Size Anomalies in US Bank Stock Returns: Your Tax Dollars at Work?

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1 Size Anomalies in US Bank Stock Returns: Your Tax Dollars at Work? Priyank Gandhi UCLA Anderson School of Management Hanno Lustig UCLA Anderson School of Management and NBER January 11, 2010 Abstract Over the last four decades, the average Gaussian-risk-adjusted return on a stock portfolio that goes long in the largest banks and short in the smallest banks is minus 7 %. Moreover, this portfolio provides US investors with insurance against recessions, even though the cash flows of large banks seem more exposed to macroeconomic risk. Using the rare events model of?, we interpret the 7% as a disaster risk premium. Recessions are periods with a high probability of a disaster. In a calibrated version of the model, we estimate an implicit recovery rate in disaster states that is 44 percentage points higher for the largest banks than for the smallest banks. If these large differences in the implied recovery rates indeed reflect the market s expectations of the government s asymmetric actions during a disaster, then the disaster risk discount for large banks represents a large hidden subsidy to large banks and a tax on small banks. First Version: May Gandhi: Anderson School of Management, University of California at Los Angeles, CA, 90034; gandhip@ucla.edu; Tel: (310) Lustig: Anderson School of Management, University of California at Los Angeles, Box , Los Angeles, CA 90095; hlustig@anderson.ucla.edu; Tel:(310) ; The authors thank Mark Garmaise, Jennifer Huang, Richard Roll, Jan Schneider, Clemens Sialm and seminar participants at UT Austin for detailed and helpful comments.

2 We don t let banks fail. We don t even let dry cleaners fail. It never occurred to us that the Americans would let Lehman fail. ECB Banker quoted by?. 1 Introduction So far, during 2009, the FDIC has shut down 95 small or mid-sized banks. Another 415 are at risk, according to a recent study by the FDIC. On the other hand, the large banks are largely considered safe now, mostly because of taxpayer support. What transpired after the collapse of Lehman in September 2008 seems to confirm the commonly held view that most policy makers are not willing to let large financial institutions fail collectively, even though they may be willing to occasionally let large individual institutions fail. 1 Our paper explores the asset pricing implications of this perceived collective bailout guarantee for large banks. If this collective guarantee is understood and priced in by stock market investors, then the stocks of large banks should respond very differently from those of small banks to changes in the probability of a large adverse shock to the US economy, irrespective of what happens to the cash flows of these banks. This is what find in the data. The traditional role of the financial sector is to channel resources from the household sector to profitable investment opportunities in the production sector. However, in recent decades, financial institutions in developed economies have increasingly sought exposure to aggregate risk, thus providing insurance against aggregate shocks to other parties 2.Figure 1 plots the profits of the US financial sector as a fraction of profits for the entire corporate sector. The financial sector s share of corporate profits has not only increased in magnitude over time, but it has also become more volatile and more pro-cyclical. The volatility of this profit share increased from 3.92% between 1948 and 1979 to 8.31% after 1979, even though the standard deviation of total non-financial real output growth actually declined over the same period from 1.8% to 1% per annum. NBER recessions (shaded areas in 1 The consensus view seems to be that letting Lehman fail in September of 2008 was a mistake. Some US policy makers have even argued that, because of institutional constraints, they had little choice in the Lehman matter. 2 Other crises linked to the financial services sector include the mutual savings bank crises, the less-developed-country debt crisis, the savings and loan crisis, failure of the Continental Illinois Bank, and the Long Term Capital Management crisis, among others. 2

3 the figure) are invariably followed by large declines in the financial sector s profit share. This increased exposure to aggregate risk on the part of commercial banks manifests itself in more volatile earnings (see???, for detailed evidence). Moreover, since the US banking sector is extremely concentrated 3, large banks have to be highly exposed to aggregate risk, given that they account for most of the bank earnings. In fact, according to the FDIC (Federal Deposit Insurance Corporation), small and mid-sized banks outperformed large banks during the two most recent US recessions according to various cash-flow-based performance measures. During the recent recession, earnings declined much faster at the largest financial institutions with assets in excess of $10 billion (see?). However, the returns on bank stocks tell a different story. By going long one dollar in a basket of large bank stocks and short one dollar in a basket of small bank stocks, investors go long 74 cents in the stock market, long 38 cents in the market for long term government bonds and short 57 cents in credit markets. Since credit spreads rise during a recession, this long-short portfolio arguably provides direct insurance against recessions, through the factor loadings. This long investment in large bank stocks and short investment in small bank stocks still consistently loses more than 7 percent per annum over the last 38 years relative to an equally risky portfolio of all (non-bank) stocks and government and corporate bonds. This benchmark portfolio controls for exposure to the standard equity and bond risk factors. In fact, when we create ten portfolios of banks sorted on size, we find that the average risk-adjusted returns decrease monotonically in the size of the portfolio of bank stocks. There is a striking, decreasing pattern in average risk-adjusted returns on the size portfolios of bank stocks. We find that the long-in-large-short-in-small bank portfolio offers indirect insurance against recession risk, above and beyond the direct insurance provided by the mimicking portfolio of stocks and bonds. This can account for the underperformance of large banks relative to the benchmark portfolio of non-bank stocks and bonds, which does not offer this type of insurance. Specifically, we find that there is a slope factor in the risk-adjusted returns of size-sorted portfolios of bank stocks; its weights decrease monotonically as we increase market capitalization. This slope factor is the second principal component of the risk-adjusted returns of size-sorted portfolios of bank stocks. It is equivalent to a long position in small bank stocks and a short position in large bank stocks. After controlling for exposure to 3 92% of earnings stem from the 115 largest commercial banks. 3

