## _wp11228

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---

### Integration of financial sector vulnerability into macro models
- Purpose:
  - Analyze whether monetary policy models should include market-based financial stability indicators (FSIs), and if so, how.
- Choice of indicator:
  - Uses distance-to-default (dtd) of the banking system estimated from contingent claims analysis (CCA) tools.
- Rationale and strengths of CCA-based FSIs:
  - Market-based FSIs summarize both the credit channel and credit risk transmission from distressed borrowers and provide information related to the quantity of credit extended and possible effects on GDP (credit expansion and the “financial accelerator”).
  - CCA uses forward-looking equity prices plus balance sheet information to infer bank asset values and volatility, then calculates probability of default or distress.
  - Advantages: equity data incorporate forward-looking market expectations absent from static indicators; high frequency equity and interest rate data allow faster updating; CCA indicators are calculated for individual banks and can be aggregated into a system-wide FSI.
- Policy intuition:
  - Default risk is non-linear and can affect future output gap in ways different from interest-rate-driven demand shocks.
  - A shock to dtd can affect volatility, skewness, and kurtosis of the discounted sum of squared deviations of inflation and GDP from target, implying welfare-relevant tail risks.
  - Reacting directly to dtd in the monetary policy rule can be preferable to only reacting indirectly via inflation and GDP gaps; alternative: react to dtd only when FSI breaches a predetermined threshold.

### Macro-financial model structure and interactions
- Model composition:
  - Integrates a CCA system-wide FSI with a five-equation dynamic, stochastic macroeconomic model used for monetary policy analysis.
  - Macro equations: output gap (IS), inflation (Phillips curve), exchange rate (interest parity), yield curve relating short and long rates, and Central Bank reaction function (Taylor rule).
  - Most equations are forward looking, with expectations of dependent variables on the right-hand side.
- Endogeneity and feedbacks:
  - The system is fully endogenous: interest rate and GDP influence banks’ equity level and volatility; distance-to-default affects the country risk premium, GDP, and the exchange rate.
  - The output gap equation includes dtd as an indicator of financial risk to test statistical significance and to understand how financial sector credit risk affects GDP.
- Empirical pedigree:
  - Macro model developed at the Central Bank of Chile at the start of fully fledged inflation targeting in 2000 and resembles the toolkit proposed by Berg, Karam, and Laxton (2006).

### Efficiency frontier analysis and policy conclusion
- Method:
  - Constructs efficiency frontiers mapping inflation and output volatilities after simulating an artificial economy with stochastic shocks drawn from a normal distribution.
- Main findings:
  - It is more efficient to include dtd in the reaction function because the central bank can reduce the volatility of both inflation and output by doing so.
  - Moving the policy interest rate more than warranted by inflation and output gaps alone can be efficient because negative shocks to asset prices and liquidity may lead to credit risk crises with negative systemic consequences for the financial system and production.
  - Very large dtd could reflect bubbles in asset prices, which usually have bitter endings (caveat).
- Literature context:
  - Related studies include Christiano et al. (2007), Bauducco et al (2008), Taylor (2008), Curdia and Woodford (2009), Kannan et al (2009), and Gertler and Karadi (2011); these often include interest rate spreads or credit growth in reaction functions.

