Measurement of Economic Tail Risk - University of Waterloo · PDF fileMeasurement of Economic...

Measurement of Economic Tail Risk

Steven Kou1,2 Xianhua Peng3

1Department of MathematicsNational University of Singapore

2Department of Industrial Engineering and Operations ResearchColumbia University

3Department of MathematicsHong Kong University of Science and Technology

S. Kou & X. Peng (NUS, CU, HKUST) Measurement of Economic Tail Risk 1 / 42

Outline

1 Introduction

2 Risk Measures with Decision Theoretical Foundation

3 Elicitablility of Risk Measures

4 Main Result

5 Robustness of Risk Measures

6 Application: Basel Accord Capital Rule


Outline

1 Introduction



4 Main Result




Measuring Tail Risk

Measuring tail risk is an important issue

For example, regulators require banks to hold certain amount ofcapital based on measurement of the tail of their loss distributions

Basel accord is concerned with measuring tail risk at high confidencelevel, e.g., 99.9% (Gordy, 2003; Basel Committee for BankingRegulation, 2012)


Measuring Risk Based on Economic Theory of Risk

Preference



Preference

Utility theory in economics are based on axioms that describe agents’risk preferences



Preference


If X is loss, then −X is gain



Preference


If X is loss, then −X is gain

We propose a new class of risk measures based on axioms that aremotivated from axioms for risk preferences, especially the Choquetexpected utility preference (Schmeidler, 1989)


Risk Measurement and Model Uncertainty

A law invariant risk measure ρ maps a distribution F to a numberρ(F )




In practice, the true F is unknown and one has to take into accountmodel uncertainty




In practice, the true F is unknown and one has to take into accountmodel uncertainty

“Economic models with explicit stochastic structures imply formalprobability statements for a variety of questions related toimplications and policy. In addition, uncertainty can come fromlimited data, unknown models and misspecification of those models”(Hansen, 2013)



The theory of elicitability (Gneiting, 2011, JASA) studies how toevaluate different forecasts for ρ(F ) when there is model uncertainty,i.e., when the true F is unknown



The theory of elicitability (Gneiting, 2011, JASA) studies how toevaluate different forecasts for ρ(F ) when there is model uncertainty,i.e., when the true F is unknown

We will identify those risk measures that are elicitable within ourproposed new class of risk measures


Model Uncertainty and Hampel Robustness

Due to model uncertainty, an external risk measure ρ should berobust with respect to model misspecification (Kou, Peng, and Heyde,2013, MOR)




Hampel robustness (1971): if the model F does not deviate muchfrom the true distribution F (in terms of Prokhorov distance), thenρ(F ) should be close to ρ(F )




Hampel robustness (1971): if the model F does not deviate muchfrom the true distribution F (in terms of Prokhorov distance), thenρ(F ) should be close to ρ(F )

We will study the robustness of risk measures


Outline

1 Introduction



4 Main Result




Risk Preference and Risk Measures

Let X1 and X2 denote the random losses of two portfolios,respectively; then −X1 and −X2 are the P&L of the two portfolios




Utility theory is about how to represent risk preference by utilityfunctional:

−X1 ≻ −X2 ⇔ U(−X1) ≥ U(−X2)




Utility theory is about how to represent risk preference by utilityfunctional:

−X1 ≻ −X2 ⇔ U(−X1) ≥ U(−X2)

The axioms for risk preferences motivate us to propose axioms for riskmeasures


Schmeidler (1989) Risk Preference

The von Neumann-Morgenstern independence axiom of the expectedutility theory:If f ≻ g , then αf + (1− α)h ≻ αg + (1− α)h, ∀f , g , h.




The independence axiom is inconsistent with various experiments,such as the Ellsberg paradox




The independence axiom is inconsistent with various experiments,such as the Ellsberg paradox

Schmeidler (Econometrica, 1989) propose the Choquet expectedutility preference: the independence axiom is replaced by thecomonotonic independence axiomIf f ≻ g and f , g , h are comonotonic, then αf + (1− α)h ≻αg + (1− α)h.


Risk Measures Based on Schmeidler (1989) Risk Preference

Axiom A1. Comonotonic independence: for all pairwise comonotonicrandom variables X ,Y , and Z and for all α ∈ (0, 1), ρ(X ) < ρ(Y )implies that ρ(αX + (1− α)Z ) < ρ(αY + (1− α)Z ).




