## 1. Market Price of Risk

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---

### Overview
- Purpose: Describe application of contingent claims analysis (CCA) and systemic CCA to the top four commercial banks in Sweden, historically and for four forward-looking stress test scenarios.
- Data and horizon:
  - Uses daily data from 2007 through 2010.
  - Stress test outputs and systemic CCA used to estimate bank-by-bank outputs for the 2011 to 2015 period.
  - Calibrated CCA model starting point: December 30, 2010.
- Outputs produced:
  - Expected losses to creditors.
  - Changes in funding costs.
  - CCA capital ratios (market value of equity to market value of assets).
  - Capital shortfall measure.
  - Historical systemic expected and tail-risk expected losses.

### Contingent Claims Analysis (CCA) framework
- Core identity:
  - Bank assets = Bank market value of equity + Bank risky debt and deposits.
  - Bank risky debt and deposits = Default-free value of debt and deposits – Expected losses to bank creditors.
- Interpretation:
  - Expected loss to bank creditors = Probability of bank default × Loss given default × exposure (default-free value of total debt and deposits).
- Comparison to accounting:
  - Traditional: accounting assets change one-for-one with book equity.
  - CCA: decline in assets leads to less-than-one-to-one decline in market equity; expected losses to creditors increase and can rise substantially in severe distress.
- Uses:
  - Quantifies impact of changes in market value of equity on borrowing costs.
  - Measures impact of changes in global/regional risk appetite (market price of risk) on expected losses, funding costs, and equity.
  - Measures effects of implicit/explicit government guarantees on borrowing costs.

### Linking CCA to stress testing (method used)
- Three methods to incorporate macro variables into bank CCA balance sheets (this note uses method 1):
  1. Use outputs from traditional stress tests (changes in PDs, LGDs, exposures, net income) as inputs into calibrated bank CCA model.
  2. Estimate historical relationship of macro factors to expected losses to creditors and project forward using stress scenarios.
  3. Estimate historical relationships of macro factors to changes in bank market value of assets.
- Process used:
  - Balance sheet stress test outputs provide annual changes in bank net income due to profits, loan losses, tax, and dividend payments from 2011 to 2015.
  - These changes map to changes in bank asset values in the CCA model, which affect asset volatility.
  - Adjusted asset and volatility are input into CCA spread formula and applied to debt due in next year to compute incremental funding cost changes year by year.
  - After accounting for fraction of funding cost transferable to customers, additional change in bank assets is computed, reinserted, and final outputs recalculated.

### Funding cost / credit spread computation
- Incremental funding cost spread formula (preserved as in source):
  -  1 ln  1 rT incrementalbase sTELBes     .
- Mechanism:
  - Incremental credit spread = scenario credit spread (function of expected loss EL, default barrier B, risk-free rate r, and T=1) minus base credit spread.
  - Incremental spread × incremental debt due each year = change in funding costs each year (debt due figures are in Appendix IV).
  - After deducting the fraction transferable to customers, the residual increases asset losses and feeds back into the CCA model.

### Transfer-to-customers assumptions (as described)
- Adverse scenario 1: 80 percent of funding cost can be transferred to customers → 20 percent of incremental funding costs subtracted from bank asset value.
- Adverse scenario 2: 85 percent of funding cost can be transferred to customers.
- Adverse scenario 3: 95 percent of funding cost can be transferred to customers.
- Base scenario: all funding costs can be transferred to customers → none of incremental funding cost is subtracted from assets.

### Systemic CCA
- Purpose: Analyze dependence between banks for the system as a whole.
- Outputs:
  - Joint expected losses to creditors (median and tail-risk).
  - Joint market value of banking system capital.
  - Market-implied government contingent liabilities to the banking system.
- Method highlights:
  - Uses extreme value theory (EVT) and Generalized Extreme Value Distribution (GEV) marginals.
  - Time-varying dependence estimated over a window of 120 working days with daily updating via the Linear Ratio of Spacings (LRS) method.
  - Multivariate VaR reported at the 95 percent percentile.

### Results for the Swedish banks — high-level findings
- Calibration: CCA models calibrated using equity market and balance sheet information (including some inputs from Moody’s KMV Credit Edge) to estimate asset level, asset volatility, default barrier, skew, kurtosis, and a volatility-adjustment parameter.
- Market price of risk projection: Global market price of risk projected for each of the four scenarios (based on historical relationships to GDP; Appendix II).
- Representative numeric outcomes:
  - By 2012, CCA capital ratio:
    - Base scenario: 5.62 percent.
    - Adverse 3 scenario: 4.93 percent.
  - Over the stress test period, CCA capital ratios range from 4.5 percent to 6 percent.
  - Historical low: CCA capital ratio reached 3 percent in 2008.
  - Pre-crisis: 8.8 percent before the crisis in 2006 as shown in Figure 1b.
- Overall assessment: Results show banks are resilient to shocks and are consistent with balance sheet stress test results.

