Statistical Methods in Financial Risk · PDF fileRisk Aggregation Statistical Methods in Financial Risk Management Lecture 3: Aggregation of Risks ... Axioms of Coherence II

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Risk Aggregation

Statistical Methods inFinancial Risk Management

Lecture 3: Aggregation of Risks

Alexander J. McNeil

Maxwell Institute of Mathematical Sciences

Heriot-Watt University, Edinburgh

2nd Workshop on Risk Analysis in Economics and FinanceGuanajuato, Mexico, 15-17 May 2013

McNeil Statistical Methods in Financial Risk Management 1 / 38

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Risk Aggregation

QRM

Alexander J. McNeil

Rüdiger Frey

Paul Embrechts

M A N A G E M E N TConcepts

Techniques

Tools

P R I N C E T O N S E R I E S I N F I N A N C E

R I S KQUANTITATIVE


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Risk Aggregation

Overview

3 Risk AggregationSome TheoryAggregation Based on CorrelationsAggregation Based on CopulasEconomic Scenario Generation Approach


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Risk Aggregation

Some TheoryAggregation Based on CorrelationsAggregation Based on CopulasEconomic Scenario Generation Approach



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Mathematical Conventions

We fix a probability space (Ω,F ,P) and a set of financial risksMdefined on this space. These risks are interpreted as portfolio orbalance sheet position losses over some fixed time horizon.We usually assume thatM is a linear space containingconstants, so that if L1,L2 ∈M,m ∈ R and k > 0 thenL1 + L2,L1 + m, kL1 ∈M.A risk measure is a mapping % :M→ R with the interpretationthat %(L) gives the amount of capital that is needed to back aposition with loss L. Examples of risk measures areValue-at-Risk (VaR), expected shortfall (ES).


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Coherent Risk Measures

A coherent risk measure is a real-valued function % onM thatsatisfies the following 4 axioms:

1 Monotonicity. For two rvs with L1 ≥ L2 we have %(L1) ≥ %(L2).2 Subadditivity. For any L1, L2 we have %(L1 + L2) ≤ %(L1) + %(L2).

This is the most debated property. Necessary for followingreasons:

Reflects idea that risk can be reduced by diversification and that “amerger creates no extra risk".Makes decentralized risk management possible.If a regulator uses a non-subadditive risk measure, a financialinstitution could reduce risk capital by splitting into subsidiaries.


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Axioms of Coherence II

3 Positive homogeneity. For k ≥ 0 we have that %(kL) = k%(L). Ifthere is no diversification we should have equality in subadditivityaxiom.

4 Translation invariance. For any a ∈ R we have that%(L + m) = %(L) + m.

Remarks:VaR is in general not coherent.ES (as we have defined it) is coherent.


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Non-Coherence of VaR: an Example

Consider portfolio of 50 defaultable bonds with independent defaults.Default probability identical and equal to 2%. Current price of bondsequal to 95, face value equal to 100.

Portfolio A: buy 50 units of bond 1.Portfolio B : buy one unit of each bond.

Common sense. Portfolio B is less risky (better diversified) thanPortfolio A. This is wrong if we measure risk with VaR !Loss of each bond equals

Li := −(100(1− Yi ) + 95) = 100Yi − 5,

where Yi = 1 if default occurs, Yi = 0 else. Yi are iid Bernoulli(0.02).


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Non-Coherence of VaR: an Example IIPortfolio A: L(A) = 50L1 and hence

VaR0.95(L(A)) = 50 VaR0.95(L1) = −250,

i.e. we may take 250 out of portfolio and still satisfy regulator.Portfolio B: L(B) =

∑50i=1 Li = 100

∑50i=1 Yi − 250 , and hence

VaRα(L(B)) = 100 qα

(50∑

i=1

Yi

)− 250.

Inspection shows that q0.95(∑50

i=1 Yi ) = 3, so that VaR0.95(L(B)) = 50,i.e. extra capital is needed to hold the portfolio.

VaR0.95(L(B)) > VaR0.95(L(A))⇒ VaR0.95

(50∑

i=1

Li

)>

50∑i=1

VaR0.95(Li )


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Portfolios and Risk Factors

Recall that we usually relate financial values and losses tounderlying risk factors or risk drivers.Each L ∈M may be of the form L = `(X) for some underlyingrandom vector of risk factor changes X and some (generallynonlinear) function `, which we sometimes call the loss impactfunction.We will consider in particular the special case where the set oflosses are linear portfolios of risk factor changes

M = L : L = m + λ′X , m, λ1, . . . , λd ∈ R (1)


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Spherical Distributions

A random vector X = (X1, . . . ,Xd )′ has a spherical distribution if

UX d= X for every orthogonal map U ∈ Rd×d

(i.e. a map satisfying UU ′ = U ′U = Id×d ).Characteristic function satisfies E(exp(it′X)) = ψ(t2

1 + · · ·+ t2d )

for function ψ of a scalar variable.Notation: X ∼ Sd (ψ)

An important property is that for any vector a ∈ Rd we have

a′X d= ‖a‖X1

where‖a‖ =

√a′a =

√a2

1 + · · ·+ a2d .


