz, use rolling one and five-year windows and require at least Ooi, and Pedersen , Appendix A.

For equity indexes, 12 and 36 observations. See, for example, De Santis and Gerard In each portfolio, securities are weighted by the ranked betas i. The portfolios are reba- w0. However, our results are very similar either way. The portfolio weights of the low- However, the amount of shrinkage affects the construction of beta and high-beta portfolios are given by the BAB portfolios because the estimated betas are used to scale the long and short sides of portfolio as seen in Eq.

By construction, we have 1n wH 1 and 1n wL 1. When we regress our portfolios on standard risk factors, To construct the BAB factor, both portfolios are rescaled to the realized factor loadings are not shrunk as above have a beta of one at portfolio formation. The BAB is the because only the ex ante betas are subject to selection self-financing zero-beta portfolio 8 that is long the low- bias.

Our results are robust to alternative beta estimation beta portfolio and that shortsells the high-beta portfolio. Data used to test the theory's portfolio predictions over a limited time period. Second, this approach is applicable even if markets are segmented. We domestic mutual funds filing with the Securities and Exchange use the world market portfolio from Asness, Frazzini, and Commission.

The holdings data run from March to Pedersen We focus our analysis on open-end, actively tests as the BAB factors earn large and significant abnormal managed, domestic equity mutual funds. Our sample selection returns in each of the asset classes in our sample. Constructing betting against beta factors Our individual investors' holdings data are collected from a nationwide discount brokerage house and contain We construct simple portfolios that are long low-beta trades made by about 78 thousand households in the securities and that shortsell high-beta securities BAB factors.

This data To construct each BAB factor, all securities in an asset class are set has been used extensively in the existing literature on ranked in ascending order on the basis of their estimated individual investors. For a detailed description of the beta. The ranked securities are assigned to one of two brokerage data set, see Barber and Odean This and if applicable completion or termination date for all estimator places more weight on the historical times series estimate when the estimate has a lower variance or when there is large dispersion takeover deals in which the target is a US publicly traded of betas in the cross section.

Pooling across all stocks in our US equity data, the shrinkage factor w has a mean of 0. The market series is monthly and ranges from We would like to thank Mark Mitchell for providing us with to For choice of risk adjustment influences the relative alpha some but not all deals, the acquirer descriptor also contribution of the long and short sides of the portfolio.

The data run market line has continued to be too flat for another four from January to March Further, Finally, we download holdings data for Berkshire Hath- our results extend internationally. We consider beta- away from Thomson-Reuters Financial Institutional 13f sorted portfolios for international equities and later turn Holding Database. The data run from March to to altogether different asset classes.

Betting against beta in each asset class ways: We consider international portfolios in which all international stocks are pooled together Table 4 , and we We now test how the required return varies in the consider results separately for each country Table 5. The cross-section of beta-sorted securities Proposition 1 and international portfolio is country-neutral, i.

As an overview of these results, below above its country median. We see that declining equities in Table 4 mimic the US results. The alpha and alphas across beta-sorted portfolios are general phenom- Sharpe ratios of the beta-sorted portfolios decline ena across asset classes. B1 in Appendix B plots the although not perfectly monotonically with the betas, Sharpe ratios of beta-sorted portfolios and also shows a and the BAB factor earns risk-adjusted returns between consistently declining pattern.

All the BAB portfo- Table 5 shows the performance of the BAB factor within lios deliver positive returns, except for a small insignif- each individual country. The BAB ratios in 18 of the 19 MSCI developed countries and portfolios based on large numbers of securities US stocks, positive four-factor alphas in 13 out of 19, displaying a international stocks, Treasuries, credits deliver high risk- strikingly consistent pattern across equity markets.

The adjusted returns relative to the standard risk factors BAB returns are statistically significantly positive in six considered in the literature. Of course, the small number of stocks in our sample in 4. Stocks many of the countries makes it difficult to reject the null hypothesis of zero return in each individual country.

Table 3 reports our tests for US stocks. We consider ten Table B1 in Appendix B reports factor loadings. The larger long investment is meant to make which is the well-known relatively flat security market the BAB factor market-neutral because the stocks that are line.

Hence, consistent with Proposition 1 and with Black held long have lower betas. The BAB factor's realized , the alphas decline almost monotonically from the market loading is not exactly zero, reflecting the fact that low-beta to high-beta portfolios. The alphas decline when our ex ante betas are measured with noise. The other estimated relative to a one-, three-, four-, and five-factor factor loadings indicate that, relative to high-beta stocks, model.

