Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Outline 1
Introduction
2
Gaps
3
Measurement
4
Variables and Factors
5
Main Result
6
Empirical Analysis
7
SIR pitfalls
2 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Background I
“Macroprudential” Stress-Testing has become important.
I
Stress-testing goal is financial system resilience with high probability.
3 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Background I
“Macroprudential” Stress-Testing has become important.
I
Stress-testing goal is financial system resilience with high probability.
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. No idiosyncratic risk: other variables are set to their expected values given Z˜ .
3 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Background I
“Macroprudential” Stress-Testing has become important.
I
Stress-testing goal is financial system resilience with high probability.
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. No idiosyncratic risk: other variables are set to their expected values given Z˜ .
2. Each BHC creates 3 BHC-scenarios tailored to its portfolio. 3. Test BHC’s capital adequacy in the scenarios. 4. BHC’s have to take “capital actions” if capital is inadequate.
3 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Background I
“Macroprudential” Stress-Testing has become important.
I
Stress-testing goal is financial system resilience with high probability.
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. No idiosyncratic risk: other variables are set to their expected values given Z˜ .
2. Each BHC creates 3 BHC-scenarios tailored to its portfolio. 3. Test BHC’s capital adequacy in the scenarios. 4. BHC’s have to take “capital actions” if capital is inadequate. I
Question: What regulatory scenarios should we choose to achieve our goals? 1. 2. 3. 4.
Which variables should we stress? In what directions? By how much should variables be stressed? How should idiosyncratic risks be accounted for? 3 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Current Regulatory Stress Scenario Formulation
I
Banking book I I
Regulators create a path for about 30 macro-variables Z˜ . Other variables X set to E (X |Z˜ ).
4 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Current Regulatory Stress Scenario Formulation
I
Banking book I I
I
Regulators create a path for about 30 macro-variables Z˜ . Other variables X set to E (X |Z˜ ).
Trading book I
Paths of many variables ( 20,000) specified by US regulator.
4 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Current Regulatory Stress Scenario Formulation
I
Banking book I I
I
Regulators create a path for about 30 macro-variables Z˜ . Other variables X set to E (X |Z˜ ).
Trading book I
Paths of many variables ( 20,000) specified by US regulator.
4 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Current Regulatory Stress Scenario Formulation
I
Banking book I I
I
Trading book I
I
Regulators create a path for about 30 macro-variables Z˜ . Other variables X set to E (X |Z˜ ).
Paths of many variables ( 20,000) specified by US regulator.
Banks exposures are not formally used to pick the scenarios.
4 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Some Areas for Improvement 1. Choosing Variables to Stress I
Macro-variables weakly explain banks P&L [Guerrieri and Welch (2012)].
5 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
5 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 I
Which of the many possible stress directions should regulators choose? May want to avoid stresses in directions where banks are hedged.
3. Choosing the Magnitude of Stresses I
How severe should scenarios be to achieve systemic risk objective?
5 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 I
Which of the many possible stress directions should regulators choose? May want to avoid stresses in directions where banks are hedged.
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
I
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 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
This Paper I
Stress scenarios designed for capital adequacy and systemic risk.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 ).
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
I
Contributions. I
Scenario choice satisfies an explicit systemic risk objective.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
I
Contributions. I I
Scenario choice satisfies an explicit systemic risk objective. Solved for Stress Direction, Magnitude, and Variable Selection.
8 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
I
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 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
9 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 ).
9 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 .
9 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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.
9 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
I
Bank i’s maximal intermediation capacity = γAi .
11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
I
Bank i’s maximal intermediation capacity = γAi .
I
Bank i’s distress = Di [ωi (XT ) + rf CIi ] ∈ [0, 1].
11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
I
Bank i’s maximal intermediation capacity = γAi .
I
Bank i’s distress = Di [ωi (XT ) + rf CIi ] ∈ [0, 1].
I
Loss of i’s capacity = γAi × Di [ωi (XT ) + rf CIi ]
11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
I
Bank i’s maximal intermediation capacity = γAi .
I
Bank i’s distress = Di [ωi (XT ) + rf CIi ] ∈ [0, 1].
I
Loss of i’s capacity = γAi × Di [ωi (XT ) + rf CIi ]
I
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
11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
I
Bank i’s maximal intermediation capacity = γAi .
I
Bank i’s distress = Di [ωi (XT ) + rf CIi ] ∈ [0, 1].
I
Loss of i’s capacity = γAi × Di [ωi (XT ) + rf CIi ]
I
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
I
Systemic Risk Systemic Impairment Threshold = ζ.
