Supplementary Appendix to Financial Intermediaries and the Cross Section of Asset Returns Tobias Adrian tobias.adrian@ny.frb.org Erkko Etula etula@post.harvard.edu Tyler Muir t-muir@kellogg.northwestern.edu December 2012 Capital Markets Function, Research and Statistics Group, Federal Reserve Bank of New York, 33 Liberty Street, New York, NY 10045 and Kellogg School of Management, Department of Finance. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System.
1 Multi-Factor Regressions Table 1 gives the results when we estimate all factors simultaneously in a joint regression. Essentially, we see the single leverage factor does about as well as a 6 factor model that also adds the Fama-French 3 factors, momentum factor, and level factor. Moreover, when we run the 6-factor model, only the leverage factor and momenutm factor are statistically significant. This is not surprising since the only portfolios the leverage factor has trouble with is the past winners portfolio. Therefore, aside from the momentum factors, the other factors do not seem to be adding additional information. The second column in Table 1 shows the results for a two factor model which includes the market portfolio. As we see, the market adds essentially no explanatory power to the leverage factor. Since the main result of our paper is that the leverage factor can, alone, explain the cross-section of asset returns, we use a one-factor model in the main text. However, there are advantages to using a two-factor model in practice that includes the market to explain more of the time-series variation in these portfolios. As we will see, one advantage of this approach is that betas with respect to our leverage factor are more stable in small samples. 2 Subsamples This section analyzes the main results of the paper across various sumbsamples as well as the behavior of leverage itself across subsamples. 2.1 Changes in Leverage Over Time One concern is whether the management of leverage by financial institutions has changed over time as regulation and practices in the financial sector has changed. For example, the broker-dealer sector typical carries much higher leverage than it used to. We document the relationship between our leverage factor and several key variables over different time periods in Table 2. 1
Most importantly, the relationship between the leverage factor and broker-dealer asset growth remains stable over the subsamples. During both periods, there is a strong positive correlation between leverage growth and asset growth which means leverage is pro-cyclical and is used to aggressively and actively adjust balance sheet size. Further, relationships with between leverage and volatility remain stable as well. This again supports the idea that leverage co-moves with funding conditions because high volatility is typically associated with tighter funding constraints. The LMP shows generally similar stability over these subsamples. Thus, we feel that the economic content of our leverage factor has not demonstrably changed over time. 2.2 Main Results Over Subsamples We show the results for our asset pricing tests over subsamples from 1968-1988 and 1989-2009. See Table 3. We find that our single factor model does not work as well across smaller subsamples and discuss the potential reasons for this below. Most notably, we find that a two-factor model that controls for the market return is robust across subsamples. The asset pricing model we estimate is: E[R e i ] = β i,lev λ lev We approximate this empirically with: 1 T T Ri,t e = α + β i,lev λlev t=1 We need to ask when, and whether, 1 T Ri,t e E[Ri e ]. Of course, this will be true T t=1 asymptotically, but this approximation can be poor in small samples and in samples with extreme realizations of returns. Similarly, in small samples leverage betas may be estimated poorly, so β i,lev β i,lev may be a poor approximation as well. Each of these issues can contribute to an apparent rejection of the model. 2