### CCA results overview and dynamics
- Relationship to Tier 1:
  - CCA capital ratios are smaller than Tier 1 ratios because RWA are lower than market value of assets; RWA often about half the level of book assets, so dividing CCA ratios by about 0.5 increases them to the same range as Tier 1.
- Reported CCA outputs:
  - (i) expected losses to creditors; (ii) bank credit spreads; (iii) CCA capital ratio; (iv) capital shortfall measure.
- Scenario floor and historical lows:
  - CCA capital ratios never fall below 4.9 percent in the scenarios examined, compared with a low point of 3.1 percent in 2009.

### Expected losses (simple sum) and trajectories (Billion SEK)
- End-2010: SEK 88 billion.
- 2012 under Adverse 1: nearly SEK 140 billion.
- 2012 under Adverse 2: SEK 180 billion.
- 2015 under Adverse 1/2: declines back to below SEK 100 billion.
- 2012 under Adverse 3: SEK 240 billion.
- 2015 under Adverse 3: SEK 140 billion.
- Peak during the crisis: SEK 375 billion in 2009.

### Credit spreads (one-year, basis points, asset-weighted)
- Start: 95 bps (implied around 2011).
- Adverse 1 (2011–2012): rise to 145 bps.
- Adverse 2 (2011–2012): rise to 200 bps.
- Adverse 3: rise to 250 bps by 2012, decline to 150 bps by 2015.
- Base case: falls to 77 bps by 2015.
- Under Adverse 1 and Adverse 2 spreads decline back under 100 bps by 2015.

### Bank-by-bank CCA capital ratios (Base scenario) — Table 1 (In percent)
- Bank 1: 2010: 6.17; 2011: 6.24; 2012: 6.31; 2013: 6.38; 2014: 6.45; 2015: 6.51
- Bank 2: 2010: 5.44; 2011: 5.48; 2012: 5.53; 2013: 5.58; 2014: 5.64; 2015: 5.69
- Bank 3: 2010: 5.56; 2011: 5.60; 2012: 5.65; 2013: 5.69; 2014: 5.73; 2015: 5.78
- Bank 4: 2010: 4.99; 2011: 5.02; 2012: 5.06; 2013: 5.10; 2014: 5.14; 2015: 5.18

### Bank-level impacts under adverse scenarios (selected figures)
- Adverse 1 (Percent change from Base Scenario; selected):
  - 2011: Exp Losses 92.06; Equity Cap -3.08; CCA Cap Ratio 6.05
  - 2012: Exp Losses 51.70; Equity Cap -2.35; CCA Cap Ratio 6.16
  - 2013: Exp Losses 22.57; Equity Cap -1.24; CCA Cap Ratio 6.30
  - 2014: Exp Losses 5.18; Equity Cap -0.37; CCA Cap Ratio 6.43
  - 2015: Exp Losses 0.76; Equity Cap -0.39; CCA Cap Ratio 6.50
- Adverse 2 (selected):
  - 2011: Exp Losses 155.11; Equity Cap -4.02; CCA Cap Ratio 5.99
  - 2012: Exp Losses 109.34; Equity Cap -2.83; CCA Cap Ratio 6.14
  - 2013: Exp Losses 73.46; Equity Cap -2.12; CCA Cap Ratio 6.26
  - 2014: Exp Losses 40.31; Equity Cap -1.65; CCA Cap Ratio 6.36
  - 2015: Exp Losses 14.56; Equity Cap -1.55; CCA Cap Ratio 6.44
- Adverse 3 (selected):
  - 2011: Exp Losses 117.80; Equity Cap -4.13; CCA Cap Ratio 5.98
  - 2012: Exp Losses 239.25; Equity Cap -7.84; CCA Cap Ratio 5.82
  - 2013: Exp Losses 160.39; Equity Cap -3.97; CCA Cap Ratio 6.14
  - 2014: Exp Losses 133.64; Equity Cap -3.37; CCA Cap Ratio 6.25
  - 2015: Exp Losses 108.69; Equity Cap -3.08; CCA Cap Ratio 6.34
- Additional highlights:
  - Bank 4 under Adverse 3 in 2012: Equity Cap -13.86 (percent change from base).
  - Individual-bank equity declines under Adverse 3 range from 3.08 percent (Bank 1) to 7.22 percent (Bank 4) in one table narrative.

