A Top-down Approach to Stress-testing Banks

Abstract

We propose a simple, parsimonious, and easily implementable method for stress-testing banks using a top-down approach that captures the heterogeneous impact of shocks to macroeconomic variables on banks’ capitalization. Our approach relies on a variable selection method to identify the macroeconomic drivers of banking variables as well as the balance sheet and income statement factors that are key in explaining bank heterogeneity in response to macroeconomic shocks. We perform a principal component analysis on the selected variables and show how the principal component factors can be used to make projections, conditional on exogenous paths of macroeconomic variables. We apply our approach, using alternative estimation strategies and assumptions, to the 2013 and 2014 stress tests of medium- and large-size U.S. banks mandated by the Dodd-Frank Act, and obtain stress projections for capitalization measures at the bank-by-bank and industry-wide levels. Our results suggest that accounting for bank heterogeneity yields expected capital shortfalls that can be over 30 percent larger than in the case where heterogeneity is ignored. Furthermore, we find that while capitalization of the U.S. banking industry has improved in recent years, under reasonable assumptions regarding growth in assets and loans, the stress scenarios continue to imply sizable deterioration in banks’ capital positions.

Appendices

Appendix

1.1 A1. LASSO selection of macroeconomic drivers

We operationalize the method that identifies the set of relevant macroeconomic variables as follows. First, we take the full panel of a banking variable of interest (PPNR or NCO in our case), Y _{i t} , and remove variation associated with its own lag and fixed effects of individual banks by running the following regression, where $\tilde {Y}_{it}$ represents the residual variation in Y _{i t} :

$$ Y_{it}=\alpha_{i}+\beta Y_{it-1}+\tilde{Y}_{it}. $$

(A.1)

This step identifies the variation of the dependent variable not explained by its own lags and is standard in the macroeconomic forecasting literature.³¹ Our selection of macroeconomic drivers on the basis of the residual variation of the dependent variable reduces the possibility that the variation of a selected macroeconomic driver picks up some of the variation in the dependent variable that could be explained by its own lag. Second, we employ the LASSO framework for variable selection on each candidate variable pool by regressing $\tilde {Y}_{it}$ on the set of K _z candidate macroeconomic variables given by the matrix Z to solve the following problem, with γ denoting the vector of coefficients associated with the candidate variables in the regression:

$$ \min_{\gamma}\left\{RSS(\gamma)+\lambda*\left( \sum\limits_{k=1}^{K_{z}}|\gamma_{k}|\right)\right\}. $$

(A.2)

Z is T×K _z where T is the number of time periods in the sample. The parameter λ imposes a penalty factor on reducing the residual sum of squares (RSS) through additional regressors. As this parameter increases, the number of elements of the vector γ set to zero increases as well, signaling that the associated variable is not useful in reducing the residual sum of squares. Since there is no general guidance for the optimal choice of λ, we conduct a grid search over λ ∈ [λ _{m i n} , λ _{m a x} ], where λ _{m i n} (λ _{m a x} ) is the minimum (maximum) value required to drop (keep) at least (at most) one variable from (in) the candidate pool. We keep all variables that appear at least 20 percent of the time over the entire grid. This implies that we only drop the macroeconomic variables that hardly ever explain a given banking variable.³² This process defines the set of relevant macroeconomic variables, Z ^∗ of size $T\times K_{z}^{*}$ where the optimal number of variables $K_{z}^{*}\leq K_{z}$ comes out of our LASSO procedure, for a given dependent variable.

1.2 A2. Mapping of macroeconomic scenarios to MPCFs

Having identified Z ^∗⊆Z via LASSO, we use the singular value decompostion:

$$ \mathbf{Z^{*}}=\mathbf{F{\Sigma} V'}, $$

(A.3)

where F and V are rotation matrices and # #Σ# # is a rescaling matrix. The columns of F are the principal component factors of Z ^∗. We denote the first principal component factor by f. Approaches of this kind have recently become popular in constructing indexes of financial stability or stress using a large number of macroeconomic or financial variables; see, for example, Kliesen and Smith (2010), and Brave and Butters (2011, 2012).

One challenge that arises in evaluating hypothetical future macroeconomic scenarios is that these projections need to be consistent with the principal components obtained from the historical series. If we simply added the scenarios as new data to the historical series, the rotation and rescaling matrices would change and the resulting principal component factors associated with each scenario would have different paths over identical historical periods. To deal with this issue, we generate scenario-invariant rotation and rescaling matrices by relying on the historical series for Z ^∗ up to the quarter prior to the first scenario period. Using these data, we obtain historical rotation (V ^∗) and rescaling (Σ^∗) matrices. Then, we obtain principal components associated with a particular scenario s, F ^s, which are consistent with the principal components of the historical data, by using V ^∗, Σ^∗, and the macroeconomic data for the relevant macroeconomic variables during the scenario, Z ^{∗, s}:

$$ \mathbf{F^{s}}=\mathbf{Z^{*,s} V^{*} {\Sigma}^{*-1}}. $$

(A.4)

This adjustment allows us to obtain a first principal component factor f (or MPCF) that is the same in the historical portion of the data, across scenarios, and a principal component f ^s for each scenario, used for forecasting purposes, which is consistent with f.