We split our sample in half and analyze the pricing results across each subsample. The results are given in Table 3. As we see, in each of the subsamples the leverage factor does not perform as well. In the earlier subsamples, 1968-1988, the cross-sectional R-squared is 38%. This drops to 11% in the later sample (1989-2009). Moreover the prices of risk are not significant and the intercept in the later sample is high at 7%. However, we should also note that the confidence interval for the R-squared is large and includes 100%, meaning we cannot rule out the possibility that our leverage factor explains the cross-section of returns but sampling error causes a low point estimate for the R-squared. It is well known that the R-squared statistic is unstable, and this is precisely the point of computing the confidence interval. First, some of this performance can potentially be attributed to average returns being a poor proxy for expected returns. If risk premia are counter-cyclical, expected returns will be high in bad times exactly when realized returns are poor, making average returns especially poor proxies for expected returns over these periods. As an example, the average excess market return in the 10 year period from 2000 to 2010 was -1.3%, suggesting a negative equity premium over this sample. Therefore, for each subsample, we also report the results from using the full sample average returns as expected return proxies. We do find that using the full sample average returns improves the model performance slightly, suggesting that poor approximation of expected returns is at least partially to blame for poor performace. Next, we assess the possibility and potential reasons for sampling error in measuring betas. Most importantly, we find that leverage does not change drastically during the dot com / tech boom and bust (Figure 1 in the main text), while stock returns exhibit a huge boom and subsequent crash. By trying to fit this large outlier in the time-series regression, we may get poor estimates of leverage betas. We take two approaches to address this. The first drops the dot com episode (1998-2002) and estimates leverage betas over the remaining sample. This substantially increases the performance of the leverage factor, increasing the 3
R-squared statistic to 30%. When using the full sample average returns, the R-squared statistic moves back to 47%, with a confidence interval of [30,100]%. This suggests that poor estimation of betas due to the dot com episode combined with poor estimation of expected returns over this subsample are the likely reasons for the models performance in the second half of the sample. Finally, a way to mitigate this estimation error is to include the market return as a second factor. This allows the market to absorb the large spike and decline in returns over the dot com episode. The two-factor model with the market return does substantially better, with an R-squared of 58%. When using the full sample average returns, the two-factor model has an R-squared of 67%. These results suggest that, in practice, it is sensible to include the market return in addition to our leverage factor, as this makes the time-series betas more reliable and alleviates a potential omitted variables problem in the first stage regression. Our point in the main text is to show that the leverage factor alone does all the work in cross-sectional pricing for the portfolios we consider, and thus we focus on the single-factor model to make this point. However, the results across subsamples emphasize that in practice including the market return will make the results more stable across subsamples. It is worth noting that the market return is not statistically significant in the cross-sectional test, even across subsamples. Therefore, including the market return in the time-series helps give more precise estimates of leverage betas, but the leverage factor is still mainly responsible for pricing the cross-section. 3 Wealth vs Leverage Table 3 shows no substantial improvement in the model when including other factors related to the return on the financial sector. In particular, many theories would imply that the return on the financial sector should also enter the intermediaries SDF. However, including the return on the broker-dealer sector, the return on the total financial sector, or the return 4