11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Systemic Risk Measure using System Assets in Distress (SAD)
I
Notation: Ai = Assets, CIi = Cap. Inj., XT = Variables , ωi = Exposure.
I
Bank i’s maximal intermediation capacity = γAi .
I
Bank i’s distress = Di [ωi (XT ) + rf CIi ] ∈ [0, 1].
I
Loss of i’s capacity = γAi × Di [ωi (XT ) + rf CIi ]
I
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
I
Systemic Risk Systemic Impairment Threshold = ζ. ψ = Prob(SADT (CI , Ω, XT ) > ζ) is a measure of systemic risk 11 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. Regulators can randomly draw X from its distribution.
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. Regulators can randomly draw X from its distribution. 2. Regulators know banks exposures to X , denoted ωi (X )
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. Regulators can randomly draw X from its distribution. 2. Regulators know banks exposures to X , denoted ωi (X ) P 3. Factors are a linear combination of X : F1,k = βi,k Xi
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
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,k = βi,k Xi Factors can be identified by subset of variables x ∈ X .
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
I
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,k = βi,k Xi Factors can be identified by subset of variables x ∈ X .
Methodology to identify F1 [Sliced Inverse Regression, Li (1991)]. 1. Make N draws of X (a function of F1 and F2 ).
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
I
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,k = βi,k Xi Factors can be identified by subset of variables x ∈ X .
Methodology to identify F1 [Sliced Inverse Regression, Li (1991)]. 1. Make N draws of X (a function of F1 and F2 ). 2. Compute banks losses ωi (X ) (a function of F1 ).
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
I
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,k = βi,k Xi Factors can be identified by subset of variables x ∈ X .
Methodology to identify F1 [Sliced Inverse Regression, Li (1991)]. 1. Make N draws of X (a function of F1 and F2 ). 2. Compute banks losses ωi (X ) (a function of F1 ). 3. Compute SAD(Ω(X )) (a function of F1 ).
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
I
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,k = βi,k Xi Factors can be identified by subset of variables x ∈ X .
Methodology to identify F1 [Sliced Inverse Regression, Li (1991)]. 1. 2. 3. 4.
Make N 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 ). Compute E (X |SAD) and ΣE (X |SAD) (functions of F1 ).
12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
How Regulators Solve for F1 I
Assumptions: 1. 2. 3. 4.
I
Methodology to identify F1 [Sliced Inverse Regression, Li (1991)]. 1. 2. 3. 4.
I
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,k = βi,k Xi Factors can be identified by subset of variables x ∈ X . Make N 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 ). Compute E (X |SAD) and ΣE (X |SAD) (functions of F1 ).
SIR:: Under approp regularity condns the principal components of Σ−1 X ΣE [X |SAD] 1. Span the same spaces as F1 . 2. Are ordered by their ability to explain systemic risk SAD. 3. F1 can be identified even if SAD is nonlinear in F1 . 12 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Variable Selection
I
SIR may in theory require enormous matrices if dimensionality of X is high because it depends on Principal Components of Σ−1 X ΣE [X |SAD]
I
If X is too high dimensional, then SIR is not feasible.
13 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Variable Selection
I
SIR may in theory require enormous matrices if dimensionality of X is high because it depends on Principal Components of Σ−1 X ΣE [X |SAD]
I
If X is too high dimensional, then SIR is not feasible. I
I
Solution: Choose x ∈ X via Correlation Pursuit (COP) (Zhong et al 2012). Methodology: Uses hypotheses tests to identify which variables are best for identifying factors to use in SIR.
13 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Choosing a Stress Scenario
I
Estimate linear statistical relation between all variables X and the factors (F1 ). X = α + F1 θ +
I
Stress-scenario formation steps. 1. Choose F1 realization. 2. Set X = E (X |F1 ) = α + F1 θ 3. SAD in the stress-scenario is SAD[Ω(E (X |F1 ))].
I
Goal: Choose the most plausible F1 for a scenario such that if banks are well capitalized for the scenario, then systemic risk is low.
19 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
Taylor expand SAD in X ωj + CIj rf : X SAD ≈ Const + Dj,1 [X ωj + CIj rf ]
(1)
j
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
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
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
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)
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
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
= I
Const + α + F1 Θ + E + Cap Inj. Equivalent (CIE)
(3)
Estimate H(.), the CDF of random variable F1 Θ + E .
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
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)
I
Estimate H(.), the CDF of random variable F1 Θ + E .
I
Find CIE ∗ such that Prob(SAD ≥ ζ) ≤ ψ.
