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Economic Quarterly— Volume 104, Number 1— First Quarter 2018— Pages 1–33  Cyclical Properties of Bank Margins: Small versus Large Banks Borys Grochulski, Daniel Schwam, and Yuzhe Zhang  T  he US banking sector is composed of few very large banks and many small ones.1 In fact, about 95 percent of all Federal Deposit Insurance Corporation-insured depository institutions are small, community banks.2 Although these small banks hold only about 15 percent of total bank assets by value, many observers, bank supervisors, and monetary policymakers share the view that small, local banks ful…ll an important role in the intermediation of credit in the US. Former Federal Reserve Governor (and current chairman) Jerome H. Powell expresses this view in his speech to the 2013 Community Banking Research Conference: My colleagues on the Board of Governors and I understand the value of having a diverse …nancial system that includes a large and vibrant contingent of community banks. By fostering the economic health and vitality of local communities throughout the country, community banks play a central role in our national economy. One important aspect of that role is to serve as a primary source of credit for  The authors would like to thank Erica Paulos, Nicholas Trachter, John Walter, John Weinberg, and Russell Wong for their helpful comments, and Sara Ho for excellent research assistance. The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve System. 1 See Section 4 for a short summary or McCord and Prescott (2014) for detailed analysis. 2 Traditionally, a community bank is de…ned as a small bank that operates in a single banking market. A single banking market is de…ned as a county in rural areas or as a metropolitan statistical area in urban areas; see, e.g., Powell (2016). A small bank is often de…ned as one with less than $1 billion in assets. Recently, however, many studies have used a $10 billion size cuto¤ instead. See Keeton et al. (2003) and Federal Deposit Insurance Corporation (2012).  DOI: https://doi.org/10.21144/eq1040101  2  Federal Reserve Bank of Richmond Economic Quarterly the small businesses that are responsible for creating a substantial proportion of all new jobs.3  More speci…cally, small banks are thought of as having access to granular information about local business conditions. In its 2012 Community Banking Study, the FDIC states that small banks [...] have specialized knowledge of their local community and their customers. Because of this expertise, community banks tend to base credit decisions on local knowledge and nonstandard data obtained through long-term relationships [...] This relationship approach to lending is particularly important to small businesses that rely on community banks for loans and other services.  Further, this granular information is thought of as containing early signals of changing business cycle conditions. In her address to the 2017 convention of the Community Bankers Association of Ohio, President Loretta J. Mester of the Federal Reserve Bank of Cleveland states: Because of their important work, community bankers are among the most knowledgeable about changes in conditions on the ground in local areas. Such information often takes much longer to show up in o¢ cial statistical reports. So I …nd the insights gained from speaking with bankers to be especially valuable as part of the mosaic of information I use in formulating my views on appropriate monetary policy.  In this article, our objective is to explore US commercial banking data and look for signs consistent with these widely held views about the special role of small banks. In particular, from these views we extract two hypotheses. First, if small banks indeed have access to some unique business cycle information not available to other …nancial intermediaries, then one could expect to see di¤erences in how small and large banks’pro…t margins react to changes in business cycle conditions. Second, if the advance information available to small banks comes from long-term relationships with local businesses, i.e., the borrowers, one could expect these di¤erences to appear on the asset side of the banks’ balance sheet rather than on the liability side. In the empirical literature on bank pro…tability, the net interest margin (NIM) is perhaps the most commonly used pro…t margin indicator. NIM is de…ned as the ratio of a bank’s net interest income and average earning assets. Net interest income, in turn, is the di¤erence 3 Similar views have been expressed by other Federal Reserve o¢ cials; see, e.g., Keeton et al. (2003) and Mester (2017).  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins  3  between interest earned on assets and interest incurred on liabilities. The cyclical properties of NIM have been studied extensively in aggregate US banking sector data. Using administrative data collected by the FDIC, Aliaga-Díaz and Olivero (2010, 2011) and Beabrun-Diant and Tripier (2015) show that the sector’s NIM is countercyclical. However, little is known about any heterogeneity in the cyclical properties of NIM between small and large banks. In the …rst part of this article, we consider the …rst of our two hypotheses. We investigate the cyclical properties of NIM in the banking sector as a whole and, separately, among small and large banks. Our baseline de…nition of a small bank follows Kashyap and Stein (2000), where small banks are de…ned as all institutions below the 95th percentile of the size distribution by assets.4 We …nd that there is a signi…cant di¤erence in the cyclical behavior of NIM at large and small banks. Among large banks, similar to the earlier estimates obtained for all banks, the cyclical component of NIM exhibits negative correlation with the cyclical component of gross domestic product (GDP). We …nd a point estimate of this correlation coe¢ cient of -0.33. Our main …nding in this article is that, by contrast, among small banks, i.e., 95 percent of all banks, NIM is positively correlated with the business cycle. Among small banks, the estimated correlation between the cyclical components of average NIM and GDP is 0.34. This empirical …nding documents a signi…cant di¤erence between small and large banks, consistent with the view that the timing of business cycle signals received by small banks is di¤erent from the aggregate statistics. It is important, however, not to overstate our result. In this article, we report a di¤erence in the correlations of small and large banks’NIM with GDP without establishing causation. In the second part of this article, we consider our second hypothesis, which points to the asset side of the balance sheet as the source of the discrepancy between the cyclical properties of small and large banks’ NIM. To investigate this hypothesis, we decompose the correlation of the net interest margin into a weighted average of correlations of the average yield on assets (interest income over average assets) and average funding costs (interest expense over average assets). We …nd that the cyclical properties of the average yield on assets are virtually identical at small and large banks. We estimate the correlation 4 This de…nition of a small bank is also consistent with the de…nition of a community bank used recently by Federal Deposit Insurance Corporation (2012). It falls in between the traditional absolute community bank size cuto¤ of $1 billion in assets and the more recently used $10 billion. Our results are fairly robust to the choice of this relative cuto¤, as we discuss in Section 5.  4  Federal Reserve Bank of Richmond Economic Quarterly  between the asset yield and GDP to be positive and virtually of the same magnitude at small and large banks. Di¤erences exist in the cyclical properties of the funding costs and in the weights with which the asset-yield and funding-cost correlations contribute to the cyclicality of NIM. While funding costs are procyclical among both small and large banks, the small banks’correlation is much lower. In particular, it is lower than the correlation of the average asset yield, consistent with the small banks’ NIM being procyclical. Large banks’NIM, in turn, is countercyclical because their funding costs are more strongly procyclical than their average asset yield. These …ndings point to the liability side of the small banks’balance sheet as the source of their procyclical pro…t margin. In this way, these …ndings seem to be at odds with the view that it is the small banks’ close relationships with their borrowers that gives small banks a special role in the intermediation of credit. Rather, these correlations point to the small banks’ relationships with their depositors. Consistent with this hypothesis, we show that small banks rely on deposits for their funding signi…cantly more than large banks, while the compositions of small and large banks’asset portfolios are less dissimilar. Further, we …nd that the di¤erence in the GDP correlations of asset income and funding costs is magni…ed among small banks by the relatively high magnitude of the weights with which these correlations contribute to the overall correlation of NIM with GDP. We attribute the magnitude of these weights to the lower volatility of NIM among small banks, which in turn can be accounted for by stronger correlation between small banks’average asset yield and funding costs. In sum, we view the fact that the cyclical behavior of net interest margins among small banks is very di¤erent from that of large banks as evidence supporting the view that small banks ful…ll a special role in the intermediation of credit in the US. However, our decomposition of the cyclicality of NIM into cyclical correlations of the asset and liability sides of the balance sheet raises an interesting question about the role of the cost of deposit funding for the behavior of net interest margins at small banks. The relative cyclical insensitivity of the small banks’ average funding costs and their strong reliance on deposit funding suggest that the “sticky” properties of deposits documented by Driscoll and Judson (2013), Drechsler et al. (2017), and others could be more pronounced among small banks. Related literature The paper in the literature most closely related to this article is Aliaga-Díaz and Olivero (2010). They show the countercyclicality of the NIM in the aggregate US