Affine Modelling of Credit Risk, Pricing of Credit Events and - - PowerPoint PPT Presentation

Affine Modelling of Credit Risk, Pricing of Credit Events and Contagion

Alain Monfort1 Fulvio Pegoraro1,2 Jean-Paul Renne3 Guillaume Roussellet4

1CREST 2Banque de France and ECB 3HEC Lausanne 4McGill University

BCB Sao Paulo Conference August 2017

Views presented here are not necessarily those of the Banque de France or of the ECB.

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Introduction

Credit risk models: pricing of cross-sections of credit-sensitive instruments (bonds and/or CDS, different entities, different maturities). Term-structure models are tractable if Et

• exp(u′wt+h)

is known in closed form (A) (wt = state vector). Credit-risk model = model of the joint dynamics of wt = [w∗

t ′, d′ t]′:

w∗

t

: real-valued common (yt) and entity-specific (xt) factors (w∗

t = [y′ t , x′ t ]′),

dt : binary default indicators (dt = [d1,t, . . . , dE,t]′). Not obvious to build general models satisfying (A) with wt = [w∗

t ′, d′ t]′.

Standard credit risk models: (over)simplifying assumptions [next slide].

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Standard credit-risk framework: Assumptions

Assumption S.1 (No systemic entities, or “no-jump condition”) {dt} does not Granger-causes {w∗

t }.

Assumption S.2 (No contagion) No contagion between entities (notation: wt = {wt, wt−1, . . . }): p(di,t|w∗

t , dj,t)

= p(di,t|w∗

t )

for i = j. Assumption S.3 (Defaults are not "priced")

[SDF]

The default events (or credit events) are not priced, in the sense that: Mt,t+1(wt+1)

• SDF between dates t and t + 1

= Mt,t+1(w∗

t+1)

• does not depend on dt

.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Standard credit-risk framework: Pricing formula

P(de,t = 1|de,t−1 = 0, w∗

t )

• def. probability

= 1 − exp(−λe,t) (≈ λe,t

• def. intensity

if small) If λe,t, rt and log(Mt−1,t) are affine in w∗

t then

date-t price of a zero-coupon bond (ZCB) issued by e: Be(t, h) = EQ

t (e −h

i=1 rt+i−1+λe,t+i ) = Et(e

−h

i=0 u′ i w∗ t+i )

(say). (1) Closed-form formula if {w∗

t } follows an affine process, i.e. if, for all u:

E exp(u′w∗

t+1)|w∗ t

• = exp

a(u)′w∗

t + b(u)

. ⇒ In the standard affine credit-risk framework, tractability is reached under (i) the no-jump condition, (ii) the absence of contagion (between entities), (iii) the absence of default-event pricing.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper

We propose a general affine credit-risk pricing model jointly allowing for:

i) Systemic entities. We break down the no-jump condition. [Collin-Dufresne, Goldstein and Hugonnier (2004)] ii) Contagion effects between entities. Economic/financial linkages. [Ait-Sahalia, Laeven and Pelizzon (2014)] iii) Pricing of credit events. Credit spread puzzle. [Gourieroux, Monfort and Renne (2014)] iv) Flexible specifications of stochastic recovery rates (RR). [Altman and Sironi (2005)]

Properties:

Our state vector –including credit-event variables– is affine ⇒ explicit pricing formulas (under flexible RR assumptions).

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Small entity Common factor (macro): 𝑧 Time t-1 DEFAULTS 𝜀, > 0 Time t 6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence Pr (𝜀, > 0) ↑↓?

6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence Pr (𝜀, > 0) ↑↓?

Impact on SDF and prices: 𝑁, ↑ 6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

Impact on SDF and prices: 𝑁, ↑

Pr (𝜀, > 0) ↑↓?

6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

Impact on SDF and prices: 𝑁, ↑ 6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

Pr (𝜀, > 0) ↑

Impact on SDF and prices: 𝑁, ↑ 6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

Pr (𝜀, > 0) ↑

Large entity Impact on SDF and prices: 𝑁, ↑ 6 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Causality scheme

Large entity Large entity Small entity Common factor (macro): 𝑧 Time t-1 Small entity Common factor (macro): 𝑧 DEFAULTS 𝜀, > 0 Time t

persistence persistence

Pr (𝜀, > 0) ↑

Large entity Impact on SDF and prices: 𝑁, ↑

Default pricing

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

This paper: Applications

1) Pricing of sovereign CDS in an Epstein-Zin endowment economy.

• Consumption affected by sovereign defaults implies that

sovereign default events (i.e. disasters) appear in the SDF.

• cannot be captured by standard credit-risk models

⇒ Measurement of sovereign credit-event premiums. 2) Pricing of quanto CDS.

