Calibrated Optimization
Perhaps the most important study of optimal capital requirements remains that prepared by the Basel Committee's Long-Term Economic Impact (LEI) working group (BCBS 2010a). Based on a review of 19 studies of historical banking crisis episodes, the study places the median estimate of losses from a crisis at 19 percent of one year's GDP if there are no permanent effects, 158 percent of GDP if there are permanent effects, and 63 percent as the median for all studies (p.
35). Based on the compilations of banking crisis episodes in Reinhart and Rogoff (2008) and Laeven and Valencia (2008), it places the frequency of banking crises in 1985-2009 at 4.1 to 5.2 percent for G-10 countries (p. 39). However, as discussed in chapter 4, expanding the horizon to 1977-2015 reduces this incidence to only 2.6 percent.To arrive at a mapping from bank capital to incidence of banking crises, the LEI working group considered the results of reduced-form econometric studies (including Barrell et al. 2010 and Kato, Kobayashi, and Saita 2010). It also incorporated findings of models treating the banking system as a portfolio of securities and examining the link between capital and default (as in Elsinger, Lehar, and Summer 2006). The resulting curve for all models shows the probability of a banking crisis falling from 4.6 percent at a capital requirement of 7 percent for tangible common equity relative to risk-weighted assets to a probability of 3 percent at a capital requirement of 8 percent, 1.9 percent at a requirement of 9 percent, and 0.3 percent at a requirement of 15 percent (see table 4.2).
On the cost side, the LEI study examined funding costs based on data for 6,600 banks in 13 countries during 1993-2007 to estimate that cost of equity. It reports an average return on equity of 14.8 percent and the cost of debt of 100 basis points above the deposit rate for short-term debt and 200 basis points for long-term debt.
Higher capital requirements impose a cost, reflecting the differential between the return on equity and the rate on long-term debt as equity is increased relative to debt. Banks are assumed to pass on fully the higher cost in lending spreads, which are calculated to rise by 13 basis points for a 1 percent increase in the ratio of capital to risk- weighted assets. The LEI considers the impact near an upper bound, in part because it makes no allowance for any M&M offset.The study then uses a set of existing models—including dynamic stochastic general equilibrium models, semistructural forecasting models, and reduced-form models, such as vector error correction models—to translate the impact of higher lending rates to changes in steady-state output. The output effect is found to be largely linear. An increase of 1 percent in the capital requirement reduces long-term output by 0.09 percent. Meeting the net stable funding ratio reduces output by an additional 0.08 percent. Overall the net benefits of higher capital ratios rise to a peak of about 1.9 percent of output as the ratio of tangible common equity to risk-weighted assets rises from 8 to 12 percent; they plateau at that level as the capital ratio is increased further to 16 percent.
A subsequent survey of the literature by the Basel Committee (BCBS 2016a) interprets the LEI findings as placing the optimum at a capital requirement of13 percent of risk-weighted assets, well above the 9.5 percent level for large banks in Basel III (7 percent plus 2.5 percent for G-SIBs). The 13 percent optimum is at the center of the 12 to 14 percent range identified in chapter 4. This similarity reflects my use of the LEI quantification of the response of the probability of crisis to the capital ratio as well as the virtually identical central estimates of crisis damage (63 percent of one year's GDP in the LEI, 64 percent in chapter 4). Key differences tend to be in offsetting directions.[28]
An important early study of optimal capital requirements is by Barrell et al.
(2009). The authors first estimate a logit function explaining the incidence of banking crises in OECD countries as a function of bank capital, the liquidity ratio, and the rate of increase in housing prices. The function shows a 22 percent probability of crisis in the United Kingdom in 2007, which could have been cut to 7 percent by a 3 percentage point increase in the ratio of capital to total assets (pp. 14-15).[29] They place the cost of crises at up to 80 percent of GDP and suggest a long-term scarring effect boosting risk premiums and hence the cost of capital. They argue that higher bank capital has economic costs, judging that M&M effects are incomplete not only because of tax incentives but also because of bankruptcy costs from fire sales under asymmetric information about fundamental values (p. 8).Barrell et al. adopt a production function approach to calculate the long-term output impact of higher lending costs stemming from higher capital requirements. Applying the National Institute Global Econometric Model (NiGEM), they estimate that a 1 percentage point increase in the capital adequacy target would reduce output by 0.08 percent in the long term.