4 Profit of U.S. Financial Sector (fraction of U.S Corporate Sector) May49 Nov54 May60 Oct65 Apr71 Oct76 Mar82 Sep87 Mar93 Sep98 Feb04 Month and Year Figure 1: Profit of financial firms as a fraction of corporate profits. This figure presents the profit of US financial sector as a fraction of profits for the entire US corporate sector. Source: NIPA Table Between 1947 and 1979 financial firms account for 14.67% of the total corporate sector profits on average in any given year. This increased to 25.50% in the post-1979 period. The increase in profits was accompanied by an increase in volatility of the profit share from 3.92% until 1979 to 8.31% in the post-1979 period. The dates are indicated on the x-axis. The dark-shaded regions represent NBER recessions and the light-shaded regions represent and the light-shaded regions represent the Less-Developed-Country debt crisis of 1982, the Mexican Peso crisis of 1994, and the Long-Term-Capital-Management crisis of 1998 respectively.

5 this slope factor, the monotonically decreasing size pattern in average risk-adjusted returns disappears. Hence, the covariance between the returns on size portfolios of banks stocks and the slope factor explains the size pattern in average risk-adjusted returns. This slope factor is highly pro-cyclical. During NBER recessions, this factor drops by an average of 3.46% per month or 41.57% per annum. Hence, by shorting small banks and investing in large banks, investors effectively buy insurance against recessions, directly through the implied exposure to standard bond and stock risk factors, by shorting credit markets and the yield curve, and indirectly, because of this slope factor in risk-adjusted returns. We use a version of the?? asset pricing model with a time-varying probability of rare events, developed by??, to interpret this slope factor. The disaster model gives rise to a slope factor in size-sorted bank Gaussian-risk-adjusted stock returns provided that the recovery rate for bank stockholders in case of a disaster realization depends on the size of the banks. The loadings of bank stock returns on this slope factor are determined by the recovery rates. The slope factor is a measure of the probability of a disaster. If we label periods with high probabilities of a disaster as recessions, then the model delivers a slope factor with the same cyclical properties as the factor we measured in the data. In this model, the average Gaussian-risk-adjusted return on the long position in the largest banks and the short position in the smallest banks of minus 7 percent per annum is the price of disaster insurance. A calibrated version of the model can match this disaster risk premium if the shareholders of the largest bank lose 44 cents less per dollar of pre-disaster cash flows than stockholders of the smallest banks. Small banks have positive exposure to credit risk, as measured by a portfolio of US investment grade bonds, and negative exposure to long term government bonds. This is in line with the standard view of financial intermediation. Banks are expected to post better performance when credit market conditions improve and they do better when the yield curve steepens, because they borrow at the short end and lend at the long end of the yield curve. However, large banks stock returns are very different. Over the last three decades, large banks have a negative or zero exposure to credit risk and a positive exposure to long-term government bonds. From the perspective of equity markets, large banks do better when credit market conditions deteriorate and they do better when the yield curve inverts. Since the yield curve inverts at the start of a recession and credit risk premia rise during re- 5

