## _wp12216

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---

### I. INTRODUCTION
- Crisis history interpreted as underestimation of risks, including probability of Black Swans and the financial system’s fragility to them.
- Post-crisis critique focused on overreliance on models that gave unwarranted confidence about downside outcomes.
- Paper proposes an intermediate approach: use existing stress-testing models plus a simple heuristic (Taleb 2011) to detect tail-risk convexities.
- Heuristic rationale:
  - Models may misestimate levels in tails due to misspecification, estimation with assumed distributions, and parameter uncertainty.
  - Instead of relying on point estimates, examine how model outcomes vary for small perturbations around a tail shock to detect convexity.
  - Analogy: an inaccurate tape measure gives unreliable level readings but can reliably detect changes over time (second-order measurement).
- Heuristic intended to provide a robust relative ranking of “fragility” to convexities even when model levels are unreliable.
- Paper organization: Section II concepts and heuristic; Section III two case studies (bank and fiscal stress tests); Section IV overview of heuristic use in stress testing; Section V conclusion.

### II. THE SIMPLE HEURISTIC TO DETECT FRAGILITY
- Purpose: detect “biases from missed nonlinearities” using a fast, model-free, probability-free heuristic (Taleb 2011).
- Heuristic scalar H definition:
  - H = ( f(α−Δ) + f(α+Δ) ) / 2 − f(α)
  - f(x) is profit or loss for level α (or a general vector for multidimensional cases).
  - Δ is a change in α, a certain multiple of the mean deviation of the variable.
- Interpretation and scaling:
  - Scale H by total capital (for banks) or GDP (for sovereigns) for comparability and ordinal ranking of fragility.
  - H = 0 (or small relative to capital) implies robustness: payoff function is linear; gains and losses around α are symmetric for deviations of size Δ.
  - H < 0 (and significant relative to capital) implies fragility when lower outcomes are adverse: additional losses from a small unfavorable shock exceed additional gains from a symmetric favorable shock.
  - If the stressed metric is one where larger values represent the adverse case (e.g., change in net debt/GDP), fragility will be represented by H > 0.
- Relation to theory:
  - Heuristic links convexity (second derivative) of mappings Y = φ(x) to tail fragility via the "fragility transfer theorem" and "fragility exacerbation theorem".
  - Convexity bias (Jensen’s inequality) can be evaluated by setting K at Infinity to assess effect on total expectation.

### III. HOW THE HEURISTIC ENHANCES STRESS TESTS
- Addresses two common shortcomings:
  - Many stress tests focus on point estimates of very few scenarios and ignore sensitivity to slightly more severe shocks.
  - Stress tests often ignore model and parameter error, masking convexities/non-linearities that lead to fragility.
- Operational advantage:
  - Heuristic probes sensitivities efficiently by averaging model results over X_t ± ΔX instead of running many additional scenarios.
  - Standardizes a scalar measure of tail convexity/fragility useful for ordinal comparisons across entities and risk drivers.
- Diagnostic interpretation:
  - Positive H (when higher values are adverse) or negative H (when lower values are adverse) signals asymmetry and potential underestimation of tail harm.
  - Antifragility is distinct from robustness: H > 0 for profit starting points implies greater volatility producing more profit.