A problematic issue is that they treat the ratio ofcapital to risk-weighted assets as virtually identical to the ratio to total assets, based on data for UK banks (pp. 25-26), yet for US and euro area banks risk-weighted assets represent only slightly more than half of total assets. Their central estimate places the optimal increase in the capital ratio at 3 percentage points (p. 42). This increase would be to about the same level as the optimal range estimated in chapter 4 if the increment is measured in capital relative to total assets.[30]
Miles, Yang, and Marcheggiano (2012) provide an approach to optimal capital requirements that is similar to that in BCBS (2010a), in underlying concepts but different in arriving at the calibrated parameters. On the cost side, the driving force is once again an increase in the lending rate imposed as a consequence of the required shift away from cheaper debt finance to more expensive equity finance.
The authors first examine the M&M offset by estimating the equity market beta coefficient for seven large UK banks using daily stock prices for 1992-2010. In their main variant, only 45 percent of the M&M offset is obtained. Based on equity unit cost of 15 percent and debt cost of 5 percent, they estimate that halving the leverage of banks from a debt to equity ratio of 30 to a ratio of 15 would boost the weighted-average cost of capital to banks by 33 basis points excluding any M&M offset but by 18 basis points after allowing for the 45 percent offset (p. 13).They then translate higher lending costs to output effects using a production function approach in which the stock of capital declines in response to a higher price of capital, after taking account of the factor share of capital and the degree of substitutability between capital and labor. Their output equation is far more transparent than the output component of such studies as BCBS (2010a). After considering that bank finance represents only one-third of corporate funding, the authors estimate that halving the debt leverage of banks would reduce long-term output by 0.15 percent. This impact corresponds to an output loss of 0.05 percent for each percentage point increase in the capital required relative to total assets.[31]
On the benefit side, the authors place the present value of output loss in a banking crisis at 140 percent of base GDP (p. 25), well above the BCBS (2010a) median figure of 63 percent. They use an unusual method for calibrating the response of crisis probability to bank capital. They assume that the proportionate decline in risk-weighted assets in a financial crisis is equal to the proportionate decline in GDP. They examine the frequency distribution of declines in GDP over the past 200 years for 31 countries, and find a higher incidence of extreme declines than would be expected. Declines of more than 15 percent occurred 1.2 percent of the time, implying that a capital ratio of 15 percent of risk-weighted assets would be insufficient to avoid banking crisis 1.2 percent of the time.
On the basis of such frequencies the authors compare marginal benefits with marginal costs. Their marginal benefits curve (p. 26) is not monotonic; it surges once again when capital ratios reach extremely high levels (40 percent), corresponding to the tail of extreme declines in GDP (e.g., war experiences). This aspect is troublesome, because in principle the gains from additional capital ought to take account of the probability of such instances already in the lower range of capital ratios. The authors do not present a single equation for benefits, or for marginal benefits, ruling out identification of a unique capital ratio at which marginal benefits begin to fall below marginal costs. In their policy recommendation, they instead emphasize that even if the upper extreme is ignored completely, the range for optimal capital requirements is 16 to 20 percent of risk-weighted assets, corresponding to 7 to 9 percent of total assets (p. 29).
Kato, Kobayashi, and Saita (2010) estimate a probit model of the probability of a banking crisis, using the episodes identified by Laeven and Valencia (2008) for 13 industrial countries. Their approach emphasizes the interaction between liquidity and capital. Their independent variables are the ratio of capital to total assets, the ratio of liquid to total assets and the corresponding liquidity ratio for liabilities, the rate of real estate inflation, and the current account balance as a percent of GDP. Their tests include the capital ratio interacted with the liquidity ratios on both the asset and liability sides. Pointing out that the US investment banks in which the government intervened had higher capital-to-assets ratios than other banks (p. 5), they stress that the Lehman shock was triggered not by undercapitalization but by liquidity stress and the inability to roll over shortterm funding, especially in the repo market (p. 19).14 They place the cost of a banking crisis at 10 percent of GDP, far lower than the model median 63 percent reported in BCBS (2010a).
