Name concentration risk and its measurement · The probability of default P n: likelihood that a...

Name concentration risk and its measurement

Luis Ortiz GraciaUniversity of Barcelona

Seminari Servei d’Estadıstica Aplicada (SEA)Bellaterra, November 29, 2017

Portfolio Credit Risk ModelingHaar Wavelets for Laplace Transform Inversion

The WA Method to Quantify LossesCredit Risk Contributions

The WA Extension to the Multi-Factor ModelConclusions

IntroductionGeneral Model SettingsThe Merton ModelThe ASRF Model and Concentration Risk

Outline

1 Portfolio Credit Risk Modeling

2 Haar Wavelets for Laplace Transform Inversion

3 The WA Method to Quantify Losses

4 Credit Risk Contributions

5 The WA Extension to the Multi-Factor Model

6 Conclusions

Luis Ortiz Gracia (UB) Name concentration risk and its measurement 2 / 67





Introduction

It is very important for a bank to manage the risks originated fromits business activities. The credit risk underlying the credit portfoliois often the largest risk in a bank.

Basel Accords laid the basis for international minimum capitalstandards. Banks became subject to regulatory capitalrequirements.

Basel II is structured in a three Pillar framework:

Pillar 1: more risk sensitive minimal capital requirements.Pillar 2: banks are allowed to calculate the economic capital (riskconcentration).Pillar 3: transparency in bank’s financial reporting.






Introduction

Concentration risks arise from an unequal distribution of loans tosingle borrowers (exposure or name concentration) or differentindustry or regional sectors (sector concentration).

Within Basell II banks may opt for the standard approach (moreconservative) or the internal rating based (IRB) approach (moreadvanced).

Merton model: basis of the Basel II IRB approach. Underhomogeneity conditions, this model leads to the ASRF model.However, this model can underestimate risks in the presence ofexposure concentration.

Credit risk managers are interested in:

How can concentration risk be quantified?How can risk measures be accurately computed in short times?How does the individual transactions contribute to the total risk?






Risk Parameters

We specify a probability space (Ω,F ,P) with filtration (Ft)t≥0

satisfying the usual conditions. We fix a time horizon T > 0 (usuallyone year).

We consider a credit portfolio consisting of N obligors.

Any obligor n is characterized by:

The exposure at default En: potential exposure measured incurrency.The loss given default Ln: magnitude of likely loss on the exposureas a percentage of the exposure.The probability of default Pn: likelihood that a loan will not berepaid.

Each of them can be estimated from empirical default data.






Risk Measures

Consider an obligor n subject to default in the fixed time horizon T .

We introduce Dn, the default indicator of obligor n,

Dn =

1, if obligor n is in default,0, if obligor n is not in default,

where P(Dn = 1) = Pn and P(Dn = 0) = 1− Pn.

Let L be the portfolio loss given by,

L =N∑

n=1

Ln,

where Ln = En · Ln · Dn.






Risk Measures

Credit risk can split in Expected Losses EL (which can be forecasted)and Unexpected Losses UL (more difficult to quantify).

Assumption 1.1

The exposure at default En, the loss given default Ln and the defaultindicator Dn of an obligor n are independent.

Denote by ELn the expectation value of Ln, therefore,

EL = E(L) =N∑

n=1

En · ELn · Pn.

Holding the UL =√V(L) as a risk capital for cases of financial distress

might not be appropriate (peak losses can be very large when they occur).






Risk Measures

Let α ∈ (0, 1) be a given confidence level, the α-quantile of the lossdistribution of L in this context is called Value at Risk (VaR). Thus,

VaRα = infl ∈ R : P(L ≤ l) ≥ α = infl ∈ R : FL(l) ≥ α,

where FL is the cumulative distribution function of the loss variable L.

VaR is the measure chosen in the Basel II Accord (α = 0.999) for thecomputation of capital requirement.

Another important risk measure is the so called economic capital ECα fora given confidence level α,

ECα = VaRα − EL.






Risk Measures: VaR

Drawbacks:

1 VaR gives no information about the severity of losses which occurwith probability less than 1− α.

2 VaR is not a coherent risk measure, since it is not sub-additive. IfL = L1 + L2 then VaRα(L) VaRα(L1) + VaRα(L2) (this factcontradicts the intuition of diversification benefits associated withmerging portfolios).






