Introduction
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Choosing Stress Scenarios for Systemic Risk Through Dimension Reduction Matt Pritsker Federal Reserve Bank of Boston
October 2015 Conference on Stress Testing and Macro-prudential Regulation: A Trans-Atlantic Assessment The views in this presentation are those of the author and not necessarily those of the Federal Reserve Board, Federal Reserve Bank of Boston, or others in the Federal Reserve System. 1 / 45
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Outline 1
Introduction
2
Gaps
3
Measurement
4
Variables and Factors
5
Main Result
6
Empirical Analysis
7
SIR pitfalls
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Main Result
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Background I
“Macroprudential” Stress-Testing has become important: SCAP, CCAR, ECB, PRA.
I
Basic U.S. methodology: 1. Create 3 common regulatory scenarios. For each scenario: I I
Generate a path Z˜ for a small number of macro-variables. Other systematic and idiosyncratic variables that affect banks are set to their expected values given Z˜ .
2. Each BHC creates 3 BHC-scenarios tailored to its portfolio. 3. Ensure each BHC’s capital is adequate in the scenarios. 4. BHC’s have to take “capital actions” if capital is inadequate. I
Issue Addressed in this Paper: 1. Stress-testing goal is financial system resilience with high probability. 2. How can stress scenario selection help accomplish this goal? 2.1 2.2 2.3 2.4
What variables should we stress? In what directions? By how much should variables be stressed? How should idiosyncratic risk be accounted for?
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Current Regulatory Stress Scenario Formulation
I
Banking book I I
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Trading book I
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Regulators create a path for about 30 macro-variables Z˜ . Banks set other variables X = E (X |Z˜ ).
Paths of many variables (30,000) specified by US regulator.
Banks exposures are not formally used to pick the scenarios.
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Some Areas for Improvement 1. Choosing Variables to Stress I
Macro-variables weakly explain banks P&L [Guerrieri and Welch (2012)].
2. Choosing Directions to Stress Variables I
Which of the many possible stress directions should regulators choose?
3. Choosing the Magnitude of Stresses I
How severe should scenarios be to achieve systemic risk objective?
4. Choosing scenarios for systemic risk I
I
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Regulatory scenarios are not chosen to satisfy an explicit systemic risk objective. Regulatory scenarios do not use banks exposures to shared vulnerabilities in scenario design. Bank-tailored scenarios do not focus on banks’ shared vulnerabilities. 5 / 45
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This Paper I
Stress scenarios designed for capital adequacy and systemic risk.
I
Main features of approach 1. There is a systemic risk objective function. 2. Banks exposures to many variables X used in scenario design. 3. Variable selection: statistics identify which variables x ∈ X are important. 4. Dimension reduction: identify systemic risk factors F1 that depend on x. 5. A stress scenarios is F˜1 and X (F˜1 ) = E (X |F˜1 ). 6. Main result: The stress sceneario is chosen so that if banks are well capitalized for it, then an approximation of systemic risk is low.
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Contributions. I I I
Scenario choice satisfies an explicit systemic risk objective. Solved for Stress Direction, Magnitude, and Variable Selection. Scenario choice accounts for idiosyncratic risk. 8 / 45
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Empirical Analysis
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Main Ideas 1. Although banks are exposed to many variables X , these depend on a smaller set of factors F = {F1 , F2 }. At a bank-level the factors are important. 2. Banks hedge some factors (F2 ) while remaining exposed to others (F1 ). 3. Systemic risk is the risk of banks experiencing joint financial distress. This can be caused by directionally similar exposures to the F1 factors. This suggests stress scenarios should be based on movement in F1 . 4. Solution Approach: Solve for scenario F˜1 and X (F˜1 ) such that if banks hold enough capital to cover losses, then for other plausible scenarios banks joint distress is low, i.e. systemic risk is low with high probability. I
Roadmap. I I I
Systemic Risk Measurement. Methodology to identify F1 . Empirical Examples. 9 / 45
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Systemic Risk Measure using System Assets in Distress (SAD)
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Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
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Bank i’s maximal intermediation capacity = γAi .
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Bank i’s distress = Di [ωi (XT ) + rf CIi ] ∈ [0, 1].