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Risk Aggregation


Elliptical Distributions

A random vector X = (X1, . . . ,Xd )′ has an elliptical distribution ifit is given by an affine transformation X = µ + AY of a sphericalvector Y ∼ Sk (ψ).Notation: X ∼ Ed (µ,Σ, ψ) where Σ = AA′.Examples: multivariate normal, multivariate t, symmetricgeneralized hyperbolic[Fang et al., 1990, Cambanis et al., 1981]A useful property is that

a′X d= ‖A′a‖Y1 + a′µ .


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A Useful Theorem

Theorem

Let X ∼ Ed (µ,Σ, ψ) and letM denote the linear portfolio set in (1).Let % be a positive-homogeneous and translation-invariant riskmeasure.

1 For any L = m + λ′X ∈M we have

%(L) = ‖A′λ‖%(Y ) + λ′µ + m

=√λ′Σλ%(Y ) + λ′µ + m

where AA′ = Σ and Y ∼ S1(ψ), i.e. a univariate sphericaldistribution (a distribution that is symmetric around 0) withgenerator ψ.

2 If %(Y ) ≥ 0 then % is subadditive onM. In particular VaRα issubadditive if α ≥ 0.5.


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Proof

1 Part (1) follows from the highlighted property of spherical/ellipticaldistribution, and the assumed properties of the risk measure:

%(L) = %(m + λ′X ) = m + %(λ′X )

= m + %(λ′µ + ‖A′λ‖Y )

= m + λ′µ + %(‖A′λ‖Y )

= m + λ′µ + ‖A′λ‖%(Y )

2 This follows from the subadditivity of the Euclidean norm.


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Risk Aggregation


Two Corollaries for Aggregation

Assume µ = E(X ) which implies E(L) = m + λ′µ. (Up to nowwe had not assumed the finiteness of moments.)The theorem implies that

%(L− E(L)) = %(λ′(X − E(X ))) =√λ′Σλ%(Y ). (2)

We will use the Theorem to derive two simple Corollaries:1 The first will be relevant to the problem of aggregating several

portfolios.2 The second will be relevant to the problem of measuring portfolio

risk by aggregating stress tests.


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Corollary 1

Let L1, . . . , Ln be portfolio losses inM. Let Lagg = L1 + · · ·+ Ln. We cancompute that

%(Lagg − E(Lagg)) =√

(λ1 + · · ·+ λn)′Σ(λ1 + · · ·+ λn)%(Y )

=

√√√√ n∑i=1

n∑j=1

λ′i Σλj%(Y )

=

√√√√√ n∑i=1

n∑j=1

λ′i Σλj√

(λ′i Σλi )(λ′

j Σλj )%(Li − E(Li ))%(Lj − E(Lj ))

=

√√√√ n∑i=1

n∑j=1

ρ(Li , Lj )%(Li − E(Li ))%(Lj − E(Lj ))


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Corollary 2

Recall the formula (2) which stated that

%(L− E(L)) = %(λ′(X − E(X ))) =√λ′Σλ%(Y ).

By considering λ = ei (the i th unit vector) we infer that

%(Xi − E(Xi )) = σi%(Y )

where σi =√

Σii .It follows that for any λ we can write

%(L− E(L)) =

√√√√ d∑i=1

d∑j=1

λiλjσiσjρij%(Y )2

=

√√√√ d∑i=1

d∑j=1

λiλjρij%(Xi − E(Xi ))%(Xj − E(Xj ))

where ρij are the correlation parameters of X .


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Aggregation of Loss Distributions

Suppose that we apply the general principle that capital iscalculated using a positive-homogeneous, translation-invariantrisk measure % so that, for any L ∈M, the economic capital isgiven by

EC = %(L)− E(L).

Now consider an enterprise with d sub-units (business lines,portfolios of assets and liabilities). Each sub-unit generates aloss or (negative) change-in-value Li over given time horizon.The aggregate loss distribution is given by

Lagg = L1 + · · ·+ Ln .

Suppose that capital calculation is decentralized and eachsub-unit calculates economic capital ECi for its ownportfolio/activities.


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Risk Aggregation


Aggregation of Loss Distributions II

How should the aggregate capital ECagg be calculated?A possible answer is given by Corollary 1.

ECagg =

√√√√ n∑i=1

n∑j=1

ρ(L,Lj ) ECi ECj (3)

But note that this has been justified by two strong assumptions:1 Underlying elliptically distributed risk factors X .2 All portfolios are linear functions of risk factors.