### IV. CASE STUDY I — BANK STRESS TESTS (12 large U.S. banks)
- Sample and baseline:
  - Sample: 12 large U.S. banks.
  - Balance-sheet baseline: end-2010 balance sheets, complemented with bank data from first half of 2011.
  - Stress horizon: 2012–2016.
  - Central macro stress: near-zero GDP path with a cumulative deviation from the WEO baseline by 10 percentage points during 2012–16; historical likelihood about 4 percent.
  - Outcome metric: Tier 1 capitalization.
  - Model features: feedback loop between banking stress and macro growth; satellite models estimate impacts of GDP on credit losses, credit growth and pre-impairment income; no recapitalization other than retained earnings; deleveraging allowed.
- Mean deviations used in experiments:
  - GDP ±1 MDev = 1.5 percentage points (cumulative); GDP ±2 MDev = 3.0 percentage points (cumulative).
  - Credit growth ±1 MDev = 3.4 percentage points; ±2 MDev = 6.8 percentage points.
  - Credit losses ±1 MDev = 0.11 percentage points; ±2 MDev = 0.22 percentage points.
  - Pre-impairment income/TA ±1 MDev = 0.17 percentage points; ±2 MDev = 0.34 percentage points.
  - Trading income/TA ±1 MDev = 0.1 percentage points; ±2 MDev = 0.2 percentage points.
- Illustrative numerical example (Bank 1):
  - Under central stress: Tier 1 ratio = 1.639 percent.
  - Under additional GDP shocks of +/- one standard deviation outcomes reported as: 0.373 and 2.835, respectively.
  - Heuristic H +/- 1 MDev (GDP) for Bank 1 = (0.373 + 2.835)/2 − 1.639 = -0.035.
- Representative numerical impacts (change in T1 capitalization and H where reported):
  - Impact of adverse scenario (change in T1 capitalization): Bank 1 -9.60; Bank 2 -13.65; Bank 3 -8.48; Bank 4 -2.82; Bank 5 -8.90; Bank 6 -15.07; Bank 7 -1.61; Bank 8 -0.56; Bank 9 2.27; Bank 10 -18.62; Bank 11 -13.31; Bank 12 -7.63.
  - GDP growth 1 MDev worse (1.5 percentage points, cumulative): Bank 1 -1.27; Bank 2 -2.71; Bank 3 -1.86; Bank 4 -0.80; Bank 5 -0.11; Bank 6 -0.24; Bank 7 -0.71; Bank 8 -0.29; Bank 9 -0.12; Bank 10 -2.01; Bank 11 -1.63; Bank 12 -1.67.
  - H +/- 1 MDev (GDP): Bank 1 -0.035; Bank 2 -0.096; Bank 3 -0.294; Bank 4 -0.019; Bank 5 0.000; Bank 6 -0.001; Bank 7 -0.066; Bank 8 0.007; Bank 9 0.000; Bank 10 -0.080; Bank 11 0.025; Bank 12 -0.053.
  - Credit growth 1 MDev more (3.4 percentage points): Bank 1 -0.99; Bank 2 -1.71; Bank 3 -1.40; Bank 4 -0.62; Bank 5 -0.03; Bank 6 -0.09; Bank 7 -0.46; Bank 8 -0.28; Bank 9 -0.02; Bank 10 -1.61; Bank 11 -1.51; Bank 12 -1.34.
  - H +/- 1 MDev (Credit growth): Bank 1 -0.043; Bank 2 -0.090; Bank 3 -0.269; Bank 4 -0.076; Bank 5 -0.001; Bank 6 -0.003; Bank 7 -0.039; Bank 8 -0.009; Bank 9 -0.001; Bank 10 -0.076; Bank 11 -0.049; Bank 12 -0.061.
  - Pre-impairment income/TA 1 MDev less (0.17 percentage points): Bank 1 -1.72; Bank 2 -4.51; Bank 3 -3.44; Bank 4 -2.06; Bank 5 -0.72; Bank 6 -1.17; Bank 7 -1.76; Bank 8 -1.00; Bank 9 -0.96; Bank 10 -1.33; Bank 11 -1.97; Bank 12 -2.66.
  - H +/- 1 MDev (Pre-impairment income/TA): Bank 1 -0.085; Bank 2 -0.291; Bank 3 -0.718; Bank 4 -0.250; Bank 5 -0.014; Bank 6 -0.144; Bank 7 -0.224; Bank 8 -0.051; Bank 9 0.000; Bank 10 -0.043; Bank 11 0.000; Bank 12 -0.162.
  - All three drivers one MDev in adverse direction (combined impact on T1): Bank 1 -4.34; Bank 2 -10.24; Bank 3 -7.58; Bank 4 -4.45; Bank 5 -0.80; Bank 6 -1.35; Bank 7 -3.86; Bank 8 -1.60; Bank 9 -1.01; Bank 10 -5.63; Bank 11 -5.29; Bank 12 -6.07.
  - H +/- 1 MDev (All three drivers): Bank 1 -0.505; Bank 2 -2.035; Bank 3 -2.098; Bank 4 -1.012; Bank 5 -0.020; Bank 6 -0.162; Bank 7 -0.924; Bank 8 -0.123; Bank 9 -0.003; Bank 10 -0.649; Bank 11 -0.212; Bank 12 -0.922.
  - Trading income/TA shocks produce much smaller H values (examples): H +/- 1 MDev (Trading income): Bank 1 -0.013; Bank 2 -0.063; Bank 3 -0.035; Bank 4 -0.001; Bank 5 -0.012; Bank 6 -0.147; Bank 7 -0.033; Bank 8 0.000; Bank 9 0.000; Bank 10 -0.011; Bank 11 0.000; Bank 12 -0.007.
- Aggregate interpretation:
  - Tail stress produces non-linear results in the majority of cases; many banks are fragile with respect to GDP growth, credit growth, credit losses and income.
  - Some banks display antifragile outcomes (positive H) owing to deleveraging and reductions in RWAs (examples: Banks 5, 8, 9, and 11 in the GDP shock case).
  - Heuristic can change ordinal vulnerability rankings versus traditional level-impact rankings and flags entities with asymmetric downside risk.