6 cessions, this means large bank stocks outperform small bank stocks (at the start of/and) during recessions. Hence, large bank stock returns provide direct recession insurance. We suspect that these risk factors co-vary with the probability of a disaster, and hence absorb part of the disaster risk premium. This can account for the anomalous loadings on the bond market factors which are more clearly cyclical than the stock market factors. As a result, our estimate of the disaster risk premium and hence the implicit government subsidy is probably too small. Hence, there is a huge gap between the cash flow and the return evidence. The stock return evidence suggests a long position in larger banks and a short position in smaller banks provides insurance against large adverse aggregate shocks to the US economy, while the cash flow evidence suggests the exact opposite. We propose a fiscal explanation of the gap between the cash flows and stock return evidence from banks. When the government provides an implicit collective bailout guarantee for large banks, but not for small banks, we expect to see exactly what we describe in the data. As the balance sheet of the average bank deteriorates when the economy enters a recession, large bank stocks can outperform small bank stocks, if the probability of a bailout increases sufficiently, even if the former has cash flows that are less exposed to recession risk. The key to activating the collective bailout clause is common variation. recent paper,? and? explore the incentives for banks in this type of environment to seek exposure to similar risk factors. 4 The government may be willing to let large banks fail individually, but not collectively. We should expect large banks to actively seek sources of common variation. This is increasingly what banks seem to have done, especially large banks. For the largest banks, the five risk factors explain nearly 60% of the time series variation in stock returns, compared to only 27% for the smallest banks. Moreover, the common variation seems to have increased. The first two principal components of returns on the 10 size-sorted portfolios of bank stocks explain only 65% of the time series variation in size-sorted portfolios of bank stocks over the entire sample, but they explain 81% of the variation between 2000 and For the largest banks, this fraction increased from 82 % over the entire sample to nearly 99 % between 2000 and In a Why look at bank stocks? Clearly, the US government and regulators are willing to let small banks fail, not so for large banks. The FDIC reports that 67 banks have failed in the last two years. All of these are small by any standard. By looking at 4 The government s guarantee creates complementarities in firm payoffs. In earlier work,? explain the currency mismatch on firm balance sheets in emerging markets endogenously by means of a bailout guarantee for the non-tradeables sector. 6

7 equity, we can compare the exposure of small and large banks to different sources of aggregate risk. This would not be possible if we focused on debt issued by banks, simply because this data is not available for small banks. Why look at portfolios of bank stocks sorted on size? The US government and US regulators may be willing to let large banks fail individually, but clearly are not willing to let large banks fail collectively. Hence, we are not interested in idiosyncratic variation in individual bank stock returns, but we are interested in common variation in large vs small bank stock returns. That is why we build portfolios of stocks sorted on the size dimension. The literature that explores the effects of bank size on performance and risk management is limited. One notable exception is?. These authors analyze the impact of size on the performance of banks as measured by accounting data. They show that increased competition and financial innovation have induced the largest banks to participate in riskier investments. They also document the poor performance of large banks, as measured by ratios of net loan charge-offs and net income to assets, as compared to that of small and medium sized banks. The impact of government regulation on equity returns for big banks has been analyzed by several authors including? and?. Our paper provides a unique perspective as it is, at least to our knowledge, one of the first that employs the standard portfolio formation strategy of?, which is common in much of asset pricing literature, to financial firms. This enables us to overcome some of the limitations encountered in the literature in analyzing the differences between big and small banks. The rest of this paper is organized as follows: Section 2 develops a simple asset pricing model with time-varying probabilities of a disaster that is consistent with our empirical observations. In Section 3 we describe the data and present summary statistics. Section 3.2 presents results from the analysis of stock returns. Section 4.2 presents the evidence from the cash-flows of banks. Section 6 concludes. 2 Model To help us interpret our empirical findings, we use a standard asset pricing model with time-varying probability of rare events or disasters. The US government and the Federal Reserve, as well as other governments, central banks around the world, stand ready to collectively bail out large financial institutions in the case of rare, large, adverse shocks to the real economy or the financial system (or both), like the Great recession of , that are perceived as a threat to the stability of 7

8 the financial system. We set up a model in which the probability of this rare event (disaster) varies over time, and we think of recessions and financial crises as periods during which this probability is elevated. 2.1 Barro-Riezs-Gabaix disaster risk We adopt a version of the models with time-varying probabilities of disasters proposed by? and?. These are extensions of the rare event models developed by??. Gaussian and disaster risk In our model, the stochastic discount factor has two components: a standard Gaussian component and a disaster component: M G t+1 M t+1 = Mt+1 G 1 in states without disaster (1) M t+1 = Mt+1 G B γ t+1 in states with disaster. denotes the representative investor s intertemporal marginal rate of substitution (IMRS) in normal times, i.e., in states without a disaster. 5 In the absence of a disaster realization, the IMRS is completely determined by Gaussian risk. Henceforth, we refer to these risk factors simply as Gaussian risk factors, to distinguish these from the disaster risk. The Gaussian risk factors are denoted f t+1. To simplify the empirical analysis, we impose linearity. Assumption 1. The Gaussian component of the SDF is linear in the risk factors: M G t+1 = b f t+1. We use β i t to denote the vector of conditional Gaussian risk factor betas for the returns on asset i, and we use λ t to denote the vector of Gaussian risk prices. We make some simplifying assumptions. Assumption 2. The conditional distribution of the Gaussian risk factors f t is independent of the disaster realization. Moreover, p t does not co-vary with the Gaussian risk factors f t. 5 In the simplest CCAPM-version of his model,? defines C t+1 C t = exp(g C ) as the growth rate of consumption in normal times, and C t+1 C t = exp(g C )B t+1 as the growth rate of the economy if there is an adverse shock, where B t+1 > 0. 8