### Capital shortfall and resilience
- Capital cushion threshold example: 4 percent of assets.
- Measure result: none of the Swedish banks has a capital shortfall in any year under any of the four stress tests when using a 4 percent threshold.
- Overall findings:
  - Expected losses increase and equity capital decreases until 2012 across adverse scenarios and then reverse trend.
  - Magnitude of changes is much smaller than in 2009 during the financial crisis.
  - Banks are found to be resilient to shocks; no bank has a capital shortfall in any year of the stress test (4 percent threshold).
  - CCA analyses are consistent with balance sheet stress test results.

### Systemic CCA analysis and tail risk
- Systemic CCA quantifies systemic financial sector risk by estimating multivariate density of expected losses (full value of the implicit put option).
- Methodological points:
  - Uses EVT and GEV marginals with parameters μj, σj, ξj.
  - Multivariate dependence function A(ω1,...,ωm) derived nonparametrically based on Pickands (1981) and Hall and Tajvidi (2000).
- Sample and peak observation:
  - Sample period: 01/01/2007-02/11/2010.
  - July and August 2009 peak: 5 percent chance of losses of SEK 200 billion over the coming year.
- Joint-loss approximations (using past dependence structure as a rough guide):
  - Base case (rough approximation): median losses of SEK 40 billion and 95th percentile at SEK 70 billion.
  - Adverse 3 (rough approximation for 2012): median around SEK 80 billion and 95th percentile around SEK 120 to 150 billion.
  - Caveat: assumes dependence structure similar to crisis periods, which may not hold.

### Conclusions and recommended future work
- Key conclusions:
  - Banks are resilient; no capital shortfalls under the stress tests (4 percent cushion).
  - Adverse scenarios increase expected losses and reduce equity until 2012, then trends reverse; changes are smaller than 2009 crisis levels.
  - System stability has increased significantly since the crisis, based on systemic risk estimates.
- Suggested future work:
  - More work on systemic risk models to analyze systemic risk under stress scenarios.
  - Focus on joint stress testing of the banking system and sovereign risk.

### Appendix I — CCA method and numerical example (values preserved exactly)
- CCA principles:
  - Assets follow a stochastic process; default occurs when assets A fall below default barrier B.
  - Equity modeled as a call option on assets with exercise price B; expected loss to creditors is a put option P on assets with exercise price B.
  - Calibration infers implied asset value A and asset volatility σA from observed equity E and equity volatility σE via Merton-type equations.
- Numerical example (values preserved exactly):
  - asset value A = $100
  - asset volatility σ = 0.40 (40 percent)
  - default barrier B = $75
  - risk free rate r = 0.05 (5 percent)
  - T = 1 (one year)
  - value of equity = $32.367
  - value of risky debt = $67.633
  - yield to maturity on the risky debt = 10.34 percent
  - credit spread = 5.34 percent or 534 basis points
  - risk-neutral default probability, RNDP = 26 percent
  - Implicit put option EP equals expected loss to bank creditors for these inputs.

### Market price of risk, Sharpe-ratio scenarios, and volatility implications
- Market price of risk relation:
  - ,AM SR     (where λ is the market price of risk, ,AM ρ is correlation of bank’s asset return with the global market, and SR is the global market Sharpe ratio).
- Empirical notes:
  - ,AM ρ usually around 0.5 to 0.7 and SR around 0.55 to 1.2 in recent years.
  - For Swedish banks: average volatility ≈ 16 percent when SR = 0.6, increases to 23 percent when SR = 1.1.
  - Incremental change in volatility empirically equals change in Sharpe ratio times 0.09.
- Four Sharpe-ratio (MPR) scenarios — Table 3 (values preserved exactly):
  - Base / Adv 1 / Adv 2 / Adv 3
  - 2010: 0.63 / 0.63 / 0.63 / 0.63
  - 2011: 0.58 / 0.79 / 0.93 / 0.84
  - 2012: 0.57 / 0.72 / 0.85 / 1.00
  - 2013: 0.57 / 0.67 / 0.78 / 0.90
  - 2014: 0.57 / 0.64 / 0.71 / 0.85
  - 2015: 0.57 / 0.63 / 0.68 / 0.80

### Incremental funding-cost calculations and iterative feedback
- Incremental spread formula (symbolic preserved):
  -  1 ln 1 rT incremental base s TELBes      .
- Procedure:
  - Compute scenario credit spread from EL, B, r, T=1.
  - Subtract base spread → incremental spread.
  - Multiply incremental spread by incremental debt due each year → change in funding costs.
  - Subtract transferred portion to customers → residual FC Aδ added to asset changes.
  - Recompute asset volatility (GC implementation uses γ parameter estimated empirically) and reinsert into CCA spread formula.
  - Final iterative outputs: expected losses to creditors; credit spreads; CCA capital ratio.
- Capital cushion metric:
  - Cushion example = 4 percent of assets; equity compared to cushion to determine shortfall (no shortfalls reported under scenarios).