banking sector both unconditionally and after conditioning on a set of monetary policy proxy variables. Our analysis here is disaggregated to small and large  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins  5  banks. We present unconditional correlations in the text and include conditional regression analysis in Appendix B. Our main …nding is the procyclicality of the average NIM among small banks. There is a large empirical literature on the pro…tability of banks.5 Most of these studies, however, use aggregate country-level data or conduct cross-country comparisons, e.g., Albertazzi and Gambacorta (2009) and Borio et al. (2017). In particular, Claessens et al. (2017) document low levels of net interest margins in the current low nominal rate environment in a cross section of banks from forty-seven countries. In the US data, Covas et al. (2015) demonstrate the di¤erence in the level of NIM between small and large banks in the last twenty years, as well as the widening of this spread since 2010. They do not systematically investigate the cyclical properties of NIM at small banks, which is our focus here. In a recent working paper, Haubrich (2018) studies cyclical properties of banks’capital ratios. Similar to our …ndings on NIM, he …nds di¤erences in the cyclical properties of small and large banks’ capital bu¤ers. Another related strand of the empirical literature on banking studies the role banks may play in the ampli…cation of macroeconomic shocks and transmission of monetary policy shocks. Most of this literature, however, studies sector aggregates without disaggregating to small and large banks. In an in‡uential exception, Kashyap and Stein (2000) examine the response of bank lending to changes in the stance of monetary policy. They …nd support for the so-called bank-lending transmission channel, where banks reduce the supply of loans to …rms in the wake of a reduction in the supply of reserves. In particular, they …nd that this e¤ect is stronger among small banks, whose balance sheets are relatively less liquid. Drechsler et al. (2017, 2018) document banks’ market power in deposit markets by showing weak pass-through of changes in shortterm interbank funding rates to deposit rates. They demonstrate a new transmission channel of monetary policy, where deposits ‡ow out of the banking system when bond yields rise, to which banks respond with a contraction in lending. They show that the pass-through is weaker in more concentrated local markets, which is consistent with our …ndings because, as shown in Meyer (2018), small banks tend to operate in more concentrated local markets. The article is organized as follows. Section 1 introduces the data used in our analysis. Sections 2 and 3 discuss the aggregate behavior 5 A separate literature surveyed in Hughes and Mester (2014) uses structural models to measure e¢ ciency in banking.  6  Federal Reserve Bank of Richmond Economic Quarterly  of the NIM. Section 4 takes a look at the size distribution of banks. Section 5 discusses the cyclical behavior of NIM among small and large banks. Section 6 decomposes the cycliclity of NIM into cyclical properties of the asset and the liability sides of the balance sheet. Section 7 discusses the heterogeneity of small and large banks’funding sources and asset portfolios. Section 8 concludes.  1.  DATA  The primary data source for our analysis is the FDIC’s Statistics on Depository Institutions (SDI) dataset. The SDI contains quarterly Call Report data for all active FDIC-reporting institutions. Our sample starts in 1992:Q4 and ends in 2016:Q2. The SDI provides the NIM along with its components: interest income, interest expense, and earning assets. This level of detail allows us to compute NIM on earning assets held by small and large institutions, discuss the interest income side and the interest expense side of NIM separately, and show di¤erences in the composition of small and large banks’balance sheets. To measure the business cycle, we use GDP reported by the Bureau of Economic Analysis (BEA). We convert GDP, and all FDIC data, into 2009:Q1 dollars using the BEA’s implicit price de‡ator.6  2.  NIM AT THE BANK AND SECTOR LEVEL  At the bank level, NIM in quarter t is de…ned as T IIi;t T IEi;t ; N IMi;t := AEAi;t where T IIi;t is the total interest income and T IEi;t is the total interest expense for bank i in period t.7 Average earning assets, AEAi;t , is the average of quarter t’s and quarter t 1’s total earning assets for bank i. As we see, the NIM represents an average spread between the rates at which a bank earns income on its assets and the rates at which it funds itself. For the banking sector as a whole, we compute NIM as an average of individual banks’NIM weighted by average earning assets: X AEAi;t i T IIi;t i T IEi;t = : (1) N IMt := N IMi;t i AEAi;t i AEAi;t i  6  In Appendix B, we also use the following covariates to check robustness of the correlations we …nd: the slope of the yield curve, the e¤ective federal funds rate, and the yield on 3-month commercial …nancial paper. These series are provided by the Federal Reserve Board. 7 The di¤erence T IIi;t T IEi;t is referred to as net interest income.  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins  7  Figure 1 NIM for All Banks  Note: Annualized percentage rate calculated by multiplying quarterly NIM by 400.  The second equality above shows that this calculation is equivalent to computing aggregate TII and TIE and dividing their di¤erence by aggregate AEA. Figure 1 plots NIM for the US banking sector from 1992:Q4 through 2016:Q2. Aggregate NIM has clearly been declining since the beginning of our sample, but with a signi…cant amount of volatility, especially around the time of the Great Recession. The negative trend in NIM seen in our sample is persistent but not permanent. Using annual data that go back to 1970, Ennis (2004) shows a trough of the ratio of aggregate net interest income to bank assets in 1975 and an upward trend between 1975 and 1992, where our sample starts.  % of Total Assets % of Interest Income  53.18 70.66  Domestic O¢ ce Loans 3.67 5.33  Foreign O¢ ce Loans 4.76 2.43  Trading Accounts 19.18 16.47  Securities  1.17 1.40  Financing Receivables  Table 1 Average Assets and Interest Income by Source  3.76 2.07  Federal Funds Sold  7.17 1.23  Balance due from Dep Inst  7.11 0.40  Other  8 Federal Reserve Bank of Richmond Economic Quarterly  % of Total Liabilities % of Interest Expense  66.76 55.64  Domestic O¢ ce Deposits 10.40 12.10  Foreign O¢ ce Deposits 6.25 8.76  Federal Funds Purchased  9.80 20.70  Demand Notes and Other  Table 2 Average Liabilities and Interest Expenses by Source  1.14 2.77  Subordinated Notes and Debentures  5.65 0.03  Other  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 9  10  Federal Reserve Bank of Richmond Economic Quarterly  Components of the NIM To provide some insight into the structure of the NIM, in Tables 1 and 2 we present average interest income and interest expense by source, as reported by banks to the FDIC. In addition, we provide the corresponding average assets and liabilities as a fraction of total earning assets and total liabilities. The percentages reported are computed as sample time averages weighted by the total size of the banking sector interest income (T IIt ).8 As we see in Table 1, loans (domestic and foreign o¢ ce loans combined) are the largest asset class, followed by securities. In the aggregate, banks in our sample earned 76 percent of their interest income from loans. Loans represented, on average, 57 percent of total assets.9 Securities generated on average 16 percent of interest income and represented 19 percent of total assets. As for liabilities, we see in Table 2 that 68 percent of interest expenses arose from deposits (domestic and foreign o¢ ce combined), which represented 77 percent of total average liabilities. Next, we turn to the cyclical properties of the NIM.  3.  CYCLICAL PROPERTIES OF THE NIM: AGGREGATE  Figure 2 plots the cyclical components of NIM and of log GDP.10 The NIM deviation from trend ranges from negative 67 basis points in 2009:Q1 to positive 32 basis points in 2010:Q1. The negative correlation between NIM and GDP is rather clearly visible in this picture. Indeed, the estimated Pearson correlation coe¢ cient, presented in Table 3, is -0.30. This estimate is statistically signi…cant with a p-value of 0.003 indicating a countercyclical relationship between NIM and GDP. This …nding is similar to Aliaga-Díaz and Olivero (2010, 2011) and Beaubrun-Diant and Tripier (2015), who also …nd a negative and statistically signi…cant correlation between NIM and GDP.11 8  The weighted time average of income from a particular asset class is calculated as T IIt ; where c refers to the asset class considered, e.g., domestic o¢ ce loans, s T IIs and t is the calendar-time quarter. 9 Table C1 provides a glossary of all variables used in the calculations. 10 We take the Hodrick-Prescott trend out from the level of aggregate NIM and from the log of real GDP. Our results do not depend on this method of detrending. In Appendix A, we discuss robustness to the alternative technique for removing trend that follows Hamilton (2016). 11 It is worth pointing out that the Great Recession does not drive this result. In the sample ending in 2005, Aliaga-Díaz and Olivero (2010) estimate this correlation at 0:237. IIc;t t T II t  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 11  Figure 2 Cyclical Components of GDP and NIM for All Banks  Note: Left scale: log deviations of GDP from trend. Right scale: percentage point deviations of NIM from trend.  In addition, Table 3 presents the correlation between NIM and GDP forwarded and lagged by one, two, four, and eight quarters. These lead-lag correlations show that, in this sample, GDP predicts NIM at durations up to one year, while NIM is not a signi…cant predictor of GDP.  4.  THE BANK SIZE DISTRIBUTION  As noted by Cuciniello and Signoretti (2015) and Ennis et al. (2016), among others, the aggregate statistics for the banking sector are primarily driven by large banks. The strong and growing concentration of assets in a small number of large banks is a well-known feature of the bank size distribution (see, e.g., McCord and Prescott [2014]).  