### V. CASE STUDY II — PUBLIC DEBT SENSITIVITY TO GROWTH SHOCKS
- Setup:
  - Central scenario: growth is 2 percentage points less than in the baseline (September 2011 WEO) per year between 2012 and 2016 (near zero real growth for most sample countries).
  - Tail scenarios: growth one or two mean deviations above or below the central scenario; mean deviations estimated over 1981–2010 for each country.
  - Purpose: analyze non-linearity in net debt response to underestimating adverse growth shocks.
- Representative numerical results (change in net debt, percentage points of GDP; source layout preserved):
  - Central shock (2% points less) 26.2 13.5 20.9 29.6 28.1 34.5 23.7
  - GDP growth 1 M Dev. less 14.0 13.1 16.1 18.5 28.9 39.6 37.5
  - GDP growth 2 M Dev. less 28.6 27.1 33.2 37.9 60.2 83.6 80.1
  - GDP growth 1 M Dev. more -13.5 -12.2 -15.2 -17.6 -26.7 -36.0 -33.3
  - GDP growth 2 M Dev. more -26.4 -23.7 -29.5 -34.3 -51.5 -68.7 -63.1
  - H: +/- 1 M Dev. 0.3 0.4 0.5 0.4 1.1 1.8 2.1
  - H: +/- 2 M Dev. 1.1 1.7 1.9 1.8 4.4 7.4 8.5
- Interpretation:
  - Positive H values indicate increased net debt/GDP when volatility around the central adverse shock rises.
  - Fragility is present in all examined cases: large negative growth shocks increase net debt disproportionately more than comparable positive shocks reduce net debt.
  - Policy implication: underestimation of adverse growth outcomes can materially understate sovereign debt dynamics.

### VI. HOW TO INTEGRATE THE HEURISTIC INTO IMF STRESS TESTS (PRACTICAL STEPS)
- Stepwise procedure:
  - Step 1: Run stress tests and obtain level results (bank capital ratios, liquidity ratios, NPL ratios, Net interest income, ROA, ROE, public debt, etc.).
  - Step 2: Construct additional scenarios X_t + ΔX and X_t - ΔX where ΔX is an estimated mean deviation of the variable X(t) over the pre-determined time period.
  - Step 3: Compute outcomes for these additional scenarios.
  - Step 4: Compute the heuristic.
  - Step 5: Draw conclusions and optionally re-iterate (either full re-iteration with different comprehensive stresses or limited re-iteration changing Δ’s).
- Diagnostic guidance:
  - If heuristic indicates fragility to higher volatility (positive when a higher outcome is adverse, negative when a lower outcome is adverse), conclude that greater adverse stress could make outcomes substantially worse and consider expanding scenario exploration.
  - Heuristic supports ordinal comparisons of fragility across risk drivers and entities and is more robust to some types of model error than single-point pass/fail outcomes.

### VII. POLICY IMPLICATIONS AND CONCLUSIONS
- The heuristic provides a simple scalar measure of convexity/fragility around a central stress that supplements conventional level-based stress-test outputs.
- Practical consequences:
  - For sovereigns: reveal limited room for countercyclical deficits when public finances are fragile to growth shocks.
  - For banks/supervisors: uncover hidden fragilities from illiquid positions, non-linear derivative exposures, or feedback loops between losses and funding costs.
- Recommendation: consider making the heuristic a standard statistic in stress-testing toolkits and use it as a diagnostic to decide when deeper tail analysis is warranted.
- Limitation: heuristic does not eliminate the need for judgment; iterations may be required and no finite iteration can guarantee full characterization of potentially complex functional forms governing outcomes.

### VIII. APPENDIX I — DETAILS ON MACROECONOMIC BANK STRESS TEST
- Bank-specific credit losses:
  - "Japan-like growth shock (i.e., a lost decade-like scenario)" results in substantial losses in banks’ loan books and on assets subject to counterparty credit risk; counterparty credit risk losses assumed to change in line with loan loss rates (in relative terms, but on a substantially lower level).
  - Housing shock: an additional “permanent” decrease of house prices by 20 percent is added, leading to LGDs of 40 percent for retail mortgages.
  - Trading income shock: trading income projected to yield losses of around 1.5 percent of total assets in 2012 and of around 1 percent in 2013.
- Credit growth and RWAs:
  - RWAs grow in line with total credit exposure; risk effects under IRB disregarded for U.S. banks; no behavioral adjustment assumed to benefit from lower RWAs over five years; remainder of RWAs held constant.
- Income and retained earnings:
  - Pre-impairment pre-tax income under the adverse growth scenario projected to drop 13 percent compared to 2010 and remain at that level throughout the projection period.
  - Assumed that 60 percent of net income is retained if net income remains positive; otherwise retained income (i.e., loss) is fully retained.
  - A tax rate of 25 percent was applied to all banks.
- Funding costs:
  - Funding costs assumed to increase conditional on capitalization level during stress — funding costs increase faster as capital falls to low levels.
  - Banks assumed to be able to pass on 50 percent of funding cost increases to their customers.
- Footnotes/methodology notes:
  - Mean deviation computed based on evolution of the median loss rate among all U.S. banks in Bankscope.
  - "20 percent is a common, conservative benchmark for housing loans (many banks use figures around 15 percent)."
  - The 2010 pre-impairment pre-tax income was very close to the average over the last 5 years except for investment banks, where trading income was more important.