Axiom A2. Monotonicity: ρ(X ) ≤ ρ(Y ), if X ≤ Y .





Axiom A3. Standardization: ρ(x · 1Ω) = x , for all x ∈ R.






Axiom A4. Law invariance: for all X and Y , ρ(X ) = ρ(Y ) if X andY have the same distribution.






Axiom A4. Law invariance: for all X and Y , ρ(X ) = ρ(Y ) if X andY have the same distribution.

The first two axioms are the axioms for the Choquet expected utility(Schmeidler, 1989) risk preferences; the last two axioms are standard for alaw invariant risk measure.


Difference from Existing Axioms

Existing class of risk measures are based on mathematical axioms insteadof axioms for risk preferences

Subadditivity axiom (Huber, 1981; Artzner, et al., 1999) is based onthe intuition that “a merger does not create extra risk”

Convexity axiom (Folmer and Shield, 2002; Frittelli and Gianin, 2002)

Comonotonic additivity (Wang, Young, and Panjer, 1997)

Comonotonic subadditivity (Kou, Peng, Heyde, 2013; Song and Yan,2009)

Comonotonic convexity (Song and Yan, 2009)


Representation

Proposition

A risk measure ρ : L∞(P) → R satisfies axioms A1-A4 if and only if thereexists a distortion function h : [0, 1] → [0, 1] (h is increasing, h(0) = 0,and h(1) = 1) such that

ρ(X ) =∫

X d(h P)

=∫ 0

−∞(h(P(X > x))− 1)dx +

∫ ∞

0h(P(X > x))dx , ∀X ∈ L∞(P).


Representation

Proposition

A risk measure ρ : L∞(P) → R satisfies axioms A1-A4 if and only if thereexists a distortion function h : [0, 1] → [0, 1] (h is increasing, h(0) = 0,and h(1) = 1) such that

ρ(X ) =∫

X d(h P)

=∫ 0

−∞(h(P(X > x))− 1)dx +

∫ ∞

0h(P(X > x))dx , ∀X ∈ L∞(P).

The main difference between this representation and that of Choquetexpected utility (Schmeidler, 1989) risk preference: the integration here iswith respect to a distorted probability ν(A) := h(P(A)), ∀A; while theintegration in Schmeidler (1989) is with respect to a general non-additiveprobability ν which may not be a distorted probability.


Example: Value-at-Risk (VaR)

If h(x) = 1x>1−α, then ρ(X ) = VaRα(X ).


Examples: Expected Shortfall (ES)

If

h(x) =

x

1−α , x ≤ 1− α,

1, x ≥ 1− α,

then ρ(X ) = ESα(X ), expected shortfall (ES) at level α.


Examples: Expected Shortfall (ES)

If

h(x) =

x

1−α , x ≤ 1− α,

1, x ≥ 1− α,

then ρ(X ) = ESα(X ), expected shortfall (ES) at level α.

ES at level α is defined as:

ESα := mean(Fα,L),

where Fα,L(·) is the α-tail distribution function of L (Rockafellar andUryasev, 2002):

Fα,L(x) :=

0, for x < VaRα(L)FL(x)−α

1−α for x ≥ VaRα(L).

Fα,L is a slight modification of the tail conditional distribution ofL|L ≥ VaRα(L).


Example: Median Shortfall (MS)

If h(x) = 1x> 1−α2 , then ρ(X ) = MSα(X ), median shortfall (MS) at level

α.




α.

Median shortfall (MS) at level α of L is defined as (Kou, Peng, andHeyde, 2013, MOR):

MSα(L) := median of the α-tail distribution of L

= median(Fα,L)




α.

Median shortfall (MS) at level α of L is defined as (Kou, Peng, andHeyde, 2013, MOR):

MSα(L) := median of the α-tail distribution of L

= median(Fα,L)

MSα(L) measures the average size of the conditional loss distributionL|L ≥ VaRα by median instead of mean.


Example: Trimmed Average VaR

Let 0 < α < β < 1, e.g., α = 99%, β = 99.9%. If

h(x) =

0, x ≤ 1− β,x−(1−β)

β−α , 1− β < x ≤ 1− α,

1, x > 1− α,

then ρ(X ) = TAVα,β(X ), the trimmed average VaR (TAV) at level α andβ.