Example

Consider two independent loans with default indicators following aBernoulli distribution B(1, p) with 0.006 ≤ p < 0.01 and exposures equalto 1. Define two portfolios A and B, each of them consisting of one unitof the above introduced loans. Then if we denote the correspondingportfolio losses by LA and LB ,

VaR0.99(LA) = VaR0.99(LB) = 0.

Now if we consider a portfolio C defined as the union of portfolios A andB and denote by LC = LA + LB . Then,

P(LC = 0) = (1− p)2 < 0.99,

and therefore,VaR0.99(LC ) > 0,

so that,VaR0.99(LC ) > VaR0.99(LA) + VaR0.99(LB).






Risk Measures: Expected Shortfall

On the contrary the Expected Shortfall (ES) enjoys the coherenceproperties.

ESα = E (L|L ≥ VaRα) ,

or alternatively,

ESα =1

1− α

∫ +∞

VaRα

xfL(x)dx ,

(fL density function of the loss variable L).






Risk Contributions

To decompose the risk measured by the VaR or the ES into individualtransactions (allocation principle, D. Tasche (2000)) define the RiskContribution to VaR (VaRC) of obligor n at confidence level α by,

VaRCα,n ≡ En ·∂VaRα∂En

,

and the Risk Contribution to ES (ESC) of obligor n at confidence levelα by,

ESCα,n ≡ En ·∂ESα∂En

.

These definitions satisfy the additivity condition,

N∑n=1

VaRCα,n = VaRα,N∑

n=1

ESCα,n = ESα.






Risk Contributions

Under appropriate conditions the marginal VaR contribution atconfidence level α of the obligor n is,

VaRCα,n = E(Ln|L = VaRα), (1)

and the marginal contribution at confidence level α to the expectedshortfall is,

ESCα,n = E(Ln|L ≥ VaRα). (2)






General Framework

Credit risk models can be divided into two fundamental classes ofmodels, structural or asset-value models and reduced-form ordefault-rate models.

Asset-value models have their origin on the famous Merton model,where the default of a firm is modeled in terms of the relationshipbetween its assets and the liabilities that it faces at the end of agiven time period.

Two famous industry models descending from the Merton approachare the KMV model (developed by Moody’s KMV) and theCreditMetrics model (developed by JPMorgan and the RiskMetricsGroup).






General Framework

Company’s debt is given by a zero-coupon bond with face value B.The model also assumes that the asset value process (Vt)t≥0 followsa geometric Brownian motion of the form,

dVt = µVVtdt + σVVtdWt , 0 ≤ t ≤ T . (3)

The solution at time T of the stochastic differential equation (3)with initial value V0 can be computed and is given by,

VT = V0e(µV− 1

2σ2V )T+σVWT .

This implies in particular that,

logVT ∼ N

(logV0 +

(µV −

1

2σ2V

)T , σ2

VT

).

The default probability of the firm by time T can be computed as,

P(VT ≤ B) = P(logVT ≤ logB) = Φ

(log B

V0− (µV − 1

2σ2V )T

σV√T

).






The Multi-Factor Merton Model

We define V(n)t to be the asset value of the counterparty n at time t ≤ T .

For every counterparty ∃Tn s.t. counterparty n defaults in the time

period [0,T ] if V(n)T < Tn.

For n = 1, . . . ,N, we define,

Dn = χV (n)T <Tn

∼ B(

1,P(V(n)T < Tn)

). (4)

Consider the borrower n’s asset-value log return rn: log(V

(n)T /V

(n)0

).

Assumption 1.2

Asset returns rn depend linearly on K standard normally distributed riskfactors X = (X1, . . . ,XK ) affecting the borrowers’ defaults in asystematic way as well as on a standard normally distributed idiosyncraticterm εn. Moreover, εn are independent of the systematic factors Xk forevery k ∈ 1, . . . ,K and the εn are uncorrelated.






The Multi-Factor Merton Model

Under this assumption and after standardization,

rn = βnYn +√

1− β2nεn.

Yn can be decomposed into K independent factors X = (X1, . . . ,XK ) by,

Yn =K∑

k=1

αn,kXk ,

K∑k=1

α2n,k = 1.