### IX. APPENDIX II — DETAILS ON PUBLIC DEBT STRESS TEST
- Debt dynamics summarized:
  - d_t = ratio of public debt to GDP.
  - pb_t = primary balance.
  - r_t = growth adjusted interest rate (function of nominal interest rate i and nominal GDP growth rate g).
  - Nominal interest rate derived from interest payments current year to end-period stock of debt previous year.
  - Primary balance depends on pb_t^base, revenue and expenditure semi-elasticity to changes in the output gap (R_ε, E_ε), and Δog_t.
- Scenario assumptions:
  - Growth shocks do not affect potential GDP and governments do not take discretionary corrective measures.
  - Partial equilibrium simulation assumes no change in nominal interest rate as a result of the growth shock.
- Key drivers affecting debt dynamics:
  - Trend growth (important for countries with projected low growth during 2012–2016).
  - Initial stock of public debt (important for highly indebted countries).
  - Size of automatic stabilizers (matters more in countries with high welfare spending).
- Revenue and expenditure elasticity data are from Girouard and André (2005); when unavailable, elasticity of one assumed for revenue and zero for expenditures.

*Source: _wp12216 - 5.0 ppts; APPENDIX I: DETAILS ON MACROECONOMIC BANK STRESS TEST; APPENDIX II: DETAILS ON PUBLIC DEBT STRESS TEST.*

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

### _wp12216 - References .............................................................................................................

### I. INTRODUCTION
- Crisis history interpreted as underestimation of risks, including probability of Black Swans and the financial system’s fragility to them.
- Post-crisis critique focused on overreliance on models that gave unwarranted confidence about downside outcomes.
- Paper proposes an intermediate approach: use existing stress-testing models plus a simple heuristic (Taleb 2011) to detect tail-risk convexities.
- Heuristic rationale:
  - Models may misestimate levels in tails due to misspecification, estimation with assumed distributions, and parameter uncertainty.
  - Instead of relying on point estimates, examine how model outcomes vary for small perturbations around a tail shock to detect convexity.
  - Analogy: an inaccurate tape measure gives unreliable level readings but can reliably detect changes over time (second-order measurement).
- Heuristic intended to provide a robust relative ranking of “fragility” to convexities even when model levels are unreliable.
- Paper organization:
  - Section II: concepts and heuristic
  - Section III: two case studies applying the heuristic to bank and fiscal stress tests
  - Section IV: overview of heuristic use in stress testing
  - Section V: conclusion

### II. REVIEW OF CONCEPTS TO ASSESS FRAGILITY
- A. The Current State of Stress Testing
  - Crisis revealed weaknesses in stress tests: poorly designed scenarios, omitted shocks, inappropriate methods, narrow coverage.
  - Two fundamental issues:
    - Many stress tests focus on point estimates of very few scenarios and pay little attention to sensitivity to slightly more severe shocks.
    - Stress tests often ignore model and parameter error, masking convexities/non-linearities that lead to fragility.
  - Developments in stress testing:
    - Expansion from solvency and market risk toward liquidity and contagion risks.
    - Liquidity stress tests less developed because liquidity risk seen as less critical pre-crisis, low-frequency/high-impact, and idiosyncratic.
    - Contagion stress tests need detailed cross-exposure data; data collection for interconnectivity is nascent.
  - Increasing model complexity increases risk of model error, potentially worsening misestimation of tail losses.
  - Open questions remain: Are tail risks captured correctly given model error, parameter stochasticity, and incorrect distributions? How do estimates respond to marginal changes in stress scenarios?