Moreover, Sharpe ratios decline monotonically low-beta stocks are likely to be larger, have higher book- from low-beta to high-beta portfolios. The BAB portfolio's leveraged low-beta stocks and that shortsells de-leveraged positive HML loading is natural since our theory predicts high-beta stocks, thus maintaining a beta-neutral portfo- that low-beta stocks are cheap and high-beta stocks are lio.

Consistent with Proposition 2, the BAB factor delivers a expensive. Specifically, the BAB Appendix B reports further tests and additional robust- factor has Fama and French abnormal returns of ness checks. In Table B2, we report results using different 0. Further adjusting window lengths to estimate betas and different bench- returns for the Carhart momentum factor, the BAB marks local, global.

We split the sample by size Table B3 portfolio earns abnormal returns of 0. Last, we adjust returns using a five- volatility Table B5 , and we report results for alternative factor model by adding the traded liquidity factor by Pastor and Stambaugh , yielding an abnormal BAB return of 0.

While the alpha of the report the result of betting against beta across equity indices BAB long-short portfolio is consistent across regressions, the separately in Table 8. Alphas of beta-sorted portfolios. This figure shows monthly alphas.

The test assets are beta-sorted portfolios. At the beginning of each calendar month, securities are ranked in ascending order on the basis of their estimated beta at the end of the previous month. The ranked securities are assigned to beta-sorted portfolios. This figure plots alphas from low beta left to high beta right. Alpha is the intercept in a regression of monthly excess return.

For equity portfolios, the explanatory variables are the monthly returns from Fama and French , Asness and Frazzini , and Carhart portfolios. For all other portfolios, the explanatory variables are the monthly returns of the market factor. Alphas are in monthly percent. Finally, in Table respect to an aggregate Treasury bond index is empirically B7 and Fig. B2 we report an out-of-sample test. We collect equivalent to ranking on duration or maturity. Therefore, pricing data from DataStream and for each country in in Table 6, one can think of the term beta, duration, or Table 1 we compute a BAB portfolio over sample period maturity in an interchangeable fashion.

The right-most not covered by the Xpressfeed Global database. Abnormal results are consistent: Equity portfolios that bet against returns are computed with respect to a one-factor model betas earn significant risk-adjusted returns. Treasury bonds The results show that the phenomenon of a flatter security market line than predicted by the standard CAPM is not Table 6 reports results for US Treasury bonds.

As before, limited to the cross section of stock returns. Consistent with we report average excess returns of bond portfolios Proposition 1, the alphas decline monotonically with beta. In the Likewise, Sharpe ratios decline monotonically from 0. Furthermore, the bond BAB portfolio deli- 18 DataStream international pricing data start in , and Xpress- vers abnormal returns of 0. Betting against beta BAB Sharpe ratios by asset class. This figures shows annualized Sharpe ratios of BAB factors across asset classes.

To construct the BAB factor, all securities are assigned to one of two portfolios: low beta and high beta. Securities are weighted by the ranked betas and the portfolios are rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio formation. The BAB factor is a self-financing portfolio that is long the low-beta portfolio and shorts the high-beta portfolio. Sharpe ratios are annualized. Because the idea that funding constraints have a sig- While the severity of leverage constraints varies across nificant effect on the term structure of interest could be market participants, it appears plausible that a five-to-one surprising, let us illustrate the economic mechanism that leverage on this part of the portfolio makes a difference could be at work.

Suppose an agent, e. One way to achieve this return target is to 4. If the agent invests in one- We next test our model using several credit portfolios and year Treasuries P1 instead, then he would need to invest report results in Table 7.

This higher assets are monthly excess returns of corporate bond indexes leverage is needed because the long-term Treasures are 11 by maturity. We see that the credit BAB portfolio delivers times more volatile than the short-term Treasuries. Hence, abnormal returns of 0.

Furthermore, alphas and his investment in one-year bonds. If the agent has leverage Sharpe ratios decline monotonically. Given According to our theory, the one-year Treasuries there- the results on Treasuries in Table 6, we are interested in fore must offer higher returns and higher Sharpe ratios, testing a pure credit version of the BAB portfolio. Each flattening the security market line for bonds. We construct test assets by going in one-year bonds. This table shows beta-sorted calendar-time portfolio returns.

