Systemic risk, credit risk and macroeconomic fluctuations

Transcription

1 Systemic risk, credit risk and macroeconomic fluctuations Fabian Lipinsky April 24, 2016 Abstract We analyse borrowers and lenders intertwined balance sheet dynamics by introducing a joint dynamic capital structure theory in an otherwise standard model of real business cycles. Leverage of borrowers and lenders arises endogenously and determines default frequencies and credit spreads. Moreover, financial intermediaries (the lenders) are subject to changes in credit and aggregate risk that strongly affect financial conditions, credit flows and macroeconomic fluctuations. We apply an econometric estimation strategy that allows to differentiate between both sources of risk. We find that while credit risk shocks associated to firms balance sheets where significantly influencing macroeconomic aggregates during the information technology crisis, aggregate risk shocks strongly affected intermediaries balance sheets and funding conditions during the financial crisis, and were the dominant force behind fluctuations in investment. We are greatful to Hélène Rey, Andrew Scott, Lucrezia Reichlin, Richard Portes, Francisco Gomes, Paolo Surico, Frédéric Malherbe, and seminar participants of London Business School for helpful comments. Mail address: London Business School, Regent s Park, London NW1 4SA; International Monetary Fund, th Street N.W., Washington, D.C

2 1 Introduction The financial crisis has drawn attention to the interconnectedness between the financial sector and real economic activity. Both borrowers and lenders balance sheet dynamics are closely connected. Deteriorating borrower balance sheets and associated increasing loan and security portfolio losses hit balance sheets of lenders hard during the financial crisis. Increased uncertainty about future credit 1 losses led to a repricing of risk and a deterioration in credit conditions, reflected in an increase of credit spreads and a reduction of credit. At the same time, funding conditions of lenders abruptly deteriorated. Uncertainty about financial intermediaries balance sheet strength lead to an increase in credit spreads of financial institutions and an outflow of funding, triggering a deleveraging process of financial institutions, with adverse consequences for the real economy. Clearly, the various feedback effects between borrowers and lenders are not separable and ask for a comprehensive but tractable framework to analyse the joint dynamics. Especially the widening of borrowers and lenders credit spreads during the financial crisis and the associated decline in economic activity has highlighted the importance of changes in risk for the allocative decisions of lenders as well as its debt holders as the marginal investors. Consequently, to be compelling a macroeconomic framework should link the balance sheet dynamics and financial conditions of borrowers and lenders, capture changes in uncertainty and dynamics in credit spreads, and explain the consequences for borrowing and lending as well macroeconomic variables such as investment, employment, output and consumption. A fast growing literature has highlighted the importance of uncertainty shocks either in the form of aggregate risk shocks or changes in the cross-sectional distribution of firms (we refer to such changes as idiosyncratic risk or credit risk interchangeably). We analyse these shocks through changes in risk-premiums charged by marginal investors. While intermediaries debt holders due to their very nature are especially concerned with the occurrence of extreme events (e.g. aggregate risk), intermediaries themselves also care about changes in the cross-section of their portfolios as the mass of defaulting borrowers determines credit losses. The question arises how such changes in uncertainty affect borrowers and lenders funding conditions, their balance sheets, as well as macroeconomic fluctuations overall. We ask which source of uncertainty, being aggregate or cross-sectional, is more important. We make four contributions to the literature. First, we develop a joint dynamic capital structure theory of borrowers and lenders that captures the interdependencies between balance sheets of both agents. It determines their capital structures, default frequencies, and associated risk premiums. Second, we include the theory in a tractable, quantitative general equilibrium macroeconomic model. Third, we estimate the model parameters and analyse how changes in risk, either aggregate or credit risk, affect the default frequencies of borrowers and lenders, their risk premiums, balance sheet dynamics and macroeconomic 1 With define credit broadly including all debt instruments such as loans, bonds, and other debt instruments. Terminology is described in detail in the appendix. 2

3 fluctuations overall. Fourth, we apply econometric techniques to quantify to which extent changes in idiosyncratic and aggregate risk were affecting macroeconomic fluctuations. Introducing a joint dynamic capital structure of borrowers and lenders modifies the standard RBC equilibrium model in the following three ways. First, firms (borrowers in our model) adjust investment and hiring taking the expected costs associated to firm and intermediary default as well as the benefits of debt financing and financial intermediation into account. Overall, general equilibrium outcomes imply that a profit discount factor enters the Euler equation for capital and the optimality condition for labour that depends crucially on leverage of borrowers and lenders as well as idiosyncratic and aggregate risk. Second, firms and financial institutions endogenously choose their capital structures. At the core of the framework is the determination of lending rates. Intermediaries face a trade-off between increasing lending rates as compensation for higher credit risk on the one hand, and tighter financial conditions of borrowers on the other hand. Third, expected default risk of borrowers and financial institutions is priced. The participation constraints of intermediaries and intermediaries creditors determine corporate credit spreads and spreads of financial institutions. We use the developed framework to analyse changes in uncertainty. We model idiosyncratic and aggregate uncertainty by assuming that after firms purchase k t units of capital at time t and hire n t+1 workers at time t+1 such capital and workers turn into ε i,t+1 z t+1 k t and ε i,t+1 z t+1 n t+1 units of effective capital and labour. The realisation of the variables ε i,t+1 and z t+1 is uncertain ex-ante. They are independently distributed random variables, normalized to have unit means. While ε i,t+1 differs among firms, z t+1 is an aggregate shock experienced by all firms. Large values of z t+1 mean a high productivity level overall. Large values of ε i,t+1 mean success of a firm compared to other firms, a low value means failure. Idiosyncratic differences may in fact be large. As a result of debt financing and wide dispersion in individual productivity ε i,t+1, each period a certain fraction of firms with productivity below a certain default threshold value ε Z t+1 defaults. In turn, default of borrowers affects financial intermediaries. Intermediaries are less concerned with default of single borrowers, because they lend to many firms, are well diversified and not subject to idiosyncratic shocks. However, changes in the mass of defaulting firms and aggregate productivity concerns them directly, because they cannot diversify away these risks. Intermediaries may default themselves if aggregate factor productivity z t+1 falls below a certain default threshold value zt+1. In that case intermediaries debt holders suffer credit losses. We denote the time period t standard deviation of log(z t ) and log(ε i,t ) with σ z,t and σ ε,t. Suppose G and F are the cumulative distribution functions of z t and ε i,t. Then G and F vary with σ z,t and σ ε,t respectively. Then G(zt+1) and F (ε Z t+1), the probability of default of intermediaries and the mass of defaulting firms, vary with σ z,t and σ ε,t respectively. Consequently, we refer to 3

