Does Banking Capital Reduce Risk? An Application of Stochastic Frontier Analysis and GMM Approach

Abstract

In this chapter, we thoroughly analyze the relationship between capital and bank risk-taking. We collect cross section of bank holding company data from 1993 to 2008. To deal with the endogeneity between risk and capital, we employ stochastic frontier analysis to create a new type of instrumental variable. The unrestricted frontier model determines the highest possible profitability based solely on the book value of assets employed. We develop a second frontier based on the level of bank holding company capital as well as the amount of assets. The implication of using the unrestricted model is that we are measuring the unconditional inefficiency of the banking organization.

We further apply generalized method of moments (GMM) regression to avoid the problem caused by departure from normality. To control for the impact of size on a bank’s risk-taking behavior, the book value of assets is considered in the model. The relationship between the variables specifying bank behavior and the use of equity is analyzed by GMM regression. Our results support the theory that banks respond to higher capital ratios by increasing the risk in their earning asset portfolios and off-balance-sheet activity. This perverse result suggests that bank regulation should be thoroughly reexamined and alternative tools developed to ensure a stable financial system.

Appendices

Appendix 1: Stochastic Frontier Analysis

Stochastic frontier analysis (SFA) is an economic modeling method that is introduced by Jondrow et al. (1982). The frontier without random component can be written as the following general form:

$$ PT{I}_i=T{E}_i\cdot f\left({\mathbf{x}}_i;b\right), $$

(13.5)

where PTI _i is the pretax income of the bank i, i = 1,..N, TE _i denotes the technical efficiency defined as the ratio of observed output to maximum feasible output, x _i is a vector of J inputs used by the bank i, f(x _i , b) is the frontier, and b is a vector of technology parameters to be estimated. Since the frontier provides an estimate of the maximum feasible output, we then can measure the shortfall of the observed output from the maximum feasible output. Considering a stochastic component that describes random shocks affecting the production process in this model, the stochastic frontier becomes

$$ PT{I}_i={e}^{v_i}\cdot T{E}_i\cdot f\left({\mathbf{x}}_i;b\right). $$

(13.6)

The shock, ${{e}}^{v_i}$, is not directly attributable to the bank or the technology but may come from random white noises in the economy, which is considered as a two-sided Gaussian distributed variable.

We further describe TE _i as a stochastic variable with a specific distribution function. Specifically,

$$ T{E}_i={e}^{-{u}_i}, $$

(13.7)

where u _i is the nonnegative technical inefficiency component, since it is required that TE _i ≤ 1. Thus, we obtain the following equation:

$$ PT{I}_i={e}^{v_i-{u}_i}\cdot f\left({\mathbf{x}}_i;b\right). $$

(13.8)

We then can describe the frontier according to a specific production model. In our case, we assume that bank’s profitability can be specified as the log-linear Cobb-Douglas function:

$$ PT{I}_i=a+{\displaystyle \sum_{h=1}^H}{b}_h \ln {x}_{i,h}+{v}_i-{u}_i. $$

(13.9)

Because both v _i and u _i constitute a compound error term with a specific distribution to be determined, hence the SFA is often referred as composed error model.

In our study, we use the above stochastic frontier with different inputs to generate the net effect of bank capital without mixing the impact of risk. The unrestricted model (without including bank equity) is

$$ \begin{array}{l} PTI\left( BVA,{\sigma}_{\mathrm{BANK}}\right)=a+{b}_1 BVA+{b}_2{(BVA)}^2+e\hfill \\ {}e=\xi -\varsigma \hfill \\ {}\xi \sim iid\;N\left(0,{\sigma}_{\xi}^2\right),\varsigma \sim iid\;N\left(0,{\sigma}_{\varsigma}^2\right)\hfill \end{array}, $$

(13.10)

where BVA is the natural logarithm of book value of assets, ξ is statistical noise, ς is systematic shortfall (under management control), and ς ≥ 0. Our restricted model is as follows:

$$ \begin{array}{l} PTI\left( BVA, BVC,{\sigma}_{\mathrm{BANK}}\right)=\alpha +{\beta}_1 BVA+{\beta}_2{(BVA)}^2+{\beta}_3 BVC+\varepsilon.\\ {}\begin{array}{l}\varepsilon =v-u\hfill \\ {}v\sim iid\;N\left(0,{\sigma}_v^2\right)u\sim iid\;N\left(0,{\sigma}_u^2\right)\hfill \end{array},\end{array} $$

(13.11)

where BVC is the natural logarithm of book value of capital, v is statistical noise, and u denotes the inefficiency of a bank considering its use of both assets and capital. The difference in the inefficiency between the restricted and unrestricted model,

$$ \delta =u-\varsigma, $$

(13.12)

is our instrumental variable. The instrumental variable for capital can be used in regressions of various measures of risk, as the dependent variable, on our instrument for capital, as the independent variable, while controlling for BHC size.