At the beginning of each calendar month, stocks are ranked in ascending order on the basis of their estimated beta at the end of the previous month. The ranked stocks are assigned to one of ten deciles portfolios based on NYSE breakpoints. All stocks are equally weighted within a given portfolio, and the portfolios are rebalanced every month to maintain equal weights.

The right-most column reports returns of the zero-beta betting against beta BAB factor. To construct the BAB factor, all stocks are assigned to one of two portfolios: low beta and high beta. Stocks are weighted by the ranked betas lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio , and the portfolios are rebalanced every calendar month.

The betting against beta factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. This table includes all available common stocks on the Center for Research in Security Prices database between January and March The explanatory variables are the monthly returns from Fama and French mimicking portfolios, Carhart momentum factor and Pastor and Stambaugh liquidity factor.

Regarding the five-factor alphas the Pastor and Stambaugh liquidity factor is available only between and Beta ex ante is the average estimated beta at portfolio formation. Beta realized is the realized loading on the market portfolio. Volatilities and Sharpe ratios are annualized. The ranked stocks are assigned to one of ten deciles portfolios.

The rightmost column reports returns of the zero-beta betting against beta BAB factor. The low- high- beta portfolio is composed of all stocks with a beta below above its country median. This table includes all available common stocks on the Xpressfeed Global database for the 19 markets listed in Table 1.

The sample period runs from January to March The explanatory variables are the monthly returns of Asness and Frazzini mimicking portfolios and Pastor and Stambaugh liquidity factor. Returns are in US dollars and do not include any currency hedging. Beta ex-ante is the average estimated beta at portfolio formation.

We compute market returns by taking and up to month t 1. One interpretation of this returns the equally weighted average of these hedged returns, and series is that it approximates the returns on a credit we compute betas and BAB portfolios as before. Abnormal 14 14 A. This table shows calendar-time portfolio returns. At the beginning of each calendar month, all stocks are assigned to one of two portfolios: low beta and high beta.

Stocks are weighted by the ranked betas, and the portfolios are rebalanced every calendar month. The zero-beta betting against beta BAB factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. The explanatory variables are the monthly returns of Asness and Frazzini mimicking portfolios.

Only non callable, non flower notes and bonds are included in the portfolios. The portfolio returns are an equal-weighted average of the unadjusted holding period return for each bond in the portfolios in excess of the risk-free rate. To construct the zero-beta betting against beta BAB factor, all bonds are assigned to one of two portfolios: low beta and high beta. Bonds are weighted by the ranked betas lower beta bonds have larger weight in the low-beta portfolio and higher beta bonds have larger weights in the high-beta portfolio and the portfolios are rebalanced every calendar month.

The explanatory variable is the monthly return of an equally weighted bond market portfolio. For P7, returns are missing from August to December The addition of the Treasury BAB factor on the test assets are credit indexes sorted by rating, ranging from right-hand side is an extra check to test a pure credit AAA to Ca-D and Distressed.

Consistent with all our previous version of the BAB portfolio. Panel A shows results for US credit indices by maturity. The test assets are monthly returns on corporate bond indices with maturity ranging from one to ten years, in excess of the risk-free rate. The sample period runs from January March Unhedged indicates excess returns and Hedged indicates excess returns after hedging the index's interest rate exposure. To construct hedged excess returns, each calendar month we run one-year rolling regressions of excess bond returns on the excess return on Barclay's US government bond index.

We construct test assets by going long the corporate bond index and hedging this position by shorting the appropriate amount of the government bond index. We compute market excess returns by taking an equal weighted average of the hedged excess returns. Panel B shows results for US corporate bond index returns by rating. Bonds are weighted by the ranked betas lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio and the portfolios are rebalanced every calendar month.

Both portfolios are rescaled to have a beta of 1 at portfolio formation. The zero-beta BAB factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. The explanatory variable is the monthly excess return of the corresponding market portfolio and, for the hedged portfolios in Panel A, the Treasury BAB factor. Equity indexes, country bond indexes, currencies, futures and country selection deliver abnormal return of and commodities 0. Table 8 reports results for equity indexes, country bond 4.

Betting against all of the betas indexes, foreign exchange, and commodities. The BAB port- folio delivers positive returns in each of the four asset classes, To summarize, the results in Tables 38 strongly sup- with an annualized Sharpe ratio ranging from 0.

We port the predictions that alphas decline with beta and BAB are able to reject the null hypothesis of zero average return factors earn positive excess returns in each asset class. Clearly, the relatively of diversification. We construct a simple equally weighted flat security market line, shown by Black, Jensen, and BAB portfolio. To account for different volatility across the Scholes for US stocks, is a pervasive phenomenon four asset classes, in month t we rescale each return series to that we find across markets and asset classes.