For the cost of additional capital, the authors apply the formula proposed by Van den Heuvel (2008): DC = D ? (Re - Rd) ? (4k/[1 - k]), where Δ indicates change, C consumption (or GDP), D deposits, R the rate of return, e equity, d debt, and k the ratio of capital to total assets.[32] [33] For Japan they place deposits at 250 percent of consumption, Re at 3.75 percent, and Rd at zero.[34] By implication, and with an illustrative value of k = 0.1, an increase in the capital ratio by 1 percentage point would reduce consumption by 0.09 percent (0.01 ? 250 ? 0.0375 = 0.094).[35] The welfare cost of additional liquidity in assets is based on the difference between average bank lending rates and the yield on liquid assets. On the liability side, the cost of raising the liquidity ratio depends on the excess of the cost of retail liabilities (e.g., deposits) over wholesale liabilities (e.g., repos). At intermediate asset- and liability-side liquidity ratios, the authors arrive at a range of 6 to 8 for the optimal ratio of capital to total assets across three states of the business cycle (p. 32). These ratios are relatively high, especially considering that the benefit of avoiding a crisis is estimated at only 10 percent of GDP. Researchers at the Bank of England (Schanz et al. 2011) use a method similar to that in chapter 4 to evaluate optimal capital ratios. They estimate that for UK banks, raising the capital ratio by 1 percent of risk-weighted assets would boost lending rates by 7.4 basis points, assuming no M&M offset. Applying a simple production function, assuming corporations have a cost of capital of 10 percent and, assuming that banks account for only one-third of financing of corporations, they estimate a reduction in output of 0.04 percent for each percentage point increase in the capital ratio. But in chapter 4 I suggest that the average corporate cost of capital is only 4.6 percent, so the proportionate increase in the cost of capital—and hence the output impact—would be about twice as high as calculated in the Bank of England study for a given increment in the average cost of capital.[36] On the benefits side, Schanz et al. estimate the lost present value from a banking crisis at 140 percent of one year's GDP—on the high side compared with the benchmark BCBS (2010a) study. They use a Merton- style structural credit risk portfolio model to relate the probability of banking crisis to the capital to risk-weighted assets ratio, calibrated to identify a crisis when two of the United Kingdom's large banks default because their capital falls below 4 percent of risk-weighted assets. Their probability mapping (p. 77) starts out far higher than that in BCBS (2010a), with a 20 percent probability of a banking crisis versus 4.5 percent, at an 8 percent capital to risk-weighted assets ratio and 3 percent probability versus 1.4 percent at a 10 percent ratio (see table 4.2). They conclude that appropriate capital requirements are “somewhere between 10 percent and 15 percent of risk-weighted assets” (p. 73). This range is consistent with the optimal range calculated in chapter 4 (12 to 14 percent of risk-weighted assets), reflecting offsetting differences in the estimates for individual components in the two studies. Gambacorta (2011) applies a vector error correction model to quarterly data for the US economy in 1994-2008 to examine the impact of changes in capital and liquidity requirements. He finds that the bank lending spread is a function of capital and liquidity requirements; the level of output is a function of the real interest rate, the bank lending spread, and government spending; bank lending to the economy is a function of the level of GDP and the lending spread; and the return on bank equity is a function of lending relative to GDP and the lending spread. One might wonder why this system would capture long-term effects accurately, with a lag structure of only three quarters, no equation directly relating output to capital and labor, no equations translating lending rates to capital stock, no explicit consideration of the share of bank lending in total finance to the economy, and no direct test of the M&M offset (although by implication it would show up in a muted influence of the capital requirement on output). Furthermore, no exogenous changes in required bank capital were imposed in this period, suggesting difficulties in attributing causality. Nonetheless, Gambacorta's key results turn out to be relatively similar to those in the calibrated model developed in chapter 4. In particular, in the long-term equilibrium, an increase in the capital requirement (tangible common equity relative to risk-weighted assets) of 6 percentage points increases the bank lending spread by 15 basis points (p. 84). The corresponding impact in the model in chapter 4 would be 13.9 basis points.