9 This second assumption means that the recession risk itself is not priced, only the disaster risk itself is. Cash flows recovery rates We consider the following specification for the (disaster component of) bank cash flow process of a bank of size i: Dt+1 i Dt i Dt+1 i Dt i ( ) = exp(g d ) 1 + ε d,i t+1 1 in states without disaster ( ) = exp(g d ) 1 + ε d,i t+1 Ft+1 i in states with disaster F i t+1 can be thought of as the recovery rate; in case the rare event is realized, a fraction F i of the dividend gets wiped out (see??). This recovery rate will vary across banks depending on size, because the realization of this rare event can trigger a collective bailout of larger banks, but not necessarily of smaller banks. To obtain a simple characterization of the disaster risk premium in banks stocks, we make the following assumption. Assumption 3. The conditional distribution of the Gaussian dividend innovations ε d,i is independent of the disaster realization. The resilience of banks is defined as: [ ] Ht i = p t E t B γ t+1 Fi 1 As the US economy enters into a recession, p t, the probability of a large adverse shock to the economy starts to increase, and the resilience of large banks Ht L increases relative to small banks Ht S if Ft+1 B > FS t+1. In fact, we assume that the recovery rate Ft n > Ft n 1 increases monotonically in size. In the interest of tractability, we assume that the recovery rates F i are constant over time, and we also assume that the size of the consumption disaster B is constant over time. Slope factor To derive a simple expression for risk premia, we abstract from variation in Gaussian betas and risk prices. Assumption 4. The conditional beta β t and the conditional risk prices λ t are constant. 9

10 Proposition 1. The expected return on asset i, conditional on no disaster realization, after adjusting for Gaussian risk exposure, becomes: E t [ R i t+1 ] = exp(r hi t), (2) where E t [ R i t+1 ] = E t[r i t+1 ] βi λ, and r denotes log R, and h i t denotes log(1 + Hi t ). The proof is in Appendix A. As is clear from eq. (2), this variation in the probability of a disaster in turn imputes common variation to the Gaussian-risk-adjusted stock returns along the size dimension, since we assumed that the recovery rate depends on size. loadings on this common [ factor ] are proportional to F i 1. To see why, note that log(1 + Ht i) p te t B γ t+1 Fi 1. We refer to this as the slope factor because the loadings depend on the recovery rates and hence size. The The conditional Gaussian-risk-adjusted multiplicative risk premium on a long-short portfolio is given by the following simple expression: ] ] log E t [ R t+1 B log E t [ R S t+1 = h S t+1 hb t+1. (3) As p t increases during recessions and financial crisis, the risk premium on this longshort portfolio becomes more negative. It effectively provides recession insurance. Average Risk-adjusted Returns and alpha s Corollary 1. In a sample without a disaster realization, the average return (in population) will be given by: E[ R i t+1 ] = exp(r hi ), where h i = E[log ( 1 + Ht) i ]. In other words, the difference in α s in a regression of returns on the Gaussian risk factors in a sample without a rare event realization is given by the difference in average resilience : log α B log α S = h S h B, (4) We refer to this as the disaster risk discount on large banks or premium on small banks. 10

11 3 Data We take this model to the data in section 3. Section 3.1 describes the data. In section we compute the average Gaussian-risk-adjusted returns on 10 sizesorted portfolios of banks stocks in the data, and we show that there is a size pattern in these α s. The average value of this Gaussian-risk-adjusted return is minus 7.12% per annum when short in the first size decile and long in the last size decile of banks. In section 3.2.2, we show that the second principal component of Gaussian-riskadjusted returns R i has loadings that depend monotonically on size. We interpret this slope factor in Gaussian-risk-adjusted returns as the common factor in returns imputed by time variation in the probability of a disaster. In the model, this disaster risk premium increases in recessions if the probability of a disaster p t increases in recessions. This is what we find in the data. 3.1 Bank stock returns We collect data on equity returns from the Center for Research in Security Prices (CRSP) for all firms with Standard Industrial Classification (SIC) codes 60, 61, and 62. Firms with these SIC codes are defined as commercial banks, non-depository credit institutions, and investment banks respectively. Henceforth, we refer to commercial banks, i.e. firms listed under SIC code 60, simply as banks and refer to commercial banks, credit institutions, and investment banks, i.e. firms listed under SIC codes 60, 61, and 62, collectively as financial firms. We exclude data for all financial firms that are inactive and/or not incorporated in the United States. We exclude financial firms not incorporated in the United States because these financial firms will be influenced by regulations applicable both in the country of operation and the country of incorporation. Since these policies vary across countries, our focus on financial firms operating and incorporated inside the United States ensures that all firms in our analysis are subject to a uniform regulatory regime. We start by focussing on portfolios of bank stocks. We employ the standard portfolio formation strategy of? for the purpose of analysis. In January of each year, we rank all bank stocks by market capitalization. The stocks are then allocated to 10 portfolios based on their market capitalization. We calculate value-weighted returns for each portfolio for each month over the next year. At the end of this exercise, we have monthly value-weighted returns for each size-sorted portfolio of banks from January 1970 to December In addition to portfolios for banks, we 11