Yn denotes the firm’s composite factor and εn the idiosyncratic shock.βn borrower n’s sensitivity to systematic risk Yn.αn,k describe the dependence of obligor n on a sector k.Now we can rewrite equation (4) as,

Dn = χrn<tn ∼ B(1,P (rn < tn)) ,

We have Pn = P(rn < tn), tn = Φ−1(Pn) and,

Pn(yn) ≡ P(rn < tn|Yn = yn) = Φ

(tn−βnyn√

1−β2n

), cond. default probability.






Conditional Default Probability

0

0.2

0.4

0.6

0.8

1

-10 -8 -6 -4 -2 0 2 4

P(y

)

y

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

P(y

)

P

y=-4y=0y=5

Figure: Conditional default probabilities.






Portfolio Loss

Purpose: find an expression for the portfolio loss variable L.

Assuming a constant loss given default equal to Ln for obligor n, theportfolio loss distribution can then be derived as,

P(L ≤ l) =∑

(d1,...,dN )∈0,1N∑Nn=1 sn·Ln·dn≤l

(N∑

n=1

sn · Ln · dn

)· P(D1 = d1, . . . ,DN = dN).

Impractical from a computational point of view for realistic portfolios(for instance N = 1000).

Remark 1.1

We present an analytical approximation for the αth percentile of the lossdistribution in the one-factor framework, under the assumption thatportfolios are infinitely fine-grained such that the idiosyncratic risk iscompletely diversified.






Portfolio Loss

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Den

sity

of L

oss

varia

ble

x

Portfolio APortfolio B

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Cum

ulat

ive

dist

ribut

ion

func

tion

of L

oss

varia

ble

x

Portfolio APortfolio B

Figure: Densities and distributions for portfolios A and B.






The ASRF Model

The Asymptotic Single Risk Factor Model (ASRF) is the model chosen inBasel II to calculate regulatory capital. It is based on the one-factorMerton model and it mainly relies in the following assumptions,

Assumption 1.3

1 Portfolios are infinitely fine-grained, i.e. no exposure accounts formore than an arbitrarily small share of total portfolio exposure.

2 Dependence across exposures is driven by a single systematic riskfactor Y . Default indicators are mutually independent conditional onY .






The ASRF Model

Theorem 1.1

Let us denote the exposure share of obligor n by sn = En∑Nn=1 En

. Then,

under assumption 1.3 the portfolio loss ratio L =∑N

n=1 sn · Ln · Dn

conditional on any realization y of the systematic risk factor Y satisfies,

L − E(L|Y = y)→ 0 almost surely as N →∞.

Under one-factor Merton model and assuming Ln to be deterministic,

E (L|Y = y) =N∑

n=1

sn · Ln · Φ(tn −

√ρny√

1− ρn

). (5)






The ASRF Model

By Theorem 1.1,

VaRα(L)− E(L|Y = l1−α(Y ))→ 0 a.s. as N →∞.

Finally,

VaRAα =

N∑n=1

sn · Ln · Φ(tn +

√ρnΦ−1(α)

√1− ρn

),

and,

VaRCAα,n = sn ·

∂VaRAα

∂sn= sn · Ln · Φ

(tn +

√ρnΦ−1(α)

√1− ρn

).






Concentration Risk

However:

Real world portfolios are not perfectly fine-grained.

The ASRF model might be approximately valid for huge portfoliosbut less satisfactory for portfolios of smaller institutions (or morespecialized).

The formula can underestimate the required economic capital.

Does not allow the measurement of sector concentration risk.

In practice: Monte Carlo simulations (robust but computationallyintensive). The variance is an issue.

Proposal: A new method based on wavelets to overcome thecomputational complexity.





Numerical Analysis of Wavelet MethodsLaplace Transform Inversion: the WA Method

Outline






6 Conclusions






The Haar System

Consider f ∈ L2(R).

Vj = g ∈ L2 : g is constant on Ij,k , k ∈ Z and Ij,k = [ k2j ,

k+12j ).

An orthogonal basis for Vj is given by the family:

φj,k(x) = 2j2φ(2jx − k), k ∈ Z, φ(x) = χ[0,1)(x).

Consider the orthogonal projector onto Vj , Pj : L2(R)→ Vj .

We can write,

Pj f =∞∑

k=−∞

cj,kφj,k ,

cj,k = 〈f , φj,k〉 = 2j2

∫Ij,k

f (x)dx .






The Haar System

Example: component P4f on [0, 1], for f (x) = sin(2πx).

