Stress Testing in a Structural Model of Bank Behavior

Transcription

1 Stress Testing in a Structural Model of Bank Behavior Dean Corbae *, Pablo D Erasmo, Sigurd Galaasen, Alfonso Irarrazabal, and Thomas Siemsen * Department of Economics, University of Wisconsin Madison Federal Reserve Bank of Philadelphia, Research Department Norges Bank, Research Department Department of Economics, Ludwig Maximilians University Munich May 18, 215 Preliminary and Incomplete! Do not distribute or cite! Abstract We present a structural model for microprudential stress testing. The bank has rational expectations and optimizes over portfolio allocation and dividend policy. It performs maturity transformation and is subject to regulatory and technological constraints. Facing these occasionally binding restrictions and uncertainty about future returns and funding opportunities, the bank has an incentive to hold a buffer stock of capital even in excess of regulatory requirements. We calibrate the model using Norwegian balance sheet data and compare it to reduced form stress testing setups. Performing a stylized stress test, we show that structural predictions for capital shortfalls deviate from their ad hoc counterparts. JEL classifications: C63, G11, G17, G21, G28 Key words: Stress testing, structural model, microprudential

2 1 Introduction State of the art models for micro and macroprudential stress testing rely on exogenous, behavioral assumptions and reduced form relationships to map macro scenarios to bank behavior. This approach may exogenously restrict behavior under stress and makes it prone to the Lucas critique. For example, the important recently developed CLASS model of the New York Fed imposes constant balance sheet growth and constant asset and liability composition. Cash flow dynamics are captured by reduced form dependencies between bank specific and macro variables (see Hirtle, Kovner, Vickery, and Bhanot, 214). In addition, stress test models often impose ad hoc dividend rules that dictate the allocation of profits between dividends and retained earnings (see for example Burrows, Learmonth, and McKeown, 212, among others). Based on the dynamic framework of Corbae and D Erasmo (214), we endogeneize portfolio allocation and dividend policy. The bank in this environment is subject to regulatory and technological constraints. We perform a stylized microprudential stress testing exercise and show that structural predictions for capital shortfalls can deviate from their ad-hoc counterparts. The approach used in recent stress testing models has an important purpose: it facilitates an otherwise daunting task the mapping of the macro scenario to the dynamics in bank variables such as allowances, interest income and losses. Modeling the idiosyncratic elasticities of bank variables to some aggregate macro variables is difficult as it requires extensive knowledge of each banks business model and operational strategies. By imposing either ad hoc restrictions or reduced form estimates of these relationships, regulators can take a stance on these dimensions of bank behavior. Moreover, these simplifying assumptions allow models a higher degree of granularity for banks balance sheets, which can generally not be found in structural (e.g. DSGE) models. Nevertheless, these behavioral rules may influence stress testing results in important ways. For example, by exogenously fixing banks balance sheet size and composition, deleveraging during stress are implicitly ruled out and capital shortfalls (capital ratio below regulatory requirements) may therefore be exaggerated. In addition, when portfolio or liability compositions are kept constant over the stress horizon, fire sales of certain asset classes are ruled out and the access to wholesale funding is guaranteed. Dividend rules, moreover, crucially affect the accumulation of bank capital, the core metric of stress tests. By basing bank behavior on ad hoc rules as well as reduced form correlations, and using this framework for policy design (e.g. capital regulation), the exercise is prone to the Lucas (1976) critique. In particular, relationships are derived from historical data and may break down under various stress scenarios. Our model provides important insights on banks intrinsic motives for portfolio and dividend choices, as well as for precautionary capital accumulation. First, a structural model makes a bank s incentives and tradeoffs for portfolio and dividend choices explicit. Second, it reflects optimal bank behavior thereby requiring less discretionary assumptions. Third, a non linear optimization problem introduces a precautionary capital motive for the bank. Facing potentially binding regulatory and technological constraints, together with uncertainty about future returns and funding opportunities, the bank has an incentive to hold a buffer stock of capital even in excess of any regulatory requirements. This precautionary motive has to be neglected in reduced form stress testing setups, as they do not provide a framework to model expectations. We therefore believe that our model is a valuable extension to state of the art stress testing literature. 1