[37] In the Gambacorta results, an increase in the capital ratio of 1 percentage point reduces steady-state output by 0.1 percent. When this impact is translated into the corresponding metric for capital relative to total assets, the impact reaches a 0.178 percent decline in output for a 1 percentage point increase in the capital requirement, close to the parameter identified in chapter 4 (ψ = 0.15). According to Gambacorta, with “moderate permanent effects” of crises, the gross benefits of crisis avoidance rise from 0.6 percent of GDP at a 7 percent risk-weighted capital ratio to 2.2 percent of GDP at 10 percent, 2.4 percent of GDP at 12 percent, and 2.6 percent at 15 percent. Costs rise linearly, from nearly zero at 7 to 0.7 percent of GDP at 15 percent. Marginal costs equal marginal benefits at a risk-weighted capital ratio of 12 percent (p. 87). Again this finding is almost the same as that in chapter 4. Yan, Hall, and Turner (2011) estimate a model of optimal capital requirements for the UK economy. On the benefit side, they estimate a probit model relating the probability of a banking crisis to the ratio of tangible common equity to risk-weighted assets, interacted with liquidity (the net stable financing ratio). Their test also includes the rate of inflation in real estate and the size of the current account deficit. A banking crisis is defined as occurring in the first quarter of2008 through the second quarter of 2010. Data for the 12 largest banks provide the capital and liquidity information. Estimated using quarterly data beginning in 1997, the model would seem to be inherently limited, given its reliance on a single country. Its results seem exceedingly binary, indicating that, whereas a 7 percent capital ratio reduces the probability of a crisis by 3.2 percent and a ratio of 8 percent reduces it by 4.6 percent, higher levels of capital have almost no additional impact reducing crisis probability. This outcome is sharply at odds with the BCBS (2010a) survey, which shows considerable additional reductions as the capital ratio reaches higher levels (see table 4.2 in chapter 4).[38] On the cost side, the study relies on a vector autoregression and vector error correction approach, which generates coefficients relating lending spreads to capital requirements and relating output to lending spreads. Curiously, this relationship is concave rather than the usual linear result.[39] More fundamentally, it seems unlikely that a vector auto regression approach would be as reliable as a calibrated capital cost and production function approach, for several reasons. The lag structure, for example, is only three quarters, far too short to capture steady-state effects. The key output equation includes only the lending spread, the real interest rate, and the rate of return on equity as explanatory variables (p. 13). There is almost no variation in the capital ratio over the period 2000-08 (p. 32).[40] Although the optimal capital ratio identified by the study (10 percent of risk-weighted assets) is not implausible, the underpinnings of the estimate may not be robust. Researchers at the Bank of England examined optimal systemic capital requirements in a framework representing “a panel of correlated Merton (1974) balance sheet models, jointly estimated using observed bank equity returns, and combined with a network of interbank exposures” (Webber and Willison 2011, 8). An iterative approach imposes contagious losses, with interbank exposures cleared using the algorithm of Eisenberg and Noe (2001). Under the assumption that M&M does not hold completely and thus that higher capital requirements impose costs on the economy, the authors minimize the total capital required for the system subject to a 5 percent target value at risk that system assets will fall below system liabilities (p. 13). Their stepwise iterative procedure involves 50,000 simulations of a model calibrated to five major UK banks in 2004-09. They use observed equity prices to estimate the expected return on banks' assets and the variance-covariance structure between the banks' asset returns. A distinctive feature of the model is that the cost-minimizing capital structure differentiates among banks, reflecting greater idiosyncratic risk at some banks than others as well as taking account of differing interbank exposure. Thus for 2008 the optimal systemic surcharge is set at 6 percent of assets for “bank 1” but only 0.8 percent for “bank 5” and an average of 2.4 percent for the five banks (p. 21). Experiments show a surprising insensitivity to further expansion of large-bank size (a doubling of bank 1's balance sheet increases its optimal capital surcharge only from 6 to 7 percent; p. 22). In research at Norway's central bank, Kragh-Sorensen (2012) estimates optimal capital ratios in a framework similar to those applied by BCBS (2010a) and Miles, Yang, and Marcheggiano (2012). However, he develops an independent estimate of the relationship between the probability of a banking crisis and the capital ratio. This estimate is based on data on problem loans to households and corporations in Norway in 1990-2011. He applies an estimated distribution to simulate the stock of problem loans for the six largest Norwegian banks, with a banking crisis identified if two or more are found to have capital ratios below 4.5 percent. It turns out that his results show much higher probabilities of banking crises at