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Loss of i’s capacity = γAi × Di [ωi (XT ) + rf CIi ]
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Percentage of economy’s intermediation capacity lost: P γAi × Di [ωi (XT ) + rf CIi ] X P = wi Di [ωi (XT ) + rf CIi ] SADT (CI , Ω, XT ) = i i γAi i
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Systemic Risk Systemic Impairment Threshold = ζ. ψ = Prob(SADT (CI , Ω, XT ) > ζ) is a measure of systemic risk 11 / 45
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How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
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Methodology to identify F [Sliced Inverse Regression, Li (1991)]. 1. 2. 3. 4. 5.
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Regulators can randomly draw X from its distribution. Regulators know banks exposures to X , denoted ωi (X ) P Factors are a linear combination of X : F1,j = βi,j Xi Factors can be identified by x ∈ X . Make draws of X (a function of F1 and F2 ). Compute banks losses ωi (X ) (a function of F1 ). Compute SAD(Ω(X )) (a function of F1 ) = SAD(F1 ). Compute E (X |SAD(F1 )). (a function of F1 ). SIR:: Under appropriate regularity conditions the principal components of Σ−1 X ΣE [X |SAD(F1 )] span the same space as F1 .
Variable Selection. (Correlation Pursuit, Zhong et al 2012) 1. Problem: If X is high dimensional, then SIR is not feasible. 2. Solution: Use COP to choose x ∈ X for SIR. 12 / 45
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Choosing a Stress Scenario
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Estimate linear statistical relation between all variables X and the factors (F1 ). X = α + F1 θ +
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Stress-scenario formation steps. 1. Choose F1 realizations. 2. Set X = E (X |F1 ) = α + F1 θ 3. SAD in the stress-scenario is SAD[Ω(E (X |F1 ))].
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Goal: Choose the most plausible F1 for a scenario such that if banks are well capitalized for the scenario, then systemic risk is low.
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SAD Approximation and Main Result I
Assume linear ωj (X ) = X ωj .
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Taylor expand SAD in X ωj + CIj rf : X SAD ≈ Const + Dj,1 [X ωj + CIj rf ]
(1)
j
=
Const +
X
Dj,1 [(α + F1 θ + )ωj + CIj rf ]
(2)
j
=
Const + α + F1 Θ + E + Cap Inj. Equivalent (CIE)
(3)
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Estimate H(.), the CDF of random variable F1 Θ + E .
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Find CIE ∗ such that Prob(SAD ≥ ζ) ≤ ψ.
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Choose F1∗ such that F1∗ Θ = −CIE ∗ − α +
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Main Result: If stress scenario is X = α + F1∗ θ, equivalent capital injected will be approx CIE ∗ , and Prob(SAD ≤ ζ) ≈≤ ψ.
P
j
Dj .
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Can SIR/COP detect the right factors-I ?
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10 countries yield curve changes (AU,CA,CHF,GE,JP,NO,NZ,SWE,UK,US) over a 2-yr horizon simulated based on a dynamic macro term-structure model [J. Wright (2011)].
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Identified first 3 principal components (PC) of all yield curve changes.
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Created bank portfolio that loaded on PC 1,2,or 3.
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X variables were zero coupon returns over 2 years, and exponentially smoothed quarterly GDP growth and inflation in all 10 countries.
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Using a different data-sample from same DGP, tested if SIR/COP identifies the PC factors banks loaded on.
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It did.
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Can SIR/COP detect the right factors-II ?
Exposure: PCA1
-31
Exposure: PCA2
26
corr=0.99
Exposure: PCA3
-11
corr=-0.91
-32
corr=0.98
25
-12
24
-13
23
-14
-34
F-cop
F-cop
F-cop
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22
-15
21
-16
20
-17
-35
-36
-37 -0.04
-0.03
-0.02
-0.01
True F
0
0.01
0.02
0.03
19 -0.15
-0.1
-0.05
0
True F
0.05
0.1
-18 -0.2
-0.15
-0.1
-0.05
True F
0
0.05
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Does SIR/COP create the right stress scenarios-I I
Want SAD to be correlated with F1 .
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Want SAD due to banks losses in the stress scenario, SAD(Ω[E (X |F1 )]) to be correlated with true SAD.
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Setting: I I
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6 Banks. Invest in zero coupon bonds of 8 countries (AU,CA,GE,JP,SWE,CHF,GB,US), 83 variables. Maturities to 30 years. Bond return distn from historical simulation: I I
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Bond and FX returns are monthly. Data from February 2000 to October 2013 = 165 observations.