1.3 A3. LASSO selection of balance sheet and income statement variables

Having obtained the optimal macroeconomic factor f that we refer to as MPCF, we estimate the following fixed-effects model:

$$ Y_{it}=\alpha_{i}+\beta Y_{it-1}+\sum\limits_{p=1}^{P}\gamma_{p} {f_{t}^{p}}+\tilde{\tilde{Y}}_{it}, $$

(A.5)

where the last term represents variation in Y _{i t} not explained by fixed effects, the lagged dependent variable, or macroeconomic factors. Letting X be the T×K _x matrix of candidate balance sheet and income statement characteristics, we regress $\tilde {\tilde {Y}}_{it}$ on X and employ the same LASSO procedure as for the macroeconomic factor construction. We solve the following problem, denoting with δ the vector of coefficients associated with the candidate variables in the regression:

$$ \min_{\delta}\left\{RSS(\delta)+\mu*\left( \sum\limits_{k=1}^{K_{x}}|\delta_{k}|\right)\right\}. $$

(A.6)

Implementing a similar grid-search procedure as the one used for the macroeconomic variables, we keep only those income statement and balance sheet variables that, for a given dependent variable, are not discarded in at least 20 percent of cases. This identifies X ^∗⊆X banking characteristics that are relevant for explaining the dynamics of Y _{i t} . We refer to the first principal component x of X ^∗ as the BPCF.

1.4 B1. Calculations for alternative measures of capitalization

We mimic CCAR requirements by calculating capital for a nine-quarter horizon, h = 1,...,9 quarters, under a stress scenario starting just after time period T.³³ We first calculate in each quarter the provision for loan and lease losses under the CCAR regulatory requirement that the allowance for loan and lease losses (ALLL) should cover at least the projected net charge-offs (NCO) for the subsequent four quarters:

$$ \widehat{ALLL}_{T+h}=\sum\limits_{\tau=1}^{4}\widehat{NCO}_{T+h+\tau}=\sum\limits_{\tau=1}^{4}\left\{L_{T}\prod\limits_{\eta=1}^{h+\tau}\left( 1+g_{T+\eta}^{L}\right)\right\} \widehat{nco}_{T+h+\tau}. $$

(B.1)

Projections for the charge-off rate (as a share of loans), $\widehat {nco}_{t}$, are constructed by forecasts from the relevant model taking the CCAR stress scenario path as given, whereas the loan growth rates, ${g_{t}^{L}}$, are given by assumptions that we detail in the article; L _T represents the level of loans the quarter before the beginning of the stress scenario. Note that to obtain the ALLL for the nine quarters in the stress scenarios, (B.1) implies that we need to forecast NCO for 13 quarters. This allows us to construct estimates for the provision for loan-and-lease losses as follows:

$$ \widehat{PROV}_{T+h}=\widehat{ALLL}_{T+h}-\widehat{ALLL}_{T+h-1}+\widehat{NCO}_{T+h}, $$

(B.2)

where the formula reflects that current NCO and ALLL increase this quantity whereas the existing ALLL at end of the previous period reduces it. We assume that profits are taxed at the statutory rate:

$$ \widehat{TAX}_{T+h}=0.35\times Max\left( 0,\widehat{PPNR}_{T+h} - \widehat{PROV}_{T+h}\right), $$

(B.3)

where

$$ \widehat{PPNR}_{T+h}=\left\{A_{T}\prod\limits_{\eta=1}^{h}(1+g_{T+\eta}^{A})\right\}\widehat{ppnr}_{T+h} $$

(B.4)

is driven by the assumptions on asset growth, ${g_{t}^{A}}$, over the stress period and the model-specific conditional forecasts for the ratio of PPNR to assets. Furthermore, we assume a constant dividend-to-assets ratio at the level of the last quarter prior to the first stress period, $\overline {div}=div_{T}$:

$$ DIV_{T+h}=\left\{A_{T}\prod\limits_{\eta=1}^{h}(1+g_{T+\eta}^{A})\right\} \overline{div} $$

(B.5)

and a constant capital adjustment (as share of assets), $\overline {k^{t1l}}=k^{t1l}_{T}$, to obtain Tier 1 leverage capital:³⁴

$$ K^{t1l}_{T+h}=\left\{A_{T}\prod\limits_{\eta=1}^{h}(1+g_{T+\eta}^{A})\right\} \overline{k^{t1l}}. $$

(B.6)

Note that the median bank in our sample paid no dividends as of 2013Q3; hence capitalization numbers derived below cannot be improved much by assuming that banks would cut back on dividends under stress.