3 Related literature The aim of this paper is to merge two strands of literature that until now coexisted: the literature on microprudential stress testing and on structural models of bank behavior. Summer (27) attributes the origins of models used today for microprudential stress testing to the quantitative risk management frameworks employed in financial institutions to assess portfolio risk exposures (see for example McNeil, Frey, and Embrechts, 25). These models fully explain exposures through a battery of exogenous, stochastic risk factors that determine idiosyncratic loss distribution of asset classes. Following this approach, state of the art micro and macroprudential stress test frameworks impose exogenous stress scenarios that drive risk characteristics of balance sheets. Two complementary approaches are used to map the exogenous macro scenario to bank specific variables: reduced form, backward looking relationships and ad hoc behavioral restrictions. 1 In the first approach, the correlations between bank specific variables and macro variables are estimated empirically, using historical dependencies. For macroprudential stress tests, Acharya, Engle, and Pierret (214) estimate a bivariate GARCH model that measures correlation between a bank s market capitalization and an equity price index to capture the bank s exposure to aggregate risk. Covas, Rump, and Zakrajcek (214) employ a dynamic panel quantile framework, which allows for non linear dynamics, to map macro variables to balance sheet and cash flow dynamics. For microprudential stress tests, the important, recently developed CLASS model (Hirtle, Kovner, Vickery, and Bhanot, 214) employs linear reduced form equations to map macro dynamics to banks cash flow (for a similar approach see Burrows, Learmonth, and McKeown, 212, among others). The second (and complementary) approach is to impose ad hoc restrictions on bank behavior. These restrictions have the advantage that they easily allow for non linear relationships. Behavioral restrictions are mostly imposed on banks dividend policy and balance sheet size and composition. For example, for Dodd-Frank act stress tests, dividends are assumed to remain constant at the pre stress scenario level (Board of Governors of the Federal Reserve System, 213). Hirtle, Kovner, Vickery, and Bhanot (214) allow for dynamic dividend adjustment during stress: dividend policy follows an exogenous error correction process with a constant divided target of 45 % of post tax income. For the 211 euro area wide stress test banks portfolio size was set to remain constant during the stress horizon (EBA, 211). Hirtle, Kovner, Vickery, and Bhanot (214) assume that banks balance sheet continue growing by 1.25 % per quarter and that portfolio shares remain constant under stress. These two approaches facilitates an otherwise daunting task the mapping of the macro scenario to dynamics of bank variables such as allowances, interest income and losses. Modeling the idiosyncratic elasticities of bank variables to aggregate macro variables is difficult as it requires extensive knowledge of banks business model and operational strategies. Imposing either ad hoc restrictions, or estimates of these relationships, allows to capture reduced form dependencies between any variables of interest. This enables richer models with a high degree of granularity on banks balance sheets and income statements, without requiring extensive information on bank specific structural parameters. However, these models are prone to misspecification and the Lucas (1976) critique. Using reduced form frameworks to derive out of sample counterfactuals requires historical relationships to be invariant to severe stress (Borio, Drehmann, and Tsatsaronis, 212) and policy shifts. This is especially a concern since correlations are not derived from past 1 For a survey on state of the art stress testing models see for example Foglia (29) or Borio, Drehmann, and Tsatsaronis (212). 2

4 stress episodes but rather from normal business cycles that can span various regulatory regimes. In addition, both ad hoc rules and reduced form estimates are silent on the role of expectations in banks behavior, thereby neglecting any anticipatory and precautionary motives. Related to that, banks structural incentives for portfolio allocation and dividend policy, the driving forces of observed data patterns, cannot be identified. Ad hoc behavioral restrictions also significantly affect stress test results. For example, by exogenously fixing banks balance sheet size and composition, frameworks implicitly rule out the possibility of deleveraging during stress, and may therefore exaggerate capital shortfalls (capital ratio below regulatory requirements). In addition, when portfolio or liability compositions are kept constant over the stress horizon, fire sales of assets are ruled out and the access to wholesale funding is guaranteed. Ad hoc dividend rules, moreover, crucially affect the accumulation of bank capital, the core metric of stress tests. There is evidence that during the recent financial crisis, US bank continued paying high dividends even though capital was running low and loan losses loomed high (Acharya, Gujral, Kulkarni, and Shin, 211). This points toward a more complex function of dividends (e.g. as a market signal) than can be captured by static or backward looking rules. We contribute to state of the are stress test literature by replacing ad hoc portfolio allocation and dividend policy by optimizing behavior derived from a structural model of banking. A growing strand of macro banking literature focuses on the effects of micro and macroprudential regulation in light of the soon to be implemented Basel III framework. However, these models mostly focus on specific regulatory requirements and policy tools and therefore restrict modeling of banks balance sheets, income statements and sources of risk to variables of interest. Thus, they do not feature the degree of granularity of balance sheet and income statement required for an informative stress test. The Basel III framework imposes liquidity requirements, prompt corrective actions and four layers of capital requirements: minimum, conservation, counter cyclical and systemic requirements (Basel Committee on Banking Supervision, 211). The important contribution by De Nicolo, Gamba, and Lucchetta (214) analyzes the interaction of these three regulatory tools in a structural, partial equilibrium model of a representative bank with a parsimonious loan profit function that does, unlike the model we are using, not explicitly model credit default, a crucial concern in stress tests. They show that an optimal minimum capital requirement exists and that liquidity requirements always reduce lending, while both are efficiency dominated by prompt corrective actions. In a medium scale DSGE model with costly state verification and bank default Clerc, Derviz, Mendicino, Moyen, Nikolov, Stracca, Suarez, and Vardoulakis (214) also find that welfare benefits are concave with respect to minimum capital requirements. Due to the complexity of the model, it is solved by first order perturbation around a steady state in which the capital requirement is binding. It is therefore not suitable for stress testing as it cannot model capital shortfalls and since shocks are assumed small enough for the model to remain in the proximity of its steady state, thus neglecting non linear crisis dynamics. Corbae and D Erasmo (214) study the effects of changes in minimum capital requirements on industry wide loan supply and competition structure in a dynamic game between a big bank with market power and a price taking fringe sector. They find that higher requirements induce a reduction in aggregate lending and a more concentrated industry with higher loan rate. Our model is closely related to theirs, however, we allow for a more granular balance sheet and maturity transformation, while abstracting for industry equilibrium by considering the single bank s decision problem. Rios-Rull, Takamura, and Terajima (214) focus on the effects of capital requirements on bank s dividend policy. In their model the misalignment between 3