Random portfolios: I I I
Some with no FX risk. Some with FX risk. Portfolios differ in pricing approxns and generation methods too. 23 / 45
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Scatter Plot and Kernel Reg SAD(X) vs F1 .
0.4 DV01 Kernel
0.35
0.3
DV01 / Kernel SAD
0.25
0.2
0.15
0.1
0.05
0 -3
-2
-1
0 Factor 1
1
2
3
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Scatter Plot SAD(X ) vs SAD[Ω(E (X |F1 )].
0.4 DV01 S.T.
0.35
0.3
DV01 / ST SAD
0.25
0.2
0.15
0.1
0.05
0 -3
-2
-1
0 Factor 1
1
2
3
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SAD(X ) vs Kernel Reg and SAD[Ω(E (X |F1 ))]. Sim. 1-10. / No FX risk
Simulation 2
Simulation 4
Simulation 5 0.45
0.35
0.4
0.4
0.4
0.35
0.35
0.1
0 -5
0 Factor 1
5
0.25 0.2 0.15
DV01 / ST / Kernel SAD
0.2 0.15 0.1 0.05 0 -5
0 Factor 1
0.25 0.2 0.15
5
0.3 0.25 0.2 0.15
0.3 0.25 0.2 0.15
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0 -5
0 Factor 1
5
0 -5
Simulation 7
0.35
0.25
0.3
0.05
Simulation 6 (k=2) 0.4
0.3
0.35
0 Factor 1
5
0 -5
Simulation 8 (k=2)
0 Factor 1
5
0 -5
Simulation 9
0.5
0.5
0.5
0.45
0.45
0.45
0.4
0.4
0.4
0.4
0.35 0.3 0.25 0.2 0.15
0.35 0.3 0.25 0.2 0.15
0.35 0.3 0.25 0.2 0.15
0.3
0.2 0.15
0.1
0.1
0.1
0.05
0.05
0.05
5
0 -5
0 Factor 1
5
0 -5
0 Factor 1
5
DV01 S.T. Kernel
0.25
0.1
0 Factor 1
5
0.35
0.05 0 -5
0 Factor 1
Simulation 10 (k=2)
0.5 0.45
DV01 / ST / Kernel SAD
0.15
0.3
DV01 / ST / Kernel SAD
0.2
DV01 / ST / Kernel SAD
0.25
0.35
DV01 / ST / Kernel SAD
0.3
DV01 / ST / Kernel SAD
0.45
0.45
DV01 / ST / Kernel SAD
0.5
0.4
0.05
DV01 / ST / Kernel SAD
Simulation 3
0.45
DV01 / ST / Kernel SAD
DV01 / ST / Kernel SAD
Simulation 1 0.4
0 -5
0 Factor 1
5
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How do stress-tests based on ASAD match up with SAD
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Choosing the magnitude of F1 based on the linear approximation of SAD (ASAD) guarantees ASAD is low with hig probability.
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But, it does not guarantee SAD will be low with high probability.
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Better to use ASAD to find directions to change F1 , and then solve for magnitude of F1 changes to satisfy systemic risk objectives.
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When multiple F˜1 choices satisfy the objective, F˜1 can be chosen based on additional criteria such as plausibility and minimization of capital costs.
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Examples of Factors Chosen by SIR
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6 banks with only interest-rate risk positions.
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6 banks with portfolios split 50% in interest rate exposures and 50% in stock market exposures.
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The figures will illustrate how one-standard deviation movements in the identified factors affect the X variables.
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The main point is the identified factors and consequent stresses are portfolio dependent. If banks alter their asset holdings, then the stress scenarios we apply to them should change.