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Risk Aggregation


Pragmatic Aggregation Rule

What happens if we use the aggregation rule (3) and theseassumptions do not hold?First the capital is calculated within each business unit byassuming a model for Li and computing ECi = %(Li )− E(Li ).Next the correlations ρ(Li ,Lj ) are determined.

Data may be sparse or unavailable and correlations might bechosen by “experts”.The correlations must form a positive semi-definite matrix to makeany kind of sense.If the distributions of Li and Lj are of different type, then there areadditional constraints on the possible correlations.

But ultimately the aggregate capital ECagg is fairly meaninglessand will generally not correspond to ρ(Lagg − E(Lagg)).


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Limits on Attainable Correlations

(linear correlation = 1) ⇒ comonotonicitycomonotonicity 6⇒ (linear correlation = 1)

We can create models where individual risks are comonotonic(undiversifiable), but have an arbitrarily small correlation.For two given distributions, attainable correlations form asub-interval of [−1,1].Upper bound corresponds to comonotonicity, lower tocountermonotonicity.The theory goes back to [Höffding, 1940] and [Fréchet, 1957].


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Example of Attainable Correlations

sigma

rho

0 1 2 3 4 5

−1.0

−0.5

0.00.5

1.0

Take X1 ∼ Lognormal(0, 1), and X2 ∼ Lognormal(0, σ2). Observe howinterval of attainable correlations varies with σ. Upper boundary representscomonotonicity. See [McNeil et al., 2005] for details.


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Conservative Upper Bounds for Capital?

Often it is argued that it is conservative to set ρ(Li ,Lj ) = 1 for allcorrelations in (3). This yields the aggregation rule

ECagg = EC1 + · · ·+ ECn . (4)

In order to be sure that this forms an upper bound for therequired capital under all possible distributions for (L1, . . . ,Ln) weshould base capital calculation on a coherent risk measure.Note that, for both VaR and ES, the formula (4) is the correctaggregation rule when L1, . . . ,Ln are comonotonic.But note that for a non-subadditive risk measure like VaR theremay be “worse” situations than comonotonicity. We may have

VaRα(Lagg) >d∑

i=1

VaRα(Li ) .


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Risk Aggregation


Aggregation of Stress Tests

A different approach to aggregation is applied in Solvency II inthe standard formula for the solvency capital requirement (SCR);see [CEIOPS, 2006, 2006], page 71, 98.In this approach the aggregation is across risk factors, notbusiness units.It is asumed that the loss distribution for the enterprise can bemodelled as L = `(X ) for a non-linear function ` and risk factorsX = (X1, . . . ,Xd )′. ` available in so-called model office software.Suppose that for each risk factor we compute

ECi = `(%(Xi )ei )− `(E(Xi )ei ) , (5)

the effect of a shock to risk factor i of size %(Xi ) relative to ashock of size E(Xi ), with all other factors unchanged.


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Risk Aggregation


Aggregation of Stress Tests II

ECi an be thought of as the “unexpected impact” of a shock torisk factor i .Now suppose that we compute overall economic capitalaccording to

EC =

√√√√ d∑i=1

d∑j=1

ρij ECi ECj , (6)

where ρij are the correlations between the risk factors X .For example, in Solvency II, we typically use % = VaR0.995.This can be justified under the strong assumptions of ellipticalrisk factors and linear portfolios using Corollary 2.


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Risk Aggregation


When Aggregation of Stress Tests is Justified

Assume that L is an element of the linear portfolio spaceMin (1) for some fixed random vector X .Assume X ∼ Ed (µ,Σ, ψ).The function ` is given by `(x) = m + λ′x .In this case

ECi = `(%(Xi )ei )− `(E(Xi )ei ) = λi%(Xi − E(Xi )) .

And, by Corollary 2, the aggregate capital EC calculatedusing (6) satisfies

EC = %(L− E(L)) .

But when the assumptions of the Theorem do not hold, theaggregate capital may not have a meaningful interpretation.


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Risk Aggregation


Aggregation of Loss Distributions

We return to aggregation of loss distributions.1 Calibrate one-dimensional marginal distributions F1, . . . ,Fn to

data on (or views about) losses in each sub-unit.2 Calibrate a copula C to describe the dependence of losses in

each sub-unit.3 Specify joint model of losses using Sklar’s Theorem

F (x1, . . . , xn) = C(F1(x1), . . . ,Fn(xn)).4 Randomly generate multivariate loss data from F :(L(j)

1 , . . . ,L(j)n ), j = 1, . . . ,m.