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

x

SineP4






The Haar System

φ is called the scaling function (approximation at certain level)

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

phi23

Figure: Scaling (φ2,3) function.






The Haar System

Example: the step function

f (x) =

−1, x ∈ [− 12 , 0],

1, x ∈ (0, 12 ],

0, otherwise.

This function is poorly approximated by its Fourier series:

-1.5

-1

-0.5

0

0.5

1

1.5

-0.4 -0.2 0 0.2 0.4

x

Step functionFourier (5 coefficients)

-1.5

-1

-0.5

0

0.5

1

1.5

-0.4 -0.2 0 0.2 0.4

x

Step functionFourier (50 coefficients)

Wavelets are more flexible: f (x) =√

22 φ1,0(x)−

√2

2 φ1,−1(x).






Laplace Transform

Define the Laplace transform of f ,

f (s) =

∫ +∞

0

e−sx f (x)dx = limτ→+∞

∫ τ

0

e−sx f (x)dx , s ∈ C.

Theorem 2.1

(Bromwich inversion integral) If the Laplace transform of f (x) exists,then,

f (x) = limk→+∞

1

2πi

∫ σ+ik

σ−ikf (s)esxds, x > 0,

where |f (x)| ≤ eΣx for some positive real number Σ and σ is any otherreal number such that σ > Σ.






The WA Method

Let f be a function in L2 ([0, 1]).

f (x) = limm→+∞

fm(x), fm(x) =2m−1∑k=0

cm,kφm,k(x),

where,

cm,k =

∫ k+12m

k2m

f (x)φm,k(x)dx ,

k = 0, . . . , 2m − 1.






The WA Method

Consider the Laplace Transform of f :

f (s) =∫ +∞

0e−sx f (x)dx , (assume f (x) = 0,∀x /∈ [0, 1]).

Wavelet Approximation (WA) method: approximate f by fm andcompute the coefficients cm,k .

f (s) =

∫ +∞

0

e−sx f (x)dx '∫ +∞

0

e−sx fm(x)dx =

=2m/2

s

(1− e−s

12m

) 2m−1∑k=0

cm,ke−s k

2m .

Change of variable z = e−s1

2m :∑2m−1

k=0 cm,kzk ' Qm(z).






The WA Method

We obtain the coefficients cm,k by means of the Cauchy’s integralformula,

cm,k '2

πrk

∫ π

0

<(Qm(re iu)) cos(ku)du, k = 0, ..., 2m − 1.

The integral can be evaluated by means of the trapezoidal rule,

I (r , k) =

∫ π

0

<(Qm(re iu)) cos(ku)du,

I (r , k; h) =h

2

Qm(r) + (−1)kQm(−r) + 2

mT−1∑j=1

<(Qm(re ihj )) cos(khj)

,

where h = πmT

and hj = jh for all j = 0, . . . ,mT .

Then,

cm,k '2

πrkI (r , k) ' 2

πrkI (r , k ; h), k = 1, ..., 2m − 1.





VaR Computation with the WA MethodNumerical Examples

Outline






6 Conclusions






The Model

Focus on the one-factor Merton model. Assume:

Ln = 100%,N∑

n=1

En = 1.

Let F be the CDF of L and fL its PDF.

rn =√ρY +

√1− ρεn,

(Y , εn i.i.d. N(0, 1)).

Conditional default probabilities, Pn(y) ≡ Φ(

tn−√ρy√

1−ρ

), tn = Φ−1(Pn).






The Approximation

Consider,

F (x) =

F (x), if 0 6 x 6 1,1, if x > 1,

Define unconditional MGF: ML(s) ≡ E(e−sL).

Assumption 3.1

Conditional Independence Framework. If the systematic factor Y isfixed, defaults occur independently because the only remaininguncertainty is the idiosyncratic risk.

Define conditional MGF:¯ML(s; y) ≡ E(e−sL | Y = y) =

∏Nn=1

[1− Pn(y) + Pn(y)e−sEn

].

Then:

ML(s) = E( ¯ML(s; y)) =∫R∏N

n=1

[1− Pn(y) + Pn(y)e−sEn

]1√2πe−

y2

2 dy .






The Approximation

Since: F ∈ L2([0, 1]) then:

F (x) ' Fm(x), Fm(x) =∑2m−1

k=0 cm,kφm,k(x),F (x) = limm→+∞ Fm(x).