5 bank manager s and bank owner s objectives induces excess dividend payment and high equity costs. While minimum capital requirements are useful for reducing moral hazard due to limited liability, conservation requirements are targeted more directly at dividend policy and are thus more efficient in restricting high dividend payments in a crisis. Our paper is also related to the contribution of Bianchi and Bigio (214). There, the authors study the effects of banks portfolio choice on the conduct of monetary policy in a DSGE model with an interbank market. Looking through the lens of their model on the recent financial crisis, they conclude that the most likely scenario that induced the observed dynamics is a initial reduction in loans supply due to an increased precautionary motive in the face of higher interbank uncertainty, which induced adverse effects onto the real economy and led to a subsequent reduction in loan demand. The setup focuses, however, on the liquidity risk in banks balance sheets due to maturity mismatch and abstracts from any credit risk, i.e. loans are considered safe. Since credit risk is an important factor for banks equity losses, our model features both liquidity risk (through stochastic deposit supply) as well as credit risk. The remainder of the paper is structured as follows: Section 2 provides some facts on bank and crises heterogeneity. Section 3 lays out the model, Section 4 shows our preliminary calibration and Section 5 conducts simulation exercises. Finally, Section 6 concludes. 2 Facts about bank and crises heterogeneity Our framework aims at understanding bank behavior during stress episods. There are two important dimensions to banks survival probabilities. First, the equity position with which banks enter stress. Second, stress severity and duration. Heterogeneity along these two dimensions are therefore key determinants of stress outcomes. This section provides some empirical facts on these two sources of heterogeneity. 2.1 Equity heterogeneity We consider the Norwegian banking sector. Norges Bank s ORBOF database provides balance sheet, income statement and interest rate data for individual Norwegian financial intermediaries. These include a total 266 banks and 47 credit institutions available for different points in time. Since 27 Norwegian banks are allowed to issue covered bonds (OMFs) through credit companies (kredittforetaks). These companies have become an important funding source for banks as well as an important source of loans for the real economy (see Raknerud and Vatne, 213). To account for the fact that banks outsourced an important part of business to credit companies, we do not consider isolated banks but merge banks with their corresponding credit companies. We call this unit of observation banking group (BG). At this level of aggregation we have full data for four relevant BGs. As can be seen in Table 1 the Norwegian banking industry is highly concentrated. The six BGs account on average for 8 % of total industry assets, 71 % of retail loans and for 76 % of commercial loans. Asset shares drop rapidly from the largest to the second largest BG and again from the third to the fourth largest. Therefore, our sample of BGs covers big, dominant groups as well as smaller fringe groups. 4

6 Table 1: Shares in aggregate market Asset Share Loan market share retail C&I BG BG BG BG accumulated Note: Shares are computed as 1997Q1 214Q2 averages, source: ORBOF Capital ratios are key measures in a stress test. They determine banks abilities to cope with losses under counterfactual stress scenarios. Table 2 shows core capital ratios both at 211Q1 (Basel III framework was announced in December 21) as 211Q1-214Q2 averages with standard deviation and for the last observation available (214Q2). The two commercial banking groups, BG1 and BG3, now hold capital ratios close to the maximum Basel III requirement (13 %), which they started building up after its announcement. The savings and loans banking groups, BG2 and BG5, now hold capital ratios above the maximum Basel III requirement. Their capital equipments were already close to or above 13 % at Basel III announcement and experienced less volatility. The smallest banking group holds highest excess equity. The two bottom rows show cross sectional mean and standard deviation. Average core capital ratio increased over time, while dispersion decreased from 2.7 pps to 1.5 pps. Table 2: Core Capital Ratios 211Q1 211Q1-214Q2 (std.dev.) 214Q2 BG (1.7) 13.6 BG (.8) 14.4 BG (2.) 13.7 BG (.6) 17.2 mean std.dev Note: Basel III was announced in December 21. Source: ORBOF 2.2 Crises heterogeneity Economic crises have multiple dimensions. They have different duration and severity, they originate in different markets and are of different international extent. Consequently, their effects on banks equity are diverse. Barro and Ursua (28) provide an empirical analysis of macroeconomic crises since 187 for multiple countries. Figure 1 shows how GDP crises differ with respect to size of contraction, duration and recovery time back to HP-trend. The red line indicates the mean of the distributions. The mean GDP contraction is 21 % with a mean duration of 2.9 years and a mean recovery time of 1.7 years. The challenge for informative stress testing is, besides the choice of framework, the choice of macro scenario. It must be server enough to impose stress on banks balance sheets, while also capturing relevant sources of future stress. 5

7 Figure 1: Crises Heterogeneity Distribution of crises size 3 4 Duration of crises 5 Duration of recovery GDP drop (%) years (peak to trough) years (trough to trend) Notes: based on 36 countries since 187. A crisis is defined as GDP drop larger than 9.5 %. Recovery time defined as years from trough back to GDP trend (λ = 6.25). Red line indicates mean. Source: Barro and Ursua (28) and WDI to extend sample until 213. We will target moments of the average Barro and Ursua (28) crisis to calibrate our macro scenario. By using GDP crises across countries instead of special financial or banking crises, we account for the uncertainty in designing relevant macro scenarios for stress testing. One the one hand, purely backward looking design is prone to the false conclusion that the next crisis will be like the last crises. On the other hand, the identification of future trouble spots is difficult. By considering diverse GDP crises, our approach is agnostic to the exact nature of a crisis. Instead, we take GDP crises characteristics as reduced form reflections of the underlying source of crisis and allow these characteristics to affect banks differently through their impact on non performing loans. 3 A model of bank behavior Our model is a modified version of Corbae and D Erasmo (214) adopted to a single bank s decision problem. In period t the bank provides loans L ts to sector s S. We follow De Nicolo, Gamba, and Lucchetta (214) and assume that loans have an exogenous 1 maturity 1+χ s such that each period a constant fraction χ s of loans L ts matures. While sector specific maturity is exogenous, the fact that the bank endogenously chooses its loan exposure to the different sector induces an endogenous aggregate loan portfolio maturity. The bank s external funding D t has one period maturity. Therefore, the bank engages in maturity transformation by converting short term funding into long term loans. Each period t is divided into to sub periods: the beginning and the end of period. Beginning of period At the beginning of period t the bank is hit by two idiosyncratic shocks: (1) a shock δ t to maximum external funding capacity and (2) a shock ρ t to external funding costs. Both shocks follow a first-order Markov process in which δ t with transition matrix G(δ t, δ t+1 ) and ρ t Υ with H(ρ t, ρ t+1 ), respectively. 6