12 also separately form portfolios for financial firms from January 1950 to December While the CRSP data are available from 1926, our analysis begins only in 1970 for banks and in 1950 for financial firms. These starting dates reflect the fact that CRSP data for earlier years did not include enough banks and financial firms to yield at least 1 security per size-sorted portfolio. Only a small fraction of all banks that operate in the US are publicly listed. For instance, for the years 2000 to 2008, data is available on CRSP for approximately 630 banks. This compares to more than 7000 FDIC-insured banks operating in the United States over the same period. This is mainly a issue for small banks. The largest 600 banks control more than 88% of all commercial bank assets in the United States 6. Most of these large banks are publicly listed. To the extent that small banks that are not publicly listed are very different from those that are, some of our results need to be qualified. Market cap Table 1 reports the total market capitalization of banks in each sizesorted portfolio as a fraction of the total market capitalization of the banking sector in January of each year. All the numbers are reported in percentages. We also report the standard deviation (σ), the minimum, the maximum and the average number of banks in the portfolio. The first panel in Table 1 shows that during , the smallest banks (those in portfolio 1) on average represented just 0.29% of the total market capitalization of all commercial banks. This compares to 49.59% represented by the largest banks (those in portfolio 10). During any year between 1970 and 1980, banks in portfolio 1 at most accounted for 0.45% of the total market cap of the commercial banking sector and at the minimum accounted for 0.18%. Table 1 clearly shows the increasing concentration of the U.S commercial banking sector. The top 10% banks account for nearly 50% of the total sector market capitalization in the 70s while they account for more than 90% during the last decade. In any given year between 1970 and 1980, on average, we have at least 9 banks per sizesorted portfolio and this increases to 64 banks for any year between 2000 and Returns Panel I in Table 26 provides mean returns for the size-sorted portfolios of banks over the entire sample. The mean monthly returns for all portfolios are annualized by multiplying by 12 and are expressed in percentages. The last column reports the difference in mean annual returns between the 10 th and the 1 st portfolio. 6 As per FDIC Bank Statistics and Data available at 12

13 Table 1: Market capitalization of size-sorted portfolios of banks Notes: This table presents the total market capitalization of firms in each size-sorted portfolio as a percentage of total market capitalization for the entire banking sector. The market values are measured in January of each year. Mean represents the average value of this percentage over the years specified in the first column. σ captures the variation in this proportion. Minimum and Maximum values indicate the range of this ratio for each portfolio. And finally N is the average number of banks in each portfolio over the same period. Size Sorted Portfolios Year Statistic A 10B 10C 10C - 1 Mean σ Minimum Maximum N Mean σ Minimum Maximum N Mean σ Minimum Maximum N Mean σ Minimum Maximum N

14 Over the entire sample, a portfolio that goes long in a basket of large banks and short in a basket of small banks on average loses 3.94% per annum. For , the annual loss on this portfolio is 5.78%. Panel II in Table 26 repeats the analysis for the means but excludes the years 2006, 2007, and If we think of the recent crisis as a realization of the rare event, then we should exclude it from the sample to measure the disaster risk premia (see eq. (4)). When we exclude the recent crisis from the sample, the return differences between large and small banks grow larger. The long large banks-short small banks portfolio loses an even larger 4.46% per annum over the entire sample and nearly 10% since 2000 on an annualized basis. This relationship between bank size and equity returns may seem consistent with the general size effect documented for non-financial firms 7, but we will show that it is actually quite different.? document that the size anomaly for nonfinancials disappears when one allows for multiple priced factors;? implement a three-factor model that includes the market, a size factor (SMB) and a value factor (HML). If the same holds true for banks, then a portfolio of large banks should not earn lower Gaussian-risk-adjusted returns after accounting for exposure to the size factor SMB as compared to a portfolio of small banks. This is not what we find. In fact, we find that the size effect for banks becomes larger when we adjust for exposure to standard risk factors. 3.2 Risk exposure In this section, we compute Gaussian-risk-adjusted returns on the size-sorted portfolios of banks stocks. We find that large banks earn even lower returns on a Gaussian-risk-adjusted basis. The lower returns on the portfolios of large banks can partly be attributed to their loadings on standard equity and bond risk factors, but this is not the complete story. We find that investors in a portfolio of large banks get indirect insurance against recession risk as well. We regress monthly excess returns for each size-sorted portfolio on the three Fama-French factors and two bond factors. Hence, the vector of Gaussian risk factors f t is 5 1. Banks manage a portfolio of bonds of varying maturities and credit risk. 8 Therefore we also include in our regressions two bond risk factors. For each portfolio i we run the following time-series regression to estimate the 7 (see???) 8 In a recent paper? also show that banks can be treated as active managers of fixed income portfolios. 14