Observe: ML(s) =∫ +∞

0e−sxF ′(x)dx = e−s + s

∫ 1

0e−sxF (x)dx .

Then:(ML(s)− e−s

)/s is the Laplace transform of F .

Apply the WA method,

Compute: cm,k .






VaR Computation

We have: F (VaRα) ' Fm(VaRα) = 2m2 · cm,k , k ∈ 0, 1, ..., 2m − 1.

Algorithm:

1 Compute Fm( 2m−1

2m ) = 2m2 · cm,2m−1 .

2 If Fm( 2m−1

2m ) > α then compute Fm( 2m−1−2m−2

2m ), otherwise compute

Fm( 2m−1+2m−2

2m ).

3 Finish after m steps storing the k value s.t. Fm( k2m ) is the closest

value to α.

In fact, Fm(ξ) = Fm( k2m ), for all ξ ∈

[k

2m ,k+12m

).

So take: VaRW (m)α = 2k+1

2m+1 .






Parameters: m = 10,mT = 2m, l = 20. MC with 5× 106 scenarios.

Portfolio 3.1

We consider N = 102 obligors, with Pn = 0.1%, En = 1, n = 1, . . . , 100,E101 = E102 = 20, ρ = 0.3 and Ln = 1.

Method VaR0.999 Relative Error

Monte Carlo 0.1500ASRF 0.0474 −68.39%Saddle Point 0.1270 −15.37%Wavelet Approximation 0.1490 −0.69%






1e-04

1e-03

1e-02

0 0.05 0.1 0.15 0.2 0.25

Tai

l pro

babi

lity

Loss level

Saddle PointMonte Carlo

ASRF

Figure: Tail probability approximation of a heterogeneous portfolio with severename concentration.






Portfolio N Pn En ρ HHI 1N

P1 100 0.21% Cn

0.15 0.0608 0.0100

P2 1000 1.00% Cn

0.15 0.0293 0.0010

P3 1000 0.30% Cn

0.15 0.0293 0.0010

P4 10000 1.00% Cn

0.15 0.0172 0.0001

P5 20 1.00% 1N

0.5 0.0500 0.0500

P6 10 0.21% Cn

0.5 0.1806 0.1000

Portfolio VaRW (8)0.999

RE(0.999, 8) VaRW (9)0.999

RE(0.999, 9) VaRW (10)0.999

RE(0.999, 10) VaRM0.999

P1 0.1934 −0.19% 0.1963 1.32% 0.1938 0.06% 0.1937P2 0.1934 1.01% 0.1924 0.50% 0.1919 0.25% 0.1914P3 0.1426 1.46% 0.1416 0.77% 0.1411 0.42% 0.1405P4 0.1621 0.24% 0.1611 −0.36% 0.1616 −0.06% 0.1617






Portfolio VaRW (8)0.999 VaR

W (9)0.999 VaR

W (10)0.999 VaRM

0.999

P1 0.2 0.4 0.7 58.3P2 1.8 3.6 7.2 571.6P3 1.8 3.6 7.2 567.6P4 18.2 36.1 72.4 1379.1

Table: CPU time (in seconds).

210


RE(0.9999, 10) VaRW (10)0.99999

RE(0.99999, 10)

P1 0.2251 −0.07% 0.2935 −1.70%P2 0.2622 −0.46% 0.3325 −1.80%P3 0.1812 −0.10% 0.2290 −1.88%P4 0.2261 −0.25% 0.2935 −1.30%

211


RE(0.9999, 10) VaRW (10)0.99999

RE(0.99999, 10)

P1 0.2251 −0.07% 0.2935 −1.70%P2 0.2622 −0.46% 0.3325 −1.80%P3 0.1812 −0.10% 0.2290 −1.88%P4 0.2261 −0.25% 0.2935 −1.30%

MC

Portfolio VaRM0.9999 VaRM0.99999P1 0.2253 0.2985P2 0.2634 0.3386P3 0.1813 0.2334P4 0.2267 0.2973






1e-04

1e-03

1e-02

1e-01

1e+00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Tai

l pro

babi

lity

Loss level

Wavelet Approximation (scale 10)Monte Carlo

ASRF

Figure: Tail probability approximation of Portfolios P1 at scale m = 10.