• Sovereign defaults affect exchange rate.
• cannot be captured by standard credit-risk models

⇒ Assessment of market-expected depreciations-at-default. 3) Ability of our model to replicate banks’ CDS spread variations observed in the aftermath of the Lehman bankruptcy.

• Usefulness of contagion effects.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Default Time Modeling

Assumption H.1 The default date of any entity e ∈ {1, . . . , E} is: τ (e) = inf

• t > 0 : δ(e)

t

> 0

• where δ(e)

t

is called credit-event variable.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Assumption H.2 (Affine) VARG dynamics of wt wt = (y′

t , δ′ t)′ : Vector Autoregressive Gamma (VARG) process (affine):

• yt | wt−1
• P

∼ Ny

j=1 γν(y)

j

  α(y)

j

+

factors

β(y)

j,y yt−1 + credit-event variables

β(y)

j,δ δt−1

• kills no-jump cond.

, µ(y)

j

   ,

• δt | yt, wt−1
• P

∼ E

e=1 γ0

  α(δ)

e

+ β(δ)

e,y yt + β(δ) e,δ δt−1

contagion , µ(δ)

e

   ,

(2) (For ease of presentation: no entity-specific factors xt: w∗

t = yt.)

γν (λ, µ): non-central Gamma distribution. γ0 (λ, µ): Gamma-zero-distribution (see MPRR (2016).

γ0 distribution

⇒ δ(e)

t

(≥ 0) can stay at 0 for prolonged periods of time (but 0 not absorbing).

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Stochastic Discount Factor

Flexible specification of the stochastic discount factor (SDF) Mt−1,t: Mt−1,t = exp −rt−1(wt−1) + θ′

w wt − ψP w,t−1(θw)

, rt−1(wt−1) = ξ0 + ξ′

1wt−1

where ψP

w,t−1(uw) is the conditional log-Laplace transform of wt.

θw = (θ′

y, θδ′)′ is the vector of risk-correction parameters.

⇒ wt also follows a VARG under Q. ⇒ If θδ = 0, then credit event risks are priced sources of risk. [see Gourieroux, Monfort and Renne (2014)]

Effects of credit-event pricing 10 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Defaultable Bond Pricing

Recovery Payment and Recovery Rate

Assumption H.3 Recovery Payment at date t + i = τ (e) of a defaultable ZCB: RR(e)

t+i

recovery rate × V(e)

t+i,h−i

recovery value (3) where RR(e)

t+i = exp

−ae − a′

w,e wt+i

• .

(4) ⇒ CDS and bond closed-form formulas for the three usual recovery payment schemes: (i) Fraction of "pre-default value" of the claim (RMV), (ii) Fraction of par (RFV), (iii) Fraction of a no-default version of the same claim (RT).

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Pricing formula

We provide pricing formula for Defaultable bonds.

schema RMV formula RFV formula

Credit Default Swaps. Two types: Currency of payoffs = currency in which underlying bonds are denominated.

schema

Currency of payoffs = currency in which underlying bonds are denominated.

schema modelling details 12 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment economy

Pricing of sovereign credit risk, focusing on the four main EA economies: France, Germany, Italy and Spain. High sovereign CDS premiums during the EA sovereign debt crisis. Semi-structural approach: Representative agent with Epstein-Zin preferences.

specification of EZ preferences

1 common factor yt, 4 country-specific factors xt (1 per country). Consumption growth ∆ct is assumed to be affine in wt = [yt, x′

t , δ′ t]′:

∆ct = µc + θ′

cwt , (ct = log(Ct)).

⇒ Credit events may impact on consumption growth (δt is part of wt).

motivation

⇒ In this model, the SDF mechanically depends on δt (pricing of credit events). [Solution method: Bansal & Yaron (2004) and Eraker & Shaliastovich (2008).]

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment economy

State-space form of the model: Measur.

• CDSo

t

= f (wt) + ηcds,t, ηcds,t ∼ N(0, Σcds) co

t

= ct + ηc,t ηc,t ∼ N(0, σ2

c)

(5) Transit.

• wt

= µw + Φwwt−1 + Σ(wt−1)εt ct = ct−1 + µc + θ′

cwt.

(6) CDSo

t : vector of CDS premiums; co t : per capita real consumption (in logs).