8 Constraint 1 (Capacity Constraint). The idiosyncratic shock to maximum funding capacity δ t induces an upper bound on external funding D t. D t δ t, (3.1) The bank is also exposed to aggregate uncertainty, captured by an exogenous GDP shock z t evolving according to a first-order Markov process with z t Z and transition matrix F(z t, z t+1 ). The GDP shock influences the share of non-performing loans on the bank s asset side. There are s + 1 endogenous state variables: the capital stock a t and the heritage loan stocks l ts. Therefore, the beginning of period state space is given by {z t, a t, {l ts } s, δ t, ρ t }. Given these states the bank makes beginning of period portfolio choices. On the liability side it decides on how many exogenous one period funding D t δ t at exogenous net costs ρ t to take on. D t has to be repaid at the beginning of period t + 1 when the new capacity shock δ t+1 materializes. External funding, together with bank capital a t, can either be invested into perpetual securities A t or into risky loans L ts. Consequently, balance sheet identity requires A t + s L ts = D t + a t (3.2) Since we assume a constant fraction χ s of loans L ts to come due at the beginning of every period, without further adjustment, the inherited loans stock l ts gradually decreases over time. The bank can decide, however, to reduce loan exposure faster than at rate χ s. In this case it must pay quadratic adjustment costs on disinvestment I ts = L ts l ts < Ψ s (I ts ) = I(I ts < )ψ s I 2 ts, s S, (3.3) where I( ) denotes the indicator function and ψ s is the cost coefficient. Marginal adjustment costs are increasing in I ts to reflect increasing reductions on loans face value if a large fraction of the loan stock has to be liquidated and sold off. These costs can capture both liquidation costs that arise when loans are sold off and fire sale costs due to sudden and large reductions in the loans stock. In contrast, increasing the loan exposure by choosing I ts does not require any additional adjustment costs. Any potential monitoring and screening costs are captured by proportional non interest expenses, c s, as shown below. Moreover, the bank is constrained in its investment choice by a regulatory capital requirement Constraint 2 (Capital Constraint). ( ) ϕ L ts + wa t a t, (3.4) s which requires to hold ϕ units of equity for each unit of risk weighted assets s L ts + wa t where w denotes the relative risk weight of securities to loans. Securities pay a safe interest of r a and bank loans generate an interest payment of r L ts. However, a fraction (1 P t+1s ) of bank loans are non performing. In that case it pays no interest and a fraction λ s has to be written down, reducing cash flow and tomorrows loan stock l t+1s. We assume that P t+1s = P(r L ts, z t, z t+1 ). 7

9 End of period End of period is initiated with the realization of the new aggregate shock z t+1. 2 This shock determines the fraction of non performing loans (1 P(rts, L z t, z t+1 )) in the bank s portfolio. At this stage, bank s cash flow is given by C t+1 = π t+1 (3.5) where π t+1 captures end of period cash profits. These are given by π t+1 = s [{ } Pt+1s rts L c s Lts Ψ s (I ts ) ] + r a A t ρ t D t κ, (3.6) where c s denotes proportional non interest expenses of loan providence such as screening and monitoring costs and κ are fixed costs. Note that there are no principal cash flows for L ts, A t and D t. Moreover, we assume that loan interest rates are floating. This is reflected in the fact that the contemporaneous interest rate r L ts applies to all loans L ts = l ts + I ts. This assumption reduces the state space, as we do not need to keep track of the whole history of loan providence. Note that Equations (3.5) and (3.6) imply that non performing loans neither pay interest nor principal. Banks now decide on their dividend policy, D t+1. They can distribute the cash flow to equityholder or retain earnings. Moreover, they have access to a short run liquidity market in which they can borrow liquidity at net costs r b. Let B t+1 < denote retained earnings and B t+1 > denote short run borrowing. Then, dividends are determined as The bank is also constrained in its dividend policy: Constraint 3 (Dividend Constraint). D t+1 = C t+1 + B t+1 (3.7) D t+1 [σd t, σd t ] (3.8) that is, we assume that contemporaneous dividend payments cannot deviate more than a factor σ (σ) above (below) previous period dividend. Constraint 3 can account for factors outside the model that induce sluggish dividend adjustment during crises (see Acharya, Gujral, Kulkarni, and Shin, 211, for the recent financial crisis). A reason for this can be asymmetric information between banks, such that dividends act as signaling device to peer institutions and the market. In this case we would have σ < 1 and σ = +. 3 If σ = and σ = +, the constraint is equivalent to ruling out seasoned equity offering, as dividends are constrained below at zero. Also, this constraint can capture the Basel III conservation requirement, which prohibits dividend payments if Tier 1 equity is below a threshold. In this case σ = σ =. Equations (3.7) and (3.8) together imply that if the bank wants to stay in the market despite negative cash flows, it has to tap the short term liquidity market (B t+1 > ) to not violate Constraint 3. If, however, the continuation value of operating in the market is low, the bank may prefer to exit the market. In contrast, if cash flow is high, the bank may not want to pay everything out as dividends but rather wants to retain some 2 We use the timing convention that all variables, which are determined after the realization of the aggregate shock z t+1 have time index t For now, we don t consider signally issues since we work in a model with perfect information. 8