4 σ z,t as systemic risk, and to σ ε,t as credit risk. We will use the terms systemic risk, macroeconomic risk, and aggregate risk as well as credit risk and idiosyncratic risk interchangeably. Both variables are realisations of stochastic processes. Thus, systemic risk is high when σ z,t is high. If credit risk σ ε,t is high, there is substantial dispersion in the outcomes across firms and intermediaries suffer higher credit losses. Our econometric analysis finds that both changes in aggregate risk and credit risk are an important source of business cycle fluctuations, because macroeconomic variables in the model respond according to the data. Both shocks generate a joint decline of investment, employment, output and consumption, even in the simple RBC model. However, while both shocks affect macroeconomic aggregates in a similar way, the dynamics are quite different. The main characteristic of changes in aggregate risk is a deterioration in intermediaries funding conditions. Following an increase in aggregate risk, the probability of intermediary default increases and successful financial intermediation becomes less likely. Financial spreads increase and investors withdraw funding. As a consequence, intermediaries increase lending spreads, less credit is extended to the economy, and leverage of borrowers declines. Changes in aggregate risk also affects intermediaries solvency, in addition to liquidity. Upon an increase in aggregate risk, tail events associated with weak aggregate productivity are more likely, unexpected credit losses increase and intermediaries capital constraint becomes tighter, triggering a deleveraging process of financial institutions. The main characteristic of changes in credit risk is a deterioration in credit conditions. Following an increase in credit risk, credit losses and the cost of screening defaulting borrowers increase, and borrowers are forced to deleverage. Such optimal behaviour of agents in the model becomes evident during crises: Borrowers reduce leverage to avoid default. Financial institutions tighten lending conditions and reduce leverage themselves due to tightening capital constraints. Credit spreads of borrowers increase as compensation for higher credit risk. While credit risk shocks have severe consequences for the real economy, intermediaries funding costs may be stable as long as agents manage to deleverage quickly enough such that systemic risk is contained. Various authors have pointed out the importance of either idiosyncratic or aggregate risk. The question arises which shock has been more important for macroeconomic fluctuations. To answer this question we estimate our model with standard Bayesian estimation techniques, and include two additional financial time series, corporate and financial credit spreads in the estimation, which allows us to pin down changes in credit and aggregate risk. While it is impossible based on macroeconomic observations alone to differentiate between different risks, the model together with financial data allows to disentangle both affects. We find that changes in aggregate risk were the main driver of fluctuations in investment during the financial crisis, while credit risk shocks were contributing strongly to the information technology crisis. Over- 4

5 all, changes in aggregate risk were more important than changes in credit risk. In checking the robustness of our findings, we were especially concerned about overestimating the importance of shocks identified with financial credit spreads, because some large financial institutions may be funded mainly with deposits or benefit from government guarantees such that estimating their funding costs with a market index would overestimate their funding pressures. Consequently, we estimate the relationship between actual credit spread data and changes in funding costs implied by the model. We find that financial credit spreads in the model capturing the entire financial system are about three times lower than observed credit spreads. On the other hand, corporate credit spreads in the data and in the model almost one-to-one correspond to each other. The credibility of our model depends on the plausibility of the dynamics as well the empirical implications. First, a variety of authors have highlighted the importance of intermediaries funding conditions and balance sheet dynamics for macroeconomic fluctuations 2. We take comfort from the fact that our model is very consistent with these findings. Moreover, our quantitative framework allows to quantify during which periods intermediaries funding conditions and balance sheet dynamics mattered most and to control for the impact of credit risk shocks. Second, empirical evidence 3 suggests that corporate credit spreads are driven by two factors, one concerning firms fundamentals and one related to the health of the financial system. Consistent with these findings, our model demonstrates that credit spreads depend on credit risk and aggregate risk, and analyses the time-varying contribution of both factors. Third, influential papers that emphasize the empirical model fit of DSGE models have demonstrated that a model of our kind fits well the data with respect to forecasting and to matching moments, and that the inclusion of financial variables further improves the model fit. Since we applied identical estimation techniques to parameters that govern credit risk and aggregate risk, it is reasonable to assume that our model would perform similarly well. 4 Fourth, several authorise have shown that RBC models are suited well to analyse changes in risk, however it has been proven difficult to generate a decline in consumption following a risk shock, as changes in risk predominantly affect the Euler equation for capital. Our model contributes to this literature by showing that increased risk also leads to a decline in consumption even in the simple RBC model, as uncertainty about future incomes also affects employment. 5 Last, we compare our model-derived measure of aggregate uncertainty with the measure of aggregate macroeconomic risk of Jurado et al. [2013]. We take comfort from the fact that both our model 2 For example see Adrian et al. [2010b], Adrian et al. [2010a], Adrian and Shin [2010]. 3 Gilchrist and Zakrajsek [2011]. 4 An interesting exercise would be to analyse to which extent the inclusion of our formulated frictions and the inclusion of financial credit spreads further improves the empirical properties and forecasting. Our model provides the basis for such an analysis. 5 For example De Fiore et al. [2011] highlight the importance of credit frictions for employment. 5