- B. A Simple Heuristic to Detect Fragility
  - Purpose: detect “biases from missed nonlinearities” using a fast, model-free, probability-free heuristic (Taleb 2011).
  - Negative convexity example: harm from adverse shocks increases disproportionately as shock size grows.
  - Empirical examples of negative convexity:
    - Société Générale forced unwind: necessity to dump roughly US$70 billion of stock index futures over three days upon discovery of rogue trading positions in January 2008.
      - Had the stock futures been dumped in ten US$7 billion increments over a longer period, price impact and losses may have been smaller.
      - Investigation: a €49 billion (US$71 billion) long position unwound between January 21 and January 23, leading to gross losses of EUR 6.4 billion.
    - Positive feedback effects: forced sales can cause price declines, triggering further liquidations and magnifying losses.
    - Fiscal dynamics: higher debt or rollover requirements can raise financing costs and trigger debt spirals.
    - Regulation-induced tail hiding: e.g., value at risk capital requirements at 5 percent can create asymmetric incentives to take positions that pay off in tail shocks at 4 percent probability.
  - Heuristic construction and interpretation:
    - A point estimate gives no sense of convexity; the heuristic averages model results over a range of shocks.
    - Heuristic scalar H is defined as:
      - H
        f(
        )f()
        2
        f()
      - f(x) is profit or loss for level  (or a general vector for multidimensional cases).
      -  is a change in , a certain multiple of the mean deviation of the variable.
    - Interpretation:
      - Scale H by total capital (for banks) or GDP (for sovereigns) for comparability and ordinal ranking of fragility.
      - H = 0 (or small relative to capital) implies robustness: payoff function is linear; gains and losses around  are symmetric for deviations of size .
      - H < 0 (and significant relative to capital) implies fragility: additional losses from a small unfavorable shock exceed additional gains from a symmetric favorable shock; volatility is harmful.
      - Note: if the stressed metric is one where larger values represent the adverse case (e.g., change in net debt/GDP), fragility will be represented by H > 0.
    - Terminology:
      - Antifragility: Taleb (2012, forthcoming) defines H > 0 for profit starting point as greater volatility producing more profit; antifragility is not the same as robustness.

- C. How the Simple Heuristic Enhances Stress Tests
  - Heuristic assesses non-linear tail risks and sensitivity of stress-test outcomes to risk-driver changes in a way more robust to model error.
  - Macro stress tests typically compute a small number of scenarios (baseline and few stresses); the heuristic efficiently probes sensitivities without numerous additional scenarios.
  - Distribution of solvency outcomes is usually left-skewed with fat left tail: limited chance of significant capitalization gains, substantial chance of sharp drops.
  - Heuristic can be standardized to measure tail risks and reveal non-linearities as banks or systems approach failure.
  - Mathematical logic (Taleb and Douady 2012):
    - Heuristic reveals tail “fragility” and model error by linking fragility (sensitivity of the tail integral/partial tail expectation below level K) to the convexity (second derivative) of the mapping function  where Y=(x).
    - "Fragility transfer theorem" and "fragility exacerbation theorem" map convexity into increased fragility; detecting convexity is therefore key.
    - Convexity bias (mis-estimation due to Jensen’s inequality) can be obtained by setting K at Infinity to assess convexity effect on total expectation.

### III. THE HEURISTIC APPLIED TO THE OUTCOME OF STRESS TESTS
- A. Purpose for the Use of the Heuristic
  - Macro stress tests usually compute a small number of scenarios (point estimates) and use these to draw policy conclusions; this hides sensitivity to small input changes from decision-makers.
  - Stress-test teams may have intuition about sensitivities, but the heuristic makes these sensitivities explicit for wider audiences.
  - Prominent recent bank stress tests (examples cited) used only a baseline and one adverse scenario, illustrating the limited area of distribution typically explored.
  - Ideally, closed-form loss functions would allow tracing complete loss arcs as function of state variables, but such closed forms are unlikely; heuristic is a practical simplifying alternative.

*Source: _wp12216 - References .............................................................................................................*

### 5.0 ppts

### _wp12216 - 5.0 ppts

### Overview of the Simple Heuristic
- Purpose: summarize fragility of stress-test outcomes by probing non-linear (convex) responses to deviations around a central stress scenario rather than relying on a single pass/fail point estimate.
- Operational implementation: run the original stress test, run two additional scenarios X_t + ΔX and X_t - ΔX where ΔX is an estimated mean deviation (MDev), compute outcomes, and calculate the heuristic as a scalar summary of asymmetry/convexity in the tails.
- Key conceptual point: fragility is defined as stresses bringing disproportionately higher harm as stress increases; the heuristic highlights fragility (positive when a higher outcome is adverse, negative when a lower outcome is adverse).