### CCA background, definitions, and computation (Section II)
- Core CCA principles:
  - (i) the values of liabilities are derived from assets;
  - (ii) liabilities have different priority (senior, subordinated, junior);
  - (iii) assets follow a stochastic process.
- Asset dynamics and default:
  - Asset return process: dA/A = Aμ dt + Aσ dε, where ε ~ N(0,1).
  - Default occurs when assets fall to or below a distress barrier (promised payments, B).
  - Risk-adjusted (risk-neutral) probability of default substitutes the risk-free interest rate r for the actual expected return μ.
- Distance-to-default (dtd) definitions:
  - Actual (drift μ): d = [ln(A/B) + (Aμ - 0.5 Aσ^2) t] / (Aσ sqrt(t)); actual PD = N(-d).
  - Risk-adjusted (drift r): d = [ln(A/B) + (r - 0.5 Aσ^2) t] / (Aσ sqrt(t)); risk-adjusted PD = N(-d).
- Practical estimation for firms with traded equity:
  - Solve simultaneously:
    - Equity pricing (Black-Scholes-Merton): E = A N(d1) - B e^{-rt} N(d2)
    - Volatility linkage: σE = (A/E) N(d1) Aσ
  - d2 is distance-to-default (dtd); compute N(-d2) as probability of default.
  - Illustrative approximation: dtd ≈ (A - DB) / (Aσ), where DB is the distress barrier.
- Bank-specific indicators derivable from CCA:
  - (i) distance-to-default;
  - (ii) risk-adjusted and actual probabilities of default;
  - (iii) expected losses (implicit put option) to depositors and debt holders;
  - (iv) potential size of public sector financial guarantees;
  - (v) sensitivity of risk indicators to changes in assets and asset volatility.
- Data and mapping issues:
  - For unlisted institutions, map accounting information to CCA indicators using relationships from listed firms (e.g., Moody’s RiskCalc).
  - Moody’s-KMV application: estimated expected default frequencies (EDFs) for over 35,000 firms and financial institutions in 55 countries.
  - Limitations: aggregation/ad hoc aggregation can mismeasure systemic risk by averaging heterogeneous agents and assuming no correlations; CCA should complement stress tests and surveillance.

### Chilean application and empirical implementation
- Treatment of banking sector:
  - Treat portfolio of banks as one “large bank.” Sample included largest banks covering:
    - approximately 50 percent of total banks’ assets,
    - 65 percent of the total amount of bonds issued by the banking system,
    - more than 80 percent of market capitalization of the banking industry.
- Data sources and construction:
  - Market capitalization and daily shares/prices from Santiago Stock Exchange.
  - Distress barrier constructed from time and saving deposits, demand deposits, Sr. Debt, and book values (linear transformation to daily data).
  - Equity volatility inferred using a conditional heteroskedasticity model with a one year horizon (GARCH(1,1)); moving-average gives similar results.
  - Foreign bank debt ignored explicitly for simplicity; represented only 7 percent of total debt in the analyzed period.
- Caveats:
  - Small banks excluded could bias results if they are riskier.
  - Interbank contagion considered limited given small share of total assets; real-time interbank settlement reduced settlement risks in Chile.
- Empirical series:
  - Distance-to-default for Chilean banking system shown for 1997 to 2006 with a three-month moving average.
  - Elevated risk identified during 1998–1999, 2000, and 3Q2002; trend of reduced risk until late 2005 leveling-off.

### Integrating dtd into the macrofinance policy model (equations and calibration)
- Two-module architecture:
  - Module 1: macro equations — output gap, Phillips curve (inflation), exchange rate (interest parity), yield curve, and a Taylor rule.
  - Module 2: distance-to-default model for the banking system (Black-Scholes-Merton based).
- Key model equations (symbols retained):
  - Output gap equation (1): ygap depends on lags/leads, r (short-run real rate), rl (long-run real rate), q (real exchange rate), and dtd; includes shock yεt.
  - Phillips curve (2): Δπt depends on expectations, output gap, real exchange rate q, and cost-push shock πεt.
  - Exchange rate (interest parity) (3): q depends on lags/leads, domestic policy rate r, foreign policy rate rf, dtd, and shock εqt.
  - Yield curve (4): rl depends on rl lag/lead and short-run r.
  - Reaction function / Taylor rule (5): r depends on lagged r, expected inflation gap, output gap, dtd, and policy shock.
- Distance-to-default module equations:
  - Asset valuation: E = A N(d1) - B e^{-rt} N(d2) (rearranged to solve for A).
  - Relation: d1^2 = d2^2 + Aσ^2 t.
  - d2 (dtd): d2 = [ln(A/B) + (r - 0.5 Aσ^2) t] / (Aσ sqrt(t)).
  - σE = (A/E) N(d1) Aσ links equity volatility to asset volatility.
- Endogeneity of equity and volatility (empirical functional forms noted in source):
  - Equity and σE modeled with feedback from GDP (ygap); making E and σE endogenous resolved counterintuitive dynamics observed when they were held constant.
- Calibration (Table 1 parameters, “Source: Authors’ estimates.”):
  - β1 = 0.1, β2 = -0.1, β3 = -0.6, β4 = -0.4, β5 = -0.5, β6 = -0.5, β7 = 0.02, β8 = 0.2
  - α1 = 0.3, α2 = 0.5, α3 = 0.05
  - δ1 = 0.3, δ2 = 0.6, δ3 = -0.04
  - ξ1 = 0.5, ξ2 = 0.45
  - ρ = 0.8, ρ1 = 0.9, θ = 1.3
  - ζ = 0.5—1.0—1.5 (three alternative values for sensitivity of policy rule to dtd)
  - γ = 0.2—0.3...—1.2 (range of γ values used)