10 earnings (B t+1 < ) to raise next periods initial capital a t+1. Short term borrowing requires collateral in form of securities. Constraint 4 (Collateral Constraint). The gross repayment of short term borrowing must not exceed contemporaneous security holdings: with r b = if B t+1. (1 + r b )B t+1 A t, (3.9) If banks do not have enough securities for covering a negative cash flow, they are forced to exit the market as Constraint 3 is violated. An alternative interpretation of Equation (3.9), on which we will drawn for the stress testing exercise, is that security holdings A t are not liquid and require liquidation costs 1 + r b if they are sold to cover negative cash flow. Following this interpretation r b is a measure for the fire sale price of securities upon stress. The higher r b, the higher is the discount, relative to face value, the bank has to accept when liquidating assets. Constraint 4 also reflects our assumption that loans on the balance sheet cannot be used as collateral for short term borrowing. Each period a fraction χ s of loans exogenously matures at the beginning of each period. Non performing loans neither generate interest nor principal repayment. Moreover, the bank writes down a fraction λ s on each dollar of non performing loan. Therefore, beginning of period t + 1 heritage loans are given by l t+1s = [1 χ s ]P t+1s L ts + [1 λ s ] (1 P t+1s ) L ts (3.1) Also, at the beginning of period t + 1, before any choice is made, the short term liquidity market is clears, i.e. B t+1 is repaid and principal repayment of performing loans occurs. Thus, beginning of next periods capital a t+1 is given by a t+1 = A t + s [l t+1s + χ s P t+1s L ts ] (1 + r b )B t+1 D t (3.11) As discussed above, retained earnings (B t+1 < ) raises a t+1. Figure 2 summarizes the timing. {z t, a t, {l ts } s, δ t, ρ t } Figure 2: Timing Assumption z t+1 {z t+1, a t+1, {l t+1s } s, δ t+1, ρ t+1 } z t+2 A t, {I ts } s, D t C t+1 A t+1, {I t+1s } s, D t+1 C t+2 stay exit B t+1, D t+1, a t+1, {l t+1s } s Due to the recursive nature of the bank s problem, we can drop time subscripts. Let 9

11 x t = x and x t+1 = x. Banks objective is to maximize expected franchise value, E t + k=t+1 β k D k, (3.12) where β is equityholders constant discount factor. The value of the bank at the beginning of the period is given by V (z, a, {l s } s, δ, ρ) = D δ A + L s a + D s ( ) ϕ L s + wa a s L s = L d s, s S max β E z zw (A, {I s } s, D, δ, ρ, z ) A,{I s} s,d s.t. The last constraint requires bank specific loan market clearing, where L d s is bank specific loan demand from sector s. In contrast to Corbae and D Erasmo (214) we assume exogenous loan demand L d s = L d s(rs L, z) with L d s/ rs L and L d s/ z. End of period value is given by { W (A, {I s } s, D, δ, ρ, z ) = max W x=1 (A, {I s } s, D, δ, ρ, z ), W x= (A, {I s } s, D, δ, ρ, z ) }, x {,1} with the exit value given by { [ ({ ) W x=1 (A, {I s } s, D, δ, ρ, z ) = max, ξ (r L s + χ s )P s + χ s [1 λ s ](1 P s) cs} Ls s ] } + (1 + r a )A (1 + ρ)d κ (3.13) where the lower bound zero implies limited liability upon exit and ξ is the salvage fraction the banks receive from liquidating assets. If limited liability kicks in, depositors are not entirely repaid. The continuation value is given by { W x= (A, {I s } s, D, δ, ρ, z ) = max D + E (δ,ρ ) (δ,ρ)v (z, a, {l s} s, δ, ρ ) } B A 1+r b D = C + B s.t. a = A + [l s + χ s P sl s ] (1 + r b )B D s l s = [1 χ s ]P sl s + [1 λ s ] (1 P s) L s, s S 1