given capital ratios for Norway than in the broader BCBS (2010a) study. For example, an increase in the capital ratio from 9 to 13 percent of risk-weighted assets reduces the probability of a banking crisis from 7.0 to 1.7 percent in his estimates for Norway (p. 6), in contrast to a reduction from 2.6 to 0.5 percent in the broader BCBS (2010a) study. Alternative estimates of the long-term output cost of higher capital requirements include one from the Norges Bank's Financial Stability Model indicating an output decline of 0.09 percent for an increase in the capital requirement of 1 percentage point at the high end and a model providing much lower cost (much higher M&M offset) at the low end. When banking crisis damage is set at a reasonable 60 percent of one year's GDP, the result is an average estimate of 19 percent for the optimal ratio of capital to risk-weighted assets across the variants, ranging from a low of 16 percent to a high of 23 percent (p. 11). The driving force behind this high range is the much higher path of the crisis probability curve than in other leading studies (BCBS 2010b, Miles, Yang, and Marcheggiano 2012). By implication, the most desirable policy would seem more likely to be to set an especially stringent capital requirement in Norway (and perhaps other countries sharing above-average fragility) rather than to place broader Basel capital targets as high as indicated in the Norway-specific study. Researchers at the Financial Services Authority examined the costs and benefits of higher capital requirements for UK banks (de-Ramon et al. 2012). Like Barrell et al. (2009), they use a logit function to relate the incidence of banking crisis to capital requirements and apply the NiGEM model to calculate output effects. They extend the crisis function by adding the current account as an explanatory variable. They add considerable detail to modeling the dynamic profile of banks' adjustments to higher capital requirements, including special attention to the cutback of total assets and change in risk composition.[41] They also take into account an expected narrowing of banks' voluntary buffers as the capital requirement increases. de-Ramon et al. estimate the long-term loss from a UK banking crisis at an ongoing 3 percent of annual GDP (implicitly equivalent to 120 percent of one year's GDP, or about twice the BCBS 2010a median, if discounted at 2.5 percent) (p. 51). They calculate that a 1 percent increase in the ratio of capital to risk-weighted assets imposes an increase of 9.4 basis points in the spread between bank deposit and lending rates (p. 31). The full Basel III package causes a short-term increase in bank lending rates of 126 basis points and a long-term increase of 67 basis points (p. 55). Higher lending rates cause lower output, working through the NiGEM model. The study's central estimate is that the Basel III regulatory reform produces an annual gross benefit to the United Kingdom of 1.29 percent of GDP at an output cost of 0.38 percent of GDP, yielding a net benefit of 0.92 percent of GDP (£11.9 billion at 2010 prices; p. 53). de-Ramon et al. report that in their stochastic simulations the net benefits of the Basel III package could reach as high as 4.9 percent of GDP at the favorable 90th percentile (p. 61), but they do not report the assumptions that would raise the net benefits this high.[42] Despite this seemingly large upside potential, the authors imply that the optimal capital requirements are not much different from Basel III: “The costs of requiring higher capital ratios beyond the [Basel III] policy package... rise faster than the benefits” (p. 54). Their conclusion is somewhat misleading in this regard. It states that “prudential standards could be raised further by up to an additional 22 percentage points in terms of banks' aggregate risk-weighted capital ratio, and still be expected to produce overall positive net benefits in the long run” (p. 65). But this statement means that fairly close to the Basel III ratio additional capital would start to impose costs that would eat away at the net benefits achieved at the Basel III level and that after an additional 22 percentage points those gains would have all been lost. Their 22 percent additional capital would correspond to the point in figure 4.3 in chapter 4 at which the benefits curve crosses the cost curve (eliminating the net benefits indicated by the vertical difference between the two curves), a point that is far to the right of the optimal capital requirement, where the slopes of the two curves are equal. Finally, a report issued in preliminary form by the Federal Reserve Bank of Minneapolis (Minneapolis Plan 2016) uses the Dagher et al. (2016) data on nonperforming loans in past crises together with simulations of the large macroeconomic model of the US Federal Reserve to estimate the optimal capital ratio, which it finds to be 23.5 percent. (Appendix 4A examines this study in detail). The study calls for a capital requirement of 38 percent of risk-weighted assets on G-SIBs in order to eliminate too big to fail (but this estimate is not obtained using optimization).