Calculating the Optimal Capital-to-Assets Ratio

Chapter 4 presents a model estimating the optimal capital-to-assets ratio, using as a key input the M&M offset coefficient estimated in chapter 3. The cost of a higher capital ratio to the economy stems from the increase in the unit cost of capital to borrowing firms and households, the consequential reduction in the economywide stock of capital, and hence the reduction of future output from its baseline path.

The chapter first develops new estimates of damage from banking crises. The analysis takes account of the fact that output may have been above potential before the crisis. It also points out that even if output does not return to its baseline trend for potential by the fifth year of the crisis, the shortfall would not be perpetual, because the forgone capital stock sac­rificed to the recession would not have had an infinite life. Based on past banking crisis episodes, and allowing a 15-year capital life following the recession to calculate lingering costs, I estimate that the typical long-term cost of a banking crisis in an advanced industrial economy has been 64 percent of one year's GDP, almost identical to the median estimate in the Basel Committee's Long-Term Economic Impact analysis (BCBS 2010a).

The next step is to identify the relationship between the probability of crisis and the level of the capital ratio. I apply the schedule of this relation­ship reported by the Basel Committee (BCBS 2010a), which is based on cross-country regression models and models of banking system contagion from interbank exposures. There is a relatively rapid drop-off in the prob­ability of crisis as the capital ratio increases. For tangible common equity, an increase in the ratio of capital to risk-weighted assets from 7 to 8 percent reduces the annual probability of crisis from 4.6 to 3.0 percent in the BCBS schedule; the probability falls to 1 percent when the capital ratio reaches 11 percent, and it declines by only another 0.1 percent (from 0.4 to 0.3 percent) when the increase is from 14 to 15 percent.

The analysis then incorporates the cost to the economy of higher capital requirements. Using the same production function framework proposed by Miles, Yang, and Marcheggiano (2012), I set the proportional output loss equal to the proportional rise in the unit cost of capital multiplied by the product of the elasticity of output with respect to capital and the elasticity of substitution between capital and labor, all divided by unity minus the elasticity of output with respect to capital. The derivative of this cost turns out to be a constant that is influenced by the shares of bank and nonbank finance, the gap between the unit cost of equity capital and the cost of debt to banks, and the M&M offset coefficient. At the central values, this con­stant turns out to be a loss of 0.15 percent of GDP for each percentage point increase in the required ratio of capital to total assets.

There is thus a straight-line upward-sloping cost curve for the capital requirement. The optimal level for the capital ratio will then be where the slope of the concave benefits curve is identical to the slope of the cost line. The central value for the M&M offset is set at 0.45, based on the results of chapter 3. Using low, central, and high variants for this parameter and six others (loss from crisis, unit cost of equity to banks, coefficient for spill­over to capital cost in nonbank lending, elasticity of output with respect to capital, elasticity of substitution between capital and labor, and crisis prob­ability curve), the analysis generates 2,187 possible outcomes for the capital ratio at which marginal benefits equal marginal costs. The median outcome is 6.9 percent of total assets; the 75th percentile outcome is 7.9 percent. The main result is thus that the optimal capital ratio is 7 to 8 percent of total assets, corresponding to 12 to 14 percent of risk-weighted assets (using the ratio of risk-weighted assets to total assets in euro area and US banks).