This portfolio t-statistics of 6. Time series tests report results for an all futures combo including all four asset classes and a country selection combo including only equity In this section, we test Proposition 3's predictions for indices, country bonds and foreign exchange.

The test assets are futures, forwards or swap returns in excess of the relevant financing rate. To construct the betting against beta BAB factor, all securities are assigned to one of two portfolios: low beta and high beta. Securities are weighted by the ranked betas lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio , and the portfolios are rebalanced every calendar month. The BAB factor is a self- financing portfolio that is long the low-beta portfolio and short the high-beta portfolio.

The explanatory variable is the monthly return of the relevant market portfolio. Panel A reports results for equity indices, country bonds, foreign exchange and commodities. Panel B reports results for all the assets listed in Tables 1 and 2. All assets includes all the assets listed in Tables 1 and 2.

Hence, our test relies on an assumption spread as a proxy of funding conditions. The sample runs that such variation of other variables does not lead to an from December the first available date for the TED omitted variables bias. To partially address this issue, spread to March The control variables are the eses for the BAB factors across asset classes. The first market return to account for possible noise in the ex ante column simply regresses the US BAB factor on the lagged betas used for making the BAB portfolio market neutral , level of the TED spread and the contemporaneous change the one-month lagged BAB return to account for possible in the TED spread.

The beta spread negatively related to the BAB returns. Consistent with the model, positive coefficient for the lagged level [Eq. Hence, the Table 9 shows that the estimated coefficient for the beta coefficient for change is consistent with the model, but the spread is positive in all specifications, but not statistically coefficient for the lagged level is not, under this interpreta- significant.

The short volatility returns is the return on a tion of the TED spread. Under this sensitivity to volatility risk. Lagged inflation is equal to the interpretation, a high TED spread could indicate that banks one-year US CPI inflation rate, lagged one month, which is are credit-constrained and that banks tighten other inves- included to account for potential effects of money illusion tors' credit constraints over time, leading to a deterioration as studied by Cohen, Polk, and Vuolteenaho , of BAB returns over time if investors do not foresee this.

These regressions include fixed effects and conditions, not as a return. Hence, the TED spread at the end of the return period is a measure of the credit conditions at that time even if the TED standard errors are clustered by date. We consistently find spread is a difference in interest rates that would be earned over the a negative relation between BAB returns and the TED following time period.

This table shows results from pooled time series regressions. The left-hand side is the month t return of the betting against beta BAB factors. To construct the BAB portfolios, all securities are assigned to one of two portfolios: low beta and high beta.

The BAB factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. The explanatory variables include the TED spread and a series of controls. Market return is the monthly return of the relevant market portfolio. Columns 1 and 2 report results for US equities.

Columns 3 and 4 report results for international equities. Columns 5 and 6 report results for all assets in our data. Asset fixed effects are included where indicated, t-statistics are shown below the coefficient estimates and all standard errors are adjusted for heteroskedasticity White, Beta compression set of dummies without intercept. In Panel C, we compute the monthly dispersion measure in each asset class and We next test Proposition 4 that betas are compressed average across assets.

All standard errors are adjusted for toward one when funding liquidity risk is high. Table 10 heteroskedasticity and autocorrelation up to 60 months. We use the volatility of Table 10 shows that, consistent with Proposition 4, the the TED spread to proxy for the volatility of margin cross-sectional dispersion in betas is lower when credit requirements.

Volatility in month t is defined as the constraints are more volatile. The average cross-sectional standard deviation of daily TED spread innovations, standard deviation of US equity betas in periods of low q spread volatility is 0. Because we are com- in volatile credit environment. The difference is statistically puting conditional moments, we use the monthly volatility significant t-statistics 2. The tests based on the other as of the prior calendar month, which ensures that the dispersion measures, the international equities, and the other conditioning variable is known at the beginning of the assets all confirm that the cross-sectional dispersion in beta measurement period.

The sample runs from December shrinks at times when credit constraints are more volatile. Appendix B contains an additional robustness check. Panel A of Table 10 shows the cross-sectional dispersion Because we are looking at the cross-sectional dispersion of in betas in different time periods sorted by the TED estimated betas, one could worry that our results was volatility for US stocks, Panel B shows the same for inter- driven by higher beta estimation errors, instead of a higher national stocks, and Panel C shows this for all asset classes variance of the true betas.

