losses given default in the presence of extreme...

Losses Given Default in the Presence of Extreme Risks

Qihe Tang[a] and Zhongyi Yuan[b]

[a] Department of Statistics and Actuarial ScienceUniversity of Iowa

[b] Smeal College of BusinessPennsylvania State University

The 19th International Congress on Insurance: Mathematics andEconomics (IME), University of Liverpool, June 24–26, 2015

Qihe Tang (University of Iowa) Losses Given Default with Extreme Risks June 24, 2015 1 / 31

Outline

1 Modeling Losses Given DefaultA Classical ModelA New Model

2 Multivariate Regular VariationMotivating DiscussionsDefinitionOn the Limit Measure

3 The Main ResultOur GoalA Heuristic ConsiderationApproximating the Distribution of LGDNumerical Studies

4 Concluding Remarks


Outline






A Classical Model for LGD

Consider a portfolio of d obligors subject to default (loans, bonds andother financial instruments).

Classical model for the LGD:

L =d

∑i=1

eiδi1(Xi>F←i (1−p))(1)

ei : positive deterministic exposure

δi ∈ (0, 1): percentage lossXi : loss variable of obligor i

p ∈ (0, 1): exogenously given default probabilityF←1 (1− p), . . . , F←d (1− p): individual default thresholds

This threshold model descends from Merton’s seminal firm-value work(Merton (1974, JF)).


A Criticism

The constant percentage loss ignores the severity of default.


Severity of Default and Loss Settlement Function

To remedy this, we introduce

severity of default: the percentage by which the loss variable exceedsthe default threshold, that is,

Si =

(Xiai− 1)+

loss settlement function: a loss function Gi that is non-decreasingwith Gi (s) = 0 for s ≤ 0 and G (∞) = 1


References

Similar ideas of introducing a loss settlement function:

Pykhtin (2003, RISK) and Tasche (2004, Working Paper) employedsuch a loss settlement function and even suggested the followingspecific form:

G (y) =(1− e−µ−σy

)+

.

Schuermann (2004, Credit Risk: Models and Management) illustratedthat the distribution of the recovery rate, and hence the loss function,contains two modes.


A New Model for LGD

Our new model for the LGD:

L(p) =d

∑i=1

eiGi

(Xi

F←i (1− p)− 1)

, (2)

where the exposures are scaled such that

d

∑i=1

ei = 1.

Compare this new model (2) with the classical model (1).


Outline






Motivating Discussions

The prevalence of rare events nowadays intensifies the urgent need forquantitatively understanding and effectively managing extreme risks in theinsurance and financial industry.

The challenges arise from both modeling the enormous sizes of the lossesand capturing their extreme dependence.

It turns out that both challenges may be dealt with in a unified frameworkcalled multivariate regular variation.


Multivariate Regular Variation

Definition

A random vector X = (X1, . . . ,Xd ) is said to have an MRV structure ifthere exists a non-degenerate limit measure ν such that, for somedistribution function F and every ν-continuous set A ⊂ [0, ∞)d away from0,

limx→∞

1

F (x)Pr (X ∈ xA) = ν (A) .

References:

de Haan and Ferreira (2006, Extreme Value Theory)

Resnick (2007, Heavy-Tail Phenomena)

T. and Yuan (2013, NAAJ) showed various commonly-used examplesincluding:

linear combinations

mixtures

Archimedean copulas


Multivariate Regular Variation

Definition

A random vector X = (X1, . . . ,Xd ) is said to have an MRV structure ifthere exists a non-degenerate limit measure ν such that, for somedistribution function F and every ν-continuous set A ⊂ [0, ∞)d away from0,

limx→∞

1

F (x)Pr (X ∈ xA) = ν (A) .

References:

de Haan and Ferreira (2006, Extreme Value Theory)

Resnick (2007, Heavy-Tail Phenomena)

T. and Yuan (2013, NAAJ) showed various commonly-used examplesincluding:

linear combinations

mixtures

Archimedean copulasQihe Tang (University of Iowa) Losses Given Default with Extreme Risks June 24, 2015 11 / 31

On the Limit Measure - Homogeneity

The definition implies that the limit measure ν is homogeneous: for someindex α ≥ 0,

ν(tB) = t−αν(B) for all B ∈ B.

Hence, we write X ∈ MRV−α.