on bank stocks does not greatly affect the models performance. We take the return on bank stocks and return on the financial sector from Ken French s industry portfolios. We do note, however, that all measures are priced with positive coeffi cients which is consistent with theory. 4 Industry Portfolios Table 5 shows the main results using the 30 industry portfolios from Kenneth French s website. All factor models essential fail on these portfolios (the leverage factor, the Fama- French factors, and momentum factors all have essentially zero explanator power). One challenge with the industry portfolios is that there is a fairly small spread in average returns to explain. Each model has a large intercept (the leverage factor is actually the lowest at 6% per annum). Each model also has a negative adjusted R-squared, meaning there is very little explanatory power for any of these models. 5
6
Table 1: Main Table: Pricing the Size, Book to Market, Momentum, and Bond Portfolios Pricing results for the 25 size and book-to-market and 10 momentum portfolios, and 6 Treasury bond portfolios sorted by maturity. Each model is estimated as E[R e ] = λ 0 + β fac λ fac. FF denotes the Fama-French 3 factors, Mom the momentum factor, PC1 the first principal component of the yield curve, LevFac our leverage factor. Panel A reports the prices of risk with Fama-MacBeth and Shanken t-statistics, respectively. Panel B reports test diagnostics, including mean absolute pricing errors (MAPE) by portfolio group, adjusted R-Squares with corresponding confidence intervals (C.I.), and a χ 2 statistic that tests whether the pricing errors are jointly zero. E[R e ] gives the average excess return to be explained. Data are quarterly, 1968Q1-2009Q4. Returns and risk premia are reported in percent per year (quarterly percentages multiplied by 4). Panel A: Prices of Risk LevFac LevFac, Mkt All Factors Intercept 0.12-0.04 0.43 t-fm 0.06-0.04 0.72 t-shanken 0.04-0.03 0.55 LevFac 62.21 60.97 44.40 t-fm 4.62 5.29 3.28 t-shanken 3.12 3.65 2.55 Mkt 5.46 5.20 t-fm 1.75 1.82 t-shanken 1.55 1.77 SMB 1.45 t-fm 0.77 t-shanken 0.75 HML 2.64 t-fm 1.15 t-shanken 1.09 MOM 7.72 t-fm 2.90 t-shanken 2.85 PC1 0.07 t-fm 0.51 t-shanken 0.41 Panel B: Test Diagnostics MAPE E[R e ] LevFac LevFac, Mkt All Factors Size B/M 7.86 1.16 1.11 0.92 MOM 5.80 1.79 1.85 1.35 Bond 1.65 0.37 0.26 0.08 Intercept 0.12 0.19 0.43 Total 6.45 1.31 1.36 1.33 AdjR2 0.77 0.78 0.87 C.I.AdjR2 [0.82, 1] [0.76,1] [0.78,0.94] T 2 (χ 2 N K ) 67.87 68.86 79.91 P-Value 0.3% 7 0.0% 0.0%
Table 2: Broker-dealer leverage is pro-cyclical. We display the correlation of U.S. brokerdealer leverage growth with a selection of variables, including the log asset growth of U.S. broker-dealers, market volatility (constructed quarterly using weekly data of the valueweighted market return), the Baa-Aaa spread, and the value-weighted stock return of the U.S. financial sector. The sample is Q1/1968-Q4/2009. Correlation of Leverage Factor with: Log Broker-Dealer Market Baa-Aaa Financials BAB Asset Growth Volatility Spread Stock Return Factor 1968-2009 0.60* -0.37* -0.16* 0.18* 0.22* 1968-1988 0.69* -0.27* 0.10 0.32* 0.07 1989-2009 0.57* -0.43* -0.45* 0.06 0.33* Correlation of LMP with: Log Broker-Dealer Market Baa-Aaa Financials BAB Asset Growth Volatility Spread Stock Return Factor 1968-2009 0.37* -0.35* -0.01* 0.48* 0.46* 1968-1988 0.45* -0.29* 0.38* 0.65* 0.31* 1989-2009 0.30* -0.39* -0.48* 0.33* 0.58* 8