### Case Study I — Bank Stress Tests (12 large U.S. banks)
- Sample and setup:
  - Sample: 12 large U.S. banks.
  - Balance-sheet baseline: end-2010 balance sheets, complemented with bank data from first half of 2011.
  - Stress horizon: 2012–2016.
  - Central macro stress: near-zero GDP path with a cumulative deviation from the WEO baseline by 10 percentage points during 2012–16; historical likelihood about 4 percent.
  - Outcome metric: Tier 1 capitalization.
  - Model features: feedback loop between banking stress and macro growth; satellite models estimate impacts of GDP on credit losses, credit growth and pre-impairment income; no recapitalization other than retained earnings; deleveraging allowed.
- Heuristic procedure applied:
  - Heuristic computed separately for risk drivers (credit growth, credit losses, trading income), for GDP shocks, and for combined shocks.
  - Mean deviations used in experiments: GDP ±1 MDev = 1.5 percentage points (cumulative), GDP ±2 MDev = 3.0 percentage points (cumulative); credit growth ±1 MDev = 3.4 percentage points, ±2 MDev = 6.8 percentage points; credit losses ±1 MDev = 0.11 percentage points, ±2 MDev = 0.22 percentage points; pre-impairment income/TA ±1 MDev = 0.17 percentage points, ±2 MDev = 0.34 percentage points; trading income/TA ±1 MDev = 0.1 percentage points, ±2 MDev = 0.2 percentage points.
- Illustrative numerical example (Bank 1):
  - Under the stress test, Tier 1 ratio = 1.639 percent.
  - Under additional GDP shocks of +/- one standard deviation outcomes reported as: .2.835 and . 373, respectively.
  - Heuristic computed as: (0.373 +2.835)/2-1.639 = -0.035 (reported heuristic H +/- 1 MDev = -0.035 for Bank 1).
- Aggregate and ranking findings:
  - The tail stress produces non-linear results in the majority of cases; many bank outcomes are fragile with respect to GDP growth, credit growth, credit losses and income, and to trading income as part of total income.
  - Fragility is especially high for banks experiencing the largest drops in Tier 1 capitalization under the central adverse scenario.
  - Some banks display antifragile outcomes (positive H) owing to deleveraging and reductions in RWAs that mitigate stress (examples: positive H for banks 5, 8, 9, and 11 in the GDP shock case).
  - Rank-order comparisons: traditional stress-test ranking and heuristic-based fragility ranking are broadly consistent but with notable exceptions (e.g., Bank 6 flagged as vulnerable by the traditional stress test but classified as less fragile by the heuristic; Banks 3, 4, 7, and 12 are more fragile per the heuristic despite not being top vulnerabilities in level-impact terms).
- Representative numerical heuristic results (selected excerpts from Table 1):
  - Impact of adverse scenario (change in T1 capitalization): Bank 1 -9.60; Bank 2 -13.65; Bank 3 -8.48; Bank 4 -2.82; Bank 5 -8.90; Bank 6 -15.07; Bank 7 -1.61; Bank 8 -0.56; Bank 9 2.27; Bank 10 -18.62; Bank 11 -13.31; Bank 12 -7.63.
  - GDP growth 1 MDev worse (1.5 percentage points, cumulative): Bank 1 -1.27; Bank 2 -2.71; Bank 3 -1.86; Bank 4 -0.80; Bank 5 -0.11; Bank 6 -0.24; Bank 7 -0.71; Bank 8 -0.29; Bank 9 -0.12; Bank 10 -2.01; Bank 11 -1.63; Bank 12 -1.67.
  - H +/- 1 MDev (GDP): Bank 1 -0.035; Bank 2 -0.096; Bank 3 -0.294; Bank 4 -0.019; Bank 5 0.000; Bank 6 -0.001; Bank 7 -0.066; Bank 8 0.007; Bank 9 0.000; Bank 10 -0.080; Bank 11 0.025; Bank 12 -0.053.
  - Credit growth 1 MDev more (3.4 percentage points): Bank 1 -0.99; Bank 2 -1.71; Bank 3 -1.40; Bank 4 -0.62; Bank 5 -0.03; Bank 6 -0.09; Bank 7 -0.46; Bank 8 -0.28; Bank 9 -0.02; Bank 10 -1.61; Bank 11 -1.51; Bank 12 -1.34.
  - H +/- 1 MDev (Credit growth): Bank 1 -0.043; Bank 2 -0.090; Bank 3 -0.269; Bank 4 -0.076; Bank 5 -0.001; Bank 6 -0.003; Bank 7 -0.039; Bank 8 -0.009; Bank 9 -0.001; Bank 10 -0.076; Bank 11 -0.049; Bank 12 -0.061.
  - Pre-impairment income/TA 1 MDev less (0.17 percentage points): Bank 1 -1.72; Bank 2 -4.51; Bank 3 -3.44; Bank 4 -2.06; Bank 5 -0.72; Bank 6 -1.17; Bank 7 -1.76; Bank 8 -1.00; Bank 9 -0.96; Bank 10 -1.33; Bank 11 -1.97; Bank 12 -2.66.
  - H +/- 1 MDev (Pre-impairment income/TA): Bank 1 -0.085; Bank 2 -0.291; Bank 3 -0.718; Bank 4 -0.250; Bank 5 -0.014; Bank 6 -0.144; Bank 7 -0.224; Bank 8 -0.051; Bank 9 0.000; Bank 10 -0.043; Bank 11 0.000; Bank 12 -0.162.
  - All three drivers one MDev in adverse direction (combined impact on T1): Bank 1 -4.34; Bank 2 -10.24; Bank 3 -7.58; Bank 4 -4.45; Bank 5 -0.80; Bank 6 -1.35; Bank 7 -3.86; Bank 8 -1.60; Bank 9 -1.01; Bank 10 -5.63; Bank 11 -5.29; Bank 12 -6.07.
  - H +/- 1 MDev (All three drivers): Bank 1 -0.505; Bank 2 -2.035; Bank 3 -2.098; Bank 4 -1.012; Bank 5 -0.020; Bank 6 -0.162; Bank 7 -0.924; Bank 8 -0.123; Bank 9 -0.003; Bank 10 -0.649; Bank 11 -0.212; Bank 12 -0.922.
  - Trading income/TA shocks produce much smaller H values (examples): H +/- 1 MDev (Trading income): Bank 1 -0.013; Bank 2 -0.063; Bank 3 -0.035; Bank 4 -0.001; Bank 5 -0.012; Bank 6 -0.147; Bank 7 -0.033; Bank 8 0.000; Bank 9 0.000; Bank 10 -0.011; Bank 11 0.000; Bank 12 -0.007.
- Interpretation:
  - Heuristic exposes non-linear fragility not always evident from level-impact tests; can change ordinal vulnerability rankings and reveal asymmetries (greater sensitivity to adverse moves than benign ones).
  - Usefulness: flag banks whose capital is asymmetrically impacted by negative shocks and identify risk drivers for targeted analysis.