0.0288 0.7913  -0.4812 0.0000  L4Q -0.5667 0.0000  L2Q -0.4855 0.0000  L1Q -0.3031 0.0030  GDP  -0.1671 0.1095  F1Q  -0.1236 0.2405  F2Q  0.0173 0.8718  F4Q  Note: Lead-lag correlations reported for one, two, four, and eight quarters. P-value reported in italics.  NIM  L8Q  Table 3 Correlation of NIM and GDP: All Banks 1992:Q4 through 2016:Q2  0.1478 0.1743  F8Q  12 Federal Reserve Bank of Richmond Economic Quarterly  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 13  Figure 3 Fraction of Total Banking Assets Held by the Largest Banks  Figure 3 displays the fraction of total assets in the US banking sector held by the largest 5, 2, and 1 percent of banks, as well as by the twenty largest institutions.12 The growing bank asset concentration is visible by all four of these measures. In 2016:Q2, 88 percent of all sector assets were held by the top 5 percent of institutions, up from 71 percent in 1992:Q4. The largest twenty banks held 62 percent of assets in 2016:Q2, up from 24 percent in 1992. In this article, we de…ne small banks as those outside the top 5 percent of the asset size distribution in any given time period (i.e., calendar quarter) in our dataset. Large banks are de…ned as those inside the …fth percentile.13 12 The number of banks reporting to the FDIC decreased from approximately 14,000 in 1992:Q4 to 6,000 in 2016:Q2. Thus, the top 5 percent of banks in 2016 equals about 280 banks. The asset size cuto¤ thresholds, also at the end of the sample, are approximately $2.9B, $9.4B, $27.3B, and $123B for, respectively, the top 5 percent, 2 percent, 1 percent, and top twenty banks, all in current dollars. 13 We also report our correlation results for the top 2 and 1 percentile cuto¤s, as well as for the top twenty banks.  14  Federal Reserve Bank of Richmond Economic Quarterly  Figure 4 NIM for Large and Small Banks (95 percent cuto )  Since large banks hold the bulk of the sector’s assets, it is clear that aggregate banking sector statistics mirror those of large banks. In particular, we should expect the cyclical properties of NIM for large banks to be similar to those previously reported for the whole banking sector. In the next section, we examine cyclical properties of NIM for small banks and show that they are quite di¤erent from the aggregate.  5.  CYCLICAL PROPERTIES OF NIM: SMALL VERSUS LARGE BANKS  We start out by presenting the time series of the level of NIM among small and large banks in our sample period. Figure 4 plots NIM levels for banks below and above the 95th percentile of the asset distribution. Note that the level of NIM at small banks is greater than the NIM at large banks in almost all quarters in the sample. This feature may re‡ect some local monopoly power of small banks in their local areas,  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 15  Figure 5 Cyclical Component of GDP and NIM by Bank Size  which could be o¤set by higher …xed cost, per dollar of assets, of smaller institutions.14 Covas et al. (2015) attribute the recent (since 2010) compression of NIM at large banks to smaller declines in funding costs relative to small banks and to declining interest income from short-term trading accounts, which makes up a greater percentage of interest income for large banks than for small banks. We now turn to the cyclical component of the NIM at small and large banks.15 Figure 5 plots the cyclical component of the NIM separately for the banks in the bottom 95 percent of the size distribution and in the top 5 percent. As we see in that …gure, the time series of the NIM at large banks tracks very closely the aggregate NIM series shown in Figure 2. This indicates that the aggregate negative correlation between the NIM and GDP also holds for large banks. However, the NIM at small banks appears to be much less volatile and its comovement with GDP is unclear from the graph. 14  See Drechsler et al. (2018). For subgroups of banks, the cyclical component on NIM is computed the same way as the aggregate NIM series, i.e., by removing the Hodrick-Prescott trend. 15  16  Federal Reserve Bank of Richmond Economic Quarterly  Table 4 Correlation of NIM and GDP by Varying Size Cuto  Size Percentile Cuto¤  Below  Above  95  0.3363 0.0009 0.2549 0.0132 0.2287 0.0266 -0.1139 0.2742  -0.3328 0.0010 -0.3361 0.0009 -0.3346 0.0010 -0.2948 0.0076  98 99 Top 20  Note: P-value reported in italics.  Table 4 presents the estimated Pearson correlation of the NIM with GDP at large and small banks, where small banks are de…ned by four di¤erent size cuto¤s. As expected, large banks have a statistically signi…cant and negative correlation with GDP at all four cuto¤s. The point estimates of the correlation coe¢ cient are close to -0.3, i.e., similar to the aggregate correlation presented in Table 3. The correlation of small banks’ NIM with GDP, however, is very di¤erent. As we see in Table 4, the average NIM in the group of all banks below the top 5, top 2, or top 1 percentile of the size distribution exhibits a statistically signi…cant and positive correlation with GDP. That is, the sign of the correlation is opposite of that of the large banks and the aggregate. If the size cuto¤ is chosen to be just the twenty largest institutions, the correlation between NIM and GDP for small banks is no longer statistically di¤erent from zero. The results presented in Table 4 highlight our main …nding: in contrast to large banks and the aggregate, small banks’NIM is procyclical. The correlations presented here are unconditional. In Appendix B, we run several regressions to test the robustness of our …ndings by conditioning on several standard covariates. After controlling for the slope of the yield curve, the level of the federal funds rate, the commercial paper spread, and an indicator of the general stance of monetary policy, our result is unchanged: the NIM at small, i.e., the majority of, banks is procyclical while at large banks, and in the aggregate series, it is countercyclical.  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 17 In the remainder of this article, we explore the sources of this difference in the cyclical properties of NIM among small and large banks. In particular, we ask if this di¤erence can be attributed to the cyclical properties of the asset side or the liability side of the banks’ balance sheet.  6.  CYCLICAL PROPERTIES OF INTEREST INCOME AND INTEREST EXPENSE  In this section, we discuss the cyclicality of NIM in terms of the cyclicality of interest income on the asset side and the cyclicality of the interest expense on the liability side of the balance sheet. For a given quarter t, we de…ne average asset yield (AAY) among T II banks as AAYt := iiAEAi;ti;t and average funding cost (AFC) as AF Ct := i T IEi;t i AEAi;t  . From equation (1), we have N IMt = AAYt  AF Ct  in every quarter t. We want to use this decomposition of the level of NIM to decompose the correlation of NIM with GDP into the corresponding correlations of AAY and AFC with GDP. Denoting these correlations by, respectively, N , A , and L , we have N  A  =  N  L A  N  L;  (2)  where N , A , and L denote the respective standard deviations.16 As we see, the NIM correlation is the di¤erence between the correlations of AAY and AFC weighted by standard deviations of AAY and AFC relative to the standard deviation of NIM.  Cyclicality of AAY and AFC among large and small banks Table 5 presents the components of this decomposition estimated for all banks in the sample as well as separately for large and small banks. We start with the following three observations. First, AAY and AFC are strongly procyclical among both large and small banks. Second, the magnitude of the correlation between AAY and GDP is virtually 16  This formula follows from  cov(AEY;GDP ) N  Y  ation of GDP.  =  A A Y N Y  L L Y N Y  =  N  =  A N  A  cov(N IM;GDP ) N L N  L,  =  Y  where  Y  cov(AAY;GDP ) N  Y  is the standard devi-  18  Federal Reserve Bank of Richmond Economic Quarterly  Table 5 Decomposition of the NIM Correlation for All, Top 5 Percent, and Bottom 95 Percent of Banks  N  All banks Large banks Small banks  -0.30 -0.33 0.33  A N  3.25 2.94 5.11  A  0.58 0.57 0.55  L N  3.31 3.00 5.02  L  0.66 0.68 0.49  Note: All p-values (not reported) are smaller than 0.001  the same among large and small banks. Third, AFC is signi…cantly less cyclical at small banks than at large ones. From these observations, we can conclude that it is the liability side of the balance sheet that drives our main result concerning the di¤erence in the cyclical properties of NIM at large and small banks. With the correlation of the average yield on assets being virtually the same, small banks’NIM is less cyclical than large banks’because small banks’ funding costs, AFC, are more stable, i.e, their comovement with the business cycle is weaker. In particular, small banks’NIM is procyclical because their average interest expenses are less strongly correlated with the business cycle than their average asset yields. Among large banks, in contrast, the funding costs are more strongly procyclical than the asset yields, implying countercyclical NIM. Table 5 presents the decomposition of the NIM correlation for small and large banks de…ned by the 95 percent cuto¤ in the bank size distribution. Our conclusion about the relative importance of AFC for explaining the di¤erences in the cyclical properties of NIM between small and large banks holds also for other cuto¤ thresholds. Table 6 presents the correlations of NIM, AAY, and AFC at small banks de…ned by four size cuto¤s. The …rst column, similar to Table 4, shows that the procyclicality of NIM among small banks becomes weaker as the de…nition of a small bank encompasses larger and larger institutions. The second and third columns show that the correlation of AAY does not change across the four de…nitions while the correlation of AFC becomes stronger, accounting for the procyclicality of small banks’ NIM becoming weaker and eventually, among all but the top twenty banks, becoming insigni…cant.17 17  The ratios A and L , which we do not report here for all size cuto¤s, do not N N depend strongly on the size cuto¤.  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 19  Table 6 Correlations of NIM, AAY, and AFC Among Small Banks De ned by Four Size Cuto s Percentile Cuto¤ 95 98 99 top 20  N  0.33 0.25 0.22 -0.11  A  0.55 0.56 0.57 0.58  L  0.49 0.51 0.53 0.59  Note: The p-value for N at top twenty banks is 0.27. All other estimates are signi…cantly di¤erent from zero.  