15 Table 2: Mean returns of banks Notes: This table presents the mean returns for each size-sorted portfolio of banks. The first column indicates the years over which mean returns were computed. The monthly mean returns have been annualized by multiplying by 12 and are expressed in percentages. 15 Size Sorted Portfolios Year A 10B 10C 10C l

16 vector of betas β i : R i t+1 R f t+1 = α i + β i M MKT t+1 + β i SMB SMB t+1 + β i HML HML t+1 (5) + β i GOV LTG t+1 + β i CRD CRD t+1 + ε i t+1, where R i t+1 is the monthly return on the ith size-sorted portfolio and R f t is the onemonth risk-free rate 9. MKT t+1, SMB t+1, and HML t+1 represent the returns on the three Fama-French factors namely the market, small minus big, and high minus low respectively. We use LTG t+1 to denote the excess returns on an index of 10- year bonds issued by the U.S Treasury as our first bond risk factor (R LTG t+1 R f t ). In addition, active participation by banks in markets for commercial, industrial and consumer loans exposes them to credit risk. We use CRD t+1 to denote the excess returns on an index of investment grade corporate bonds, maintained by Dow Jones, as our second bond risk factor (R CRD t+1 R f t ). Since all of the risk factors in f t are traded returns, the estimated residuals in the time series regression are the estimated Gaussian-risk-adjusted returns R i t+1, and the estimated intercept α is the average disaster risk premium, i.e. the residual risk premium after taking out the compensation for Gaussian risk Gaussian risk exposure Table 17 provides the results of the regression specified in equation (5). The table reports the regression coefficients for each size-sorted portfolio along with their statistical significance and the adjusted R 2. The first column indicates the dates over which the regressions are carried out. For the columns, titled 10-1, 9-2, and 8-3 the dependent variable is defined as R 10 t+1 R1 t+1, R9 t+1 R2 t+1 and R 8 t+1 R3 t+1 respectively. The standard errors are adjusted for heteroskedasticity and auto-correlation using Newey-West. The first panel in Table 17 shows the regression results over the entire sample from 1970 to α indicates the intercepts for this regression. The intercepts have been annualized by multiplying by 12 and are expressed in percentages. The computed intercepts decrease nearly monotonically with bank size and are not statistically different from zero. This is however, not true for intercept of the the difference portfolio in the last 3 columns. A long-short position that goes long one dollar in a portfolio of the largest banks and short one dollar in a portfolio of the smallest 9 Data for the risk-free rate and the Fama-French factors was collected from Kenneth French s website at 16

17 banks loses 7.12% over the entire sample. Thus, a portfolio of large banks underperforms a portfolio of small banks by even more after adjusting for standard risk factors. Similarly, the average Gaussian-risk-adjusted return on a 9 minus 2 position is % per annum, and % per annum for the 8 minus 3 portfolio. These results are consistent with those in? who also document the relative Gaussian-risk-adjusted under-performance of an index of large banks as compared to an index of small banks over a much shorter sample from 2004 to Equity factors The second row of panel 1 in Table 17 reports the coefficient on excess market return, β M, for each size-sorted portfolio. The loading on market risk increases monotonically with bank size for every sub-period of analysis. Over the entire sample, a portfolio of large banks has a market β of 1.13 as compared to a β of 0.39 for a portfolio of small banks. This may be partly due to differences in leverage. Over the entire sample, from 1970 to 2008, the largest banks were 2.9 times more exposed to market risk as compared to the smallest banks. By , large bank stocks were nearly 3.6 times more exposed to market risk than small banks. Thus large banks have collectively increased exposure to market risk over time. As a result, the long-short position described above (i.e. long $1 in large banks and short $1 in small banks) will be long the market by 74 cents. This net exposure to market risk increases by 17 cents over time to 91 cents during Nearly all of this increase is attributable to an increase in the aggregate risk exposure by the largest banks. The loadings on SMB (β SMB ) and HML (β HML ) also depend systematically on size. We first look at the exposure to the size factor. Contrary to what one expects to find, over the entire sample, the loading on SMB t+1 actually increases from 0.38 for the 1 st portfolio to 0.47 for the 9 th portfolio, and then it drops to 0.02 for the 10 th portfolio. Clearly, the common variation in banks stock returns along the size dimension is very different from that in other industries. The same pattern holds true for the loadings on HML which increase from 0.39 for the 1 st portfolio to 0.61 for the 9 th portfolio and then it drops to 0.55 for the 10 th portfolio. This result seems to suggest that bank exposure to risk as measured by SMB and HML increases with size, but only up until some given size threshold is attained. Bond factors There is a clear size pattern in the loadings on the bond risk factors as well. β GOV, the slope coefficient on the excess return on an index of 10-year bonds issued by the U.S Treasury, is negative and statistically insignificant for small 17

