Appendix I: Details on Macroeconomic Bank Stress Test
Appendix II: Details on Public Debt Stress Test
The following standard debt dynamic equations illustrate the main parameters influencing debt dynamics:
dt represents the ratio of public debt to GDP, pbt is the primary balance, and rt is the growth adjusted interest rate. The growth adjusted interest rate is 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 (pbt) depends on the primary balance under the baseline (pbtbase), revenue and expenditure semi-elasticity to changes in the output gap (εR, εE),31 and the change in the output gap between the baseline and different scenarios (Δogt). All scenarios assume that growth shocks do not affect potential GDP and governments do not take any discretionary corrective measures to smooth their impacts.32 As a consequence, growth shocks affect debt ratios through the size of automatic stabilizers and changes in the GDP base.
Three macroeconomic variables will therefore affect each country’s debt dynamics: trend growth, the size of the initial (pre-shock) stock of public debt, and the size of the automatic stabilizers. Trend growth would particularly matter for countries with projected low growth rates during the period 2012–2016. For these countries, a negative growth shock would lead to a significantly higher build-up of public debt than in high growth countries. The initial stock of public debt would be particularly important for highly indebted countries, notably those that experienced a surge in their debt ratios as a result of the crisis. The size of automatic stabilizers matters more in countries with particularly high welfare spending, as the relationship between tax revenues and economic activity tends not to vary greatly across countries.
Borio, Claudio, Drehmann, Mathias, and Kostas Tsatsaronis, 2012, “Stress-testing macro stress testing: does it live up to expectations?,” BIS Working Paper no.369, January.
Financial Stability Board, 2011, “Understanding Financial Linkages: A Common Data Template for Global Systemically Important Banks: Consultation Paper,” October 6, 2011.
Financial Stability Board Secretariat and IMF, 2009, “The Financial Crisis and Information Gaps: Report to the G-20 Finance Ministers and Central Bank Governors,” October 29, 2009.
Girouard, Nathalie and Christophe André, 2005, “Measuring Cyclically-Adjusted Budget Balances for OECD Countries,” OECD Economics Department Working Paper No. 434 (Paris: Organization for Economic Cooperation and Development).
Ong, Li Lian and M. Cihak, 2010, “On Runes and Sagas: Perspective on Liquidity Stress Testing Using an Iceland Example,” IMF Working Paper 10/156 (Washington: International Monetary Fund).
Schmieder, Christian, Puhr, Claus and Maher Hasan, 2011, “Next Generation Balance Sheet Stress Testing,” IMF Working Paper 11/83 (Washington: International Monetary Fund).
Schmieder, Christian, Hesse, Heiko, Neudorfer, Benjamin, Puhr, Claus and Stefan W. Schmitz, 2012, “Next Generation System-Wide Liquidity Stress Testing,” IMF Working Paper 03/12 (Washington: International Monetary Fund).
Taleb, Nassim N., 2011, “A Map and Simple Heuristic to Detect Fragility, Antifragility, and Model Error”, NYU-Poly working paper, SSRN.
Vitek, Francis, and Tamim Bayoumi, 2011, “Spillovers from the Euro Area sovereign debt crisis: A macroeconometric model based analysis,” CEPR Discussion Paper, 8497.
This paper benefited from comments by Gianni De Nicolo and Christopher Towe.
Arguably, these were not “Black Swan” events as both sub-prime losses and Greek sovereign distress were in principle foreseeable, but in both cases, the magnitude of the outcomes would at least have been regarded as fairly extreme tail events a year or two prior to their full flowering.
These conditions include that loss distributions are monomodal, that the bias in the incorrect model (compared to the true model) does not change signs, and higher differences do not carry opposite signs.
The G-20 has called for “the IMF to investigate, develop, and encourage implementation of standard measures that can provide information on tail risks.” See IMF/FSB (2009), recommendation #3.
Taleb and Douady (2012) develop a theorem that proves how a nonlinear exposure maps into tail-sensitivity to volatility and model error and produce a transfer function expressing fragility as a direct result of nonlinearity.
The most notable example is in Iceland, where stress tests were performed just before its liquidity crisis (Ong and Cihak, 2010). See also Borio, Drehmann and Tsatsaronis (2012) for a critical review of the early warning properties of stress tests.
Given the conceivably virtually unlimited number of dimensions that could be covered by stress tests, scenarios will hardly ever be realized precisely as assumed by the stress tests.
Note that if exposure to an event is negative (e.g., a short position), then the concave payoff structure shown in the diagram would actually become convex, and in fact would be exactly the negative of the concave payoff function.