Example: Trimmed Average VaR

Let 0 < α < β < 1, e.g., α = 99%, β = 99.9%. If

h(x) =

0, x ≤ 1− β,x−(1−β)

β−α , 1− β < x ≤ 1− α,

1, x > 1− α,

then ρ(X ) = TAVα,β(X ), the trimmed average VaR (TAV) at level α andβ.

Trimmed average VaR (Kou, Peng, Heyde, 2013, MOR)

TAVα,β(X ) :=1

β − α

∫ β

αF−1X

(u)du.


Outline

1 Introduction



4 Main Result




Forecasting Objective Function and Best Forecasting

Suppose one wants to forecast the realization of a random variable Yusing x




The forecasting error is measured by

ES(x ,Y ) =∫

S(x , y )dF (y ),

whereS(x , y ) : R

2 → R

is the forecasting objective function.




The forecasting error is measured by

ES(x ,Y ) =∫

S(x , y )dF (y ),

whereS(x , y ) : R

2 → R

is the forecasting objective function.

The best forecast corresponding to the forecasting objective functionS is

x∗(F ) = argminx

∫

S(x , y )dF (y )


Forecasting Objective Function and Best Forecast

For example,

For S(x , y ) = (x − y )2, the best forecast is given by

x∗(F ) = mean(F ) = argminx

∫

S(x , y )dF (y ).


Forecasting Objective Function and Best Forecast

For example,

For S(x , y ) = (x − y )2, the best forecast is given by

x∗(F ) = mean(F ) = argminx

∫

S(x , y )dF (y ).

For S(x , y ) = |x − y |, the best forecast is given by

x∗(F ) = median(F ) = argminx

∫

S(x , y )dF (y ).


Elicitability of a Statistical Functional

A statistical functional ρ maps a distribution F to a number ρ(F )




Suppose we want to forecast ρ(F ) but do not know the true F





We can use different models for F and compute ρ under thesemodels: which model gives the more accurate forecast for ρ(F )?





We can use different models for F and compute ρ under thesemodels: which model gives the more accurate forecast for ρ(F )?

A statistical functional ρ is elicitable if one can find a forecastingobjective function S such that for any F , the best forecast withrespect to S is exactly ρ(F ) (Gneiting, JASA, 2011).


Elicitability of a Single-valued Statistical Functional

Definition

A single-valued statistical functional ρ(·) is elicitable with respect to aclass of distributions F if there exists a forecasting error function S(x , y )such that

ρ(F ) = minx | x ∈ argminx

∫

S(x , y )dF (y ), ∀F ∈ F .

ρ is elicitable means that one can find a forecasting objective functionS such that, for any F , the minimization of the forecasting errorcorresponding to S elicits ρ(F ).


Elicitability of a Single-valued Statistical Functional

Definition

A single-valued statistical functional ρ(·) is elicitable with respect to aclass of distributions F if there exists a forecasting error function S(x , y )such that

ρ(F ) = minx | x ∈ argminx

∫

S(x , y )dF (y ), ∀F ∈ F .

ρ is elicitable means that one can find a forecasting objective functionS such that, for any F , the minimization of the forecasting errorcorresponding to S elicits ρ(F ).

If ρ is not elicitable, then one cannot find such a forecasting objectivefunction S .


MS Is Elicitable But ES Is Not

Proposition

(i) For any α ∈ (0, 1), MSα (as a single-valued statistical functional) iselicitable, and the corresponding forecasting error function is

Sα(x , y ) =

1+α2 |x − y |, x ≤ y ,

1−α2 |x − y |, x ≥ y .

(ii) For any α ∈ (0, 1), ESα is not elicitable (Gneiting, 2011, JASA).


MS Is Elicitable But ES Is Not

Proposition

(i) For any α ∈ (0, 1), MSα (as a single-valued statistical functional) iselicitable, and the corresponding forecasting error function is

Sα(x , y ) =

1+α2 |x − y |, x ≤ y ,

1−α2 |x − y |, x ≥ y .

(ii) For any α ∈ (0, 1), ESα is not elicitable (Gneiting, 2011, JASA).