### Stochastic simulations, impulse responses, and policy experiments
- Simulation setup:
  - Shocks include 100 basis points shock to GDP and to inflation.
  - Simulations run in Dynare: artificial economy simulated over 200 periods repeatedly; average standard deviations computed between periods 100 and 120 across repetitions to build efficiency frontiers.
- Stylized impulse responses:
  - Cost-push (inflation) shock: output falls, interest rate initially increases, dtd falls (increased financial vulnerability), exchange rate increases; interactions can yield counterintuitive outcomes if dtd not modeled endogenously.
  - Positive output shock: GDP and inflation increase, interest rates increase, exchange rate falls; system returns to equilibrium in about four years.
  - Responses are more volatile when dtd is not included in the policy reaction function.
- Efficiency frontier experiments and key quantitative insights:
  - Policy rule includes standard Taylor terms plus dtd weight ζ. Three ζ values: 0.5 (weak), 1.0 (medium), 1.5 (strong).
  - Larger ζ shifts efficiency frontiers closer to the origin (lower volatilities of inflation and output), with diminishing marginal gains beyond ζ = 1.0.
  - Endogeneity experiment: increasing GDP feedback to bank equity (raising coefficient in equation (10) from 0.01 to 0.1) and equity volatility (raising coefficient in equation (11) from 1 to 1.5) amplifies gains from reacting to dtd — larger reductions in volatility observed.
  - Exchange rate risk-premium sensitivity experiment: increasing coefficient of dtd in interest parity from 0.04 to 0.5 shows similar improvements when policy reacts to dtd; stronger weight on dtd yields lower inflation volatility in that calibration.
  - Pass-through experiment: raising exchange rate-to-inflation pass-through from 0.05 to 0.7 makes reacting to dtd more efficient for reducing output volatility without increasing inflation volatility.
- Summary:
  - Including dtd in the monetary policy rule generally reduces both inflation and output volatility.
  - Stronger central bank reaction to dtd (higher ζ) produces greater stabilization benefits with diminishing returns.
  - Greater endogeneity of financial indicators to macro variables, higher pass-through, or larger dtd effect on exchange rates all increase the efficiency gains from including dtd.

### Conclusions and suggested extensions
- Main conclusion:
  - Integrating CCA-derived financial sector risk indicators (dtd) into a macroeconomic policy model is practical and can improve policy efficiency; including dtd in the policy reaction function tends to reduce volatilities of inflation and output.
- Policy implication:
  - Moving the policy interest rate beyond what is warranted by inflation and output gaps can be efficient when negative asset/liquidity shocks could trigger systemic credit crises with broader macro effects.
- Suggested refinements/extensions (nonexhaustive):
  - (i) explore other financial sector risk indicators;
  - (ii) analyze combinations of financial scenarios (strong, normal, fragility);
  - (iii) change dynamics of the macro model;
  - (iv) adopt a more micro-founded general equilibrium macro model;
  - (v) use the framework to analyze countercyclical capital requirements and other macroprudential policies;
  - (vi) include risk-adjusted GDP measures in the monetary policy framework.