6 derived measure and a completely independent, empirical measure of aggregate risk move reasonably well together. We also applied simple VARs to confirm both the model implied responses of risk shocks to aggregate variables, as well as our generated responses of borrowers and lenders credit spreads. Related literature Our paper is related to several strands of literature, including the macroeconomic effects of changes in (i) borrower balance sheets and corporate credit spreads and (ii) lender balance sheets and financial intermediary asset prices, and both in association with changes in uncertainty. Uncertainty and borrower balance sheets The seminal works of Kiyotaki and Moore [1997] and Bernanke et al. [1998] have shown that the conditions of borrower balance sheets matter for macroeconomic fluctuations. Christiano et al. [2014] and Gilchrist et al. [2010] emphasize the impact of changes in idiosyncratic risk on firms funding conditions, while Bloom et al. [2012] analyse it without balance sheet frictions. Gourio [2013] and Bloom [2009] emphasize the impact of aggregate risk, with and without balance sheet frictions. Jermann and Quadrini [2009] highlight the impact of financial shocks that tighten borrowers financing constraint. Building on these frameworks, we contribute to the existing literature by (a) determining the dynamic capital structure choice of borrowers and lenders 6, (b) analysing the impact of lenders balance sheet conditions for borrowers funding conditions and credit spreads, and (c) differentiating between idiosyncratic and aggregate uncertainty shocks. Uncertainty and lender balance sheets Gertler et al. [2010] and Gertler et al. [2012] apply the financial accelerator mechanism of Bernanke et al. [1998] to financial intermediaries. Gertler and Kiyotaki [2013] further develop the model to include household liquidity shocks according to Diamond and Dybvig [1983]. He and Krishnamurthy [2014], He and Krishnamurthy [2013], Brunnermeier and Sannikov [2014], and Adrian and Boyarchenko [2012] emphasize the importance of intermediaries balance sheets, liquidity and occasionally binding capital constraints. Nuño and Thomas [2012] highlight the importance of changes in aggregate risk. We contribute to the literature by (a) explicitly modelling the impact of borrower balance sheet strength on lenders balance sheets, (b) capturing the feedback effects between borrowers and lenders, and (c) quantifying the importance of risk shocks for macroeconomic fluctuations. While we don t model liquidity directly, dynamics in our model associated to systemic risk shocks mimic very closely the dynamics related to liquidity dry-ups, including an abrupt increase in financial credit spreads and 6 For example, in the financial accelerator models, equity is a state variable and not chosen by agents. 6

7 an outflow of funding. Moreover, our paper offers a discrete-time theoretical macroeconomic framework for quantifying systemic risk. Risky banking literature There is a large literature modelling the failure of financial institutions. 7 Default in this literature is often related to idisyncratic risk associated to financial institutions balance sheet. We contribute to this stream of literature by analysing financial sector default due to aggregate risk. Econometric strategy A large body of literature has analysed empirically implications of changes in corporate and financial credit spreads. Instead we follow closely an influential contribution by Christiano et al. [2014] in identifying risk shocks. We contribute to the literature by (a) differentiating between idiosyncratic and aggregate risk shocks, (b) deciphering the relative importance of both shocks, and (c) analysing the joint balance sheet dynamics of borrowers and lenders. Structure of the paper The paper is structured as follows. Section 2 describes the model. Section 3 describes the solution and the general equilibrium conditions. The model is then taken to the data. Section 4 describes the data and the key parameters that were estimated. Section 5 describes the estimation results, the dynamics of the model, as well as the overall results related to the importance of credit and aggregate risk. Section 6 concludes. In addition, a technical appendix and an annex with the figures and tables is included at the end of the paper. 7 For example see Krasa and Villamil [1992], Hirakata et al. [2011], Hirakata et al. [2013], Zeng [2013], Benes and Kumhof [2011] and Jin et al. [2011] 7

8 2 The model The standard RBC model (can be decentralized such that it) consists of a household and an all equity financed firm. We add to the standard framework a financial sector in form of financial intermediaries that lend to firms and borrow themselves from households. As a result, firms and financial intermediaries choose actively their capital structure. Consequently, our model consists of a representative household, firms (the borrowers), financial intermediaries (the lenders), and a government. We begin by describing households in subsection 2.1, and describe in detail firms and financial intermediaries in subsection 2.2 on financial frictions. The government and market clearing are defined as logic prescribes after equilibrium conditions are derived in subsection Standard part of the model - households There is a representative household that maximises lifetime utility E 0 t=0 βt U t, subject to a budget constraint. The utility function takes the standard CRRA form with habit formation: U t = z c,t ( (c t ηc t 1 ) 1 γ 1 γ ) n 1+φn t z n,t τ n 1 + φ n The household sends n t members to work. Its derives utility from consumption c t and dis-utility from sending members to work (or utility from home production). Two shocks, a preference shock z c,t and a labour dis-utility shock z n,t are included. The two shocks absorb a large part of fluctuations that are not explained by financial variables, when the model is taken to data. The preference shocks absorbs any residual variance in the Euler equation for capital. While the labour dis-utility shock absorbs any residual variance in the first order condition with respect to labour. As a consequence, the shocks are very powerful, and if one interprets them as residuals or wedges that need further explanation, they provide a rigorous check for any model. The household maximises utility subject to the following budget constraint. c t + d g,t + d t + T t = w t n t (1 τ) + R t 1 d g,t 1 + R D,t d t 1 + b H t + Π t It consumes c t, invests d g,t in government bonds, and provides funds d t to financial institutions, and pays lump sum taxes T t. It receives labour income at the wage rate w t from each member that is working net of taxes, receives the gross risk-free rate R t 1 on government bonds, and earns income R D,t on funds intermediated by financial institutions. It receives transfers b H t from the government, and owns all firms and financial institutions; Π t = div F,t + div B,t denotes combined dividend income of these sectors. The first order conditions with respect consumption, labour, government bonds, and saving in the financial 8

