Structural credit risk models and systemic capital

Transcription

1 Structural credit risk models and systemic capital Somnath Chatterjee CCBS, Bank of England November 7, 2013

2 Structural credit risk model Structural credit risk models are based on the notion that both debt and equity are fundamentally related. They trade similar risks related to the firm s assets. They use information from equity markets to infer the value of a firm s debt via the structure of it s balance sheet. Because they capture balance sheet dynamics they can be used to run comparative statics experiments.

3 Merton model Simplest approach introduced by Merton (1974) is to consider a firm with a capital structure comprised of two basic elements: a fixed amount of zero coupon debt and equity If the firm s assets were insufficient to pay the face value of debt when it fell due, the company would default. Equity holders would receive nothing and bond holders, would recover whatever the firm s assets were after paying bankruptcy costs. The value of the companies assets are assumed to follow a geometric Brownian motion.

4 Merton Model We can therefore calculate the value of the equity today, using the Black-Scholes formula: Where E 0 V0 d rt N( d1) De N( 2) d 1 ln( V 0 D) ( r v T 2 v 2) T d 2 d 1 v T

5 Merton Model The risk-neutral probability that the company will default on the debt is N(-d 2 ). To calculate this we require V 0 and σ V. Neither of which is directly observable. However if the company is publically traded then we can observe E 0. This means we can use the Black-Scholes equation as a condition that must be satisfied by V 0 and σ V. We also have an estimate for σ E.

6 Merton Model Using Ito s lemma we can state also that: E E E V 0 VV 0 Where E / V is the delta of the equity. We can prove that this delta is: E E 0 N d1) ( V v 0 Hence we have two equations containing V 0 and σ V which we can solve simultaneously.

7 Merton Model Crucially from the above we can relate the unknown volatility of asset values to the observable volatility of equity. V V E 0 0 E V 1 E

8 Balance sheet: evolution summary Merton (1974) Debt Assets Equity Assets Liabilities

9 Balance sheet: evolution summary Merton (1974) Market Value Face Value Debt Assets Residual Equity Assets Liabilities

10 Selected prices of call options on Google stock in Sep 2008 closing price $430 Maturity Date Exercise Price ($) Price of Call Option ($) December March

11 Selected prices of call options on Google stock in Sep 2008 closing price $430 Value of call $430 0 $430 Share price

12 Balance sheet: evolution summary Merton (1974)

13 Balance sheet: evolution summary Merton (1974) Market Value Face Value Debt Assets Residual Equity Assets Liabilities

14 Balance sheet: evolution summary Merton (1974)

15 Setup of a structural credit risk model Assume an underlying process for the value of a firm s assets. E.g. GBM Default occurs when the asset value falls below some threshold.

16 Introduction (1) Banking regulation has historically focused on risk at the level of individual banks But in an interconnected system, banks that appear healthy individually may present a material threat to system solvency: Common non-bank exposures: solvency positions move together in response to macrofinancial developments Interbank exposures: possibility of contagious defaults following one or more fundamental defaults Capital requirements to offset spillover costs

17 Introduction (2) The model captures two channels of systemic credit risk: (i) the risk that banks fail simultaneously because of correlations in the values of their assets; and (ii) the risk that banks fail contagiously because of direct balance sheet interlinkages between banks, assuming that contagious bankruptcy carries a fixed deadweight cost ( as a proportion of assets) in addition to the loss given default associated with fundamental uncertainty.

18 Introduction (3) The logic of the model is as follows: A bank whose failure would ex post impose contagious costs on other participants in the financial system is required ex ante to hold more capital thereby reducing its ex ante probability of failure and lowering the ex ante expected total cost to other banks of its presence in the system Motivates a system-wide risk-management type approach to calibrating individual banks systemic capital requirements

19 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results - benchmark calibration - comparative statics Summary

20 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results - benchmark calibration - comparative statics Summary

21 Broad approach Three key building blocks: (1) A structural credit risk model of an interconnected banking system (set of coupled Merton-type models) (2) A systemic policymaker s objective for a particular characteristic of that system (3) An optimisation algorithm over banks capital buffers to achieve the systemic policymaker s chosen objective In practice, there is a modelling trade-off between realism in (1) and (2) vs. numerical tractability in (3)

22 Broad approach Ingredients can be summarised in a couple of diagrams Individual banks balance sheets System structure (example for three banks) Assets Liabilities Non-bank assets Non-bank debt Inter-bank assets Inter-bank debt Capital

23 Overview of the model Each bank holds assets outside of the banking system of and is assumed to have a single issue of zero-coupon debt outstanding to non-banks with a face value of that falls due for repayment at time. In addition, each bank may have an aggregate interbank asset against the other banks in the system of and an interbank liability of. Like debt to non-banks, interbank debt is also assumed to have a maturity.

24 Overview of the model The configuration of interbank obligations is described by an matrix : (1) Where and. Banks are also partly financed by equity: bank has a capital ratio, where is the nominal value of capital issued by a bank. Capital is assumed to be comprised exclusively of common equity.