On the Limit Measure - Tail Dependence

The limit measure ν carries all asymptotic dependence information of X inthe upper-right tail:

limx→∞

1

F (x)Pr

(d⋂

i=1

(Xi > x)

)= ν (1, ∞] .

Asymptotic dependence:

ν (1, ∞] > 0 means that X exhibits large joint movements.

If ν is concentrated on a straight line, then the components of X areasymptotically fully dependent, of which comonotonicity is a specialcase.

Asymptotic independence:

If ν (1, ∞] = 0, then X does not exhibit large joint movements.


Outline






Our Goal

Recall the LGD given in (2):

L(p) =d

∑i=1

eiGi

(Xi

F←i (1− p)− 1)

.

We study Pr (L(p) > l) as p ↓ 0 for arbitrarily fixed l ∈ (0, 1).

The default probability p being small means that the portfolioconsists of assets of good credit quality.

The extreme scenario p ↓ 0 reflects the excessive prudence inregulation guidelines for investors.

Certain frameworks, such as the Prudent Person InvestmentPrinciples (PPIP) required by EU Solvency II, require the investor toact prudently.

However, the consideration of small p may cause standard simulationprocedures to break down.


A Heuristic Consideration

Intuitively, for L(p) > l to happen, at least one of the obligors needs toexperience a loss over its threshold, which has probability p.

Depending on the value of l , it may require multiple obligors to default,which under asymptotically dependent still has a probability of order p.

Thus, for the asymptotically dependent case, we expect that as p ↓ 0 theprobability Pr (L(p) > l) decays to 0 at rate p.


An Exploratory Numerical Study

d = 5 obligors

each loss Xi distributed by Pareto(α, θi )

X possesses a Gumbel copula

C (u) = exp

−(

5

∑i=1

(− ln ui )r)1/r

each Gi specified to be a uniform distribution over (0, yG )

α = 2, θi = i , and r = 5

p ranges from 0.001 to 0.05

l = 0.3

N = 106 simulations


Figure: The decaying rate of the estimated probability Pr (L(p) > l) as papproaches 0 (with yG = 3 in (a) and yG = 10 and (b)).

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r(L(

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Approximating the Distribution of LGD

Theorem (T. and Yuan (2015, working))

Assume that X ∈ MRV−α with limit measure ν, and that all thecomponents of X are comparable in the sense that, for some ci > 0,

limx→∞

Fi (x)

F (x)= ci .

Then it holds for every l ∈ (0, 1) that

limp↓0

Pr (L(p) > l)

p= ν

(c1/αBl

), (3)

where Bl ={y ∈ [0, ∞]d : ∑di=1 eiGi (yi − 1) > l

}.


Numerical Studies

To numerically check the accuracy of the asymptotic approximationprovided by relation (3), we follow the modeling specifications before:

α = 1

r = 5

yG=2

. . .


Example 1a: d = 3, l = 0.1, N = 105

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Example 1b: d = 3, l = 0.1, N = 106

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Example 2a: d = 3, l = 0.3, N = 105

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Example 2b: d = 3, l = 0.3, N = 106

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Example 3a: d = 3, l = 0.5, N = 105

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Example 3b: d = 3, l = 0.5, N = 106

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Example 4a: d = 5, l = 0.1, N = 106

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Example 4b: d = 5, l = 0.3, N = 106

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Example 4c: d = 5, l = 0.5, N = 106

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Outline






Concluding Remarks

We proposed a new static model for the LGD which reflects theseverity of default.

We studied the limit distribution of the LGD as the default probabilitybecomes small (meaning that the portfolio consists of assets of goodcredit quality).

In the studies we applied the sophisticated concept of MRV fromEVT.

Our numerical studies show that the approximation for thedistribution of the LGD is accurate at least for small portfolios.

Thank You for Your Attention!!


Concluding Remarks

We proposed a new static model for the LGD which reflects theseverity of default.

We studied the limit distribution of the LGD as the default probabilitybecomes small (meaning that the portfolio consists of assets of goodcredit quality).

In the studies we applied the sophisticated concept of MRV fromEVT.

Our numerical studies show that the approximation for thedistribution of the LGD is accurate at least for small portfolios.

Thank You for Your Attention!!


Modeling Losses Given DefaultA Classical ModelA New Model

Multivariate Regular VariationMotivating DiscussionsDefinitionOn the Limit Measure

The Main ResultOur GoalA Heuristic ConsiderationApproximating the Distribution of LGDNumerical Studies

Concluding Remarks

losses given default in the presence of extreme...

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