(3)
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
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)
I
Estimate H(.), the CDF of random variable F1 Θ + E .
I
Find CIE ∗ such that Prob(SAD ≥ ζ) ≤ ψ.
I
Choose F1∗ such that F1∗ Θ = −CIE ∗ − α +
P
j
(3)
Dj .
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SAD Approximation and Main Result I
Linearize banks exposure ωj (X ) = X ωj .
I
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)
I
Estimate H(.), the CDF of random variable F1 Θ + E .
I
Find CIE ∗ such that Prob(SAD ≥ ζ) ≤ ψ.
I
Choose F1∗ such that F1∗ Θ = −CIE ∗ − α +
I
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 .
20 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Can SIR/COP detect the right factors-I ?
I
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)].
I
Identified first 3 principal components (PC) of all yield curve changes.
I
Created bank portfolio that loaded on PC 1,2,or 3.
I
X variables were zero coupon returns over 2 years, and exponentially smoothed quarterly GDP growth and inflation in all 10 countries.
I
Using a different data-sample from same DGP, tested if SIR/COP identifies the PC factors banks loaded on.
I
It did.
22 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
-33
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
23 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Does SIR/COP create the right stress scenarios-I I
Want SAD to be correlated with F1 .
I
Want SAD due to banks losses in stress scenarios based on F1 , to be correlated with true SAD.
I
Setting: I I
6 Banks. Invest in zero coupon bonds of 8 countries (AU,CA,GE,JP,SWE,CHF,GB,US), 83 variables.
24 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Does SIR/COP create the right stress scenarios-I I
Want SAD to be correlated with F1 .
I
Want SAD due to banks losses in stress scenarios based on F1 , to be correlated with true SAD.
I
Setting: I I
I
6 Banks. Invest in zero coupon bonds of 8 countries (AU,CA,GE,JP,SWE,CHF,GB,US), 83 variables. Maturities to 30 years.
24 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Does SIR/COP create the right stress scenarios-I I
Want SAD to be correlated with F1 .
I
Want SAD due to banks losses in stress scenarios based on F1 , to be correlated with true SAD.
I
Setting: I I
I I
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
Bond and FX returns are monthly. Data from February 2000 to October 2013 = 165 observations.
24 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Does SIR/COP create the right stress scenarios-I I
Want SAD to be correlated with F1 .
I
Want SAD due to banks losses in stress scenarios based on F1 , to be correlated with true SAD.
I
Setting: I I
I I
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
I
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. 24 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
28 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
True SAD(X ) vs SAD based on losses in stress scenario.
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
30 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
32 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Do stress-tests and capital injections based on ASAD achieve goal of low SAD with high probability
I
No.
33 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Do stress-tests and capital injections based on ASAD achieve goal of low SAD with high probability
I
No. Choosing the magnitude of F1 based on the linear approximation of SAD (ASAD) guarantees ASAD is low with hig probability.
I
But, it does not guarantee SAD will be low with high probability.
33 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Do stress-tests and capital injections based on ASAD achieve goal of low SAD with high probability
I
No. Choosing the magnitude of F1 based on the linear approximation of SAD (ASAD) guarantees ASAD is low with hig probability.
I
But, it does not guarantee SAD will be low with high probability.
I
Better to use ASAD to find directions to change F1 , and then solve for magnitude of F1 changes to satisfy systemic risk objectives.
I
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.
33 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Examples of Factors Chosen by SIR
I
6 banks with only interest-rate risk positions.
I
6 banks with portfolios split 50% in interest rate exposures and 50% in stock market exposures.
I
The figures will illustrate how one-standard deviation movements in the identified factors affect the X variables.
I
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.
34 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
35 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
36 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SIR Pitfall: Symmetry
I
SIR can have difficulty detecting factors when SAD is a symmetric function of X, or of the factors. Example 1: When SAD = X 2 , then E (X |SAD) = 0. In this case, SIR has trouble detecting how SAD is related to X . Example 2: If there are 6 large banks exposed to a single factor, and 3 are long the factor, and 3 are symmetrically short, SIR has trouble identifying the factor.
I
Solution: using scatter plots of simulated P&L for the banks, compute SAD using P&L from positively or negatively correlated banks only, and identify F1 from that.
40 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
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
41 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
SIR Simulations with occasional symmetry 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
42 / 46
Introduction
Gaps
Measurement
Variables and Factors
Main Result
Empirical Analysis
SIR pitfalls
Conclusions
I
Presented A New Approach for choosing stress-scenarios.
I
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.
I
Very preliminary results appear promising.
I
More work is needed.
46 / 46