9 sector are: λ t = (c t ηc t 1 ) γ βηe t ( zc,t+1 z c,t (c t+1 ηc t ) γ) (1) n φn t w t (1 τ) =z n,t τ n (2) λ t 1 =E t (M t+1 R t ) (3) 1 =E t (M t+1 R D,t+1 ) (4) M t =β λ t z c,t (5) λ t 1 z c,t 1 λ t denotes the Lagrange multiplier associated to the budget constraint, divided by z c,t, and M t+1 denotes the stochastic discount factor of households. 2.2 Financial frictions Structure of economy Firms and financial intermediaries build the core of the economy. Firms hire workers and invest, and produce consumption and capital goods. Firms are located on states or islands. On each island operates a large amount of firms. The entire universe of firms (e.g. the stock market) is owned by households, which delegate management to an asset manager. The asset manager injects capital in firms, poles incomes from all firms together at the end of the period, distributes dividends, and invests again. In addition to capital, firms receive loans and bonds from financial intermediaries. Intermediaries play an essential role in the economy, and channel funds from households to firms. There exists only one financial intermediary for each island. While intermediaries lend to firms, they are itself borrowers. We refer to lending households as debt holders of financial intermediaries. Similar to the producing sector, the whole financial sector is managed by an asset manager, which poles incomes from all intermediaries, injects capital, and distributes dividends to households. 8 Default of firms and intermediaries Firms and intermediaries in our model may default if the value of their assets falls below the value of their liabilities. In this section, we define exactly what happens in case of default and under which circumstances firms and intermediaries may default. The intuition is quite simple. In the introduction 8 The island set-up is chosen to facilitate aggregation. However, it only affects the aggregate resource constraint. An almost equivalent set-up would be to consider only one large set of firms and a single financial institution. The intermediary would represent the entire financial system and be subject to aggregate uncertainty (instead of island-varying aggregate uncertainty in our set-up). Both set-ups are identical (including all first-order conditions), except for the aggregate resource constraint. Our estimations show that the impact of financial frictions on the resource constraint is negligable. Consequently, we can conclude that both set-up are simialar, and that our framework is situable to analyse systemic risk. 9

10 we explained that the factors of production are subject to idiosyncratic and aggregate risk. Whether a firm defaults ultimately depends on idiosyncratic productivity. Even if aggregate productivity is low, a well-performing firm can still succeed. Intermediaries on the other hand are well diversified. They are not subject to idiosyncratic risk for the benefit of investors. However, intermediaries may default themselves as they are subject to aggregate risk. Costs associated to systemic failure We model the consequences of systemic default by assuming that financial intermediation is severely hampered when an intermediary defaults, and that lower quality of financial intermediation leads to a drop in island productivity. Formally, we implement this idea, by assuming that firms factors of production experience an efficiency gain or loss equal to ε i,t+1 z t+1 µ z,t+1 k t and ε i,t+1 z t+1 µ z,t+1 n t+1. 9 We denote the net effect on productivity with Z t+1 z t+1 (1 + µ z,t+1 ) and call it net factor productivity. We assume that the efficiency add-on and hence net factor productivity depend crucially on the health of the financial system. An intermediary defaults if aggregate island productivity z t+1 falls below the default threshold value zt+1. We assume that net factor productivity is linked to intermediary default, and takes either a high z N or a low value z E, depending on intermediary default. Consequently, the expected value of net factor productivity is: E t (Z t+1 ) = (1 G(z t+1))z N + G(z t+1)z E = z N G(z t+1)(z N z E ) Our assumptions lead to a cost of systemic default equal to a loss in productivity of z N z E. The superscripts N and E refer to normal and extreme times. A. Firms Firms factors of production Firms acquire raw capital k t and hire workers n t+1 for production. Effective capital k t+1 and labour ñ t+1 at time t + 1 are: k t+1 = ε i,t+t Z t+1 k t ñ t+1 = ε i,t+1 Z t+1 n t+1 Firms produce according to a Cobb Douglas production function. Effective output is equal to ỹ t+1 = k t+1(a α t+1 ñ t+1 ) 1 α. They also own the undepreciated part of the capital stock (1 δ) k t+1. Firms effective assets at the end of the period are equal to Ãt+1 ỹ t+1 + (1 δ) k t+1. After substitution of the factors of production, firms effects assets are: Ã t+1 = ε i,t+1 Z t+1 A t+1 9 It is a gain or loss, depending on whether µ z,t+1 is greater or smaller than zero. 10