18 Table 3: Loadings for equity and bond factors for size-sorted portfolio of banks Notes: This table presents the estimates from OLS regression of monthly excess returns of each size-sorted portfolio of banks on Fama-French and bond risk factors. R i t+1 R f t+1 = αi + β i M MKT t+1 + β i SMB SMB t+1 + β i HML HML t+1 + β i GOV LTG t+1 + β i CRD CRD t+1 + ε i t+1, i = 1, 2,...10.Ri t+1 is the monthly return on the i th size-sorted portfolio, R f t+1 is the one-month risk free rate, and MKT t+1, SMB t+1, and HML t+1 are the three Fama-French factors namely the market, small minus big, and high minus low respectively. LTG t+1 is the excess return on an index of long-term government bonds and CRD t+1 is the excess return on an index of investment-grade corporate bonds. Statistical significance is indicated by *, **, and *** at the 10%, 5% and 1% levels respectively. α s have been annualized by multiplying by 12 and are expressed in percentages. The standard errors were computed after adjusting for heteroskedasticity and auto-correlation. Size Sorted Portfolios Year Coefficient A 10B 10C α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β GOV β CRD R

19 Table 4: Loadings for equity and bond factors for size-sorted difference portfolios of banks Notes: This table presents the estimates from OLS regression of monthly excess returns of the difference portfolios of banks on Fama-French and bond risk factors. R i t+1 R f t+1 = αi + β i M MKT t+1 + β i SMB SMB t+1 + β i HML HML t+1 + β i GOV LTG t+1 + β i CRD CRD t+1 + ε i t+1, i = 1, 2,...10.Ri t+1 is the monthly return on the i th size-sorted portfolio, R f t+1 is the one-month risk free rate, and MKT t+1, SMB t+1, and HML t+1 are the three Fama-French factors namely the market, small minus big, and high minus low respectively. LTG t+1 is the excess return on an index of long-term government bonds and CRD t+1 is the excess return on an index of investment-grade corporate bonds. Statistical significance is indicated by *, **, and *** at the 10%, 5% and 1% levels respectively. α s have been annualized by multiplying by 12 and are expressed in percentages. The columns indicate the difference between the estimates for the 10C th 1 st, 10B th 1 st, 10A th 1 st, 9 th 2 nd, and the 8 th 3 rd portfolios. The standard errors were computed after adjusting for heteroskedasticity and auto-correlation. Size Sorted Portfolios Year Coefficient 10C B A α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β LTG β CRD R α β M β SMB β HML β GOV β CRD R

20 banks and is positive and almost always statistically significant for large banks. The loadings vary monotonically in size. A $1 long position in large banks and a $ 1 short position in small banks not only results in a net exposure of 38 cents to longterm government bonds over the entire sample, but this exposure also increases to 76 cents over Thus large bank stocks relatively out-perform small bank stocks when excess returns on long term government bonds are high. Over , an 1% increase in excess returns of long term government bonds results in a 0.38% increase in equity returns of large banks versus those of small banks. On the other hand, the loadings on the credit risk factor, β CRD, are negative for large banks and positive for small banks. A long-large-banks-short-small-banks position results in a net negative exposure to credit markets of 57 cents over and this increases to a net negative exposure of $.68 in the subsample and even $1.39 by Most of this is due to large banks, whose loading on the credit risk proxy decreases from over the entire sample to during These coefficients imply that a 1% fall in excess returns on an index of investment-grade bonds results in a 1.39% increase in excess equity returns for a portfolio of long-large-banks-short-small-banks over This is puzzling. Why should large banks load negatively or not-at-all on proxies for credit risk? One can argue that better access to markets for securitization allows large banks to more effectively manage exposure to credit risk. This does not, however, explain the negative or statistically insignificant coefficients on proxies of credit risk, especially considering the fact that the securitization process requires these banks typically hold some portion of the first-loss tranche. In any case, because of the credit risk exposure, the long in big, short in small banks portfolio outperforms during US recessions. Overall, because of the exposure to government bond and credit markets, the portfolio that goes long in large and short in small banks offers insurance to investors against large, adverse shocks to the US economy. This is the direct recession insurance effect that is captured by the standard factors. However, the Gaussianrisk-adjusted returns suggest we may be missing factors Size effect in average risk-adjusted returns Since all the factors are traded returns, we can interpret the intercepts as the average Gaussian-risk-adjusted returns on investments in portfolios of banks stocks. There is a clear size pattern in these average Gaussian-risk-adjusted returns. They vary quasi-monotonically from 3.75 percent per annum on the first portfolio to percent per annum on the last 20