Volatility of AAY and AFC among large and small banks In the decomposition given in Equation (2), the NIM correlation H depends on the correlations of asset yields and funding costs, but also on the weights NA and NL that multiply these correlations. The estimated decomposition presented in Table 5 lets us make the following two observations about these weights. First, in each decomposition (i.e., for all, large, and small banks), the weights on the asset yields and funding costs are similar. Clearly, this means the standard deviations of AAY and AFC, A and L , are similar in each group of institutions. Second, in the decomposition for large banks, the weights are slightly smaller than in the decomposition for all banks, while in the decomposition for small banks, the weights are much larger. The size of these weights, clearly, a¤ects the magnitude of the NIM correlation. In particular, the di¤erences in the small and large banks’ weights account for the large di¤erence in the correlation of small and large banks’NIM despite the di¤erence in the correlation of AFC being relatively small (and AAY virtually nonexistent). In this section, we ask if the large weights in the decomposition for small banks are due to high volatility of asset yields and funding costs or due to low volatility of small banks’NIM. Table 7 provides estimates of the standard deviation, i.e., volatility, of NIM, AAY, and AFC for all banks in the sample as well as for large and small banks separately.18 Clearly, small banks have lower volatility on all three measures. This leads us to conclude that the weights for 18  In this section, we present the results for largest 5 percent and smallest 95 percent of institutions. The estimates using the top 2 percent and the top 1 percent size cuto¤s are similar.  20  Federal Reserve Bank of Richmond Economic Quarterly  Table 7 Standard Deviations of NIM and its Components for All, Top 5 Percent, and Bottom 95 Percent of Banks  N  All banks Large banks Small banks  A  0.37 0.44 0.19  1.21 1.28 0.97  L  1.24 1.31 0.95  Note: Values reported multiplied by 103 . All p-values (not reported) are smaller than 0.001  the small banks are large because their NIM is less volatile, despite the fact that the standard deviations of their asset yields and funding costs are lower. Further, Table 7 shows that the standard deviation of NIM equals about a third of the standard deviation of asset yields or funding costs among large banks but only about a …fth among small banks. This suggests that asset yields and funding are more strongly correlated with each other for small banks than they are for large ones. Using the identity 2 N  =  2 A  +  2 L  2  AL A L ;  we can compute the correlation between average asset yields and average funding costs, AL , from the estimates of the standard deviations N , A , and L given in Table 7. We …nd that, indeed, AAY and AFC are more closely correlated at small banks. The computed correlation is AL = 0:955 for all banks, 0:942 for large banks, and 0:981 for small banks.  7.  BALANCE SHEET COMPOSITION AT SMALL AND LARGE BANKS  Having shown di¤erences in the large and small banks’ cyclical properties of NIM, AAY, and AFC, we now ask if these di¤erences are re‡ected in the composition of the average balance sheet of small and large banks.19 Section 2 introduced the main classes of assets and liabilities held by banks. On the one hand, it is possible that small and large banks’ 19 Debbaut and Ennis (2014) provide a detailed description of the average balance sheet of large US bank-holding companies between 2005 and 2011.  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 21  Table 8 Breakdown of Assets by Class as a Percent of Total Assets  Asset Cuto¤  Domestic O¢ ce Loans  Foreign O¢ ce Loans  Trading Accounts  Securities  Other  Aggregate  53.18  3.67  4.76  19.18  19.21  64.01 51.10 64.16 49.89 63.75 48.60 63.00 43.53  0.05 4.37 0.11 4.74 0.16 5.19 0.43 6.86  0.04 5.66 0.08 6.16 0.14 6.75 0.55 8.89  23.37 18.37 23.11 18.00 22.86 17.59 21.25 17.15  12.53 20.50 12.54 21.21 13.09 21.87 14.77 23.57  95 98 99 Top 20  Below Above Below Above Below Above Below Above  Note: Numbers reported are percents, averaged over time.  NIM, AAY, and AFC di¤er in their business cycle correlations because assets and liabilities held by large and small banks have inherently di¤erent business cycle properties. On the other hand, it is possible that assets and liabilities of small and large banks are homogeneous but are simply held by small and large banks in di¤erent proportions. Although a full investigation of these alternatives is beyond the scope of this article, we take a step toward it by presenting the average balance sheet composition of small and large banks. Table 8 presents average asset portfolios for large and small banks over our sample period, where small banks are de…ned by four separate size cuto¤s.20 The asset classes shown are domestic o¢ ce loans, foreign o¢ ce loans, trading accounts, securities, and other. The other class consists of balances from depository institutions, federal funds sold, and lease …nancing receivables.21 We observe that small banks hold a larger proportion of domestic o¢ ce loans and securities than large banks, which in turn hold relatively more foreign o¢ ce loans, trading accounts, and other. The di¤erences in the asset composition, however, are not large. The average bank allocates 53 percent of assets to domestic o¢ ce loans, the largest asset class across the board, while the average small bank (below the 95 percent cuto¤) and the average large bank (top 5 20 The small/large banks’ average portfolio is a size-weighted sample-period average of the portfolios of all banks below/above the respective cuto¤. 21 In Table 1, these subclasses are broken out separately but only reported for the aggregate (i.e., all banks) average portfolio.  22  Federal Reserve Bank of Richmond Economic Quarterly  percent) allocate, respectively, 64 and 51 percent of assets to domestic o¢ ce loans. Further, these allocations do not vary with the size cuto¤ by much. Di¤erences between large and small banks are more pronounced on the liabilities side of the balance sheet, presented in Table 9. Looking at the 95 percent size cuto¤, domestic o¢ ce deposits account for 91 percent of total liabilities for small banks and only 62 percent for large banks. Large banks hold signi…cant amounts of foreign o¢ ce deposits, which are virtually unheld by small banks. Still, total deposits constitute a much smaller fraction of liabilities at large banks than at small. Further, the di¤erences in the composition of liabilities between small and large banks depend strongly on the size cuto¤. The average bank in the top percentile of the size distribution allocates less than 59 percent of liabilities to the domestic deposit class. These observations lead us to conclude that balance sheet composition is an important factor behind the di¤erences in the cyclical properties of asset yields, liability costs, and, consequently, net margins at small and large banks. Section 6 shows that small and large banks di¤er in the cyclical properties of their funding costs but are virtually homogeneous with respect to the cyclical properties of their asset returns. Correspondingly, Table 9 shows signi…cant di¤erences between small and large banks in the composition of liabilities, while Table 8 shows that the di¤erences in the composition of assets are much smaller. While composition does not explain everything, these observations suggest it is important. In particular, the high proportion of funding obtained from domestic deposits seems to be important for the lower cyclicality of small banks’cost of funding and, therefore, also for the observed procyclicality of their NIM.  91.09 62.10 88.23 60.35 85.14 58.82 78.56 55.25  95  0.14 12.37 0.30 13.41 0.53 14.66 2.09 18.50  10.40  Foreign O¢ ce Deposits  Note: Numbers reported are percents, averaged over time.  Top 20  99  98  66.76  Aggregate Below Above Below Above Below Above Below Above  Domestic O¢ ce Deposits  Asset Cuto¤  2.20 7.03 3.33 7.12 4.44 7.03 5.89 6.60  6.25  Federal Funds Purchased  5.44 10.63 6.67 10.73 8.08 10.54 10.58 9.03  9.80  Demand Notes  Table 9 Breakdown of Liabilities by Source as a Percent of Total Liabilities  0.04 1.35 0.10 1.45 0.20 1.55 0.58 1.69  1.14  Subordinated Notes and Debentures  1.09 6.52 1.37 6.94 1.61 7.40 2.30 8.93  5.65  Other  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 23  24 8.  Federal Reserve Bank of Richmond Economic Quarterly  CONCLUSIONS  In this article, we analyze the comovement of bank NIMs with the business cycle both at the sector level and disaggregated for large and small banks. We …nd that the cyclical component of NIM among large and small banks responds di¤erently to business cycle ‡uctuations. Specifically, while the average NIM at large banks (the top 5 percent of the size distribution by assets) is negatively correlated with the business cycle, the average NIM at small banks (the bottom 95 percent of the size distribution) is positively correlated with GDP. Due to the high degree of concentration of asset holdings in the banking sector, the aggregate, sector-wide correlation is nearly the same as that of the largest 5 percent of institutions, standing at about -0.3. The correlation computed for the bottom 95 percent banks is of the opposite sign and nearly the same magnitude, i.e., it stands at about +0.3. To our knowledge, our …ndings on small banks are novel to the literature. In an Appendix, we show our results to be robust to the detrending method and introducing controls for the stance of monetary policy. We consider this …nding to be broadly supportive of the widely held view that small banks occupy a special role in the intermediation of credit. However, when we decompose the cyclical properties of NIM into the asset and liability sides of the balance sheet, we …nd the liability side to be the driver. This points us to attribute the small banks’special role to their ability to keep their funding costs relatively insensitive to the business cycle rather than their ability to extract business-cyclerelevant information from their long-term relationships with borrowers.  