The non-elicitability of ES “may challenge the use of the ES as apredictive measure of risk, and may provide a partial explanation forthe lack of literature on the evaluation of ES forecasts.” (Gneiting,2011, JASA)


Outline

1 Introduction



4 Main Result




Elicitability of Risk Measures Based on Choquet Expected

Utility

Theorem

Let F be the set of all distributions with compact support on R. Let ρ(·)be a risk measure that satisfies the axioms A1-A4. Then, ρ(·) is elicitablewith respect to F if and only if either one of the following two cases holds:



Utility

Theorem


(i) There exists α ∈ (0, 1] such that ρ(F ) = MSα(F ), ∀F .



Utility

Theorem


(i) There exists α ∈ (0, 1] such that ρ(F ) = MSα(F ), ∀F .

(ii) ρ(F ) = mean(F ), ∀F .


Sketch of Proof

First, we show that the necessary condition for a statistical functionalρ to be elicitable is that ρ has convex level sets, i.e., ρ(F1) = ρ(F2)implies that ρ(F1) = ρ(λF1 + (1− λ)F2), ∀λ ∈ (0, 1).


Sketch of Proof


The key step is to show that only three kinds of statistical functionalshave convex level sets


Sketch of Proof



MSα, α ∈ (0, 1].


Sketch of Proof



MSα, α ∈ (0, 1].Mean functional ρ(F ) = mean(F ).


Sketch of Proof



MSα, α ∈ (0, 1].Mean functional ρ(F ) = mean(F ).There exists a c ∈ (0, 1) such that ρ(F ) = cq−α (F ) + (1− c)q+α (F ),where q−α (F ) := infx | F (x) ≥ α and q+α (F ) := infx | F (x) > α.


Sketch of Proof




The major difficulty: the only property of h(·) that we can use in theproof is that it is increasing; it can have any kind of discoutinuities.


Sketch of Proof




The major difficulty: the only property of h(·) that we can use in theproof is that it is increasing; it can have any kind of discoutinuities.

Lastly, we show that ρ(F ) = cq−α (F ) + (1− c)q+α (F ) is not elicitableby extending the main propositon in Thomson (1979, Journal ofEconomic Theory, Eliciting production possibilities from awell-informed manager)


Characterization of Median Shortfall

Median shortfall is the only risk measure that




captures tail risk;




captures tail risk;

is elicitable;




captures tail risk;

is elicitable;

has the decision theoretical foundation of Choquet expected utility.


Outline

1 Introduction



4 Main Result




Robustness of Risk Measures

A risk measure ρ is robust if it is insensitive to model misspecification(Hampel, 1971).




Suppose the true loss distribution is F and we want to calculate ρ(F )





However, F is unknown; hence, we have to use a model F toapproximate F . What we can compute is ρ(F ).





However, F is unknown; hence, we have to use a model F toapproximate F . What we can compute is ρ(F ).

ρ is insensitive to model misspecification means that: if themisspecified model F does not deviate much from the true F , thenρ(F ) should be close to ρ(F ).


MS Is Robust But ES Is Not

Proposition

Let α ∈ (0, 1). The following hold:

(i) For any F such that MSα(F ) is continuous at α, MSα is robust at F .

(ii) For any F , ESα is not robust at F .



Proposition




ES is not robust implies that even if the model F is only a littledifferent (in terms of Prokhorov distance) from the true distributionF , ESα(F ) can be very different from ESα(F ).



Proposition




ES is not robust implies that even if the model F is only a littledifferent (in terms of Prokhorov distance) from the true distributionF , ESα(F ) can be very different from ESα(F ).

ES is not robust implies that ES is too sensitive to model risk.