### Appendix IV — Bank-level figures (Billion SKr)
- Nordea Bank AB 470.97 14.14 92.65 18.65 50.9
- Skandinaviska Enskilda Banken AB 328.83 327.82 98.02 91.82 79.7
- Svenska Handelsbanken AB 438.15 540.24 65.54 59.34 65.4
- Swedbank AB 328.63 335.62 97.82 97.12 74.5

*Source: _cr11286 - 1. Market Price of Risk (PDF chapter).*

### 1. Market Price of Risk ................................................................................................

### 1. Market Price of Risk

### Overview
- Purpose: Describe application of contingent claims analysis (CCA) and systemic CCA to the top four commercial banks in Sweden, historically and for four forward-looking stress test scenarios.
- Data and horizon:
  - Uses daily data from 2007 through 2010.
  - Stress test outputs and systemic CCA used to estimate bank-by-bank outputs for the 2011 to 2015 period.
  - Calibrated CCA model starting point: December 30, 2010.
- Outputs produced:
  - Expected losses to creditors.
  - Changes in funding costs.
  - CCA capital ratios (market value of equity to market value of assets).
  - Capital shortfall measure.
  - Historical systemic expected and tail-risk expected losses.

### Contingent Claims Analysis (CCA) framework
- Core identity (risk-adjusted balance sheet):
  - Bank assets = Bank market value of equity + Bank risky debt and deposits.
  - Bank risky debt and deposits = Default-free value of debt and deposits – Expected losses to bank creditors.
- Interpretation of expected losses in CCA:
  - Expected loss to bank creditors relates to total debt and deposits on the full bank balance sheet.
  - Expected loss can be viewed as: Probability of bank default × Loss given default × exposure (represented by the default-free value of total debt and deposits).
- Comparison to traditional accounting:
  - Traditional: accounting assets change one-for-one with book equity.
  - CCA: decline in assets leads to less-than-one-to-one decline in market equity; expected losses to creditors increase and can rise substantially in severe distress.
- Uses of CCA:
  - Quantifies impact of changes in market value of equity on borrowing costs.
  - Measures impact of changes in global/regional risk appetite (market price of risk) on expected losses, funding costs, and equity.
  - Measures effects of implicit/explicit government guarantees on borrowing costs.

### Linking CCA to stress testing
- Three methods to incorporate macro variables into bank CCA balance sheets:
  1. Use outputs from traditional stress tests (changes in PDs, LGDs, exposures, net income) as inputs into calibrated bank CCA model. (This note uses this first method.)
  2. Estimate historical relationship of macro factors to expected losses to creditors and project forward using stress scenarios.
  3. Estimate historical relationships of macro factors to changes in bank market value of assets (e.g., MKMV global correlation).
- Process used in this study:
  - Balance sheet stress test outputs provide annual changes in bank net income due to profits, loan losses, tax, and dividend payments from 2011 to 2015.
  - These changes map to changes in bank asset values in the CCA model, which affect asset volatility.
  - Adjusted asset and volatility are input into CCA spread formula and applied to debt due in next year to compute incremental funding cost changes year by year.
  - After accounting for fraction of funding cost transferable to customers, additional change in bank assets is computed, reinserted, and final outputs recalculated.

### Funding cost / credit spread computation (as presented)
- Incremental funding cost spread formula (preserved as in source):
  -  1 ln  1 rT incrementalbase sTELBes     .
- Mechanism:
  - Credit spread from the scenario (function of expected loss, EL, bank default barrier, B, risk-free rate, r, and time horizon T equal to one year) minus the base credit spread gives the incremental spread.
  - Incremental spread × incremental debt due each year (data in Appendix IV) = change in funding costs each year.
  - After deducting the fraction transferable to customers, the residual increases asset losses and feeds back into the CCA model.

### Systemic CCA
- Purpose: Analyze dependence between banks for the system as a whole.
- Outputs:
  - Joint expected losses to creditors (median and tail-risk).
  - Joint market value of banking system capital.
  - Market-implied government contingent liabilities to the banking system (as in Sweden 2010 Article IV SIP).
- Application: Individual bank losses used with systemic CCA to estimate median and tail-risk systemic losses historically and under scenarios.