1e-04

1e-03

1e-02

1e-01

1e+00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Tai

l pro

babi

lity

Loss level


ASRF






The Expected Shortfall and the Risk ContributionsNumerical Examples

Outline






6 Conclusions






The Expected Shortfall

By definition: ESα(E ) = 11−α

∫ +∞VaRα(E)

xfL(E , x)dx .

Integrating by parts,

ESα(E ) =1

1− α

(1− αVaRα(E )−

∫ 1

VaRα(E)

F (E , x)dx

)' ESW (m)

α (E ),

where,

ESW (m)α (E ) ≡ 1

1− α

1− αVaRW (m)α (E )− 1

2m2 +1

cm,k(E )− 1

2m2

2m−1∑k=k+1

cm,k(E )

.






VaR Contributions

Taking into account that F (E ,VaRα) = α then,

VaRCα,i ≡ Ei ·∂VaRα∂Ei

= −Ei ·∂F (E ,VaRα)

∂Ei

fL(E ,VaRα). (6)

Recall that: F (E , x) '∑2m−1

k=0 cm,k(E )φm,k(x).

Numerator: ∂F (E ,x)∂Ei

'∑2m−1

k=0∂cm,k (E)∂Ei

φm,k(x).

Evaluating this expression in VaRα: ∂F (E ,VaRα)∂Ei

' 2m2∂cm,k (E)

∂Ei.

Finally, VaRCα,i ' VaRCW (m)α,i , where VaRC

W (m)α,i ≡ C · Ei ·

∂cm,k (E)

∂Eiand C

is a constant such that∑N

i=1 VaRCW (m)α,i = VaRW (m)

α .






ES Contributions

Take partial derivatives w.r.t. the exposures,

ESCα,i ≡ Ei ·∂ESα∂Ei

= Ei ·1

1− α

(−α∂VaRα

∂Ei+∂VaRα∂Ei

F (E ,VaRα)−∫ 1

VaRα

∂F (E , x)

∂Eidx

)' ESC

W (m)α,i ,

where,

ESCW (m)α,i ≡ −Ei ·

1

2m2· 1

1− α·

1

2

∂cm,k(E )

∂Ei+

2m−1∑k=k+1

∂cm,k(E )

∂Ei

.






Parameters: m = 10,mT = 2m. Contributions: MC with 108 scenarios.

Portfolio 4.1

This portfolio has N = 10000 obligors with ρ = 0.15, Pn = 0.01 andEn = 1

n for n = 1, ...,N, as Portfolio P4.

Method l/2− n2 l/2− n1 α = 0.9999 α = 0.99999

VaRMα 0.2267 0.2973

VaRAα 0.1683 (−25.76%) 0.2322 (−21.91%)

VaRW (10)α (20) 10 10 0.2261 (−0.25%) 0.2935 (−1.30%)

VaRW (10)α (20, 4 · 10−1) 9 0 0.2261 (−0.25%) 0.2944 (−0.97%)

VaRW (10)α (20, 6 · 10−1) 7 0 0.2261 (−0.25%) 0.2944 (−0.97%)

Method l/2 − n2 l/2 − n1 α = 0.99 α = 0.999 α = 0.9999

ESMα 0.1290 0.1895 0.2553

ESW (10)α (20) 10 10 0.1290 (−0.02%) 0.1895 (−0.01%) 0.2556 (0.12%)

ESW (10)α (20, 4 · 10−1) 9 0 0.1289 (−0.12%) 0.1895 (−0.01%) 0.2556 (0.12%)

ESW (10)α (20, 6 · 10−1) 7 0 0.1289 (−0.11%) 0.1896 (0.00%) 0.2559 (0.25%)






1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

50 100 150 200 250

Ris

k co

ntrib

utio

n to

99%

ES

Obligor number

Wavelet ApproximationMonte Carlo

Figure: The 250 biggest risk contributions to the ES at 99% confidence levelfor the Portfolio 4.1.






1e-07

1e-06

9800 9850 9900 9950 10000

Ris

k co

ntrib

utio

n to

99%

ES

Obligor number


Figure: The 250 smallest risk contributions to the ES at 99% confidence levelfor the Portfolio 4.1.






1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

50 100 150 200 250

Ris

k co

ntrib

utio

n to

99.

9% E

S

Obligor number


Figure: The 250 biggest risk contributions to the ES at 99.9% confidence levelfor the Portfolio 4.1.






1e-07

1e-06

9800 9850 9900 9950 10000

Ris

k co

ntrib

utio

n to

99.