ES is Sensitive to Modeling Assumption of Tail Heaviness

ESα and MSα for Laplace and T-distributions, α ∈ [99%, 99.9%]

−7 −6 −5 −4 −3

2

3

4

5

6

7

8

9

Expected shortfall for Laplacian and T distributions

log(1−α)

ES

α

LaplacianT−3T−5T−12

−7 −6 −5 −4 −3

2

3

4

5

6

7

8

9

Median shortfall for Laplacian and T distributions

log(1−α)

MS

α

LaplacianT−3T−5T−12

1.44 0.75


Comparing MS with ES for S&P 500 Daily Loss

Data: daily return of S&P 500 Index during 1/2/1980–11/26/2012




We fit the data with two IGARCH(1, 1) model similar to the model ofRiskMetrics:

Model 1: IGARCH(1, 1) with conditional distribution being Gaussian

rt = µ + σtǫt , σ2t = βσ2

t−1 + (1− β)r2t−1

ǫtd∼ N(0, 1)







t−1 + (1− β)r2t−1

ǫtd∼ N(0, 1)

Model 2: the same as model 1 except that conditional distribution is tν:

ǫtd∼ tν







t−1 + (1− β)r2t−1

ǫtd∼ N(0, 1)

Model 2: the same as model 1 except that conditional distribution is tν:

ǫtd∼ tν

We compare the 1-day MS and ES on 11/26/2012 under the twomodels.



Table: A portfolio of S&P500 stocks worth 1,000,000 dollars on 11/26/2012

αES MS ES2−ES1

MS2−MS1− 1

ES1 ES2 ES2 − ES1 MS1 MS2 MS2 −MS198.0% 21337 23918 2581 20483 22011 1529 68.8%98.5% 22275 25530 3254 21441 23564 2123 53.3%99.0% 23546 27863 4317 22738 25807 3070 40.6%99.5% 25595 32049 6454 24827 29823 4996 29.2%

l



Table: A portfolio of S&P500 stocks worth 1,000,000 dollars on 11/26/2012

αES MS ES2−ES1

MS2−MS1− 1

ES1 ES2 ES2 − ES1 MS1 MS2 MS2 −MS198.0% 21337 23918 2581 20483 22011 1529 68.8%98.5% 22275 25530 3254 21441 23564 2123 53.3%99.0% 23546 27863 4317 22738 25807 3070 40.6%99.5% 25595 32049 6454 24827 29823 4996 29.2%

l

ES is much more sensitive to model misspecification than MS


Outline

1 Introduction



4 Main Result




Basel 2.5 Risk Measure for Trading Book

The Basel committee implemented the Basel 2.5 risk measure fortrading books in 2009

ct = max

VaRt−1(L),k

60

60

∑i=1

VaRt−i (L)

+max

sVaRt−1(L),ℓ

60

60

∑i=1

sVaRt−i (L)

.

sVaRt−i (L) is called the stressed VaR of L on day t − i , which iscalculated under the scenario that the financial market is undersignificant stress.


A New Proposal by BCBS

Basel Committee on Banking Supervision, May, 2012. Fundamentalreview of the trading book.



Basel Committee on Banking Supervision, May, 2012. Fundamentalreview of the trading book.

BCBS proposes to replace VaR by ES.



The main reason that the Basel committee proposes to replace VaRby ES is that ES “better captures tail risk.”




However, ES is not the only risk measure that captures tail risk.





If we want to capture the tail risk at level α, i.e., the size of the lossbeyond VaRα, we can use ESα, or MSα.






ESα > MSα?






ESα > MSα? Sometimes yes, sometimes no, as it is not clear whichone is bigger, median or mean.


MS is Easy to Implement

Proposition

For any L and α ∈ (0, 1),

MSα(L) = VaR 1+α2(L).



Proposition



MS is easy to compute and backtest.



Proposition



MS is easy to compute and backtest.

No need to change the existing codes in banks. Just adjust the levelfrom 99% to 99.5%.


MS or ES?


MS or ES?

Both MS and ES capture tail risk.


MS or ES?


MS is elicitable but ES is not elicitable.


MS or ES?



MS is robust but ES is not robust.


MS or ES?




MS is easy to implement but ES is much more difficult to implement.


MS or ES?




MS is easy to implement but ES is much more difficult to implement.

MS is better than ES.


Summary

We propose a new risk measure called median shortfall, whichmeasures the median of the tail conditional loss distribution.


Summary



captures tail risk;is elicitable;has the decision theoretical foundation of Choquet Expected Utilitytheory.


Summary




Median shortfall is robust and easy to implement.


Summary





In contrast, ES is neither elicitable nor robust, and is difficult toimplement.


Summary





In contrast, ES is neither elicitable nor robust, and is difficult toimplement.

MS may be a better alternative than ES for trading book capitalrequirement.


Thank you!


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