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 25  APPENDIX  A: DETRENDING METHOD APPENDIX: A: DETRENDING METHOD In this article, we detrend our time series data with the HodrickPrescott (HP) …lter. In this section of the Appendix, we consider an alternative detrending method proposed in Hamilton (2016), which de…nes the cyclical component as the deviation of actual data from their predicted values based on a linear projection from their own lags. For quarterly data, we follow Hamilton (2016) in using an eight-quarter ahead projection from a regression including four lags. Our results are robust to using this de…nition of the cyclical component of GDP and NIM. The estimated aggregate correlation coe¢ cient is -0.4484 with the p-value of 0.0000, which shows a stronger negative correlation than the one we report in Table 3. Similarly, when using the Hamilton method of detrending and recomputing the correlation between the cyclical components of GDP and NIM for small and large banks separately, i.e., the correlations presented in Table 4, we get estimates consistent with those obtained using the HP …lter up to the 98th percentile asset cuto¤, which indicates robustness of our results to the method of detrending. Likewise, our results concerning the asset- and liability-adjusted measures of NIM are robust. Detailed estimates of these correlations with the Hamilton detrending method are available upon request.  B: COVARIATES  APPENDIX:  B: COVARIATES  In this section, we use a multivariate regression to check robustness of our correlation results to the stance of monetary policy. We consider the following four control variables: the slope of the yield curve, measured as the di¤erence in the yields on 10-year and 3-month Treasury securities; the level of the e¤ective federal funds rate; the spread on 3-month …nancial commercial paper over the 3-month Treasury yield;  26  Federal Reserve Bank of Richmond Economic Quarterly  and a simple indicator variable marking periods of contractionary, expansionary, and neutral monetary policy. The policy-stance indicator is constructed, similar to Ennis et al. (2016), via two dummy variables: one indicating periods of increasing rates and the other periods of decreasing rates. In addition, the post2007 period of zero nominal rates and large reserves outstanding is classi…ed as a loose monetary policy stance. The e¤ective federal funds rate path is presented in Figure B1 along with color-coded periods of contractionary, expansionary, and neutral monetary policy stances. We run two regression speci…cations, with each speci…cation consisting of independent regressions for large and small banks. In the …rst speci…cation, presented in Table B1, we regress the cyclical component of the NIM on the cyclical component of GDP, one of our four monetary policy proxy variables, and an interaction term.22 In all cases, the coe¢ cient on the GDP variable is negative for large banks and positive for small banks, con…rming our unconditional correlation results. When using the e¤ective federal funds rate or the commercial paper spread, these coe¢ cients are statistically signi…cant at the 5 percent level. When using the slope of the yield curve or our dummy variables as proxies for the stance of monetary policy, some coe¢ cients become statistically insigni…cant at the 5 percent level. In our second speci…cation, presented in Table B2, we regress NIM on GDP and four lags of the proxy variable of interest. In all cases for small banks, the coe¢ cient on GDP is positive and statistically signi…cant at the 1 percent level, indicating that the positive correlation seen earlier is robust to controlling for monetary policy. Regarding large banks, the coe¢ cients on GDP are all negative and at least statistically signi…cant at the 10 percent level.23 The coe¢ cient is signi…cant at the 5 percent level when using the dummy variables for periods of increasing (decreasing) federal funds rate and is signi…cant at the 1 percent level when using the e¤ective federal funds rate and the spread on commercial …nancial paper. Overall, these results indicate to us that the conclusions reached in the main body of the paper are statistically robust to including controls for the stance of monetary policy. 22  As noted by Borio et al. (2017), among others, the impact of interest rates on the NIM can be nonlinear. By including the interaction term, we can capture potential changes in the slope coe¢ cient. 23 This conclusion is not overturned by the positive coe¢ cient on the interaction term with the credit spread variable in speci…cation (6) of the regression because the credit spread variable contains only the cyclical component of the spread, and, thus, it is very close to zero on average with small dispersion.  0.00298 (0.0165) 94 0.112  1.820* (0.970) -0.000466 (0.00831) 0.264 (0.468)  (1) Small Banks  -0.0732** (0.0330) 94 0.148  -0.490 (2.183) 0.0341** (0.0161) -1.567 (1.081)  (2) Large Banks  0.000308 (0.0112) 94 0.122  0.00121 (0.00299) -0.328 (0.267)  3.124** (1.276)  (3) Small Banks  0.0106 (0.0260) 94 0.116  -0.00520 (0.00755) 0.307 (0.613)  -5.721** (2.498)  (4) Large Banks  -0.000734 (0.00753) 94 0.130  -0.00281 (0.0415) 4.186 (3.342)  2.291*** (0.741)  (5) Small Banks  Note: Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1  Observations R-squared  Constant  2.FFR decrease #GDP  1.FFR increase #GDP  2.FFR decrease  ComPprFFRSpreadcyc #GDP 1.FFR increase  ComPprFFRSpreadcyc  FFR #GDP  FFR  Yield Curve Slop #GDP  Yield Curve Slope  GDP  Variables  Table B1 Regression of Cyclical NIM on Cyclical GDP and other Covariates with Interaction Terms  -0.00571 (0.0160) 94 0.251  -0.183 (0.123) 20.62** (8.274)  -5.084*** (1.368)  (6) Large Banks  0.00239 (0.0232) -0.0361** (0.0169) 3.668** (1.790) 0.629 (1.460) 0.0154 (0.0130) 94 0.211  0.430 (0.837)  (7) Small Banks  -0.0509 (0.0552) -0.00832 (0.0428) 5.867 (4.572) 0.857 (3.534) 0.00956 (0.0356) 94 0.124  -6.520** (2.883)  (8) Large Banks  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 27  Note: Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1  Table B2 Regression of Cyclical NIM on Cyclical GDP and other Lagged Covariates  28 Federal Reserve Bank of Richmond Economic Quarterly  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 29  Figure B1 Effective Federal Funds Rate  30  Federal Reserve Bank of Richmond Economic Quarterly  C: GLOSSARY Table C1 Glossary of Variables  Assets  Variable Name  FDIC Variable ID  Total Loans and Leases Total Loans & Leases held in Foreign O¢ ces Lease Financing Receivables Lease Fin. Receiv. held in Foreign O¢ ces Total Securities Trading Account Assets Cash Balances due from Depository Instit. Federal Funds Sold and Reverse Repo Total Assets Earning Assets  lnlsgr lnlsgrf ls lsfor sc trade chbal frepo asset ernast  Domestic O¢ ce Loans Foreign O¢ ce Loans Liabilities  Interest Income (II)  Interest Expense (IE)  Formula  lnlsgr-ls-(lnlsgrf-lsfor) lnlsgrf-lsfor  Deposits held in Domestic O¢ ces Deposits held in Foreign O¢ ces Subordinated Debt Other Borrowed Funds Federal Funds Purchased and Repo  depdom depfor subnd idobrmtg frepp  II from Domestic O¢ ce Loans II from Foreign O¢ ce Loans II from Lease Financing Receivables II from Securities II from Trading Account Assets II From Cash Balances due from Dep Insts II from Federal Funds Sold & Reverse Repo Other Interest Income Total Interest Income  ilndom ilnfor ils isc itrade ichbal ifrepo iothii intinc  IE from Deposits held in Domestic O¢ ces IE from Deposits held in Foreign O¢ ces IE from Subordinated Notes & Debentures IE from Other Borrowed Funds IE from Federal Funds Purchased & Repo Total Interest Expense  edepdom edepfor esubnd ettlotmg efrepp eintexp  Net Interest Income Net Interest Margin  nim nimy  intinc-eintexp n im e rn a st  Grochulski, Schwam, Zhang: Cyclical Properties of Bank Margins 31  REFERENCES Albertazzi, Ugo, and Leonardo Gambacorta. 2009. “Bank Pro…tability and the Business Cycle.” Journal of Financial Stability 5 (December): 393–409. 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Powell, Jerome H. 2016. “Trends in Community Bank Performance Over the Past 20 Years.” Speech at the “Community Banking in the 21st Century” Fourth Annual Community Banking Research and Policy Conference, St. Louis, Missouri, September 29.  Economic Quarterly— Volume 104, Number 1— First Quarter 2018— Pages 35–52  Self-Insurance and the Risk-Sharing Role of Money Tsz-Nga Wong  “The inherent vice of capitalism is the unequal sharing of blessings.” -Winston Churchill address to the House of Commons, 1945  M  oney is well-acknowledged as a social construct to overcome the lack of coincidence of wants in a society. On top of its transactional role, the literature of monetary theory has also pointed out the role of self-insurance following the fact that money can be a vehicle for precautionary saving.1 Less is mentioned about money as a social construct to promote risk sharing among individuals. In this review, I will provide a simple mathematical model to illustrate this new insight. The model can be solved in closed form with paper and pencil. The material here is borrowed from some recent studies.2 In my “toy model”there are a lot of agents; each receives a constant ‡ow of endowment but also faces uncertainty about the timing and the number of “liquidity shocks”: when the shock hits, the agent needs to spend a big chunk of endowment. In real life, the liquidity shock I have bene…ted from the comments of John Weinberg, Felix Ackon, Felipe Schwartzman, and Nicholas Trachter. The views expressed in this article are those of the author and do not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System. 1 See the seminal paper of Bewley (1980) and its corrigendum Bewley (1983) for the error in the existence proof pointed out by Hellwig (1982). It is well-known that precautionary motive can lead to e¢ ciency loss due to excessive saving; see Aiyagari (1994) and Davila et al. (2012). 2 I select and simplify some results from Rocheteau et al. (forthcoming); see therein for a general treatment. The discrete time setting in the introduction comes from Rocheteau et al. (2015). The discussion of perfect self-insurance in a monetary economy borrows from Wong (2016). See Lagos et al. (2017) for a recent survey on the literature of monetary theory; see Rocheteau and Nosal (2017) for a textbook introduction. The general welfare property of money with the risk-sharing role is still an open question; see Wallace (2014) for a conjecture.  