### Results for the Swedish banks (high-level findings reported)
- Calibration: CCA models for each of the four banks calibrated using equity market and balance sheet information (including some inputs from Moody’s KMV Credit Edge) to estimate parameters: bank asset level, asset volatility, debt distress barrier, skew, kurtosis, and a volatility-adjustment parameter.
- Scenario inputs:
  - Balance sheet stress test outputs provide annual changes in bank net income and loan losses for 2011 to 2015 under four stress test scenarios (agreed with authorities under the FSAP Update).
  - Global market price of risk projected for each of the four scenarios (based on historical relationships to GDP; Appendix II).
- Representative numeric outcomes reported:
  - By 2012, CCA capital ratio:
    - Base scenario: 5.62 percent.
    - Adverse 3 scenario: 4.93 percent.
  - Over the stress test period, CCA capital ratios range from 4.5 percent to 6 percent.
  - Historical low: CCA capital ratio reached 3 percent in 2008 at the height of the crisis.
- Temporal behavior:
  - CCA capital ratio for the four banks in aggregate declines initially for the three adverse scenarios, but starts increasing sooner in Adverse 1 and Adverse 2 than in Adverse 3.
- Overall assessment from CCA and systemic CCA:
  - Results show banks are resilient to shocks and are consistent with balance sheet stress test results.

### Appendices and figures referenced (topics only)
- Appendix I: Contingent Claims Analysis (formulas and technical details).
- Appendix II: CCA with the Market Price of Risk (projection of market price of risk and incorporation into CCA).
- Appendix III: Systemic CCA.
- Appendix IV: Funding Cost Using CCA and Debt Due Each Year.
- Figures and Tables (selected mentions in source):
  - Figure 1a. CCA Capital Ratio (Equity Market Capitalization/Market Value of Assets).
  - Figure 1b. CCA Capital Ratio 2006 to 2015.
  - Table 1. CCA Capital Ratios for Base Scenario.
  - Table 3. Market Sharpe Ratio Scenarios.
  - Figure 6. Global Sharpe Ratio; Figure 7. GDP and Sharpe Ratio (inverted axis); Figure 8. Market Sharpe Ratio Scenarios.

*Source: _cr11286 - 1. Market Price of Risk (PDF chapter), using content from the provided document.*

### 8.8 percent before the crisis in 2006 as shown in Figure 1b.

### _cr11286 - 8.8 percent before the crisis in 2006 as shown in Figure 1b.

### CCA results overview and dynamics
- CCA capital ratios are smaller than Tier 1 ratios because risk-weighted assets (RWA) are lower than market value of assets; in many cases, RWA are about half the level of book assets, so CCA capital ratios, when divided by about 0.5, increase to the same range as Tier 1.
- CCA outputs reported: (i) expected losses to creditors; (ii) bank credit spreads; (iii) CCA capital ratio (market value of equity divided by market value of assets); and (iv) a capital shortfall measure.
- CCA capital ratios never fall below 4.9 percent in the scenarios examined, compared with a low point of 3.1 percent in 2009.

### Expected losses (simple sum) and trajectories
- Simple sum of expected losses to bank creditors:
  - End-2010: SEK 88 billion.
  - 2012 under Adverse 1: nearly SEK 140 billion.
  - 2012 under Adverse 2: SEK 180 billion.
  - 2015 under Adverse 1/2: declines back to below SEK 100 billion.
  - 2012 under Adverse 3: SEK 240 billion.
  - 2015 under Adverse 3: SEK 140 billion.
  - Peak during the crisis: SEK 375 billion in 2009.

### Credit spreads (one-year, basis points, asset-weighted)
- Weighted spreads path:
  - Start: 95 bps (implied around 2011).
  - Adverse 1 (2011–2012): rise to 145 bps.
  - Adverse 2 (2011–2012): rise to 200 bps.
  - Adverse 3: rise to 250 bps by 2012, decline to 150 bps by 2015.
  - Base case: falls to 77 bps by 2015.
  - Under Adverse 1 and Adverse 2 spreads decline back under 100 bps by 2015.