Table 3: Subsamples: Pricing the Size, Book to Market, Momentum, and Bond Portfolios Pricing results for the 25 size and book-to-market and 10 momentum portfolios, and 6 Treasury bond portfolios sorted by maturity. Each model is estimated as E[R e ] = λ 0 + β fac λ fac. We split our sample in half and analyze the pricing results on two subsamples, 1968-1988 and 1989-2009. We show the results from our one-factor model and a two-factor model that adds the market. Further, the column Full E[R] usese the full sample (1968-2009) to estimate expected returns, rather than using the realized returns over that subsample. This is because average returns can be a poor proxy for expected returns, especially in small subsamples. Finally, for the subsample from 1989-2009, we explore the effect of dropping the dot com episode (1998-2002) as our leverage factor does not change drastically, yet stocks have a large upward and then downward swing during this period. Panel A: 1968-1988 1-fac 2-fac 1-fac, FE 2-fac, FE Intercept -0.52 0.53 0.65 1.42 t-shanken -0.30 0.24 0.38 0.70 Mkt 3.11 3.47 t-shanken 0.61 0.70 LevFac 20.67 44.83 19.84 37.49 t-shanken 1.28 2.50 1.23 2.25 MAPE 2.18 1.55 1.74 1.74 AdjR2 0.38 0.56 0.40 0.51 C.I.AdjR2 [0.22, 0.78] [0.44,1] [0.26,0.80] [0.42,1] Pr(R2,alpha) 0.03 0.00 0 0 Panel B: 1989-2009 1-fac 2-fac 1-fac, FE 2-fac, FE No tech No tech, FE Intercept 7.21 1.70 6.54 0.68 6.90 4.76 t-shanken 2.16 1.89 1.85 0.86 1.49 0.95 Mkt 6.25 6.04 t-shanken 1.47 1.35 LevFac 17.77 45.00 22.89 59.82 35.04 46.00 t-shanken 0.83 3.40 1.04 2.50 1.69 2.05 MAPE 2.62 1.79 2.70 1.50 2.17 2.35 AdjR2 0.11 0.58 0.17 0.67 0.30 0.47 C.I.AdjR2 [0,1] [0.46,1] [0.04,1] [0.50,1] [0.16,1] [0.30,1] Pr(R2,alpha) 0.29 0.00 0.15 0.00 0.12 0.07 9
Table 4: Main Table: Pricing the Size, Book to Market, Momentum, and Bond Portfolios Pricing results for the 25 size and book-to-market and 10 momentum portfolios, and 6 Treasury bond portfolios sorted by maturity. Each model is estimated as E[R e ] = λ 0 + β fac λ fac. FF denotes the Fama-French 3 factors, Mom the momentum factor, PC1 the first principal component of the yield curve, LevFac our leverage factor. Panel A reports the prices of risk with Fama-MacBeth and Shanken t-statistics, respectively. Panel B reports test diagnostics, including mean absolute pricing errors (MAPE) by portfolio group, adjusted R-Squares with corresponding confidence intervals (C.I.), and a χ 2 statistic that tests whether the pricing errors are jointly zero. E[R e ] gives the average excess return to be explained. Data are quarterly, 1968Q1-2009Q4. Returns and risk premia are reported in percent per year (quarterly percentages multiplied by 4). Panel A: Prices of Risk LevFac LevFac, All Fin LevFac, Bank LevFac, B-D Ret Intercept 0.12-0.11-0.12-0.23 t-shanken 0.04-0.08-0.08-0.16 LevFac 62.21 60.99 61.17 60.62 t-shanken 3.12 3.48 3.67 3.63 Fin Ret 10.75 t-shanken 1.73 Banks 8.77 t-shanken 1.27 Broker-Dealer Ret 8.02 t-shanken 0.84 Panel B: Test Diagnostics MAPE E[R e ] LevFac LevFac, Fin LevFac, Bank LevFac, B-D Ret Size B/M 7.86 1.16 1.13 1.12 1.09 MOM 5.80 1.79 1.84 1.83 1.84 Bond 1.65 0.37 0.28 0.27 0.26 Intercept 0.12 0.11 0.12 0.23 Total 6.45 1.31 1.18 1.18 1.38 AdjR2 0.77 0.77 0.78 0.77 C.I.AdjR2 [0.82, 1] [0.76, 1] [0.76, 1] [0.76, 1] T 2 (χ 2 N K ) 67.87 68.80 69.14 69.50 P-Value 0.3% 0.0% 0.0% 0.0% 10
Table 5: Main Table: Pricing the 30 Industry Portfolios Pricing results for the industry portfolios. Each model is estimated as E[R e ] = λ 0 + β fac λ fac. FF denotes the Fama-French 3 factors, Mom the momentum factor, LevFac our leverage factor. Panel A reports the prices of risk with Fama-MacBeth and Shanken t-statistics, respectively. Panel B reports test diagnostics, including mean absolute pricing errors (MAPE) by portfolio group, adjusted R-Squares with corresponding confidence intervals (C.I.), and a χ 2 statistic that tests whether the pricing errors are jointly zero. E[R e ] gives the average excess return to be explained. Data are quarterly, 1968Q1-2009Q4. Returns and risk premia are reported in percent per year (quarterly percentages multiplied by 4). Panel A: Prices of Risk LevFac FF FF Plus Mom Intercept 6.26 9.52 10.12 t-fm 2.03 1.91 2.01 t-shanken 2.03 1.90 1.96 LevFac 1.55 t-fm 0.10 t-shanken 0.10 Mkt -2.82-3.50 t-fm -0.51-0.63 t-shanken -0.51-0.61 SMB -0.75-1.03 t-fm -0.31-0.42 t-shanken -0.31-0.42 HML -0.97-0.60 t-fm -0.35-0.22 t-shanken -0.35-0.21 MOM -3.75 t-fm -0.68 t-shanken -0.67 PC1 t-fm t-shanken Panel B: Test Diagnostics MAPE E[R e ] LevFac FF FF Plus Mom Industries 6.43 1.61 1.58 1.56 Intercept 6.26 9.52 10.12 Total 7.87 11.10 11.68 AdjR2-0.03-0.03-0.03 C.I.AdjR2 [0, 0.12] [0, 0.22] [0, 0.26] T 2 (χ 2 N K ) 35.00 35.43 32.97 P-Value 0.17 0.10 0.13 11