### APPENDIX I — Extensions of the Merton model (summary)
- Limitations of original Merton model:
  - (i) default only at debt maturity; (ii) fixed default barrier; (iii) constant risk-free rate; (iv) constant asset volatility.
- First-passage/barrier-based extensions:
  - Black and Cox (1976) allow default prior to maturity if assets cross a barrier; practical default often occurs above strict Merton condition due to covenant breaches or liquidity issues.
  - Distress barrier can be the present value of promised payments; KMV sets barrier equal to short-term debt + promised interest next 12 months + half of long-term debt.
- KMV / MKMV framework:
  - Iteratively solve for asset volatility to produce distance-to-default, then map to empirical default probabilities (EDF).
  - KMV_DD includes the real drift Aµ; Merton distance-to-default uses r for the asset drift.
  - MKMV estimates actual default probabilities; risk-neutral default probabilities derived from correlation with market, market Sharpe Ratio, and time horizon.
- Stochastic interest-rate extensions:
  - STV model (Shimko, Tejima, Van Deventer, 1993) includes Vasicek interest rate and correlation with asset returns; Longstaff and Schwartz (1995) extend Black and Cox with stochastic rates.
- Equity-options and volatility-skew approaches:
  - CreditGrades includes asset diffusion, first passage-time default, stochastic default barrier; later modifications incorporate equity derivatives.
  - Research links volatility skew from equity options with CDS spreads (Hull et al., 2004); alternative implementations derive spreads and risk-neutral default probabilities from equity option implied volatilities.
- Capital barriers and central bank relevance:
  - Define a minimum capital barrier above default (example: default barrier plus 8 percent of market value of assets).
  - Region between minimum capital barrier and default barrier corresponds to probability of falling below minimum capital but not default.
  - “Capital barrier put option” measures implicit loss related to liquidity/support needs; useful for quantifying potential central bank interventions prior to default.
- Systemic risk:
  - Contingent claims models can be extended to portfolios of financial institutions and include correlation/dependence structures for systemic risk assessment (IMF Global Financial Stability Report, 2009, April 1).

*Source: _wp11228 — Section II; APPENDIX I; References*

### References .............................................................................................................

### _wp11228 - References

### Integration of financial sector vulnerability into macro models
- The paper analyzes whether monetary policy models should include market-based financial stability indicators (FSIs), and if so, how.
- Market-based FSIs summarize both the credit channel and credit risk transmission from distressed borrowers and provide information related to the quantity of credit extended and possible effects on GDP (credit expansion and the “financial accelerator”).
- Market-based FSIs capture reduced financial soundness when borrowers default: lower value of risky debt, lower banking sector assets, and higher banking asset volatility.
- The paper uses distance-to-default (dtd) of the banking system as the chosen market-based FSI, estimated from contingent claims analysis (CCA) tools.
- CCA uses forward-looking equity prices plus balance sheet information to infer bank asset values and volatility, then calculates probability of default or distress.
- Advantages of CCA-based FSIs:
  - Equity data incorporate forward-looking market expectations absent from static indicators such as nonperforming loan ratios and provisioning.
  - High frequency of equity and interest rate data allow faster updating of risk measures.
  - CCA indicators are calculated for individual banks and can be aggregated into a system-wide FSI.

### Macro-financial model structure and interactions
- The paper integrates a CCA system-wide FSI with a five equation dynamic, stochastic macroeconomic model used for monetary policy analysis.
- The macro model specifications:
  - Equation for the output gap (IS).
  - Equation for inflation (Phillips curve or aggregate supply).
  - Equation for the exchange rate (interest parity condition).
  - Yield curve relating short and long-run interest rates.
  - Central Bank reaction function (Taylor rule).
- Model pedigree and context:
  - Macro model was developed at the Central Bank of Chile at the start of fully fledged inflation targeting in 2000 (García, Herrera, and Valdés, 2002).
  - It closely resembles the toolkit proposed by Berg, Karam, and Laxton (2006).
- Most equations are forward looking, including expected levels of dependent variables on the right-hand side.
- Endogeneity and feedbacks:
  - The system is perfectly endogenous: interest rate and GDP influence banks’ equity level and volatility; distance-to-default affects the country risk premium, GDP, and the exchange rate.
  - The output gap equation includes dtd as an indicator of financial risk to test statistical significance and to understand how financial sector credit risk affects GDP.

### Rationale for including dtd in policy rules
- Distance-to-default is highlighted as a good indicator of financial sector conditions because default risk is non-linear and can affect future output gap in ways different from interest-rate-driven demand shocks.
- A shock to dtd can affect volatility, skewness, and kurtosis of the discounted sum of squared deviations of inflation and GDP from target, implying welfare-relevant tail risks.
- Low probability large deviation events matter disproportionately to welfare; thus risk indicators have a role in monetary policy and central bank interest rate setting.
- The paper argues that reacting to dtd directly in the monetary policy rule can be preferable to only reacting indirectly via inflation and GDP gaps, because indirect reaction may miss tail and distributional effects of financial distress.
- An alternative policy design mentioned: the central bank could react directly to financial risk only when the FSI breaches a predetermined threshold.