9% E

S

Obligor number


Figure: The 250 smallest risk contributions to the ES at 99.9% confidence levelfor the Portfolio 4.1.






Method l/2 − n2 l/2 − n1 α = 0.99 α = 0.999∑Nn=1 ESCM

α,n 0.1290 0.1892∑Nn=1 ESC

W (10)α,n (20) 10 10 0.1293 0.1891∑N

n=1 ESCW (10)α,n (20, 10−4) 10 4 0.1293 0.1890

Method CPU time (in seconds)

VaRW (10)α (20, 6 · 10−1) 25.3

VaRW (9)α (20, 6 · 10−1) 12.7

VaRW (8)α (20, 6 · 10−1) 6.4

ESCW (10)α,n (20, 10−4) 622.5






Portfolio 4.2

This portfolio has N = 1001 obligors, with En = 1 for n = 1, ..., 1000,and one obligor with E1001 = 100. Pn = 0.0033 for all the obligors andρ = 0.2.

Method CPU time (in seconds)

VaRW (10)α (64, 5 · 10−1) 5.6

ESCW (10)α,n (64, 10−4) 78.4






1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Tai

l pro

babi

lity

Loss level


ASRF

Figure: Tail probability approximation for Portfolio 4.2.






Portfolio 4.3

This portfolio has N = 100 obligors, all them with Pn = 0.01, ρ = 0.5and exposures (as in Glasserman (2005)),

En =

1, n = 1, ..., 20,4, n = 21, ..., 40,9, n = 41, ..., 60,16, n = 61, ..., 80,25, n = 81, ..., 100,






0e+00

2e-03

4e-03

6e-03

8e-03

1e-02

1e-02

10 20 30 40 50 60 70 80 90 100

Ris

k co

ntrib

utio

n to

99.

9% V

aR

Obligor number

Wavelet ApproximationMC Confidence Interval 99%

Figure: VaR contributions at 99.9% confidence level for Portfolio 4.3 using theWA method and GH integration formulas with 64 nodes and ε = 10−4.






0e+00

2e-03

4e-03

6e-03

8e-03

1e-02

1e-02

1e-02

10 20 30 40 50 60 70 80 90 100

Ris

k co

ntrib

utio

n to

99.

9% E

S

Obligor number


Figure: Expected Shortfall contributions at 99.9% confidence level forPortfolio 4.3 using the WA method and GH integration formulas with 64 nodesand ε = 10−4.






0e+00

5e-03

1e-02

1e-02

2e-02

10 20 30 40 50 60 70 80 90 100

Ris

k co

ntrib

utio

n to

99.

99%

ES

Obligor number


Figure: Expected Shortfall contributions at 99.99% confidence level forPortfolio 4.3 using the WA method and GH integration formulas with 64 nodesand ε = 10−4.





Outline






6 Conclusions





The Model

Multi-factor Gaussian copula:

Wn = aTn Y + bnZn. (7)

Multi-factor t-copula:

Wn =

√ν

V

(aT

n Y + bnZn

). (8)

The computation of characteristic functions (or Laplace transforms) arenot affordable by numerical quadrature.





Outline






6 Conclusions





Conclusions

New method for Laplace Transform inversion based on Haarwavelets. Particularly well suited for stepped-shape functions, whereother state-of-the-art methods fail.

Computation of the VaR and the ES risk measures under theone-factor Merton model. Very accurate and fast, even in thepresence of severe name concentration. Rel. err. < 1%.

The WA method computes the entire distribution of losses withoutextra computational time (CDO pricing).

Accurately computation of the risk contributions to the VaR and theES. Rel. err. < 1%.

Extension to the multi-factor Merton model (to account for sectorconcentration).





References

J.J. Masdemont and L. Ortiz-Gracia (2014). Haar wavelets-basedapproach for quantifying credit portfolio losses. QuantitativeFinance, 14(9), 1587–1595.

L. Ortiz-Gracia and J.J. Masdemont (2014). Credit riskcontributions under the Vasicek one-factor model: a fast waveletexpansion approximation. Journal of Computational Finance,17(4), 59–97.

G. Colldeforns-Papiol, L. Ortiz-Gracia and C.W. Oosterlee (2017).Quantifying credit portfolio losses under multi-factor models.Submitted for publication, available at www.ssrn.com.


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