21 portfolio. A long in the 10 th and short in the 1 st portfolio investment produced a Gaussian-risk-adjusted return of minus 7.12 percent per annum between 1970 and This return is statistically significant at the 5 percent level. It is also important to note that including other factors like the? aggregate liquidity factor, the VIX volatility index, or macro factors, such as the rate of industrial production growth, in the vector of Gaussian risk factors does not change these average returns significantly. Subsamples The three middle panels in Table 17 report the same multiple regression results for different sub-samples. The difference in Gaussian-risk-adjusted returns between the 10 th and short in the 1 st portfolio declines somewhat in the recent decades, but that effect is due to the increase in the difference in credit risk betas. Between , the effective exposure of the long-short portfolio is minus 1.39, while the Gaussian-risk-adjusted return is only minus 4.17 %. This explains the reduction in the average Gaussian-risk-adjusted returns. While forming the portfolios for our empirical analysis above, so far, we include the stock market returns for banks for the years 2006, 2007, and However, the model implies that we should exclude observations of the actual rare event from the sample when estimating average Gaussian-risk-adjusted returns. Not surprisingly, the results are actually stronger when we exclude these years from the sample. The bottom panel in Table 17 reports the regression coefficients, statistical significance, and the adjusted R 2 for each of the size-sorted portfolio of banks for the regression specified in equation (5), excluding 2006, 2007, and Our main result still holds. The average risk adjusted return on the long-large-short-small-banks portfolio is larger: % over the sample, % for the 9-minus-one portfolio, and % for the 8-minus-3 portfolio. These are all significant at the 5 % level. This is not surprising. By excluding the rare event itself from our sample, the average realized excess return in the sample should increase. Moreover, the negative credit exposures in the long-short positions are much smaller: the 10-minus-one loading on the credit risk factor drops by 20 basis points in absolute value. This is not surprising. The Gaussian risk factors in the data are correlated with p t, the probability of a rare event realization. As the probability of a disaster increased during the recent crisis, credit spreads increases and large banks outperformed small banks as a result. In the model, we assumed that p t is orthogonal to f t, but this assumption is obviously violated in the data. This implies that our estimates of the disaster risk premium in small bank stocks is likely to be 21

22 biased downwards because part of the effect of variation in p t is absorbed by the Gaussian risk factors themselves. This is the likely explanation for the anomalous negative loadings of large bank returns on the credit factor and the large positive loadings on the long bond return. In fact, these negative credit loadings disappear when we exclude NBER recessions from our sample (see Table 16 in the appendix) Disaster risk exposure The size pattern in the average Gaussian-risk-adjusted returns suggests that we may be missing risk factors. To investigate this further, we look at the time-series properties of the Gaussian-risk-adjusted returns the residuals of the time series regression in equation (5). There is a slope factor in the time series of the Gaussianrisk-adjusted-returns that explains the pattern in average Gaussian-risk-adjusted returns. We compute the residuals from the time series regression of returns of each sizesorted portfolio of banks on the equity and bond risk factor in 5. We extract the loadings for the principal components (w 1, w 2 ) of these regression residuals and we report the results in Table 5. This table only shows the loadings for the first two principal components. The first principal component is a banking industry ( level ) factor with roughly equal weights on all 10 portfolios. The second principal component is a slope factor that loads positively on portfolios of small banks and negatively on portfolios of large banks. The loadings vary monotonically in size. This is a candidate risk factor because the loadings line up with the average Gaussian-risk-adjusted returns that we want to explain. Together, these two principal components explain 65% of the residual variation over the entire sample and this fraction increases steadily to nearly 81% of the residual variation during Next, we take our (T 10) matrix of estimated residuals, ǫ t, formed above and multiply it by the (10 10) loadings of the principal components, to construct the asset pricing factors. The weights (w 1, w 2 ) are re-normalized to (ŵ 1, ŵ 2 ) so that they sum to 1. This results in a (T 10) linear combination of the residuals. We focus on the first two principal components, denoted PCt 1 = ŵ 1 ǫ t and PCt 2 = ŵ 2 ǫ t. Thus for each month, the residuals of each of the 10 portfolios from the above regression are multiplied by their corresponding re-normalized weights in the second principal component and added together. PC 2 is the Gaussian-risk-adjusted return on a portfolio that is long small banks and short large banks. We refer to PC 2 as a slope factor. The weights of the port- 22