To investigate this possibility, in our sample. Each calendar month, we compute cross- we run simulations under the null hypothesis of a constant sectional standard deviation, mean absolute deviation, and standard deviation of true betas and test whether the inter-quintile range of the betas for all assets in the measurement error in betas can account for the compres- universe. We assign the TED spread volatility into three sion observed in the data.

B3 shows that the compres- groups low, medium, and high based on full sample sion observed in the data is much larger than what could breakpoints top and bottom third and regress the times be generated by estimation error variance alone.

Naturally, series of the cross-sectional dispersion measure on the full while this bootstrap analysis does not indicate that the 18 18 A. This table reports results of cross-sectional and time-series tests of beta compression. Panels A, B and C report cross-sectional dispersion of betas in US equities, international equities, and all asset classes in our sample. Each calendar month we compute cross sectional standard deviation, mean absolute deviation, and inter quintile range of betas.

In Panel C we compute each dispersions measure for each asset class and average across asset classes. The row denoted all reports times series means of the dispersion measures. P1 to P3 report coefficients on a regression of the dispersion measure on a series of TED spread volatility dummies. TED spread volatility is defined as the standard deviation of daily changes in the TED spread in the prior calendar month.

We assign the TED spread volatility into three groups low, neutral, and high based on full sample breakpoints top and bottom one third and regress the times series of the cross-sectional dispersion measure on the full set of dummies without intercept. The dependent variable is the monthly return of the BAB portfolios. The explanatory variables are the monthly returns of the market portfolio, Fama and French , Asness and Frazzini , and Carhart mimicking portfolios, but only the alpha and the market betas are reported.

Market betas are allowed to vary across TED spread volatility regimes low, neutral, and high using the full set of dummies. All standard errors are adjusted for heteroskedasticity and autocorrelation using a Bartlett kernel Newey and West, with a lag length of sixty months. The right-most column shows that the measurement error, we cannot rule out all types of difference between low- and high-credit volatility environ- measurement error.

Controlling Panels D, E, and F report conditional market betas of for three or four factors yields similar results. The results for the BAB portfolio returns based on the volatility of the our sample of international equities Panel E and for the credit environment for US equities, international equities, average BAB across all assets Panel F are similar, but they are and the average BAB factor across all assets, respectively.

The explanatory variables are the monthly and statistically significant even when we control for the returns of the market portfolio, Fama and French time-varying market exposure. This means that, if we mimicking portfolios, and Carhart momentum hedge the BAB factor to be market-neutral conditional on factor. Market betas are allowed to vary across TED the TED spread volatility environment, then this condi- volatility regimes low, neutral, and high using the full tionally market-neutral BAB factor continues to earn set of TED dummies.

Testing the model's portfolio predictions volatility environment. To understand this test, recall first that the BAB factor is market neutral conditional on the The theory's last prediction Proposition 5 is that information set used in the estimation of ex ante betas more-constrained investors hold higher-beta securities which determine the ex ante relative position sizes of the than less-constrained investors.

Consistent with this pre- long and short sides of the portfolio. Hence, if the TED diction, Table 11 presents evidence that mutual funds and spread volatility was used in the ex ante beta estimation, individual investors hold high-beta stocks while LBO firms then the BAB factor would be market-neutral conditional on and Berkshire Hathaway buy low-beta stocks.

However, the BAB factor was constructed Before we delve into the details, let us highlight a using historical betas that do not take into account the effect challenge in testing Proposition 5. Whether an investor's of the TED spread and, therefore, a high TED spread volatility constraint is binding depends both on the investor's ability means that the realized betas will be compressed relative to to apply leverage mi in the model and on its unobser- the ex ante estimated betas used in portfolio construction.

For example, while a hedge fund could Therefore, a high TED spread volatility should increase the apply some leverage, its leverage constraint could never- conditional market sensitivity of the BAB factor because the theless be binding if its desired volatility is high especially long side of the portfolio is leveraged too much and the short if its portfolio is very diversified and hedged.

Indeed, Table 10 shows that Given that binding constraints are difficult to observe when credit constraints are more volatile, the market beta of directly, we seek to identify groups of investors that are Table 11 Testing the model's portfolio predictions, This table shows average ex ante and realized portfolio betas for different groups of investors. Panel A reports results for our sample of open-end actively- managed domestic equity mutual funds as well as results a sample of individual retail investors.

Panel B reports results for a sample of leveraged buyouts private equity and for Berkshire Hathaway.

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