11 We refer to A t y t + (1 δ)k t 1 as end-of-period raw assets, and to y t = k α t 1(a t n t ) 1 α as output, which is common across all firms. Firms capital structure and default Firms finance expenditures partly with debt b t and partly with own resources. Debt has to be repaid at the state contingent interest rate R b,t Total liabilities at the end of the period are equal to B t+1 = R b,t+1 b t. Firms default if the value of its effective assets falls below the value of liabilities Ãt+1 < B t+1, or idiosyncratic productivity falls below the default threshold ε N t+1 or ε E t+1, depending on intermediary default: Firms optimisation problem ε i,t+1 < ε Z t+1 B t+1 Z t+1 A t+1 ε Z t+1/(z t+1 z t+1) = ε N t+1 B t+1 z N A t+1 ε Z t+1/(z t+1 < z t+1) = ε E t+1 B t+1 z E A t+1 Firms choose capital, debt and labour, internalizing the effect on interest rates, as well as the effect on intermediaries funding structure and associated funding costs. Capital k t, debt b t, and intermediary funding d t are chosen in period t. Labour n t+1, borrowing rates R b,t+1 and intermediary funding costs R d,t+1 are chosen in period t+1. Firms objective function is: { ( max k t + b t + E t Mt+1 (V t+1 w t+1 n t+1 )(1 τ) )} k t, b t, d t, n t+1, R b,t+1, R d,t+1 V t+1 = (1 G(zt+1)) + G(z t+1) ε N t+1 ε E t+1 (ε i,t+1 z N A t+1 B t+1 )f(ε i,t+1 )dε i,t+1 (ε i,t+1 z E A t+1 B t+1 )f(ε i,t+1 )dε i,t+1 Firms discount future income with the stochastic discount factor M t+1 of households, its ultimate owners. Total income net of wages w t+1 n t+1 is taxed at rate τ > Expected future cash-flows depend on the health of the financial system. In normal times, island productivity is z N. If the intermediary defaults, island productivity falls to z E. Substituting (x) x 0 ε i,t+1f(ε i,t+1 )dε i,t+1 10 This feature is not strictly necessary. It can be shown that state contingent interest rates actually increase the capacity of the system to withstand adverse shooks, and act as a shock absorber. 11 The implications of deductibility are discussed below. 11

12 and making use of the cumulative distribution function F (ε i,t+1 ), the expression for V t+1 can be simplified: V t+1 = ((1 G(z t+1))((1 (ε N t+1))z N A t+1 B t+1 (1 F (ε N t+1))) + G(z t+1)((1 (ε E t+1))z E A t+1 B t+1 (1 F (ε E t+1))) Aggregating over all firms, and over all islands, on behalf of the asset manager, firms budget constraint is: B. Financial intermediaries div F,t + k t b t = (V t w t n t )(1 τ) Financial intermediaries provide loans and bonds b t to non-financial firms at gross lending rate R b,t+1 = 1 + i b,t+1. Intermediaries total exposure at the end of the period is B t+1 R b,t+1 b t. Most of the borrowers perform well, but some default. The value of intermediaries loan portfolio at the end of the period without efficiency gain is: B t+1 (1 F (ε z t+1)) + (ε z t+1)z t+1 A t+1 (1 µ) B t ε z t z t A t Lenders receive interest plus principle on performing loans, and claim the assets of defaulting firms. µ > 0 denotes the fraction of assets that is lost in case of default, e.g. the amount of screening costs associated to monitor borrowers. We simplify this expression conveniently by substituting out firms end of period assets and obtain the standard expected loss formula: B t+1 (1 δ(ε z t+1)) B t+1 (1 F (ε z t+1)) + (ε z t+1)z t+1 A t+1 (1 µ) Intermediaries receive their exposures net of expected losses EL t+1 δ(ε z t+1)b t+1. The expected loss rate is equal to the product of probability of default F (ε z t ) and loss given default. 12 Intermediaries finance their activities partly with capital and partly with debt. They raise funding d t from households at gross funding rate R d,t+1 = 1 + i d,t+1. Intermediaries total liabilities at the end of the period are D t+1 R d,t+1 d t. Taking intermediaries assets and liabilities together, end-of-period revenues are: B t+1 (1 δ(ε z t+1)) D t+1 Intermediaries default if aggregate productivity falls below z t+1, and losses exceed unexpected losses UL t+1 δ(ε t+1)b t+1 such that the value of assets falls 12 δ(ε z t+1) F (ε z t+1) ( 1 (εz t+1 ) ) ε z t+1 F (1 µ) (εz t+1 ) 12

13 below the value of liabilities. The following identities uniquely determine intermediaries default productivity threshold z t : B t (1 δ(ε t )) = D t B t ε t zt A t After aggregate productivity z t+1 realises, the factors of production experience an efficiency gain or loss. Intermediaries portfolios appreciate or lose in value depending on whether intermediaries operate normally or default. The value of intermediaries loan portfolio at the end of the period including efficiency gains is: B t+1 (1 F (ε Z t+1)) + (ε Z t+1)z t+1 A t+1 (1 µ) Intermediaries objective function is: b t + d t + E t ( M t+1 z t+1 ) (B t+1 (1 δ(ε Z t+1)) D t+1 )(1 τ)dg(z t+1 ) 0 Similarly to firms, intermediaries earnings are taxed at rate τ. Intermediaries operate if its objective function is greater or equal to zero. 13 The benefits of financial intermediation are in diversification. Financial institutions lend to a large universe of firms, but their loan portfolio does not depend on idiosyncratic firm factors. Expected losses only depend on the end-of-period corporate default threshold ε Z t+1 = B t+1 /Z t+1 A t+1. However, it crucially depends on aggregate factors. The benefit of diversification was arguable overestimated during the boom in securitized financial products in the U.S. between We show that an important driver behind the increase in funding costs and dry-up of funding for financial institutions, and the following recession, was aggregate nondiversifyable risk. Suppose an adverse event, a low realisation of z t+1, becomes more likely, then intermediaries probability of default increases, no matter how well diversified a financial institutions is. Considering that net productivity takes a high value when intermediaries operate normally, and making use of the cumulative distribution function of z t+1, it becomes evident that financial institutions discount future earnings by the survival probability 1 G(zt+1): ) ) b t + d t + E t (M t+1 (1 G(zt+1)) (B t+1 (1 δ(ε N t+1)) D t+1 (1 τ) 0 The equation for the default threshold acts in the model like a borrowing constraint. Let λ V t+1 be the Lagrange multiplier assigned to it, then the borrowing constraint enters in the final optimisation problem as: λ V t+1[b t+1 (1 δ(ε t+1)) D t+1 ] 13 Interestingly, the participation constraint equally holds if intermediaries (instead of firms) maximise borrowing d t and lending b t, subject to firms zero profit condition. Instead of optimising firms, financial intermediaries could also be the optimizing agent. 13