12 4 Calibration Our calibration approach follows Corbae and D Erasmo (214) closely. One period corresponds to a quarter. The bank in the model corresponds to a banking group in the data. A banking group includes both the retail banking units as well as the credit companies which emerged in Norway in 27 and have since become an important funding source for banking groups (see Raknerud and Vatne, 213). We allow for two sectors s S = {retail, C&I}. The data is extracted from the Norges Bank ORBOF database, which provides information about individual Norwegian banks balance sheets, income statements and interest rates. All parameters are in real terms. We deflate using total CPI index. Let variables with superscript i denote variables calibrated to bank group i and variables without superscript aggregate variables. Exogenous shock processes The aggregate shock, z Z = [z H z L z C z R ], follows a four state Markov process. We need to calibrate the state vector Z and the transition matrix F(z, z) R 4 4. In our model, z is the only source of aggregate fluctuations. Therefore, it captures normal business cycle fluctuations, as well as the aggregate component of the stress scenario. We allow for two states to capture normal fluctuations: a high state, z H, and a low state, z L. These states and their transition probabilities are calibrated to capture the normal Norwegian business cycle. There is one crisis state, z C, and on recovery state, z R, which captures a smooth transition out of crises. Theses states can be calibrated to reflect any exogenous stress scenario. The calibration we provide here is to illustrate the mechanics of our model. Since crisis observations in Norway alone are limited, we rely on the Barro and Ursua (28) disaster dataset, which captures boom bust cycles for 36 countries between 187 and 28. We extend the data until 213 and identify GDP peaks and troughs using the method suggested in Barro (26). A crisis is defined as a GDP contraction of larger or equal 9.5 %. We derive the crisis calibration from the average of the 177 international crises observations (see Section 2). Figure 3 shows the log GDP series for Norway between 187 and 213 together with identified peaks and troughs. From the 144 Norwegian annual observations, 118 are normal business cycle years and 26 are crises years. The average contraction from a business cycle peak to a non crises trough is 2.58 % in Norway. We normalize z H to unity and set z L = z H.258 =.9742 to match the average contraction. Turning to the crisis state, we find that the average GDP contraction from peak to crisis trough is 2.56 %. Since the normal business cycle peak is identified by z H we set z C = z H.256 = To calibrate the recovery state z R, we measure the average recovery time from crisis trough back to GDP trend. We find that it takes on average years to recover back to trend. We identify z R as the average GDP level after half the recovery time: z R =

13 Figure 3: Norwegian GDP log Index (26=1) GDP Trend Peak Trough Crisis Notes: Peak and troughs identified by peak trough method of Barro (26). Data from Barro and Ursua (28) and extended using WDI database. Let P ij denote the probability of switching from state i to j. For the transition matrix F(z, z) we impose the following zero restrictions: P HH P HL F(z, z) = P LH P LL P LC P CC P CR, P RL P RR i.e. from z H only z L can be reached, the only way into a crisis is through z L, the recovery state z R can only be reached from the crisis state and from the recovery state only z L can be reached. To derive the switching probabilities between normal times state we follow Barro and Ursua (28) and estimate these probabilities as the ratio of normal times boom bust cycles (13) over normal time years (118). Then, P HL = P LH = 13/118 =.112 and P HH = = As in Barro and Ursua (28) we calibrate the probability of leaving normal times and entering a crisis as the ratio of crises observations over normal time years. In our dataset we have 544 yearly observations including 515 crises years over 177 crises and 4925 normal time years. Therefore P LC = 177/4925 =.359 and P LL = 1 P LH P LC = Along the same line, the probability of leaving a crisis and starting a recovery is estimated as the ratio of crises observations of crises year, i.e. P CR = 177/515 = Thus, P CC = 1 P CR = Finally, we calibrate the recovery persistence to match the average recovery duration (trough to trend) of years in the data. Since the expected recovery duration is given by 1 1 P RR, we have P RR =.66 and P RL =.34. We transform the annual probabilities to quarterly probabilities, P Q ij, through P ii = calibration. ( P Q ii) 4. Figure 4 summarizes our crisis 12

14 Figure 4: Crisis Calibration duration: 2.9 yrs recovery: 3. yrs z H z L recession: 2.6 % z R contraction: 21 % recovery growth: 19 % z C trough For the idiosyncratic deposit supply shock δ i = [δh i δi N δi H ], we proceed similar. First, we derive the quarterly volume of external funding, d i t, by summing up bank i s outstanding deposits, bonds and commercial papers. We estimate the following relationship for the period 1987Q4 214Q2: log(d i t) = (1 ρ d )k + ρ d log(d i t 1) + k 1 t + k 2 t 2 + k 3t + v d t, (4.1) where t denotes a time trend, k 3t are quarterly fixed effects and vt d N(, σd 2 ). We use the estimates ˆρ d and ˆσ d to discretize log(d i t) with Tauchen and Hussey (1991). Thereby, we set the middle state δn i equal to bank group i s average share of external funding in the aggregate Norwegian banking market. The discretization algorithm also provides us with G i (δ, δ). For the shock to bank i s external funding cost we calibrate the state vector ρ Υ i = [ρ i H ρi N ρi L ] and the corresponding transition matrix Hi (ρ, ρ) R 3 3 to match a rough cost measure. As shown in Section 2 a bank s stable external funding sources consist by a large part of deposits and bonds. The latter are mainly covered bonds issued by a banking group s credit companies. Therefore, we approximate total funding costs using a weighted average of deposit costs and interest expenses on bonds and commercial paper. We construct a quarterly series for interest expanses on bonds and commercial paper by dividing interest expenses on bonds and commercial paper in a given quarter by the total stock of bonds and commercial paper in this quarter. We weight costs on deposits and costs on external funding by their shares in the banking liability side. We then identify ρ i N with average funding costs between 21Q4 to 214Q2, ρi L with funding costs one standard deviation below mean and ρ i H with funding costs one standard deviation above mean. The maximum likelihood estimator for transition probability H i (ρ k, ρ l ), i.e. from state l to k, is the ratio of observations funding costs switched from state l to k to the number of state l observations. External calibration We calibrate net return on securities, ri a, to match the average ratio of interest income from bonds and certificates in a given quarter to the total stock 13