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Factor shocks for random bond portfolio
AUD
−2 −4 −6 −8 5
Estimated yield curve shift, bps
Estimated yield curve shift, bps
−11
−10 −11 −12 −13 0
5
maturity
10
Estimated yield curve shift, bps
−10 −12 −14
5
maturity
10
15
maturity
10
−8
5
maturity
−6 −7 −8
0
5
10
15
maturity
10
15
10
15
JPY
0
−7
0
−5
−9
15
GBP
−0.5 −1 −1.5 −2 −2.5
0
5
maturity
USD
−8
−8
0
5
−6
−9
15
SEK
−6
0
−5
−9
−16
−9
−12
15
−8
−14
−8
−10
EUR
−7
Estimated yield curve shift, bps
maturity
10
−7
Estimated yield curve shift, bps
0
CHF
−4
Estimated yield curve shift, bps
0
−10
CAD
−6
Estimated yield curve shift, bps
Estimated yield curve shift, bps
2
−10 −12 −14 −16 −18 −20
0
5
10
15 maturity
20
25
30
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Factor shocks for random bond and stock portfolio
AUD
−6 −8 −10 −12
−5
−10
0
5
maturity
10
Estimated yield curve shift,bps
−4 −6 −8 −10 0
5
maturity
5
10
15
maturity
10
−2 −4 −6 −8 −10
15
GBP
−4 −6 −8 −10 0
5
maturity
0
5
10
maturity
10
15
10
15
JPY
0.5
0
0 −0.5 −1 −1.5 −2 −2.5 −3
15
USD
5
−2
−12
0
−2
−12
15
SEK
0
−10
2
0
−15
−8
−12
15
Estimated yield curve shift,bps
Estimated yield curve shift,bps
10
EUR
5
Estimated yield curve shift,bps
maturity
−6
Estimated yield curve shift,bps
5
−4
0
5
maturity Stock
2 1
0
Estimated Return, %
0
CHF
0
Estimated yield curve shift,bps
−4
−14
CAD
−2
Estimated yield curve shift,bps
Estimated yield curve shift,bps
0 −2
−5
−10
0 Factor 1 Factor 2
−1 −2 −3 −4
−15
0
5
10
15 maturity
20
25
30
−5 AUD
CAD
CHF
EUR GBP Major Stock Index
JPY
SEK
USD
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SIR Pitfall: Symmetry
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SIR can have difficulty detecting factors when SAD is a symmetric function of X, or of the factors. Example: When SAD = X 2 , then E (X |SAD) = 0. In this case, SIR has trouble detecting how SAD is related to X .
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In our use of SAD, identifying factors using subsets of banks breaks the symmetry and helps identify factors.
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SIR Simulations with occasional symmetry
Simulation 1
0.5 0.45
Simulation 2
0.5 NS Kernel ST
0.45
0.4 0.35
Simulation 3
0.5 NS Kernel ST
Simulation 4
0.45 NS Kernel ST
0.45
0.4
0.4
0.4
0.35
0.35
0.35
0.3
Simulation 5
0.45 NS Kernel ST
NS Kernel ST
0.4 0.35 0.3
0.2 0.15
−2
0 Factor 1
2
Simulation 6
0.55
SAD
0.25
0.25
0.25
0.2
0.2
0.2
0.2
0.15
0.15
0.45 NS Kernel ST
0.5
0.3
0.25
0.1 −4
4
SAD
0.3
0.15
0.1 0.05 −4
SAD
SAD
SAD
0.3 0.25
0.15
−2
0 Factor 1
2
0.1 −4
4
Simulation 7 (k=2)
0.5 NS Kernel ST
0.4
0.1
−2
0 Factor 1
2
0.05 −4
4
Simulation 8 (k=2)
0.4 NS Kernel ST
0.45
0.1
−2
0 Factor 1
2
4
0.05 −4
Simulation 9 (k=2)
0.7 NS Kernel ST
0.35
−2
0 Factor 1
2
4
Simulation 10 NS Kernel ST
0.6
0.45 0.4
0.3
0.35
0.25
0.25 0.2
0.5 0.4
SAD
SAD
SAD
SAD
0.3
0.3 0.25
0.3
SAD
0.35 0.4 0.35
0.2
0.3
0.15
0.2
0.25
0.2 0.15
0.2
0.1
0.15
0.15 0.1 0.05 −4
−2
0 Factor 1
2
4
0.05 −4
−2
0 Factor 1
2
4
0.1 −5
0.1
0 Factor 1
5
0.05 −4
0.1
−2
0 Factor 1
2
4
0 −2
0
Factor 1
2
4
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SIR Simulations with occasional symmetry corrected Simulation 2
0.5
NS Kernel ST
0.45 0.4
SAD
0.35 0.3 0.25 0.2 0.15 0.1 −2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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Conclusions
I
Presented A New Approach for choosing stress-scenarios.
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Contributions: 1. Stress-scenarios are chosen so that resulting capital requirements keep systemic risk low with high probability. 2. Variables for stress-testing are selected based on their ability to explain systemic risk. 3. Stress factors are created based on their ability to explain systemic risk. 4. Systemic risk scenarios are created from the factors. This is a natural wa to choose stress-directions.
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Very preliminary results appear promising.
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More work is needed.
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