### Case Study II — Public Debt Sensitivity to Growth Shocks
- Setup:
  - Central scenario: growth is 2 percentage points less than in the baseline (September 2011 WEO) per year between 2012 and 2016 (near zero real growth for most sample countries).
  - Tail scenarios: growth one or two mean deviations above or below the central scenario; mean deviations estimated over 1981–2010 for each country.
  - Purpose: analyze how underestimating adverse growth shocks could produce disproportionately larger underestimation of countries' debt dynamics (non-linearity).
- Main findings:
  - Impact of tail growth shocks on net debt is non-linear in all cases examined; all outcomes in the sample are fragile.
  - Large negative growth shocks increase net debt (percentage points of GDP) disproportionately more than comparable positive shocks reduce net debt.
  - Asymmetry implies a higher cost to underestimating growth shocks than to overestimating them.
- Representative numerical results from Table 3 (change in net debt, percentage points of GDP; presentation follows source layout):
  - Central shock (2% points less)26.213.520.929.628.134.523.7
  - GDP growth 1 M Dev. less  14.013.116.118.528.939.637.5
  - GDP growth 2 M Dev. less  28.627.133.237.960.283.680.1
  - GDP growth 1 M Dev. more  -13.5-12.2-15.2-17.6-26.7-36.0-33.3
  - GDP growth 2 M Dev. more  -26.4-23.7-29.5-34.3-51.5-68.7-63.1
  - H: +/- 1 M Dev.0.30.40.50.41.11.82.1
  - H: +/- 2 M Dev.1.11.71.91.84.47.48.5
- Interpretation:
  - The positive H values reported indicate increased risk (increase in net debt/GDP) when volatility around the central adverse shock rises.
  - Policy implication: symmetrical stress around a chosen central adverse shock can reveal fragility; underestimation of adverse growth outcomes can lead to materially larger debt outcomes than implied by linear models.

### How to Integrate the Heuristic into IMF Stress Tests (practical steps)
- Stepwise procedure:
  - Step 1: Run stress tests and obtain level results (bank capital ratios, liquidity ratios, NPL ratios, Net interest income, ROA, ROE, public debt, etc.).
  - Step 2: Construct additional scenarios X_t + ΔX and X_t - ΔX where ΔX is an estimated mean deviation of the variable X(t) over the pre-determined time period.
  - Step 3: Compute outcomes for these additional scenarios.
  - Step 4: Compute the heuristic.
  - Step 5: Draw conclusions and optionally re-iterate (either full re-iteration with different comprehensive stresses or limited re-iteration changing Δ’s).
- Diagnostic use:
  - If heuristic indicates fragility to higher volatility (positive when a higher outcome is adverse, negative when a lower outcome is adverse), conclude that greater adverse stress could make outcomes substantially worse and consider expanding scenario exploration.
  - Heuristic supports ordinal comparisons of fragility across risk drivers and entities and is more robust to some types of model error than single-point pass/fail outcomes.

### Policy implications and conclusions
- The heuristic provides a simple scalar measure of convexity/fragility around a central stress that supplements conventional level-based stress-test outputs.
- Practical consequences:
  - For sovereigns: reveal limited room for countercyclical deficits when public finances are fragile to growth shocks.
  - For banks/supervisors: uncover hidden fragilities from illiquid positions, non-linear derivative exposures, or feedback loops between losses and funding costs.
- Recommendation: consider making the heuristic a standard statistic in stress-testing toolkits and use it as a diagnostic to decide when deeper tail analysis is warranted.
- Limitation: heuristic does not eliminate the need for judgment; iterations may be required and no finite iteration can guarantee full characterization of potentially complex functional forms governing outcomes.