DOI: https://doi.org/10.21144/eq1040102  36  Federal Reserve Bank of Richmond Economic Quarterly  captures unexpected expenditures like car accidents and medical expenses (a top reason for bankruptcy in America). Liquidity shock is idiosyncratic, so it comes early for some agents but late for others; some agents experience only a few shocks over a long period of time, but some agents are hit often.3 Agents can store their endowments, thus in principle, agents always can self-insure against their liquidity shocks with private storage. In the design of the toy model, I do not force agents to use money— agents choose between money and storage— so the model has the potential to explain the fundamental reason for the emergence of money rather than merely begging the question. Furthermore, there is always coincidence of wants: every agent owns and consumes the same goods. If money is socially useful in this model, then it must be due to other reasons— in particular, I will show mathematically that when individuals hold money, it helps share others’liquidity risks.  1.  MODEL  As a benchmark, I begin an economy without money. The economy is populated by a unit measure of risk-neutral agents. I …rst consider the time discrete and the horizon in…nite. I need an in…nite horizon for money to circulate in the latter section; otherwise, if the economy will terminate at a …xed date, then no agent will exchange his goods for money on the date before (since money has no future use), and hence on the date before before— by backward induction, no one will ever hold money in a …nite horizon.4 Timing. The timing follows the literature of banking theory (e.g. Diamond and Dybvig [1983]). Each period has two stages. In the …rst stage, some agents are hit by liquidity shocks: when the shock hits, the agent needs to cover an expense of y > 0 units of goods, otherwise she is subject to an increasing loss for the amount falling short. Mathematically, it is conveniently captured by a quadratic loss function, L (y) = 0:5A min (y y; 0)2 , where A > 0 captures the marginal loss and y is the amount of goods the agent can raise to cover the liquidity shock. I simply refer to y as early consumption as in the literature. Liquidity shocks are i.i.d. and occur with probability > 0. In the second stage, each agent receives h units of goods and consumes 3 In a neat setting, Scheinkman and Weiss (1986) cleverly assume shocks are stochastic but alternate between two types of agent. This captures the redistribution e¤ect, still the model remains highly tractable. Lippi et al. (2015) found that with this redistribution channel, the optimal monetary policy can be countercyclical. 4 The backward induction argument hinges on the discreteness of time. In the continuous time, money can circulate in a …nite horizon, but the monetary equilibrium, if it exists, is typically nonstationary. It is out of the scope of this review.  Wong: Self-Insurance and the Risk-Sharing Role of Money  37  c , pro rata to the length of period. In the absence of technologies enforcing and monitoring actions, debt contracts, either across stages or across periods, are not incentive feasible. However, agents can selfinsure against the liquidity shocks by storing any unconsumed goods in the second stage, subject to the depreciation rate . In sum, the period-utility function is "L (y) + c , where " 2 f0; 1g is the indicator of the liquidity shock with Pr (" = 1) = . The discount factor across periods is exp ( r ). The economy starts in the second stage of period 0. The order of events does not quite matter when time becomes continuous. Value. The choices of action are made with the following consideration. Contingent on the history of liquidity shocks, Ht f"1 ; "2 ; :::"t g, agents choose the level of consumption and storage prepared for potential early consumption. Denote such a contingent plan as the functions, C and Y , of history such that ct = C Ht and yt = Y Ht . Also, denote at as the level of storage brought to the second stage of t. Starting in the second stage of t, the value of storage is the expected discounted sum of the future utility and loss, which is given by ( ) 1 X Vt = max C Ht + Et e r s f"s L [Y (Hs )] + C (Hs ) g , C;Y  s=t+1  given at :  With bounded endowments and depreciating storage, the level of storage is always bounded. Together with the fact that the loss function, L (y), is bounded, the value of storage, Vt , cannot explode to in…nity (positive or negative), and hence it satis…es the asymptotic boundary condition that lims!1 Et exp ( r s) Vs = 0 almost surely. Then there exists a value function, V (a), such that the value of storage, Vt , can be recursively expressed by the following Bellman equation Vt = V (a) = max c 0 c;a ;y  +e  r  L (y) + V a0  y  + (1  ) V a0 (1)  s.t. at = a; a0 = (1 ) (h c 0; y 2 0; a0 :  c  + a) ;  Denote the solutions to (1) as c (a) and y (a). Instead of making use of the entire history, Ht , the current level of storage, at = a, is su¢ cient information for decision-making such that the agent’s choices are given by ct = c (a) and yt = y (a). This recursive structure will be useful for analyzing the agent’s in…nite-horizon problem.  38  Federal Reserve Bank of Richmond Economic Quarterly  The economic meaning of the Bellman equation (1) is as follows. According to (1), the typical agent chooses her consumption, c, the next-period storage (after depreciation applies), a0 , and the early consumption, y, in order to maximize her expected discounted continuation value in the next period. The budget identity speci…es that the next-period storage is equal to the current income net of consumption multiplied by the gross depreciation factor. With probability , the agent receives a liquidity shock for early consumption. The maximal early consumption the agent can draw on is her entire level of storage, i.e. y a0 . The choice of y balances the trade-o¤ between early consumption and leaving some storage for the future. With probability 1 , the agent is not hit by a liquidity shock and enters the second stage with a0 . In sum, to self-insure against the liquidity shocks, the agent wants to maintain a su¢ cient level of storage at the cost of giving up current consumption and wasting resources to depreciation. Self-insurance. Agents who are frequently hit by liquidity shocks consume less, hold less storage, and, in a vicious cycle, can become more vulnerable to future liquidity shocks. Should they share the liquidity risks, if possible? Actually, when their endowment, h , is su¢ ciently large, agents can achieve perfect self-insurance with constant storage. In this case, the marginal value of storage is constant at V 0 (a) = 1 for all a, and there is no need to share liquidity risks across agents in the economy. This case is well-studied in the literature after Lagos and Wright (2005).5 Here, I am interested in the case otherwise, but then the solution to (1) will feature occasionally binding constraints. To ease the analysis, I take the period length to zero and the time becomes continuous. In this case, the ‡ow of endowment, h , is so small compared to the magnitude of the liquidity shock, y, that agents can never achieve perfect self-insurance. The rate of storage in the continuous time is given by  a_  lim  !0  a0  a  =h  c  a:  5 The appearance of perfect self-insurance does not necessarily depend on the riskneutral assumption. Wong (2016) illustrates examples with strictly concave utility functions. See the discussion therein for details. The key to perfect self-insurance is that agents can reach the target level of storage immediately after shocks, either by adjusting labor supply, consumption, portfolio, or a combination of these. Of course, achieving perfect self-insurance does not mean the …rst best.  Wong: Self-Insurance and the Risk-Sharing Role of Money  39  In the continuous time, the value function solves the following HamiltonJacob-Bellman (HJB) equation instead6 : rV (a) =  max  c 0;y2[0;a]  c + V 0 (a) (h  c  a) +  [L (y) + V (a  y)  V (a)] ;  (2) where V 0 (a) denotes the …rst derivative of V (a). Unlike the case of perfect self-insurance, generically V 0 (a) is varying in a.7 The economic meaning of the HJB equation is as follows. When agents maximize their utility, the ‡ow of the agent’s value, rV (a), is equal to the ‡ow of the consumption utility, c, the rate the value changes due to storage, V 0 (a) a, _ and the expected change in the value due to the liquidity shocks, [L (y) + V (a y) V (a)]. For the later derivation of the distribution, de…ne (a) as the maximal level of storage such that after a liquidity shock the agent will keep a units of storage, given by (a)  max d s.t. d = y (d) + a;  (3)  which means that the preshock storage, , is equal to the sum of the early consumption, y ( ), and the postshock storage, a. In general, there can be multiple levels of preshock storage that lead to the same postshock storage; for example, when the agent always draws her entire storage for the early consumption, any level of preshock storage will lead to zero postshock storage. That is why in the de…nition of , I always pick the maximal one. I adopt the convention that (a) = 1 if no solution to (3) exists. Closed-form solutions. In general, the value function, V (a), is the solution to the delayed di¤erential equation (DDE), (2), satisfying the asymptotic boundary condition. Here, it is straightforward to verify 6  Heuristically, the HJB equation can be derived as follows. Rearranging (1), I have  V (a0 ) a0  V (a) a0 a  a  = max c + e  r  L (y) + V a0  y  V a0  +  e  r  1  V a0  When ! 0; I have a0 ! a. The right side above becomes max fc + [L (y) + V (a y) V (a)]g rV (a). The …rst term on the left side becomes V 0 (a) at the limit. The second term converges to h c a. Collecting these terms, I have the HJB equation (2). It is only a heuristic derivation because it begs the question of proving the existence of V (a) and, more challenging, that V (z) is twice di¤erentiable for the HJB to be well-de…ned. The complete proof is given by Rocheteau et al. (forthcoming), which utilizes the techniques of the viscosity solution. 7 Suppose V 0 (a) = v for all a. The HJB equation becomes r (va + constant) = v (h  a) + max (1 c 0  v) c + max  y2[0;a]  [L (y)  vy] :  The right side is linear in a only if the constraint y a never binds. Then the …rstorder condition implies that the agent can always …nance a constant early consumption y (a) = L0 1 (v) after any history, which is impossible.  :  40  Federal Reserve Bank of Richmond Economic Quarterly  that the value function admits the following closed-form solution: v2 2 V (a) = a + v1 a + v0 , for a a ; (4) 2 where vi , i = 0; 1; 2, are constant given by A ; v2 = r+ +2 Ay v2 h ; v1 = r+ + v1 h A 2 v0 = y ; r 2r a = (v1 1) =v2 : The optimal choices for consumption and early consumption are functions of storage given by8 0, for a < a ; h a , for a = a : y (a) = a, for a a : c (a) =  (5) (6)  The rate of storage, a_ = sa (a), is given by sa (a) = h  c (a)  a:  (7)  The typical pattern of storage is illustrated by Figure 1. Whenever a liquidity shock hits, the agent will draw all her storage available for early consumption and then “restart”the accumulation. The agent accumulates her storage toward the target level a . During this process, she stores all her endowment and consumes nothing. By doing so, she can build up the target level of storage the quickest. Once she reaches the target, she starts consumption at the rate that maintains her storage at the target, i.e., the endowment after deducting depreciation. The target level of storage balances current consumption and self-insurance for future liquidity shocks: maintaining a higher target gives better protection against the liquidity shocks, but it wastes more resources to depreciation and leads to lower consumption. Remark. The model has a closed-form solution because it is designed to make use of the following properties. Suppose the value function, V (a), is a polynomial of degree n and the consumption function, c (a), is of degree nc . Notice that the second term on the right side of HJB equation (2) involves a product of a term of degree n 1 and a sum of degrees of 0; 1; and nc , so in general the resulting product is a term 8 For these to hold, I have implicitly assumed the parameterization such that a 2 [0; y], c (a ) 0, and L0 (a ) v1 . It requires, for example, the marginal loss, A, to be su¢ ciently high such that an agent prefers to fully deplete all of her storage rather than leave some for the future. See Rocheteau et al. (forthcoming) for details.  Wong: Self-Insurance and the Risk-Sharing Role of Money  41  Figure 1 A Time Path of Storage  of degrees of max (n; n 1 + nc ). If the consumption utility is linear, then the consumption is bang-bang so nc = 0 almost everywhere. If the equilibrium features full depletion, i.e., y (a) = a, then the last term of (2) is of degree max (nL ; n). Setting nL = 2, a closed-form solution of V with N = 2 will match the degree on the both sides of (2). Distribution. In this economy, agents have di¤erent histories of liquidity shocks, so they are di¤erent from one another in the level of storage. Are there a lot of agents poor in storage and hence severely exposed to the liquidity shocks? If so, then there will be potential social gain from risk sharing by having an alternative market structure. To answer this question, I need to know the distribution of storage in this economy. Denote Fa (a) 2 [0; 1] the share of agents with weakly less than a units of storage, also known as the distribution function. In the continuous time, the distribution function simply solves the following Kolmorgorov forward equation (KFE): sa (a) Fa0 (a) =  [Fa [ (a)]  Fa (a)] ; for all interior a,  (8)  where Fa0 (a) denotes the …rst derivative of F (a)— the density function. The mechanical meaning of the KFE is as follows. Consider the group  42  Federal Reserve Bank of Richmond Economic Quarterly  of agents with storage weakly less than a. Let’s call this group of agents A. The size of A is Fa (a) by de…nition. I want to check the change in the size of A after a very short amount of time, = 0. For any member of A with storage strictly less than a, i.e., those with a0 where a0 < a, their storage will increase by sa (a0 ) units if they do not receive any liquidity shock during the spell of . When is very small, the level of storage after is given by a0 + sa (a0 ) a. If they do receive a liquidity shock, then according to the solution (6) they draw all their storage, y (a0 ) = a0 , for the early consumption, and hence their postshock storage is zero. In either case, the level of storage after is still weakly less than a, so they remain in A and the size of A does not change. Now, consider the “border” agents to A with exactly a0 = a. The size of the border agents is, roughly speaking, given by the density function, Fa0 (a) @a. In the continuous time, the out‡ow of the border agents is sa (a) Fa0 (a): the rate they are leaving the border group, sa (a) =@a, multiplied by the size of the border group, Fa0 (a) @a. On the other hand, there are also agents who did not belong to A until the liquidity shocks. These agents must have storage strictly greater than a before the liquidity shock but less than a after the shock. The size of this group of potential “immigrants” is Fa [ (a)] Fa (a). In the continuous time, the in‡ow to A from the potential immigrants is [Fa [ (a)] Fa (a)]: by the law of large numbers, there is a share of the potential immigrants hit by liquidity shocks. According to the KFE (8), the distribution of a is stationary when the out‡ow is equal to the in‡ow for any A. In general, the distribution function, Fa (a), is the solution to the DDE (8) satisfying the boundary conditions Fa (0) = 0 and Fa (a) = 1 for all a a : there is no agent with storage less than 0 or strictly greater than a . Since agents always draw all the storage for early consumption, i.e., y (a) = a, the real balances before a shock are given by (a) = 1 for all a > 0. It is straightforward to verify that the distribution function follows a truncated beta distribution: ( = , if a < a ; 1 1 ha Fa (a) = 1, if a a : Figure 2 illustrates the distribution Fa (a). The distribution is smooth everywhere except at a = a with the point mass 1 Fa (a ). Once they reach the target a , agents stop accumulating further storage so there is a positive measure of agents with exactly a units of storage. The aggregate storage is given by " # Z a = +1 h E (a) = [1 Fa (a)] da = 1 1 a : + h 0  Wong: Self-Insurance and the Risk-Sharing Role of Money  43  Figure 2 The Distribution Function  The following proposition summarizes how the fundamentals change the aggregate storage. Proposition 1 The aggregate storage, E (a), is increasing in the magnitude of the liquidity shock, y. Proof. It follows the fact that a is increasing in y. Proposition 1 states that liquidity shock raises the aggregate storage. This is because when agents expect bigger liquidity shocks, they want to prepare a higher level of storage on average for self-insurance. Welfare formula. The (utilitarian) welfare of this economy is the total sum of the utility and loss across agents, which is given by Z 1 Wa = [c (a) + L [y (a)]] dFa (a) : 0  The …rst term is given by Z 1 c (a) dFa (a) = c (a ) [1 0  = +1  Fa (a )] = h 1  h  a  = h ( + ) E (a) :  Since the economy features full depletion, the second term is given by Z 1 Z i A a Ah L [y (a)] dFa (a) = (y a)2 dFa (a) = [y E (a)]2 + V AR (a) : 2 0 2 0  44  Federal Reserve Bank of Richmond Economic Quarterly  Collecting these terms, the welfare can be written as the following mean-variance formula: ( + )2 Wa = h ( + ) y+ 2 A  A y 2  + A  2  A V AR (a) : 2 (9) The following proposition summarizes how the welfare depends on the distribution. E (a)  Proposition 2 The welfare of the self-insurance economy, Wa , is (a) negatively related to the dispersion of individual storage, V AR (a), and (b) positively related to the aggregate storage, E (a), if and only if E (a) y ( + ) = ( A). Proposition 2 states that dispersion and aggregate level of storage are su¢ cient statistics for welfare. Inequality in storage reduces welfare because it means there are a lot of storage-poor agents exposed to the liquidity shocks. The e¤ect of the aggregate storage on the welfare is not monotone. When the aggregate storage is small, such that E (a) y ( + ) = ( A), an increase in the aggregate storage allows better protection against the liquidity shocks on average and raises the welfare. However, when the aggregate storage is large, such that E (a) y ( + ) = ( A), an increase in the aggregate storage diverts too many resources from consumption on average, which reduces the welfare. This threshold is increasing in the marginal loss of the liquidity shock, A, because agents need more protection against the liquidity shock when its loss becomes more costly. For a similar reason, the threshold is increasing in the probability of the liquidity shock, , but decreasing in the depreciation rate, .  2.  EFFICIENCY LOSS TO SELF-INSURANCE  In this economy there is unit measure of uncountably many agents. One advantage to a mass society is that it has the critical mass to eliminate individual shocks by pooling resources. What should agents do collectively if they can coordinate for the best of themselves? The situation is the same as when there is one “representative” agent, the planner, solving the following problem max fc + L (y )g s.t. c ;y  h  c + y :  The planner maximizes the total ‡ow of the consumption utility, c, and minimizes the total ‡ow of the loss to liquidity shocks, L (y). The  Wong: Self-Insurance and the Risk-Sharing Role of Money  45  resource constraint faced by the planner is that the total consumption and early consumption are not greater than the total endowments pooled by agents. The planner’s solution, also known as the …rst best, is given by y c  = y = h  A  1  y  A  1  :  In the …rst best, agents pay a premium y to insure the liquidity shocks by guaranteeing y units of early consumption. Compared with the …rst best, there are two sources of e¢ ciency loss in the self-insurance economy. On one hand, when agents self-insure with storage, the average consumption is lower, E [c (a)] < c , because some resources are wasted to depreciation. On the other hand, self-insurance by building up storage is a slow process due to depreciation and limited endowment. Compared with the …rst best, the self-insurance economy features less protection against the liquidity shocks on average, E [a] < y , especially to the agents with less storage. It echoes Proposition 2 that the welfare is increasing in the inequality of storage. In the …rst best, every agent can perfectly share the liquidity risk and there is no inequality.  