According to an investigation into the scandal by Société Generale’s own General Inspection department, a €49 billion (US$71 billion) long position on index futures was discovered on January 20 then unwound between January 21 and January 23, leading to gross losses of EUR 6.4 billion. See Mission Green: Summary Report, Société Generale General Inspection Department http://www.societegenerale.com/sites/default/files/documents/Green_VA.pdf.
Traders’ compensation systems generally provide extremely asymmetrical incentives since traders will receive large bonuses for highly risky trades that pay off, but an equivalent amount cannot be clawed back from them should the trade result in large losses, since their salaries will be bounded at zero. Such asymmetric incentives can lead traders to take on more risk than they would if there were a symmetrical incentive scheme.
Note that if the stress test involves something where a larger result represents the adverse case, such as in the change in the net debt/GDP ratio examined in Section III.C, fragility will be represented by H > 0.
Taleb (2012, forthcoming) posits the opposite situation. If we start with profits, and H>0, then greater volatility leads to a more profitable outcome than lower volatility. Such a situation is termed “antifragility.” This is not the same as robustness, since with robustness, higher volatility provides neither significant harm nor benefit.
For example, the most prominent recent bank stress tests, the Supervisory Capital Assessment Program (SCAP) and the subsequent Comprehensive Capital Analysis and Review in the United States, and the two sets of published stress tests conducted by the European Banking Authority (EBA) used only a baseline scenario and one adverse stress scenario.
Of course, ideally, losses would be derived in a closed-form expression that would allow the stress tester to trace out the complete arc of losses as a function of the state variables, but it is exceedingly unlikely that such a closed-form expression could be tractably derived, hence, the need for the simplifying heuristic.
The mathematical logic that (1) the heuristic reveals tail “fragility” and (2) that it also reveals model error is demonstrated in Taleb and Douady (2012) as follows. The definition of fragility below a certain level K is the sensitivity of the tail integral—the partial tail expectation—between minus infinity and K to changes in parameters, particularly the lower mean deviation. By the “fragility transfer theorem”, such sensitivity for a variable Y is caused by the second derivative of the function φ, such that Y=φ(x) hence a direct result of the convexity of such function. By the “fragility exacerbation theorem”, the increase in fragility is mapped as a direct effect of such convexity. So it becomes a matter of detecting the convexity, hence the heuristic. Furthermore, parameter imprecisions in a model are considered as fragilizing if they are capable of causing an increase in the left tail. Taleb and Douady (2012) also shows that the convexity bias, that is the mis-estimation of the effect of Jensen’s inequality can be obtained by setting K at Infinity, to take the effect of the convexity on the total expectation.
More specifically, scenarios with a cumulative deviation from average growth rates by 10 percentage points (independent from when the deviation occurs) have been used to compute the likelihood for the occurrence of the zero-growth scenario.
One of the virtues of the heuristic is that it may reveal non-linearities in the stress testing model, e.g. arising from feedback loops, even when such non-linearities were not explicitly or intentionally built into the stress testing model. This could reveal either a true non-linearity, or a need to refine the model.
GDP affects credit losses, income and credit growth through the satellite models, which makes scenarios 1–4 complementary to the scenarios 17–20.
By way of a numerical example, the heuristic for Bank 1 under the 1 standard deviation GDP shock (-0.035) is computed as follows. Under the stress test, the Tier 1 ratio is 1.639 percent and under additional GDP shocks of +/- one standard deviation is .2.835 and . 373, respectively. Thus, the heuristic is calculated as: (0.373 +2.835)/2-1.639 = -0.035, the same as the computation using the changes of capitalization as shown in Table 1: (-1.27–1.2)/2.
This impact explains the positive H for banks 5, 8, 9, and 11 in the case of the GDP shock.
Under the hypothesis that the impact of a growth shock on debt is linear, the impact should be similar when a growth shock is augmented and reduced by a similar constant.
For example, if the non-linearity arises from traders hiding risks in the tails (as posited in Section II.B.) then a relatively small delta plus and minus around the 5 percent probability level could reveal such a non-linearity.
There will be no obvious procedure to specify when the iterations should stop, since the functional form of the relationship between outcomes and stressors may be almost limitlessly complex. Rather, the iteration of the procedure will be used to expand the stress tester’s knowledge about behavior in the tails, but the stress tester should never have a pretence to complete knowledge of the functional form governing how the outcome will react to different stressors.
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 pre-impairment income includes all sources of operating income other than trading income, i.e., interest income, commission and fee income, etc.
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.
Revenue and expenditure elasticity to the output gap are from Girouard and André (2005). When not available, an elasticity of one is assumed for revenue and zero for expenditures.
This is a partial equilibrium simulation that also assumes no change in the nominal interest rate as a result of the growth shock.