5 Compute corresponding aggregate lossesL(j) =

∑ni=1 L(j)

i , j = 1, . . . ,m.6 Estimate risk measures using appropriate sample statistics

(e.g. estimate VaR using empirical quantile).


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Risk Aggregation


Pros and Cons of Copula Approach

Copulas are a better theoretical tool for aggregation. They allowus to dispense with implausible assumptions. They avoid theconsistency requirements imposed by working with linearcorrelations.Implicitly aggregation based on the Gauss copula has been usedin insurance for years. For example @RISK by Palisade softwareimplicitly uses the Gauss copula to perform Monte Carlo riskanalysis.However, calibration remains a problem. Copula parameters areusually inferred from matrices of rank correlations, but these mayalso be based on sparse data and/or expert judgement?Like the standard formula approach, the copula approachrequires the exogenous specification of parameters determiningthe dependence model.


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Calibrating Gauss copula with Spearman’s rho

Suppose we assume a meta-Gaussian model for X with copulaCGa

P and we wish to calibrate the correlation matrix P.It can be shown (Theorem 5.36 in MFE) that

ρS(Xi ,Xj ) =6π

arcsinρij

2≈ ρij ,

where the final approximation is very accurate.This suggests we estimate P by the matrix of pairwiseSpearman’s rank coefficients RS of the losses in the differentbusiness lines.Thus we need an estimate of (or expert opinion on) Rs.This is essentially the @RISK approach.


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Sample Rank Correlations

Consider iid bivariate data (X1,1,X1,2), . . . , (Xn,1,Xn,2). Thestandard estimator of ρτ (X1,X2) is

1(n2

) ∑1≤i<j≤n

sgn [(Xi,1 − Xj,1) (Xi,2 − Xj,2)] ,

and the estimator of ρS(X1,X2) is

12n(n2 − 1)

n∑i=1

(rank(Xi,1)− n + 1

2

)(rank(Xi,2)− n + 1

2

).


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Calibrating t Copula with Kendall’s tau

Suppose we assume a meta t model for X with copula C tν,P and we

wish to calibrate the correlation matrix P. The theoretical relationshipbetween Spearman’s rho and P is not known in this case, but arelationship between Kendall’s tau and P is known.It follows from Proposition 5.37 in book that

ρτ (Xi ,Xj ) =2π

arcsin ρij ,

so that a possible estimator of P is the matrix R∗ with componentsgiven by r∗ij = sin(πr τij /2) This may not be positive definite, in whichcase R∗ can be transformed by the eigenvalue method(Algorithm 5.55 in MFE) to obtain a positive definite matrix that isclose to R∗.


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Risk Aggregation


Economic Scenario Generation (ESG) Approach

In the economic scenario generation approach the correlationsare endogenous and result from specifying the mutualdependence of risks across the enterprise on common riskdrivers or factors.Let XF denote the set of common risk drivers and Xi the riskdrivers that are only relevant to sub-unit i . We assume

Li = `i (XF ,Xi ), i = 1, . . . ,d ,= `i (common factors, idiosyncratic factors) .

These models are invariably handled by Monte Carlo, i.e. therandom generation of scenarios for the factors.


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Advantages

The ESG approach is much more “principles-based” thanapproaches based on correlations or copulas.It provides a natural framework for risk-based allocation of capitalto business units which allows risk-based performancemeasurement (RAROC).It also provides a framework for sensitivity analyses with respectto common factors and model risk studies with respect to modelassumptions.Tail dependence may be introduced by using heavy-tailedcommon risk drivers XF .


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References For Further Reading

For Further Reading

Cambanis, S., Huang, S., and Simons, G. (1981).On the theory of elliptically contoured distributions.J. Multivariate Anal., 11:368–385.

CEIOPS, 2006 (2006).Draft advice to the European Commission in the Framework ofthe Solvency II project on Pillar I issues - further advice,Committee of European Insurance and Occupational PensionsSupervisors.

Fang, K.-T., Kotz, S., and Ng, K.-W. (1990).Symmetric Multivariate and Related Distributions.Chapman & Hall, London.


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References For Further Reading

For Further Reading (cont.)

Fréchet, M. (1957).Les tableaux de corrélation dont les marges sont données.Ann. Univ. Lyon, Sciences Mathématiques et Astronomie, SérieA, 4:13–31.

Höffding, W. (1940).Massstabinvariante Korrelationstheorie.Schriften des Mathematischen Seminars und des Instituts fürAngewandte Mathematik der Universität Berlin, 5:181–233.

McNeil, A., Frey, R., and Embrechts, P. (2005).Quantitative Risk Management: Concepts, Techniques andTools.Princeton University Press, Princeton.


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Statistical Methods in Financial Risk · PDF fileRisk Aggregation Statistical Methods in Financial Risk Management Lecture 3: Aggregation of Risks ... Axioms of Coherence II