### Bank-by-bank CCA capital ratios (Base scenario) — Table 1 (In percent)
- Bank 1:
  - 2010: 6.17
  - 2011: 6.24
  - 2012: 6.31
  - 2013: 6.38
  - 2014: 6.45
  - 2015: 6.51
- Bank 2:
  - 2010: 5.44
  - 2011: 5.48
  - 2012: 5.53
  - 2013: 5.58
  - 2014: 5.64
  - 2015: 5.69
- Bank 3:
  - 2010: 5.56
  - 2011: 5.60
  - 2012: 5.65
  - 2013: 5.69
  - 2014: 5.73
  - 2015: 5.78
- Bank 4:
  - 2010: 4.99
  - 2011: 5.02
  - 2012: 5.06
  - 2013: 5.10
  - 2014: 5.14
  - 2015: 5.18

### Bank-level impacts under adverse scenarios (selected percent-change figures and CCA ratios)
- Adverse 1 (selected cells reported as Percent change from Base Scenario and CCA Cap Ratio):
  - 2011: Exp Losses 92.06; Equity Cap -3.08; CCA Cap Ratio 6.05
  - 2012: Exp Losses 51.70; Equity Cap -2.35; CCA Cap Ratio 6.16
  - 2013: Exp Losses 22.57; Equity Cap -1.24; CCA Cap Ratio 6.30
  - 2014: Exp Losses 5.18; Equity Cap -0.37; CCA Cap Ratio 6.43
  - 2015: Exp Losses 0.76; Equity Cap -0.39; CCA Cap Ratio 6.50
- Adverse 2 (selected):
  - 2011: Exp Losses 155.11; Equity Cap -4.02; CCA Cap Ratio 5.99
  - 2012: Exp Losses 109.34; Equity Cap -2.83; CCA Cap Ratio 6.14
  - 2013: Exp Losses 73.46; Equity Cap -2.12; CCA Cap Ratio 6.26
  - 2014: Exp Losses 40.31; Equity Cap -1.65; CCA Cap Ratio 6.36
  - 2015: Exp Losses 14.56; Equity Cap -1.55; CCA Cap Ratio 6.44
- Adverse 3 (selected):
  - 2011: Exp Losses 117.80; Equity Cap -4.13; CCA Cap Ratio 5.98
  - 2012: Exp Losses 239.25; Equity Cap -7.84; CCA Cap Ratio 5.82
  - 2013: Exp Losses 160.39; Equity Cap -3.97; CCA Cap Ratio 6.14
  - 2014: Exp Losses 133.64; Equity Cap -3.37; CCA Cap Ratio 6.25
  - 2015: Exp Losses 108.69; Equity Cap -3.08; CCA Cap Ratio 6.34
- Additional bank-level adverse scenario excerpts (Bank 1–Bank 4 splits) show expected losses and equity declines larger under Adverse 3 in 2012, with equity declines up to:
  - Bank 4 under Adverse 3 in 2012: Equity Cap -13.86 (percent change from base).
  - Individual-bank equity declines under Adverse 3 range from a low of 3.08 percent (Bank 1) to 7.22 percent (Bank 4) in one table narrative.

### Capital shortfall and resilience
- Definition used: capital cushion threshold example equal to 4 percent of assets.
- Measure result: none of the Swedish banks has a capital shortfall in any year under any of the four stress tests when using a 4 percent threshold.
- Overall findings:
  - Expected losses increase and equity capital decreases until 2012 across adverse scenarios and then reverse trend.
  - Magnitude of changes is much smaller than in 2009 during the financial crisis.
  - Banks are found to be resilient to shocks; no bank has a capital shortfall in any year of the stress test (4 percent threshold).
  - CCA analyses are consistent with balance sheet stress test results.

### Systemic CCA analysis and tail risk
- Systemic CCA quantifies systemic financial sector risk by estimating multivariate density of expected losses (full value of the implicit put option).
- Methodological points:
  - Uses extreme value theory (EVT) and Generalized Extreme Value Distribution (GEV) marginals.
  - Accounts for time-varying dependence structure; marginals and dependence estimated over a window of 120 working days with daily updating via the Linear Ratio of Spacings (LRS) method.
  - Multivariate VaR reported at the 95 percent percentile.
- Sample and peak observation:
  - Sample period: 01/01/2007-02/11/2010.
  - July and August 2009 peak: 5 percent chance of losses of SEK 200 billion over the coming year.
- Joint-loss approximations (using past dependence structure as a rough guide):
  - Base case (rough approximation): median losses of SEK 40 billion and 95th percentile at SEK 70 billion.
  - Adverse 3 (rough approximation for 2012): median around SEK 80 billion and 95th percentile around SEK 120 to 150 billion.
  - Caveat: this assumes a dependence structure similar to points in the crisis, which may not hold in the future.

### Conclusions and recommended future work
- Key conclusions:
  - Banks are resilient; no capital shortfalls under the stress tests.
  - Adverse scenarios increase expected losses and reduce equity until 2012, then trends reverse; changes are smaller than 2009 crisis levels.
  - System stability has increased significantly since the crisis, based on systemic risk estimates.
- Suggested future work:
  - More work on systemic risk models to analyze systemic risk under stress scenarios.
  - Focus on joint stress testing of the banking system and sovereign risk.