15 of bonds and certificates in the same quarter. We take the time average from 1997Q1 to 214Q2. To calibrate the proportional net non interest expanses, c i, to we use the ratio of wage and other staff costs minus net provision income over total gross lending. This choice of items reflects the assumption that to some degree staff costs and provision incomes are proportional to a banking group s share of loan providence and that these items should therefore not be allocated to fixed costs κ. Due to a lack of data, we cannot calibrate loss given default, λ i, by sector but instead assume that it is identical between retail and commercial sector. We therefore match total loss on lending. In the model total loss on lending is given by s (1 Pi st+1)l i stλ i s. To calibrate λ i consistently, we first derive a time series measure for (1 Pst+1) i as the ratio of new non performing loans by sector over gross lending by sector. 4 We then calibrate λ i as the ratio of total loss on lending over total non performing loans ( s (1 Pi s)l i s) taking the 1997Q1 to 214Q2 time average. In contrast to Corbae and D Erasmo (214) our model does not feature an entrepreneurial sector. Instead we calibrate the inverse demand equation ris(l L is, z) as well as the entrepreneurial risk reaction function in normal times P is (ris, L z ) directly from the data. To this end, we assume (1 P i st+1) = g(r L ist, z ) (4.2) r L ist = f(l i st, L i st, z), (4.3) where rist L denotes the loan rate of banking group i to sector s in quarter t, L ist is detrended loan supply of i to s in t and L ist is detrended loan supply of all other banking groups except i to s in t. z the continuous counterpart to the discretized shock z derived from HP-filtered GDP. (1 Pst+1) i is the fraction of non performing loans of i in s at end of t constructed as discussed in footnote 4. To account for stationarity of our model, we detrend loan supply from i to s by subtracting the aggregate linear market trend in sector s between 1987Q4 and 214Q2 from from time series L i st and L i st. By including L i st to Equation (4.3) we can, to some degree, capture the competitive environment with which banking group i interacts. Using least square regression we estimate Equation (4.2) as (1 P i st+1) = α + α 1 r L ist + α 2 z t+1 + u i st, (4.4) based on normal times observations. Therefore, this estimated equation is used in the model to describe the dynamics of non performing loans in states z H and z L. Non performing loans in crisis states z C and z R are calibrated based on banking crisis date in Laeven and Valencia (212). Over all documented crises, average peak non performing loans are 14.4% of total loans. For this simulation exercise, we assume that peak non performing loans occur on impact, when entering a crisis. Therefore 1 P i st+1(z = z L, z = z C ) = When staying in a crisis for multiple periods, non performing loans are assumed to be 5 % lower than on impact: 1 P i st+1(z = z C, z = z C ) = When leaving the crisis trough and entering a recovery non performing loans are calibrated as follows: we linearly regress log peak non performing loans from Laeven and Valencia (212) for each identified crisis on peak output loss, implied by HP-filtered real 4 ORBOF only provides data on new non performing loans from 21Q4 on. Therefore, we impute a time series going back until 1997 by computing the fraction of new non performing loans in the stock of non performing loans for the quarters available, take time average and then assume this fraction to be the same for the quarters where no data is available. 14

16 GDP (λ = 6.25). Using this exponential dependency, we derived the implied reduction in peak non performing loans when going from z C to z R. This gives us 1 P i st+1(z = z C, z = z R ) = To calibrate Equation (4.3) we use the estimates found in Akram (214). There, the author shows that the aggregate long run relationship between loan demand and real interest rate and GDP for Norway can be described by 5 r L t,c&i =.4726 z t.2141 log (L t,c&i ) (4.5) r L t,retail =.7368 z t.2632 log (L t,retail ) (4.6) From these equations we trace out the idiosyncratic demand curves, banking group i faces. To this end we adjust the semi elasticity of loan demand to target the average net interest margin, defined as r L is r D is, of banking group i. We adjust the constant to target i s loan market share in the aggregate market. We first adjust the sector specific aggregate demand curves from above to account for our normalization of the aggregate state, z t. Specifically, we adjust the constant to match the average aggregate real loan rate and the average aggregate loan to GDP ratio, Ls between 21Q4 and 214Q2. This determines sector specific aggregate loan demand L st (r L ts, z t ). We then derive the group specific constant by computing the average loan market share (over the same time period) of group i in sector s, m i s, and subtracting the implied loan demand that all other banks face, (1 m i s) [ Ls z ], from aggregate loan demand. Thus, idiosyncratic loan demand is given by L i st = L st (r L ts, z t ) (1 m i s) [ Ls z ], {i, s} (4.7) Adjustment costs are only paid if the bank liquidates the loan stock faster then the rate of maturity. This happens exclusively during crisis times, because in normal times the return on lending is positive. Therefore, the adjustment cost coefficient is set to target the relative variance of aggregate loans of bank i to GDP variance during the Norwegian banking crisis of We set fixed cost κ to target the average annual equity return of bank i. We calibrate to maturity parameters χ retail and χ C&I to an average maturity of 1 years and 4 years respectively. Since we consider a model with perfect information, we choose σ = and σ = +, such that Constraint 3 is equivalent to ruling out seasoned equity offerings as dividend is constrained below by zero. The two remaining parameters ξ and β are calibrated as in Corbae and D Erasmo (214): ξ =.7 and β =.987 (adjusting for quarterly instead of annual calibration). Table 3 shows the calibration for the a large Norwegian banking group. 5 see his equations (4) and (5) plus (6). 15