### Efficiency frontier analysis and policy conclusion
- The paper constructs efficiency frontiers mapping inflation and output volatilities after the artificial economy is hit with stochastic shocks drawn from a normal distribution.
- General conclusion:
  - It is more efficient to include dtd in the reaction function because the central bank can reduce the volatility of both inflation and output by doing so.
  - Moving the policy interest rate more than warranted by inflation and output gaps alone can be efficient because negative shocks to asset prices and liquidity may lead to credit risk crises with negative systemic consequences for the financial system and production.
- Caveat:
  - Very large dtd could reflect bubbles in asset prices, which usually have bitter endings.

### Literature context
- Growing literature includes financial variables in monetary policy rules in addition to inflation and output gaps. Examples (non exhaustive): Christiano et al. (2007), Bauducco et al (2008), Taylor (2008), Curdia and Woodford (2009), Kannan et al (2009), and Gertler and Karadi (2011).
- These studies include in the reaction function variables such as the interest rate spread or the growth of a credit aggregate.

*Source: _wp11228 - References .............................................................................................................*

### Section II presents the background of CCA distance-to-default (dtd), and discusses the data

### _wp11228 - Section II presents the background of CCA distance-to-default (dtd), and discusses the data

### Background: Contingent Claims Analysis (CCA) and distance-to-default
- CCA uses forward looking information to build risk indicators for the banking system and combines balance sheet information with option-pricing tools to construct marked-to-market balance sheets.
- Core principles of the contingent claims approach:
  - (i) the values of liabilities are derived from assets;
  - (ii) liabilities have different priority (senior, subordinated, junior);
  - (iii) assets follow a stochastic process.
- Default occurs when assets fall to or below a distress barrier (promised payments, B).
- Asset return process: dA/A = Aμ dt + Aσ dε, where Aμ is the drift, Aσ is asset return standard deviation, and ε ~ N(0,1).
- Risk-adjusted (risk-neutral) probability of default substitutes the risk-free interest rate r for the actual expected return μ; risk-adjusted default probability is larger when μ > r.
- Distance-to-default (dtd) definitions:
  - Actual (drift μ): d = [ln(A/B) + (Aμ - 0.5 Aσ^2) t] / (Aσ sqrt(t)); actual PD = N(-d).
  - Risk-adjusted (drift r): d = [ln(A/B) + (r - 0.5 Aσ^2) t] / (Aσ sqrt(t)); risk-adjusted PD = N(-d).
- Conceptual identity for risky debt:
  - Risky debt + Guarantee against default ≡ Risk-free debt
  - Risky debt ≡ Risk-free debt - Guarantee against default
- Guarantees can be modeled as put options on collateral assets with exercise price equal to debt face value.

### Calculating implied assets, asset volatility, and dtd
- For firms with traded equity, solve two equations (Black-Scholes-Merton equity pricing and equity-asset volatility relation) for the two unknowns: asset value A and asset volatility Aσ.
  - Equity pricing (call option): E = A N(d1) - B e^{-rt} N(d2)
  - Volatility linkage: σE = (A/E) N(d1) Aσ, with d1^2 relation: d1^2 = d2^2 + Aσ^2 t
- d2 is precisely distance-to-default (dtd = d2).
- Practical iterative solution: solve simultaneously for A, Aσ, d1, d2 and compute N(-d2) as the probability of default.
- Illustrative approximation: dtd ≈ (A - DB) / (Aσ), where DB is the distress barrier; this expresses dtd as the number of standard deviations from current assets to the distress barrier (KMV reference).

### Bank-specific risk indicators and data considerations
- Required market inputs: market capitalization (E), equity volatility (σE), distress barrier (derived from book values of deposits and debt).
- Risk indicators derived for each bank/institution:
  - (i) distance-to-default;
  - (ii) risk-adjusted and actual probabilities of default;
  - (iii) expected losses (implicit put option) to depositors and debt holders;
  - (iv) potential size of public sector financial guarantees;
  - (v) sensitivity of risk indicators to changes in assets, asset volatility, etc.
- For unlisted institutions, map accounting information to CCA indicators using relationships from listed firms (e.g., Moody’s RiskCalc).
- Moody’s-KMV application: estimated expected default frequencies (EDFs) for over 35,000 firms and financial institutions in 55 countries.
- Limitations: aggregation/ad hoc aggregation can mismeasure systemic risk by averaging heterogeneous agents and assuming no correlations; CCA should complement stress tests and surveillance.