### Appendix I — Contingent Claims Analysis (method)
- CCA principles:
  - Values of liabilities (equity and debt) are derived from assets.
  - Liabilities have different priority (senior and junior claims).
  - Assets follow a stochastic process; default occurs when assets A fall below default barrier B.
- Asset decomposition: A = E + D.
- Equity is modeled as a call option on assets with exercise price equal to B; expected loss to creditors is modeled as a put option P on assets with exercise price equal to B.
- Option formulas and calibration notes provided, with time horizon T usually between one and five years.
- Calibration uses observed equity value E, equity volatility σE, and distress barrier to infer implied asset value A and asset volatility σA via standard Merton-type equations.

*Source: _cr11286 - 8.8 percent before the crisis in 2006 as shown in Figure 1b.*

### 31.      Once the asset value and asset volatility are known, expected losses to bank creditors

### _cr11286 - 31.      Once the asset value and asset volatility are known, expected losses to bank creditors

### Expected-losses framework and put-option interpretation
- Asset value A and asset volatility σ, together with default barrier B, time horizon T, and discount rate r, determine the implicit put option EP, which equals the expected loss to bank creditors.
- Yield to maturity on risky debt defined by: yT DBe = B.
- Credit spread s = y – r. Using the relationships in the text, the credit spread can be written as:
  - 1 1 E rT s P TBe       (as expressed in the source).
- The spread can also be written as a function of the risk-neutral default probability (RNDP) and LGD:
  - sT = ln(1 – RNDP × LGD) / (–1)   (expressed in source as:  1 ln  1sRNDP   LGD T  ).

### Numerical example (values preserved exactly)
- Assumptions:
  - asset value A = $100
  - asset volatility σ = 0.40 (40 percent)
  - default barrier B = $75
  - risk free rate r = 0.05 (5 percent)
  - T = 1 (one year)
- Resulting valuations and risk metrics:
  - value of equity = $32.367
  - value of risky debt = $67.633
  - yield to maturity on the risky debt = 10.34 percent
  - credit spread = 5.34 percent or 534 basis points
  - risk-neutral default probability, RNDP (= 2()Nd) = 26 percent
- Implicit put option EP equals expected loss to bank creditors for these inputs.

### Decomposition of expected loss EP
- EP can be decomposed into three components:
  - (i) risk-neutral default probability (RNDP)
  - (ii) loss given default (LGD)
  - (iii) default-free value of debt (B)
- The decomposed expression in the text (preserved symbolically) is:
  -    1 2 2 1 rT E rT RNDP LGD P Nd A Be e NdBe Nd           

### Impact of asset changes, volatility feedback, and funding costs
- Asset changes (profits, loan losses, taxes, dividends) change asset value A and thus asset volatility σA.
- For BSM geometric Brownian motion, relationship when asset declines: initial A times initial volatility = changed A times new volatility (source provides BSM relation symbolically).
- For the Gram-Charlier (GC) implementation used (allowing skewness and kurtosis), revised volatility after an asset change Aδ is computed using:
  - AAA A A A            (symbolic expression from text; parameter γ is calculated empirically).
- γ is estimated empirically from observed relationship between implied asset values and implied asset volatility over a recent historical period (about one year).

### Market price of risk (MPR), Sharpe ratio scenarios, and systemic volatility
- Market price of risk is the wedge between actual and risk-neutral default probabilities and is given in the model by:
  - ,AM SR     (where λ is the market price of risk, ,AM ρ is correlation of the bank’s asset return with the global market, and SR is the global market Sharpe ratio).
- MKMV estimates ,AM ρ usually around 0.5 to 0.7 and SR around 0.55 to 1.2 in recent years (as reported).
- For Swedish banks: average volatility is around 16 percent (annualized) when SR = 0.6, and increases to 23 percent when SR = 1.1. Empirically, the incremental change in volatility equals change in Sharpe ratio times 0.09 (measured as a fraction).
- Four stress-test scenarios map to four Sharpe-ratio (MPR) scenarios. Table 3 scenarios (values preserved exactly):
  - Base / Adv 1 / Adv 2 / Adv 3
  - 2010: 0.63 / 0.63 / 0.63 / 0.63
  - 2011: 0.58 / 0.79 / 0.93 / 0.84
  - 2012: 0.57 / 0.72 / 0.85 / 1.00
  - 2013: 0.57 / 0.67 / 0.78 / 0.90
  - 2014: 0.57 / 0.64 / 0.71 / 0.85
  - 2015: 0.57 / 0.63 / 0.68 / 0.80