3.  SHARING RISKS WITH MONEY  Now consider an intrinsically useless object called money. Maintained by a central bank, the stock of money, Mt , grows at the rate , where 2 [0; ]. The central bank does not consume anything or withhold any resources, so the simplest way of injecting new money to the economy is the helicopter drop: the central bank creates and transfers a lump sum Mt of money to every agent. It is also the same as the policy where the central bank purchases endowments from agents with newly printed money and then transfers all the purchased endowments to agents. In other words, the central bank keeps printing for agents increasing amounts of paper, so-called money. How can it change the economy? Potentially, there is a market where agents can buy or sell endowment with money (they are not forced to do so). Denote as t the real price of money in terms of goods, i.e., each unit of money can buy t units of goods and its negative growth rate, _ t = t , is simply the in‡ation rate: the loss rate of the real purchasing power of money. The real price of money is determined by agents’net demand and the central bank’s supply. The situation where money cannot be exchanged for anything is captured by t = 0. I …rst guess (and verify later) that agents will no longer store any endowment but will hold money instead. In the  46  Federal Reserve Bank of Richmond Economic Quarterly  continuous time, the budget constraint is given by m _t=  h + Mt  ct  :  t  That is, the change in money holding, m _ t , is equal to the monetary amount of income not consumed, (h + Mt ct ) = t . In the stationary equilibrium, if it exists, the total purchasing power of money should be constant such that t Mt remains the same over time. It implies _  t  = :  t  In other words, in‡ation is always a monetary phenomenon in the sense that the change in the real price of money is driven by the increase of the money supply. Denote the individual real balances as zt = t mt and the aggregate real balances as Z = t Mt , then the budget constraint in the stationary equilibrium is given by z_t = h + Z  ct  zt :  (10)  That is, the change in the real balances of money is the income not consumed, h + Z ct , minus the loss of real balances due to in‡ation, zt . Similar to the previous section, the value function of holding money solves the following HJB equation rW (z) =  max  c 0;y2[0;z]  c + W 0 (z) (h + Z  c  z) +  [L (y) + W (z  y)  (11) It is straightforward to verify that the value function admits the following closed-form solution: w2 2 z + w1 z + w0 , for z z (12) W (z) = 2 where wi , i = 0; 1; 2, are constant, given by A w2 = ; r+ +2 Ay w2 (h + Z) w1 = ; r+ + w1 (h + Z) A 2 w0 = y ; r 2r z = (w1 1) =w2 : The optimal choices for consumption and early consumption are functions of real balances given by 0, for z < z ; h+ Z z , for z = z : y (z) = z, for z z : c (z) =  (13) (14)  W (z)] :  Wong: Self-Insurance and the Risk-Sharing Role of Money  47  The rate of accumulating real balances, z_ = sz (z), is given by sz (z) = h + Z  c (z)  z:  (15)  The distribution is given by the following KFE sz (z) Fz0 (z) =  [1  Fz (z)] ; for all interior z.  (16)  The closed-form solution to the distribution function of real balances is given by ( ) = 1 1 , for z < z ; z h+ Z Fz (z) = ; if > 0; 1, for z z : or 1 exp h z , for z < z ; Fz (z) = ; if = 0; 1, for z z : where Z is the …xed point solving " h+ Z 1 Z = E (z) = +  = +1  1  h+ Z  z  #  :  (17)  The following lemma shows that Z is well-de…ned. Lemma 1 There exists a unique solution Z to (17). Proof. The …xed point Z exists because the left side of (17) is smaller than the right side for Z = 0 but becomes larger than the right side for Z = 1, so there must exist some Z where the left side is equal to the right side. The …xed point is also unique because the right side, as a function of Z, has a slope less than unity. Equation (17) captures the market-clearing condition for money: the left side captures the money supply Z = t Mt in real terms, and the Rright side captures the aggregate money demand by agents, E (z) zdFz (z). Similar to Proposition 1, the following proposition establishes that the liquidity shock increases the money demand. Proposition 3 The aggregate real balances of money, E (z), is increasing in the magnitude of the liquidity shock, y. Proof. The left side of (17) is a function of Z with a unit slope. The right side has a slope less than unity. An increase in y shifts up the right side via z , so the …xed point Z increases. Like storage, agents in this economy hold money in order to selfinsure against the liquidity shocks. Therefore, the more severe the magnitude of the liquidity shock, the more money agents hold. The value of money is measured by the equilibrium price, which is given by Z Z = exp ( t) : t = Mt M0  48  Federal Reserve Bank of Richmond Economic Quarterly  Indeed, the fact that t > 0 veri…es that money circulates in this economy. Finally, think of the in‡ation rate, _ t = t , as the depreciation rate of a new storage technology, which is preferred by agents to the original storage technology, since the depreciation rate with money is _ = = lower, i.e. . It veri…es the premise that the use of t t money will “crowd out” individual storage. Having shown that money circulates despite the fact that money is intrinsically useless and there is always a coincidence of wants, now I want to check whether or not the economy is better o¤ after the introduction of money. To do so, I need to compare the welfare to the self-insurance economy. The welfare in the monetary economy is given by Z 1 [c (z) + L [y (z)]] dFz (z) : Wz = 0  The …rst term is given by Z 1 c (z) dFz (z) = h 0  E (z) ;  where I have made use of the fact that Z = E (z) in the equilibrium. Since the economy features full-depletion, the second term is given by Z 1 i Ah [y E (z)]2 + V AR (z) : L [y (z)] dFz (az) = 2 0 Collecting these terms, the welfare can be written as the similar meanvariance formula: A A Wz = c [y E (z)]2 V AR (z) : (18) 2A 2 2 Proposition 4 The welfare of the monetary economy is (a) negatively related to the dispersion of real balances, V AR (z), and (b) positively related to the aggregate real balances, E (z), if and only if E (z) y . Similar to the self-insurance economy, inequality in money holding reduces the welfare because it means there are a lot of money-poor agents exposed to the liquidity shocks. The aggregate real balances of money raise the welfare if and only if the aggregate real balances are less than the …rst-best level of early consumption, E (z) y , i.e., when agents do not hold more money on average than they should in the …rst best. Now I am ready to conclude the social role of money with the following proposition.  Wong: Self-Insurance and the Risk-Sharing Role of Money  49  Proposition 5 Welfare is higher under the monetary economy than the self-insurance economy, i.e., Wz > Wa . R Proof. Notice that welfare is also equal to Wa = r V (a) dFa (a) R and Wz = r W (z) dFz (z), i.e., the average value in the economy. It is straightforward, although tedious, to check the closed-form solutions that V (x) < W (x) and z < a . Intuitively, the fact that money depreciates at a lower rate and agents receive an additional transfer Z means that the agent with wealth x has higher value in the monetary economy, for any x. Finally, from the KFE the distribution functions are given by Z a log [1 Fa (a)] = dx; 0 sa (x) Z z dx: log [1 Fz (z)] = 0 sz (x) Given the fact that sz (x) > sa (x), I have 1 Fz (x) > 1 Fa (x), i.e. there are more wealthy agents in the monetary economy. Combining the factsR that 1 Fz (x) > R1 Fa (x) and V (x) < W (x), I prove that Wa = r V (a) dFa (a) < r W (z) dFz (z) = Wz . If the in‡ation rate is not too high that , then the monetary economy is always more socially desirable than the self-insurance economy with private storage only. But what is the optimal in‡ation rate? In general, it is not zero, and …nding the optimal rate is a quantitative exercise involving the following trade-o¤. A higher in‡ation hurts the return of money (in‡ation as a tax to money holding) and hence discourages its use as a precautionary saving. A higher in‡ation, however, enables generous monetary transfers and promotes social sharing of liquidity risks. The second e¤ect dominates when h is low (self-insurance is low because income is low) or is high (liquidity shock comes too frequent to maintain a su¢ cient level of self-insurance).  4.  CONCLUSION  As rightly pointed out by Winston Churchill in my opening quotation, capitalism that builds on money and market inevitably results in unequal sharing of blessing and misery. A lesson from this review is, however, that money can well be our best response to mitigate unequal sharing of blessing and misery, so economy without money is even worse. How can money improve social welfare? My model illustrates three channels. Firstly, the usage of money mimics, though not perfectly, the …rst-best allocation. When agents hold and accumulate money, they sell their endowments to others who want to sell money. Let’s call the former the seller (of endowment) and the latter the buyer  50  Federal Reserve Bank of Richmond Economic Quarterly  (of endowment). In the monetary economy, these buyers are exactly the ones who are hit by the liquidity shocks. When the buyers buy endowments and sell money to the sellers in the market, resources are e¢ ciently transferred from nonshocked agents to the shocked agents— the monetary mechanism of risk sharing. Why do the sellers want to buy money? Because when they sell their endowments, the sellers actually buy an insurance against the future liquidity shocks, and the insurance premium is essentially the consumption forgone for acquiring money in the market. It is exactly what should happen in the …rst best, illustrated in Section 2. Secondly, money allows agents to save with other agents, instead of decentralized storage. And saving in money gives higher returns to agents, comparing the loss due to in‡ation with the loss due to depreciation. It avoids wasting resources to depreciation and allows more resources for consumption. Finally, agents who are poor in money are the unlucky ones frequently hit by the liquidity shocks. 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