### Chile application and empirical implementation
- Strategy: treat portfolio of banks as one “large bank.” Sample included largest banks covering:
  - approximately 50 percent of total banks’ assets,
  - 65 percent of the total amount of bonds issued by the banking system,
  - more than 80 percent of market capitalization of the banking industry.
- Data and implementation details:
  - Market capitalization and daily shares/prices from Santiago Stock Exchange.
  - Distress barrier constructed from time and saving deposits, demand deposits, Sr. Debt, and book values (linear transformation to daily data).
  - Equity volatility could not be inferred from options (no such derivatives in Chile); used a conditional heteroskedasticity model with a one year horizon (GARCH(1,1) used, moving-average gives similar results).
  - Foreign bank debt ignored explicitly for simplicity; represented only 7 percent of total debt in the analyzed period.
- Caveats:
  - Small banks excluded could bias results if they are riskier.
  - Interbank contagion considered limited given small share of total assets; introduction of real-time interbank settlement reduced settlement risks in Chile.
- Empirical series:
  - Distance-to-default for Chilean banking system shown for 1997 to 2006 with a three-month moving average.
  - Identified elevated risk during 1998–1999 (Asia crisis), 2000 (internet bubble collapse), and 3Q2002 (Brazilian presidential elections). Trend of reduced risk until a leveling-off in late 2005.

### Integrating dtd into a macrofinance policy model
- Model architecture: two modules
  - Module 1: macro equations — output gap, Phillips curve (inflation), exchange rate (interest parity), yield curve, and a Taylor rule.
  - Module 2: distance-to-default model for the banking system (Black-Scholes-Merton based).
- Key model equations (symbols as in source):
  - Output gap equation (1): ygap depends on lags/leads, r (short-run real rate), rl (long-run real rate), q (real exchange rate), and dtd; includes shock yεt.
  - Phillips curve (2): inflation Δπt depends on expectations, output gap, real exchange rate q, and cost-push shock πεt.
  - Exchange rate (interest parity) (3): q depends on lags/leads, domestic policy rate r, foreign policy rate rf, dtd (risk indicator), and shock εqt.
  - Yield curve (4): rl depends on rl lag/lead and short-run r.
  - Reaction function / Taylor rule (5): r depends on lagged r, expected inflation gap, output gap, dtd, and policy shock; including dtd in rule may improve welfare if financial stability affects output.
- Distance-to-default module:
  - Asset valuation (6): E = A N(d1) - B e^{-rt} N(d2) rearranged to solve for A.
  - d1 and d2 definitions:
    - d1^2 relation: d1^2 = d2^2 + Aσ^2 t (equation 7 in source: 1^2 = d^2 + Aσ^2 t)
    - d2 (distance-to-default) given by equation (8): d2 = [ln(A/B) + (r - 0.5 Aσ^2) t] / (Aσ sqrt(t))
  - Link between equity volatility and asset volatility (9): σE = (A/E) N(d1) Aσ
- Endogeneity of equity and its volatility:
  - Modeled feedback from GDP (ygap) to bank equity E and equity volatility σE:
    - Equation (10): E_t = 1 - ρ_E * ygap_t ? (exact functional form in source: (1) 1           1                        0.01* t EEygapρ = − +  — presented as: 1           1                        0.01* t t EEygapρ = − +)
    - Equation (11): σE dynamics: 14 0.1 3* (  ) () / 3 Etttt r ygap ygap ygap σ +− = + − + +  — (presented as in source; σE depends on r, ygap and lags/leads)
  - Empirical note: initially holding E and σE constant produced counterintuitive dynamics; making E and σE endogenous resolved these issues.
- Model calibration and parameters (Table 1):
  - β1 = 0.1, β2 = -0.1, β3 = -0.6, β4 = -0.4, β5 = -0.5, β6 = -0.5, β7 = 0.02, β8 = 0.2
  - α1 = 0.3, α2 = 0.5, α3 = 0.05
  - δ1 = 0.3, δ2 = 0.6, δ3 = -0.04
  - ξ1 = 0.5, ξ2 = 0.45
  - ρ = 0.8, ρ1 = 0.9, θ = 1.3
  - ζ = 0.5—1.0—1.5 (three alternative values for sensitivity of policy rule to dtd)
  - γ = 0.2—0.3...—1.2 (range of γ values used)
  - Table noted as “Source: Authors’ estimates.”