### Incremental funding-cost calculations and feedback loop
- Incremental spread formula (from text, preserved symbolic form):
  -  1 ln 1 rT incremental base s TELBes      .
- Incremental credit spread = scenario credit spread (function of EL, B, r, T=1) minus base credit spread.
- Incremental spread multiplied by incremental debt due each year yields change in funding costs each year (debt due figures are in Annex 4 per source).
- A portion of increased funding cost may be passed to customers (assumptions in Appendix 4); the remainder increases bank asset changes, leading to revised volatility and recalculation in the CCA model.
- Adjusted asset and adjusted volatility are reinserted into the CCA spread formula; funding-cost-induced asset changes FC Aδ are added and volatility revised again.
- Final iterative outputs include:
  - (i) expected losses to creditors
  - (ii) credit spreads
  - (iii) CCA capital ratio = market value of equity (scenario output) / market value of assets (same scenario)
- The model thus internalizes changes in profits, loan losses, taxes, dividends, associated volatility changes, market price of risk shifts, and incremental funding-cost feedbacks.

### Capital shortfall metric and stress-test finding
- CCA-based capital cushion defined as threshold above default barrier (example given: cushion = 4 percent of assets).
- The amount of market value of equity is compared to the capital cushion:
  - If equity > capital cushion → no capital shortfall.
- Stress-test result reported: no bank has a capital shortfall under any scenario in any year when using a cushion equal to 4 percent of assets.

### Appendix highlights
- Modeling default risk:
  - Asset return process: / AA dttdA A μ σ ε  (symbols preserved). Actual probability of default expressed as 2, ()Nd μ (with D2 and d formulas given symbolically in source).
  - Risk-neutral probability uses r (risk-free rate) as drift; risk-neutral default probability is larger than actual default probability when μ > r.
- Market Price of Risk box:
  - CAPM relation and beta decomposition preserved as in source; MPR computed as ,AM SR λ ρ  and λ = (μA – r)/σA (expressed in source).
  - Empirical notes: ,AM ρ around 0.5–0.7; SR around 0.55–1.2 in recent years.
- Systemic CCA and EVT:
  - Use of extreme value theory to model joint tails; limiting distribution converges to a Generalized Extreme Value distribution (GEV) with parameters μj (location), σj (scale), ξj (shape) as in source.
  - Multivariate dependence function A(ω1,...,ωm) derived nonparametrically by expanding Pickands (1981) bivariate logistic method and adjusting margins per Hall and Tajvidi (2000).
  - Systemic CCA allows marginal contribution of individual institutions to joint tail risk to be quantified (contribution derived as partial derivative of multivariate density as described in source).
- Appendix IV:
  - Table 4: Sweden: Debt Due Each Year (source: Bloomberg) — debt due figures used in incremental funding-cost calculations are reported in Annex 4 (per source).

* _cr11286 - 31.      Once the asset value and asset volatility are known, expected losses to bank creditors_*

### 48.      The incremental funding cost spread formula is

### _cr11286 - 48.      The incremental funding cost spread formula is

### Incremental funding cost spread formula
- The incremental funding cost spread formula is:
  
  1
  ln  1
  rT
  incrementalbase
  sTELBes
  
  
  
  
- This formula is multiplied by the incremental debt due each year.
- These incremental additional funding costs (positive or negative) are adjusted for the percent of funding cost that can be transferred to customers.

### Transfer to customers and scenario assumptions
- As described in the balance sheet stress test TN:
  - For Adverse scenario 1, 80 percent of funding cost can be transferred to customers so only 20 percent of the incremental funding costs is subtracted from the bank asset value.
  - For Adverse scenario 2, 85 percent of funding cost can be transferred to customers and for Adverse scenario 1, 95 percent can be transferred.
  - For the base scenario it is assumed all funding costs can be transferred so none of the incremental funding cost is subtracted from assets.

### Adjustment process and model outputs
- Once the funding cost adjustments are made:
  - The expected losses to bank creditors of the banks are re-estimated.
  - These provide outputs of the CCA.
  - Outputs are then aggregated to produce results for the Systemic CCA model.

### Bank-level figures (Billion SKr)
- Nordea Bank AB 470.97 14.14 92.65 18.65 50.9
- Skandinaviska Enskilda Banken AB 328.83 327.82 98.02 91.82 79.7
- Svenska Handelsbanken AB 438.15 540.24 65.54 59.34 65.4
- Swedbank AB 328.63 335.62 97.82 97.12 74.5

*Source: _cr11286 - 48.      The incremental funding cost spread formula is*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/scr/2011/_cr11286.pdf_