17 Table 3: Parameter Calibration large Norwegian banking group Parameter Calibration Target z G good state of the world 1 normalization z B bad state of the world.9742 Norwegian business cycle z C bad state of the world.7898 Barro and Ursua (28) z R bad state of the world.8729 Barro and Ursua (28) GDP data ρ H high cost state.58 funding cost measure ρ N high cost state.33 funding cost measure ρ L low cost state.8 funding cost measure δ H high funding state.1275 funding measure δ N medium funding state.85 funding measure δ L low funding state.425 funding measure β discount factor bank.987 Corbae and D Erasmo (214) r a net securities return.52 int. inc. bonds and sec. r b short term borrowing.52 r a (Corbae and D Erasmo, 214) c net non interest exp.14 net non-interest expanses ψ adjustment costs 2.1 Var(Loans)/Var(GDP) Crisis,Norway λ loss given default.1796 total loss to lending κ fixed cost of operation.1 return om equity ξ salvage value.7 Corbae and D Erasmo (214) χ retai maturity parameter 1/14 average maturity of 1 years χ C&I maturity parameter 1/5 average maturity of 4 years [σ, σ] dividend constraint [, + ] no SEO F(z, z) = H(ρ, ρ) = G(δ, δ) = The estimation results from equation (4.4) with p-values in parenthesis are (1 P retail,t ) =.4 (.72) (1 P C&I,t ) =.57 (.13) (.) (.) r L retail,t.635 (.) r L C&I,t.2545 (.) [z t+1 1] [z t+1 1] 16

18 5 Quantitative Analysis This section features preliminary stress testing results, derived from a one sector version of the model without C&I lending. Using the calibration discussed in Section 4, we derive quantitative results from our model s policy functions as follows: 1. Given the preliminary state of our calibration we currently target two moments in state z L : a RoE of 12 % and a net interest margin of 2.3 pps. 2. We seed the simulation at z L steady state values for bank i s equity and capital ratio. Given that there is no positive crisis probability in z H, seeding the simulation in z L allows for immediate crisis. 3. In exercises without idiosyncratic uncertainty, idiosyncratic shocks are assumed to remain in their normal states (ρ i N, δi N ). In exercises with idiosyncratic uncertainty, the shocks are initial seeded in their normal states. 4. We simulate 12 periods (3 years) of the aggregate and idiosyncratic shocks, 1, times. For the counterfactual stress scenario, we impose the following regulatory capital regime: { 13. %, if z {z h, z L } ϕ(z) = 4.5 %, if z {z C, z R } The 13 % requirement in normal times corresponds to the maximum requirement under Basel III, with a 4.5 % minimum requirement, a 2.5 % conservation requirement, a 3.5 % systemic requirement and a 2.5 % countercyclical requirement. 5.1 Normal Times Analysis Given our preliminary calibration, Table 4 shows currently targeted and non targeted moments for this BG. 5.2 Stress Test Table 4: Simulated Moments Moment Model Data s.s. z L 214Q2 targeted NIM RoE non targeted core capital ratio Notes: Source: ORBOF Only Aggregate Uncertainty As discussed in Section 2 two key dimensions affect stress outcomes: equity position upon stress entry and stress duration. In our macro scenario both dimensions are driven by the probabilistic crisis nature. 17

19 Facing the adverse macro scenario, the sufficiency of a bank s contemporaneous equity equipment is of ample importance for regulators to prevent bank default that imposes costs on deposit insurance and tax payers. Our framework provides two metrics of stress outcomes: first, the exit probability of a bank, i.e. E[W x=1 ( ) > W x= ( )], and second, conditional on survival, the equity depletion the bank experienced during stress. In contrast to reduced form stress tests, our framework identifies the bank s structural incentives for equity accumulation. The reasons for equity accumulation in our framework are twofold: (1) the bank is exogenously required through regulation to hold a fraction ϕ of RWA as equity, but (2) even in absence of any regulatory requirement the bank chooses to hold positive equity to protect and maximize its charter value both before and during stress. Table 5 shows stress test outcomes. Given an initial capital ratio of 13 % our model predicts a counterfactual survival probability facing an average Barro and Ursua (28) crisis of 96 % and a 4 % default probability. The bank will on average exit after 24.8 crisis periods (6.2 years) and will survive an average of 12.9 crisis periods (3.2 years). Table 5: Stress Test Outcomes exit survival Frequency.4.96 E(duration) E(equity loss) Note: baseline calibration for 13 % normal times capital requirement. The bank exits when equity is fully run down. Conditional on survival, the bank is expected to suffer a equity loss of 36 %, leaving the stress horizon with a strongly depleted capital position. Generally the bank can take two different exit routes: first, it can decide to exit immediately after observing end of period cash flow. In this case the bank pays no further dividend and receives the salvage value subject to limited liability (Equation 3.13). The alternative route is to stay in the market to liquidate as much assets as possible. To this end, the bank uses short term borrowing not only to cover the negative contemporaneous cash flow, but to liquidate securities (B = A/(1 + r b )). According to Equation (3.11) next period capital is only a = s [l s + χ s P s L s ] D. 6 Next period, capital is completely invested into securities and loans are liquidated. At the end of this period, the bank exits. This way, the bank is able to pay out dividends longer. On the other hand, for each additional period to enter, it has to pay fixed cost κ. If fixed costs are sufficiently low, the bank will choose to liquidate capital before leaving the market, resulting in a 1 % equity loss upon exit. Figure 5 shows an exit path. 6 Note that a must be positive, as otherwise staying in the loan market would not be in the choice set. 18