### Stochastic simulations, impulse responses, and policy analysis
- Simulation setup:
  - Shocks: 100 basis points shock to GDP and to inflation analyzed in impulse responses.
  - Simulations run in Dynare: artificial economy simulated over 200 periods repeatedly; average standard deviations computed between periods 100 and 120 across repetitions to build efficiency frontiers.
- Impulse response stylized outcomes:
  - Cost-push (inflation) shock: output falls, interest rate initially increases, dtd falls significantly (increased financial vulnerability), exchange rate increases; interactions can lead to counterintuitive outcomes if dtd not modeled endogenously.
  - Positive output shock: GDP and inflation increase, interest rates increase, exchange rate falls; system returns to equilibrium in about four years.
  - Responses are more volatile when dtd is not included in the policy reaction function.
- Efficiency frontier experiments:
  - Policy rule includes standard Taylor terms plus dtd with weight ζ. Three ζ values used: 0.5 (weak reaction), 1.0 (medium), 1.5 (strong).
  - Key finding: larger ζ (stronger monetary reaction to dtd) shifts efficiency frontiers closer to the origin (lower volatilities of inflation and output), with diminishing marginal gains beyond ζ = 1.0.
  - Endogeneity experiment: increasing GDP feedback to bank equity (raising coefficient in equation (10) from 0.01 to 0.1) and equity volatility (raising coefficient in equation (11) from 1 to 1.5) amplifies gains from reacting to dtd — larger reductions in volatility observed.
  - Exchange rate risk-premium sensitivity experiment: increasing coefficient of dtd in interest parity from 0.04 to 0.5 shows similar improvements when policy reacts to dtd; stronger weight on dtd yields lower inflation volatility in that calibration.
  - Pass-through experiment: raising exchange rate-to-inflation pass-through from 0.05 to 0.7 makes reacting to dtd more efficient for reducing output volatility without increasing inflation volatility; higher pass-through raises trade-offs making dtd inclusion more valuable.
- Summary of simulation insights:
  - Including dtd in the monetary policy rule generally reduces both inflation and output volatility.
  - Stronger central bank reaction to dtd (higher ζ) produces greater stabilization benefits with diminishing returns.
  - Greater endogeneity of financial indicators to macro variables, higher pass-through, or larger dtd effect on exchange rates all increase the efficiency gains from including dtd in the policy rule.

### Conclusions and suggested extensions
- Main conclusion: integrating CCA-derived financial sector risk indicators (dtd) into a macroeconomic policy model is practical and can improve policy efficiency; including dtd in the policy reaction function tends to reduce volatilities of inflation and output.
- Policy implication: moving the policy interest rate beyond what is warranted by inflation and output gaps can be efficient when negative asset/liquidity shocks could trigger systemic credit crises with broader macro effects.
- Nonexhaustive list of suggested refinements/extensions for future research:
  - (i) explore other financial sector risk indicators;
  - (ii) analyze combinations of financial scenarios (strong, normal, fragility);
  - (iii) change dynamics of the macro model;
  - (iv) adopt a more micro-founded general equilibrium macro model;
  - (v) use the framework to analyze countercyclical capital requirements and other macroprudential policies;
  - (vi) include risk-adjusted GDP measures in the monetary policy framework.

*Italic: Source — _wp11228 - Section II presents the background of CCA distance-to-default (dtd), and discusses the data*

### APPENDIX I. EXTENSIONS OF THE MERTON MODEL

### APPENDIX I. EXTENSIONS OF THE MERTON MODEL

### Overview
- The original Merton Model relies on several restrictive assumptions: (i) default can occur only at the maturity date of the debt; (ii) there is a fixed default barrier; (iii) there is a constant risk-free rate; and, (iv) asset volatility is constant.
- Numerous extensions relax these assumptions to better capture real-world default behavior, liquidity constraints, stochastic interest rates, and layered liability structures.