14 Aggregation across the financial sector results in the resource constraint of the sector: div B,t + b t d t = (1 G(z t ))(B t (1 δ(ε N t )) D t )(1 τ) C. Intermediaries debt holders Financial intermediaries raise funding from households. Households participation constraint is: ( )) d t + E t M t+1 ((1 G(zt+1))D t+1 + G(zt+1)B t+1 (1 δ(ε E t+1))(1 µ B ) 0 Debt holders receive the principal amount plus interest in normal times. However, when a financial institution defaults, investors claim the institution s assets. The fraction (1 µ B ) < 1 denotes the recovery value of financial assets. In this paper we focus on the case that µ B = τ This could be rationalized with the fact that recovery values tend to be higher for financial institutions than for non-financial corporates. The opposite argument would be that costs are very high in case of distress of the financial system. However, the impact of financial distress is already captured in a fall of aggregate productivity from z N to z E. We choose µ B = τ for simplicity. It would be interesting to explore the case of µ B > τ. 14

15 3 Joint capital structure choice 3.1 Optimisation problem Agents chose capital k t, debt b t, and funding d t in period t, and labour n t+1, lending rates R b,t+1, funding rates R d,t+1, and the intermediary default threshold z t+1 in period t+1, subject to the intermediary borrowing constraint and the participation constraints of financial intermediaries and their debt holders. The first order conditions with respect to b t and d t imply that the participation constraints are binding. We can substitute them into firms optimisation problem and obtain the following objective function: k t + E t (M t+1 ((v t+1 A t+1 w t+1 n t+1 )(1 τ) + λ V t+1(b t+1 (1 δ(ε t+1)) D t+1 ) ) + τd t+1 (1 G(zt+1))) v t+1 ((1 G(z t+1))z N (1 µ (ε N t+1)) + G(z t+1)z E (1 µ (ε E t+1))) Firms adjust their factors of production taking into account expected costs of default, related to systemic default and defaulting firms, the intermediary financial constraint, as well as the benefits of financial intermediation. Expected costs of default (variable v t+1 ) Financial intermediaries default with probability G(zt+1). In case of systemic default productivity falls to a lower level z E < z N. In addition, each period a fraction of firms default causing another loss equal to µ. The corporate default threshold that determines the number of defaulting firms varies with aggregate productivity, and is equal to ε N t = B t /(z N A t ) in normal times and to ε E t = B t /(z E A t ) under stress. Financial constraint (2nd term) The intermediary defaults if its assets fall below the value of its liabilities, or portfolio losses increase above unexpected losses δ(ε t+1)b t+1 as a result of productivity z t+1 falling below the default threshold zt+1. The value of ε t = B t /(zt A t ) uniquely pins down the value of zt in general equilibrium. Benefit of financial intermediation (3rd term) Incomes of firms and intermediaries are taxed at rate τ. Taking firms, intermediaries and creditors together, deductibility results in an overall expected benefit of financial intermediation equal to τd t+1. Such benefit is only realized when a intermediary does not default, and occurs with probability (1 G(z t+1)). 15

16 3.2 Optimality The first order condition with respect to the lending rate R b,t and the funding rate R d,t determine corporate and financial intermediary leverage. Corporate leverage Agents choose R b,t and hence corporate leverage φ F t B t /A t trading-off financial stability reflected in a relaxation of the financial constraint against tighter financial conditions for firms. Tigher financial conditions result in more corporate defaults and costs associated to it. ( ) + λ V t b t 1 (1 δ(ε t )) B t δ (ε t ) ε t R b,t A t ( (1 G(z t ))z N µ (ε N t ) εn t R b,t + G(z t )z E µ (ε E t ) εe t R b,t ) (1 τ) = 0 Intermediary leverage Agents choose R d,t and hence intermediary leverage φ B t D t /B t trading-off the benefit of financial intermediation against costs associated to higher systemic risk. The marginal contribution to systemic risk is equal to g(zt ) = G (zt ). The first order condition with respect to R d,t determines the expected value λ V t of intermediated funds: τ(1 G(zt )) λ V t = 0 The first order condition with respect to zt institutions: determines leverage of financial + λ V t B t ( δ (ε t )) ε t zt g(zt )τd t g(zt )A t (z N (1 µ (ε N t )) z E (1 µ (ε E t )))(1 τ) = 0 Capital and labour Agents internalize that acquiring more capital and hiring more workers increases production and end-of-period assets, and consequently lowers the default barrier ε t / A t < 0, and reduces expected costs of default. The optimality conditions with respect to capital and labour are: ( ( 1 + E t M t+1 v t+1 A k,t+1 (1 τ) λ V t+1b t+1 δ (ε t+1) ε t+1 k t ( A t+1 (1 G(zt+1))z N µ (ε N t+1) εn t+1 + G(z k t+1)z E µ (ε E t+1) εe ) )) t+1 (1 τ) t k t = 0 A t ( (1 G(z t ))z N µ (ε N t ) εn t n t + (v t A n,t w t )(1 τ) λ V t B t δ (ε t ) ε t n t + G(z t )z E µ (ε E t ) εe t n t ) (1 τ) = 0 16