25 Overview of the model Total system assets: Total face value of debt liabilities:. Each bank s assets evolve according to a geometric Brownian motion with ex-ante fixed coefficients : (2) Where shocks may be correlated across banks,

26 Overview of the model Based on these correlated asset dynamics, the solvency positions of banks are checked at date. There are two types of default in the model, labelled fundamental and contagious. If, after simulating forward the above diffusion process, the assets of a given bank,, are below its (fixed) debt liabilities at time bank is declared fundamentally insolvent. In this case, its loss given default is endogenously given by the difference between the level of assets at the point at which solvency is assessed and the face value of its debt falling due.

27 Overview of the model The fundamental default of bank triggers losses for other banks in the network that have extended its interbank loans. In some cases, clearing of the interbank network may result in a second bank,, defaulting even though it may be above the solvency threshold if it had not made this loss on its interbank exposure to bank. This represents a contagious failure of bank. In this case, the assets of bank are marked down from the level reached under the diffusion process in equation (2) by an exogenously chosen bankruptcy cost of 10%. The interbank positions of other banks in the network are then re-evaluated.

28 Overview of the model This process is repeated until there are no further rounds of contagious default in the banking network. It presents a mechanism by which losses initially borne by one bank can be transmitted and amplified through an interconnected banking system. Denoting the value of each bank s assets after network clearing by total losses in the banking system at debt maturity are thus given by:

29 Overview of the model Other more realistic descriptions of banks balance sheets could, in principle, be used inside the network model we just described. For example, the liability side of banks balance sheets could also be made dynamic and banks could seek to target a long-run leverage ratio in response to asset and liability shocks. The choice made here to base the evolution of banks balance sheets on Merton (1974) is pragmatic. The policymaker s optimisation problem namely, to achieve a tolerable measure of systemic risk for the lowest compatible level of capital in the system as a whole is solved numerically.

30 Quantifying risks to systemic solvency The asset value dynamics in equation (2) imply a log-normal distribution of asset returns: An estimate of the asset drift rate can be obtained from the mean of this distribution:

31 Quantifying risks to systemic solvency Using the well-known functional form of the normal distribution, we derive a log-likelihood function for banks asset returns,, the value of which is determined by a single unknown parameter, : This likelihood function can be used to estimate the vector of drift rates on banks assets and the unknown contemporaneous variance-covariance structure between banks asset returns These parameters can, in turn, be used to simulate the distributions of losses borne by individual banks and at the level of the system overall at the time when debt is due to be repaid.

32 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results - benchmark calibration - comparative statics Summary

33 Quantifying systemic solvency risk Core of the model follows closely Elsinger et al (2006) Non-bank assets Dynamics Non-bank assets Non-bank debt... correlated across banks Corresponding distribution of returns Inter-bank assets Inter-bank debt Capital Interbank claims Log-likelihood function (iid) where

34 Quantifying systemic solvency risk So depending on the simulated paths for the value of nonbank assets, any given bank might fundamentally default And this can lead to contagious defaults of other banks, depending on the structure of the interbank network Captures the two drivers of system-wide credit risk mentioned in the Introduction, i.e. Common non-bank exposures: solvency positions move together in response to macrofinancial developments Interbank exposures: possibility of contagious defaults following one or more fundamental defaults

35 Quantifying systemic solvency risk Can use the model to build up the distribution of losses at the level of the system as a whole, evaluated at some future point in time The properties of this aggregate loss distribution (constructed bottom-up) are one metric of system stability Adjusting banks capital buffers will alter the location and shape of this loss distribution...

36 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results - benchmark calibration - comparative statics Summary

37 Achieving a particular target for systemic solvency risk Systemic policymaker is concerned with the resilience of the system as a whole, but is also mindful of efficiency Trade-off If capital is relatively costly compared to debt, might want to minimise total banking system capital subject to being able to meet a chosen system risk constraint (Highly) non-linear constrained optimisation problem

38 Achieving a particular target for systemic solvency risk Systemic policymaker is concerned with the resilience of the system as a whole, but is also mindful of efficiency Trade-off If capital is relatively costly compared to debt, might want to minimise total banking system capital subject to being able to meet a chosen system risk constraint (Highly) non-linear constrained optimisation problem Capital across banks (endogenous) Systemic risk constraint (exogenous choice of definition and target) But this measure is an output of the model...