### First-passage and barrier-based extensions
- Black and Cox (1976) introduce a “first passage time” model relaxing assumptions (i) and (ii): default can occur prior to debt maturity if assets fall below a specified barrier function for the first time.
- In practice, default typically occurs at asset values higher than the strict Merton condition (assets < required debt payments) because of covenant breaches, inadequate liquidity (assets cannot be sold to meet payments), or strategic sovereign decisions to default to induce renegotiation.
- To capture these conditions, specify a market value of total assets that triggers default: the “distress barrier.”
  - The distress barrier can be viewed as the present value of the promised payments discounted at the risk-free rate.
  - The KMV approach sets the barrier level equal to the sum of: the book value of short-term debt, promised interest payments for the next 12 months, and half of long-term debt (see Crouhy, et. al. (2000) and KMV (1999, 2001)).

### KMV / VK (MKMV) framework, distance-to-default, and mapping to default probabilities
- The KMV model (based on the VK model — Vasicek and Kealhofer) incorporates multiple layers of liabilities and confidential features.
- MKMV’s EDF (expected default frequency) credit measure is calculated by iteratively solving for asset volatility, producing a distance-to-default that is then mapped to empirical default probabilities using a database of real-world default frequencies.
- The MKMV distance-to-default and CEDF (cumulative expected default probabilities) are given in the source as:

  
  
  2
  0
  ln//2
  tAA
  A
  KMV
  ABt
  DDf
  t
  
  
  
  
  
  
  
  

  
  ()
  tKMV
  CEDFf   DDt

- Note: the KMV definition of KMV_DD includes the real drift of the asset, Aµ, whereas the Merton distance-to-default uses r for the asset drift.
- MKMV estimates actual default probabilities; risk-neutral default probabilities are calculated from the correlation of the implied asset with the market, the market Sharpe Ratio, and time horizon.

### Stochastic interest-rate extensions
- The Merton Model has been extended to include stochastic interest rates, relaxing assumption (iii).
- Shimko, Tejima, and Van Deventer (1993) include a Vasicek interest rate term structure model and allow correlation between asset return and interest rate. This two-factor model (asset and interest rate) is commonly called the STV model.
- Longstaff and Schwartz (1995) extend the Black and Cox (1976) model by adding stochastic interest rates in a manner similar to STV.

### CreditGrades, equity-options approaches, and volatility skew
- The CreditGrades model (2002) includes asset diffusion, first passage-time default, and a stochastic default barrier; it was later modified to incorporate equity derivatives (Stamicar and Finger 2005).
- Recent research links the volatility skew implied by equity options with CDS spreads (Hull et al., 2004). They establish a relationship among implied volatility of two equity options, leverage, and asset volatility.
  - This provides an alternative implementation of Merton’s Model to derive spreads and risk-neutral default probabilities directly from equity option implied volatilities.
- A related approach using several equity options is discussed in Zou (2003).

### Capital barriers, liquidity support, and central bank relevance
- Authorities frequently provide financial support (liquidity and credit) before a firm reaches the “default” barrier.
- Define a minimum capital barrier (capital barrier) in addition to the default barrier. Example: default barrier plus 8 percent of market value of assets could be used as a “minimum 8 percent capital” barrier.
- The region between the minimum capital barrier and the default barrier represents the probability of falling below minimum capital but not as far as default.
  - The value of this area is calculated as: implicit put option below the minimum capital barrier minus the implicit default put option.
  - Termed the “capital barrier put option” or “capital barrier expected loss.”
- Relevance to central banks:
  - The capital barrier put option is a measure of loss directly related to liquidity support/financial support needed to restore bank asset levels above the minimum capital level.
  - This metric helps quantify potential central bank interventions prior to actual default.

### Systemic risk applications
- Contingent claims models can be extended to assess systemic risk in portfolios of financial institutions, including modeling correlation or dependence structures among institutions (IMF Global Financial Stability Report, 2009, April 1).

*Source: APPENDIX I. EXTENSIONS OF THE MERTON MODEL, _wp11228 - APPENDIX I. EXTENSIONS OF THE MERTON MODEL*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11228.pdf_