17 3.3 General equilibrium conditions The general equilibrium conditions can be conveniently simplified, as shown in the technical appendix. Total end-of-period raw assets, liabilities of firms and financial intermediaries were defined as A t y t + (1 δ)k t 1, B t R b,t b t 1, and D t R d,t d t 1. We define R k,t A k,t = α yt k t 1 + (1 δ). Capital and labour The Euler equation for capital and the first order condition with respect to labour are: 1 = E t (M t+1 ϕ A t+1r k,t+1 ) (6) ϕ A t (1 α) y t n t = w t (1 τ) (7) Financial frictions lead to a discount factor ϕ A t and contemporaneous hiring: that affects future investment ϕ A t v t (1 τ) + τφ F t φ B t (1 G(z t )) The discount factor varies with corporate leverage, intermediary leverage, credit risk, and systemic risk ϕ A t = ϕ A( φ F t, φ B t, σ ε,t, σ z,t ). Credit spreads The participation constraints of intermediaries and creditor are binding, and determine lending and funding rates. Together with household s government bond pricing equation (1 = E t (M t+1 R t )) they determine credit spreads: ( ) 1 = E t M t+1 ϕ B t+1r b,t+1 (8) ( ) 1 = E t M t+1 ϕ C t+1r d,t+1 (9) The corporate risk premium depends on expected and unexpected losses and the probability of systemic default. The Intermediary risk premium depends on expected losses comprising the probability of default and loss given default χ, as shown in the appendix: ϕ B t ((1 G(z t ))(1 δ(ε N t ))+G(z t )(1 δ(ε E t )))(1 τ) + τφ B t (1 G(z t )) ϕ C t 1 G(z t )χ(z t ) While ϕ B t depends on the same factors as ϕ A t, we can show that the intermediary risk premium depends almost entirely on intermediary leverage and systemic risk ϕ C t = ϕ C( ) φ B t, σ z,t. Equations (4) and (9) imply that households gross savings rate is R D,t = ϕ C t R d,t. 17

18 Leverage Credit B t and corporate leverage φ F t are determined by the corporate default threshold equations: ( ) + τ(1 G(zt )) 1 F (ε t ) µf(ε t )ε t ( ) (10) µ (1 G(zt ))f(ε N t )ε N t + G(zt )f(ε E t )ε E t (1 τ) = 0 ε N t = φf t z N (11) ε E t = φf t z E (12) Intermediary funding D t and intermediary leverage φ B t are determined by the intermediary default threshold equations: ( ( +τ (1 G(zt )) (ε t )(1 µ) + µf(ε t )(ε t ) 2) ) g(zt )φ F t φ B t ( ) (13) g(zt ) z N (1 µ (ε N t )) z E (1 µ (ε E t )) (1 τ) = 0 1 δ(ε t ) = φ B t (14) ε t = φf t z t (15) Government and market clearing The government finances spending with tax revenues and lump sum taxes such that income equals expenditures. T t + w t n t τ + π τ,t = g t + g τ,t + b H t + R t 1 d g,t 1 d g,t The government receives tax income w t n t τ and π τ,t from households and firms and financial intermediaries respectively, in addition to lump sum taxes T t. Without loss of generality we assume that the government receives the default costs (it can be shown that then π τ,t = A t (1 ϕ A t )). It spends a share φ g of these revenues on public investment g τ,t, and distributes the remainder b H t to households. g τ,t = φ g A t (1 ϕ A t ) b H t = (1 φ g )A t (1 ϕ A t ) Overall, total public investment spending (G t g t + g τ,t ) is stochastic. g t = g ss z g,t (16) 18

19 Taking all agents together, the aggregate resource constraint is equal to: c t + i t + g t + φ g A t (1 ϕ A t ) = y t (17) i t = k t (1 δ)k t 1 (18) y t = k α t 1(a t n t ) 1 α (19) In case the government invests all revenues (φ g = 1), financial frictions have maximum impact on the resource constraint: c t + k t + g t = ϕ A t (y t + (1 δ)k t 1 ) In case the government distributes all tax income from firms and intermediaries to households (φ g = 0), financial frictions don t enter at all: c t + i t + g t = y t 19

20 4 Estimation The model is estimated with standard Bayesian estimation techniques according to An and Schorfheide [2007]. The intuition behind the estimation strategy is quite simple. The exogenous shocks are fit to the data by estimating the parameters of the model such that the model optimally matches the data. Our measure of optimality is the likelihood of observing the realized data. We use four macro-economic and two financial variables to identify the main parameters of the model, as well as the exogenous shocks. The RBC model defined in this paper without financial frictions would have four shock processes, and could be estimated for example with data for consumption, investment, output and employment. Instead, we include two additional shock processes in our model, idiosyncratic risk and aggregate risk, and include two additional financial variables in our estimation, the corporate credit spread and the financial credit spread, to estimate how the described financial frictions affect macroeconomic variables. Intermediary debt holders especially care about extreme events and aggregate risk, because intermediaries are well diversified and not subject to idiosyncratic risk. This is confirmed by our model. Conditional on the intermediary default threshold zt, financial spreads almost entirely depend on aggregate risk. As a result, changes in aggregate risk can be inferred from financial credit spreads through the pricing equation of intermediary debt. Similarly, conditional on aggregate risk, changes in credit risk can be inferred from corporate credit spreads through the pricing equation of corporate debt. In the following, we will describe the data, the measurement equations, the calibrated parameters, and the parameters that will be estimated. 4.1 Data The macroeconomic and financial variables that were used for the estimation are shown in Figure 1. The data covers the period between 1991.Q1 to 2015.Q1. The four macro-economic variables are real quarter-on-quarter percentage changes of GDP, investment, employment and consumption, controlled first for population growth, and then demeaned, as in Smets and Wouters [2007] and Christiano et al. [2014] 15. GDP, investment, and consumption are taken from the Bureau of Economic Analysis. The data for investment is constructed as the sum of total fixed investment and durable consumption. The series for consumption is constructed as the sum of non-durable consumption and consumption services. Employment and population growth are taken from the Bureau of Labor Statistics. The corporate credit spread is computed as the percentage difference between Moody s Baa Rated Corporate Bond Yield Index and the constant 10- year treasury maturity bond yield. The financial credit spread is computed as the percentage difference between Citi s Financial Institutions Yield Index and 15 Alternatively, hp-filtered or quadratically de-trended variables could be taken. The methods of detrending are still discussed intensively, for example see Canova [2012]. 20