*Source: Authors (content unit: _wp12216 - 5.0 ppts).*

### APPENDIX I:  DETAILS ON MACROECONOMIC BANK STRESS TEST

### APPENDIX I:  DETAILS ON MACROECONOMIC BANK STRESS TEST

### Bank-specific credit losses
- Scenario: "Japan-like growth shock (i.e., a lost decade-like scenario)" results in substantial losses in banks’ loan books and on assets subject to counterparty credit risk; counterparty credit risk losses were assumed to change in line with loan loss rates (in relative terms, but on a substantially lower level).  
- Housing shock: an additional “permanent” decrease of house prices by 20 percent is added, leading to LGDs (Loss Given Default, i.e., 1 minus the recovery rate) of 40 percent for retail mortgages.  
- Trading income shock: trading income is projected to yield losses of around 1.5 percent of total assets in 2012 and of around 1 percent in 2013 (worst case trading results during 2012 and 2013, in line with very severe historical cases).

### Credit growth and risk-weighted assets (RWAs)
- Credit growth estimation: through satellite models reflecting GDP growth; credit growth becomes slightly negative, providing banks some breathing space to digest losses.  
- RWA dynamics:
  - RWAs grow in line with total credit exposure (accounting for credit growth on the one hand and losses on the other).  
  - Risk effects (as under the IRB approach) are disregarded because U.S. banks are not under the IRB (yet).  
  - No behavioral adjustment assumed to benefit from lower RWAs (for example, replacing loans by securities) — a conservative assumption over the five year period.  
  - The remainder of the RWAs (i.e., for market risk and credit risk) are held constant.

### Income and retained earnings
- Pre-impairment pre-tax income under the adverse growth scenario:
  - Projected to drop 13 percent compared to 2010 and to remain at that level throughout the projection period, reflecting the quasi-zero growth path.  
  - Note: pre-impairment income includes all sources of operating income other than trading income (i.e., interest income, commission and fee income, etc.).  
- Retention and taxes:
  - Assumed that 60 percent of net income is retained (in line with empirical evidence) if net income remains positive; otherwise retained income (i.e., loss) is fully retained.  
  - A tax rate of 25 percent was applied to all banks.

### Funding costs
- Funding cost response: funding costs assumed to increase conditional on capitalization level during stress — funding costs increase faster as capital falls to low levels, reflecting empirical evidence.  
- Pass-through to customers: banks are assumed to be able to pass on 50 percent of funding cost increases to their customers, given competition pressures.  
- Funding costs introduce a dynamic element to the stress test. Further methodological details are referenced to Schmieder, et. al. (2012).

### Footnotes and methodological notes (as presented)
- The mean deviation has been computed based on the evolution of the median loss rate among all U.S. banks in Bankscope.  
- "20 percent is a common, conservative benchmark for housing loans (many banks use figures around 15 percent)."  
- "The loss for a specific bank will depend on the intensity of its trading business."  
- The 2010 pre-impairment pre-tax income was very close to the average over the last 5 years except for investment banks, where trading income was more important.

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### APPENDIX II: DETAILS ON PUBLIC DEBT STRESS TEST

- Debt dynamics are illustrated by standard debt dynamic equations where:
  - d_t represents the ratio of public debt to GDP.  
  - pb_t is the primary balance.  
  - r_t is the growth adjusted interest rate (a function of the nominal interest rate i and the nominal GDP growth rate g).  
  - The nominal interest rate is derived from the ratio of interest payments during the current year to the end-period stock of debt during the previous year.  
  - The primary balance (pb_t) depends on the primary balance under the baseline (pb_t^base), revenue and expenditure semi-elasticity to changes in the output gap (R_ε, E_ε), and the change in the output gap between the baseline and different scenarios (Δog_t).  
- Scenario assumptions:
  - Growth shocks do not affect potential GDP and governments do not take any discretionary corrective measures to smooth their impacts.  
  - Partial equilibrium simulation that also assumes no change in the nominal interest rate as a result of the growth shock.  
- Key drivers affecting each country’s debt dynamics:
  - Trend growth (particularly important for countries with projected low growth rates during the period 2012–2016).  
  - The size of the initial (pre-shock) stock of public debt (particularly important for highly indebted countries).  
  - The size of automatic stabilizers (matters more in countries with particularly high welfare spending).  
- Revenue and expenditure elasticity data are from Girouard and André (2005); when not available, an elasticity of one is assumed for revenue and zero for expenditures.

*Source: _wp12216 - APPENDIX I:  DETAILS ON MACROECONOMIC BANK STRESS TEST*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12216.pdf_