Appendix 2: Generalized Method of Moments

Hansen (1982) develops generalized method of moments (GMM) to estimate parameters that its full shape of the distribution function is not known. The method requires that a certain number of moment conditions were specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the true values of the parameters. The GMM method then minimizes a certain norm of the sample averages of the moment conditions.

Suppose the error term ε _t = ε(x _t , θ) is a (T × 1) vector that contains T observations of the error term ε _t , where x _t includes the data relevant for the model and θ is a vector of N _β coefficients. Assume there are N _H instrumental variables in an (N _H × 1) column vector, h _t and T observations of this vector form a (T × N _H ) matrix H. We define

$$ {\mathbf{f}}_t\left(\theta \right)\equiv {\mathbf{h}}_t\otimes \boldsymbol{\upvarepsilon} \left({\mathbf{x}}_t,\theta \right). $$

(13.13)

The notation ⊗ denotes the Kronecker product of the two vectors. Therefore, f _t (θ) is a vector containing the cross product of each instrument in h with each element of ε. The expected value of this cross product is a vector with N _ε N _Η elements of zeros at the parameter vector:

$$ \mathrm{E}\left[{\mathbf{f}}_t\left({\theta}_0\right)\right]=0. $$

(13.14)

Since we do not observe the true expected values of f, thus we must work instead with the sample mean of f,

$$ {\mathbf{g}}_t\left(\theta \right)\equiv {T}^{-1}{\displaystyle \sum_{t=1}^T}{\mathbf{f}}_t\left(\theta \right)={T}^{-1}{\displaystyle \sum_{t=1}^T}{\mathbf{h}}_t{\boldsymbol{\varepsilon}}_t\left(\theta \right)={T}^{-1}\ {\mathbf{H}}^{\mathbf{\prime}}{\boldsymbol{\varepsilon}}_t\left(\theta \right). $$

(13.15)

We can minimize the quadratic form

$$ {\mathbf{Q}}_T\left(\theta \right)\equiv {\mathbf{g}}_T{\left(\theta \right)}^{\prime }{\mathbf{W}}_T{\mathbf{g}}_T\left(\theta \right), $$

(13.16)

where W _T is an (N _H × N _H ) symmetric, positive definite weighting matrix. We then find the first-order condition is

$$ {\mathbf{D}}_T{\left({\widehat{\theta}}_T\right)}^{\prime }{\mathbf{W}}_T{\mathbf{g}}_T\left({\widehat{\theta}}_T\right)=0, $$

(13.17)

where D _T (θ _T ) is a matrix of partial derivatives defined by

$$ {\mathbf{D}}_T\left({\theta}_T\right)=\partial {\mathbf{g}}_T\left({\theta}_T\right)/\partial {\theta}^{\prime }. $$

Note the above problem is nonlinear; thus, the optimization must be solved numerically.

Applying the asymptotic distribution theory, the coefficient estimate ${\widehat{\theta}}_T$ is

$$ \sqrt{T}\left({\widehat{\theta}}_T-{\theta}_0\right)\overset{d}{\to }N\left(0,\Omega \right), $$

(13.18)

where Ω = (D ₀′WD ₀)⁻¹ D ₀′WSWD ₀ (D ₀′WD ₀)⁻¹. D ₀ is a generalization of M _HX in those equations and is defined by D ₀ ≡ E[∂f(x _t , θ ₀)/∂θ ₀]. S is defined as

$$ \mathbf{S}\equiv \underset{T\to \infty }{ \lim}\mathrm{Var}\left[{T}^{1/2}{\displaystyle \sum_{t=1}^T}\ {\mathbf{f}}_t\left({\theta}_0\right)\right]=\underset{T\to \infty }{ \lim}\mathrm{Var}\left[{T}^{1/2}{\mathbf{g}}_T\left({\theta}_0\right)\right]. $$

(13.19)

The GMM estimators are known to be consistent, asymptotically normal, and efficient in the class of all estimators that do not use any extra information aside from that contained in the moment conditions. For more discussion of the execution of the GMM, please refer to Campbell et al. (1997) and Hamilton (1994).