39 Achieving a particular target for systemic solvency risk Optimisation algorithm to identify a unique level and distribution of optimal capital is the main innovation: (a) Optimise over the total level of system capital, holding fixed banks relative shares, to achieve the objective (b) Adjust the composition of capital across banks until a configuration is found that leads to (c) Reduce system capital a small amount, allocating to banks pro-rata (d) Repeat (b)-(c) until it is no longer possible to further reduce system capital and meet the risk objective, such that from below. Could augment this with a bank-specific minimum capital restriction (but it might not bind, depending on calibration)

40 Key steps in the model Hidden Stage 1 Stage 2 Guess unobserved asset return volatility for each bank, i Re-compute asset values for each bank using the new diagonal elements of the full variance-covariance matrix Optimise over the i to maximise the l i ( i, A i ) Invert standard Merton (1974) model to calculate, from observed equity values and guess i, corresponding asset values for each bank A i (t=0) over time series T Calculate value of likelihood function of asset returns for each bank over time window T, l i ( i, A i ) Calculate realised variancecovariance matrix between banks asset returns, Option A (a) Solve for minimum level of system capital that can be allocated across banks such that the chosen VaR 1-z (C) const. constraint is met from below. Option B (b) Optimise over the distribution of equity capital across banks, given a fixed level for the system as a whole, such that VaR 1-z (C) is minimised Perform Cholesky decomposition of variance-covariance matrix and use to simulate forward correlated asset values for the panel of banks at debt maturity A i (t= ) Calculate asset value shortfall below promised debt liabilities for each bank and aggregate across banks (with and without the network clearing model) Numerically integrate the shortfall distribution including network effects to find the z-th percentile, VaR 1-z (C) (a) This is the optimisation under consideration here. (b) This is closer in spirit to Gauthier et al (2010), but is not investigated here.

41 Calibration details Hidden UK data for five (four) banks, 2004 H H1 Banks balance sheets based on published accounts Debt to non-banks = Total liabilities - Large exposures (regulatory returns) - Shareholders funds exc. minority interests Defined as exposures to other banks that exceed 10% of eligible capital Fundamental default: LGD is endogenous, determined by the asset value dynamics discussed earlier Contagious default: Assets marked down by 10% from the level reached endogenously under the diffusion dynamics Weighted-average debt maturity = 1 year

42 Calibration details Hidden Parameters for asset dynamics across banks (drift rate vector and variance-covariance matrix) estimated by maximum likelihood using market data Estimated correlation matrices between banks asset returns

43 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results - benchmark calibration - comparative statics Summary

44 Results: benchmark calibration Evolution of aggregate losses in the banking system over time (a)(b)(c) Grey: Basic Blue: Correlations Red: Correlations and network Per cent Benchmark aggregate loss distribution over time (based on banks actual capital levels) Tails become significantly elongated during the crisis as the aggregate balance sheet weakens (a) Percentiles of aggregate loss distribution across a panel of UK banks: based on stand-alone balance sheets (grey); accounting for correlation between asset returns across banks (blue); and accounting for asset return correlation and explicit interbank exposures between firms, assuming contagious default carries a deadweight cost of 10% of assets (red). (b) Before performing the optimisation experiment. (c) Loss expressed as a fraction of system-wide debt liabilities.

45 Results: benchmark calibration Distribution of aggregate losses in the banking system (2009 H1) (a)(b)(c) Grey: Basic Blue: Correlations Snapshot of benchmark aggregate loss distribution for 2009 H1 (based on banks actual capital levels) Red: Correlations and network Loss (per cent) (a) Aggregate loss distribution across a panel of major UK banks: based on stand-alone balance sheets (grey); accounting for correlation between asset returns across banks (blue); and accounting for asset return correlation and explicit interbank exposures between firms, assuming contagious default carries a deadweight cost of 10% of assets (red). (b) Before performing the optimisation experiment. (c) Loss expressed as a fraction of system-wide debt liabilities. Bimodality corresponds to splitting of outcomes for the system - Contagious defaults contained (first mode) - Wave of contagious defaults (second mode)

46 Results: benchmark calibration System loss distributions pre- and post-optimisation (2009 H1) (a)(b)(c)(d) Benchmark aggregate loss distribution for 2009 H1 (for optimised systemic capital requirements) Optimised (95th percentile) Tail risk mitigated (but, of course, not eliminated) Pre-optimisation Loss (per cent) (a) Accounting for asset correlation and explicit interbank exposures between firms, assuming contagious default carries a deadweight cost of 10% of assets. (b) Circles show location of 95 th percentile. (c) Before performing the optimisation experiment. (d) Loss expressed as a fraction of system-wide debt liabilities.

47 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results Summary

48 Organisation Broad approach Quantifying systemic solvency risk Achieving a particular target for systemic solvency risk Results - benchmark calibration - comparative statics Summary

49 Summary System-wide risk management approach to calibrating banks systemic capital requirements Interconnected panel of structural credit risk models embedded inside a numerical optimisation algorithm Solves for a unique level and distribution of capital across banks commensurate with a chosen measure of system resilience i.e. bank-specific systemic capital requirements Large, highly interconnected and complex (high contagious LGD) banks are shown to pose the greatest threat to system resilience and are required in the model to hold more capital ex ante to mitigate that risk

50 Selected literature Elsinger, Lehar and Summer (2006) Gauthier, Lehar and Souissi (2010) Tarashev, Borio and Tsatsaronis (2010) Adrian and Brunnermeir (2009) Webber and Willison (2011)