We calculate the path for equity as:

$$ \widehat{EQ_{T+h}}=\widehat{EQ_{T+h-1}}+\widehat{PPNR}_{T+h} - \widehat{PROV}_{T+h}-\widehat{TAX}_{T+h}-DIV_{T+h} $$

(B.7)

and Tier 1 capital as:

$$ \widehat{T1C}_{T+h}=\widehat{EQ_{T+h}}-K^{t1l}_{T+h}. $$

(B.8)

Finally, we obtain the Tier 1 leverage ratio by dividing Tier 1 capital by A ^aa, adjusted average assets:³⁵

$$ \widehat{t1lr}_{T+h}=\frac{\widehat{T1C}_{T+h}}{A^{aa}_{T}\prod\limits_{\eta=1}^{h}(1+g_{T+\eta}^{A})}, $$

(B.9)

where for simplicity we assume that adjusted average assets grow at the same rate as quarter-end assets, an assumption that roughly holds in practice.

Similarly, we can calculate the total risk-based capital ratio. The main differences with T1LR is that this ratio is normalized by risk-weighted assets, A ^rw, and requires a different adjustment from equity. In particular, the adjustment factor is given by:³⁶

$$ K^{tr}_{T+h}=\left\{A^{rw}_{T}\prod\limits_{\eta=1}^{h}\left( 1+g_{T+\eta}^{Arw}\right)\right\}\overline{k^{tr}}, $$

(B.10)

where the growth rate of risk-weighted assets, $g_{t}^{rw}$, is allowed to differ from that of assets. Total risk-based capital is given by:

$$ \widehat{TRC}_{T+h}=\widehat{EQ_{T+h}}-K^{tr}_{T+h}. $$

(B.11)

The total risk-based capital ratio then is given by:

$$ \widehat{trcr}_{T+h}=\frac{\widehat{TRC}_{T+h}}{A^{rw}_{T}\prod\limits_{\eta=1}^{h}\left( 1+g_{T+\eta}^{Arw}\right)}. $$

(B.12)

B2. Calculations of capital shortfall

Using the projected capitalization positions, we construct measures of expected capital shortfall as an approximation to the amount of capital necessary to recapitalize banks under severe stress. First, we construct a measure of bank-quarter expected shortfall relative to a regulatory threshold ρ _r , r = {1, 2, 3}, for the T1LR as:

$$ ES^{\rho_{r}, t1lr}_{i,T+h}=\left\{A^{aa}_{T}\prod\limits_{\eta=1}^{h}(1+g_{T+\eta}^{A})\right\} Max(0,\rho_{r}-\widehat{t1lr}_{T+h}). $$

(B.13)

For each bank i, we can obtain the maximum T1LR shortfall over the course of the stress testing exercise:

$$ ES^{\rho_{r}, t1lr}_{i}=Max\left( ES^{\rho_{r}, t1lr}_{i,T+1},\ldots,ES^{\rho_{r}, t1lr}_{i,T+H}\right). $$

(B.14)

We construct similar measures for the TRCR as:

$$ ES^{\rho_{r}, trcr}_{i,T+h}=\left\{A^{rw}_{T}\prod\limits_{\eta=1}^{h}(1+g_{T+\eta}^{Arw})\right\} Max(0,\rho_{r}-\widehat{trcr}_{T+h}) $$

(B.15)

for the path of expected capital shortfalls for an individual bank and

$$ ES^{\rho_{r}, trcr}_{i}=Max\left( ES^{\rho_{r}, trcr}_{i,T+1},\ldots,ES^{\rho_{r}, trcr}_{i,T+H}\right) $$

(B.16)

for each bank’s maximum TRCR shortfall over the stress path. We then construct the maximum capital shortfall for each bank as the largest of the T1LR and TRCR shortfalls:

$$ ES^{\rho_{r}}_{i}=Max\left( ES^{\rho_{r}, t1lr}_{i},\;ES^{\rho_{r}, trcr}_{i}\right). $$

(B.17)

Finally, aggregating across banks, we can calculate a measure of shortfall for all banks in our sample (industry shortfall):

$$ ES^{\rho_{r}}=\sum\limits_{i=1}^{N} ES^{\rho_{r}}_{i}. $$

(B.18)