21 the constant 10-year treasury maturity bond yield. The spreads are also demeaned. For the process of demeaning, the years 2008 and 2009 of the financial crisis were excluded. 4.2 Measurment equations The measurement equations link the observed variables to variables in the model. The measurement equations for the macroeconomic variables are trivial. Output, investment, employment, and consumption correspond one to one to each other. However, the question arises to which extend the credit spreads reflect adequately lending and funding conditions in the economy. We are especially concerned about overestimating the importance of shocks identified with financial credit spreads, because some large intermediaries may be funded mainly with deposits or benefit from government guarantees such that spreads are low even during crises. Funding costs implied by an index would overestimate their funding pressures. Consequently, we estimate the relationship between observed financial credit spreads (F CS t ) and observed corporate credit spreads (CCS t ) with their model counterparts by estimating sensitivity coefficients 1/κ C and 1/κ F, which link the data (right-hand side) to the model variables (left-hand side): (E t (R b,t+1 ) R t (R b,ss R ss )) 100 = CCS t κ C (E t (R d,t+1 ) R t (R d,ss R ss )) 100 = F CS t κ F Our prior believe is that both parameters equal unity, κ C = 1 and κ F = 1, because to our knowledge previous papers haven t estimated them. Similarly to demeaning the data, we subtract steady-state spreads from the model spreads. This way differences in steady-state spread levels between the data and the model do not influence our estimations. 4.3 Calibrated parameters Table 1 shows calibrated parameters. We chose standard values where possible. We choose µ = 0.4 such that the recovery value 1 µ is close to the average recovery value of senior secured bonds (50 percent) and senior secured loans (70 percent), as reported by Moody s. We set the tax rate τ = 0.2 similar to capital gains taxes not to overestimate the impact of taxation, since in the model the tax rate does not only affect net incomes but also capital gains. Last, as described in the model section on financial intermediaries we set z N and z E close to the expected value of z t+1 in the steady state, conditional on being above or below z t+1, such that their dispersion does not overduely amplify the response to aggregate risk shocks. 21

22 4.4 Estimated parameters Four different sets of parameters were estimated: coefficients of autocorrelation, standard deviations of the various shocks, parameters that fit the model to financial data ( financial paramaters ), and parameters that govern adjustment costs. Shock processes All exogenous shocks (preferences z c,t, labour dis-utility z n,t, mean technology a t, exogenous spending z g,t, credit risk σ ε,t and aggregate risk σ z,t ) follow an auto-regressive process of order one: ln(z c,t ) =ρ c ln(z c,t 1 ) + σ c ɛ c,t (20) ln(z n,t ) =ρ n ln(z n,t 1 ) + σ n ɛ n,t (21) ln(a t ) =ρ y ln(a t 1 ) + σ y ɛ a,t (22) ln(z g,t ) =ρ g ln(z g,t 1 ) + σ g ɛ g,t (23) ln(σ ε,t ) =(1 ρ ε ) ln(σ ε,ss ) + ρ ε ln(σ ε,t 1 ) + σ ε ε ε,t (24) ln(σ z,t ) =(1 ρ z ) ln(σ z,ss ) + ρ z ln(σ z,t 1 ) + σ z ε z,t (25) The shocks ɛ j,t are standard normally distributed. We estimate the autocorrelation coefficients ρ j and the standard deviations σ j of the shocks. Financial parameters We estimate steady state credit risk and aggregate risk σ ε,ss and σ z,ss, spread sensitivities κ C and κ F and government behaviour φ g. Adjustment cost and habit persistence Agents adjust their capital structures dynamically to offset adverse consequences arising from default. The model might overestimate their ability to respond to adverse shocks. As a consequence, in addition to relatively standard capital adjustment costs and habit persistence η, we also estimate leverage adjustment costs. The three parameters φ K, φ ε, and φ z determine capital, corporate leverage and intermediary leverage adjustment costs. The higher φ K, φ ε, and φ z are, the more difficult it is to adjust. Functional form of adjustment costs We assume that firms pay resources C(x } t ) out of their budget to adjust the variables x t, with x {k, ε N, ε, ε E, z. We applied various adjustment cost functions and found little impact on adjusting leverage until we applied adjust- 22

23 ment costs of the following quadratic form: ( S ( C(x t ) = i x,t S i x,t i x,t 1 i x,t i x,t 1 ) ) ( i x,t φ x 1 i x,t 1 i x,t = x t δ x,t x t 1 We assume that φ ε N = φ ε = φ ε E = φ ε, and δ ε N,t = δ ε,t = δ ε E,t = δ z,t 10 6 to avoid division by zero in the steady state. Investment takes the standard form i k,t = k t (1 δ)k t 1. It can be shown that marginal adjustment cost are then equal to C (k t ) for capital and C (x t ) for the other variables. ) 2 C (k t ) = c k,t E t (M t+1 c k,t+1 (1 δ)) C (x t ) = c x,t E t (M t+1 c x,t+1 ) The variable c x,t is defined as: c x,t S ( i x,t i x,t 1 ) + S ( i x,t i x,t 1 ) ( ( i x,t i x,t+1 E t M t+1 S i x,t 1 i x,t )( i x,t+1 i x,t ) 2 ) We show in the technical appendix how the various adjustment cost affect the optimality conditions. 23