Some of the model parameters are calibrated, the remaining ones are estimated by QML (Extended Kalman filter).

calibrated parameters 14 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment economy

Model Fit

2008 2012 2016 50 100 150

FRA

Maturity: 2 years 5 years 2008 2012 2016 20 40 60 80

DEU

2008 2012 2016 100 200 300 400 500

ITA

2008 2012 2016 100 200 300 400 500

SPA

2008 2012 2016 0.48 0.49 0.50 0.51 0.52 0.53

consumption (log)

• Data

Model

First four charts: dashed lines = model-implied CDS spreads (bps); solid lines = data. Last chart: co

t (log p.c. real consumption) with model-implied counterpart (ct). 15 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment-economy

Model-implied Q- and P-DP (in percentage points) at the 5-year horizon

2008 2010 2012 2014 2016 5 10 15

FRA

Probability: Risk−neutral Physical 2008 2010 2012 2014 2016 2 4 6 8

DEU

2008 2010 2012 2014 2016 10 20 30 40

ITA

2008 2010 2012 2014 2016 10 20 30 40

SPA

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment economy

Term Structure of CDS Spreads

20 40 60 80 120 50 100 150 200

FRA, 2011−11−25

Maturity (in months)

• Data

Model Model (physical) 20 40 60 80 120 20 40 60 80 100

DEU, 2011−11−25

Maturity (in months)

• 20

40 60 80 120 200 400 600

ITA, 2011−11−25

Maturity (in months)

• ●
• 20

40 60 80 120 100 300 500

SPA, 2011−11−25

Maturity (in months)

• ●
• 20

40 60 80 120 10 30 50 70

FRA, 2016−07−29

Maturity (in months)

• ●
• 20

40 60 80 120 10 20 30

DEU, 2016−07−29

Maturity (in months)

• ●
• 20

40 60 80 120 50 100 150

ITA, 2016−07−29

Maturity (in months)

• 20

40 60 80 120 20 60 100

SPA, 2016−07−29

Maturity (in months)

• 17 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign defaults and exchange rates

Previous illustration: e-denominated CDS. Large share of (associated) European CDS are denominated in $. $-CDS: better protection against a potential severe depreciation of the e in case of euro-area sovereign credit event. Indeed: a depreciation of the e leads to an increase of the notional of the $-denominated CDS expressed in e. ⇒ If the defaults of euro-area state members tend to be accompanied by euro depreciations, we expect dollar-denominated CDS to have higher spreads than euro-denominated ones. Data are consistent with this: quanto CDS ($ − e) are mostly positive (see next plots).

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign defaults and exchange rates

2008 2010 2012 2014 2016 20 40 60 FRA 1 year Data Model 2008 2010 2012 2014 2016 20 40 DEU 1 year 2008 2010 2012 2014 2016 40 80 ITA 1 year 2008 2010 2012 2014 2016 40 80 120 SPA 1 year 2008 2010 2012 2014 2016 20 40 60 2 years 2008 2010 2012 2014 2016 20 40 2 years 2008 2010 2012 2014 2016 40 80 2 years 2008 2010 2012 2014 2016 40 80 120 2 years 2008 2010 2012 2014 2016 20 40 60 80 5 years 2008 2010 2012 2014 2016 20 40 5 years 2008 2010 2012 2014 2016 40 80 5 years 2008 2010 2012 2014 2016 40 80 120 5 years 20 60 10 years 20 40 60 10 years 40 80 10 years 40 80 120 10 years

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign defaults and exchange rates

Model augmented with the e-$ exchange rate ⇒ prices of quanto CDS. Assumption: st = log(FXt) = χ + y2,t + u′

s,δδt,

(7) where y2,t is an additional component of w∗

t (w∗ t = [y1,t, y2,t, x′ t ]′).

y2,t is parameterized so as to match the sample mean, variance and autocorrelation of a measure of real exchange rate over our estimation period. us,δ calibrated so as to minimize square pricing errors on quanto CDS. ⇒ According to the results, on average: Sovereign defaults in France, Germany, Italy and Spain would respectively trigger euro depreciations of 18%, 20%, 9% and 12%. 50% of the variances of observed quanto CDS accounted for.

alternative specification 20 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Contagion: Banks’ CDS after Lehman bankruptcy

2007 2008 2009 2010 200 400 600 800 1000

Citigroup

CDS maturity: 2 years 10 years 2007 2008 2009 2010 100 200 300

Barclays

2007 2008 2009 2010 50 100 150

Deutsche Bank

2007 2008 2009 2010 200 400 600

Goldman Sachs

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Contagion: Banks’ CDS after Lehman bankruptcy

x(1)

1,t'

x(2)

1,t'

x(3)

1,t'

yt' δ(1)

t'

δ(2)

t'

δ(3)

t'

x(1)

2,t'

x(2)

2,t'

x(3)

2,t'

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Contagion: Banks’ CDS after Lehman bankruptcy

20 40 60 80 100 10 20 30 40 50 (a) Common factor yt 20 40 60 80 100 20 60 100 140 (b) Entity−specific factors x1,t (e) entity 1 entity 2 entity 3 20 40 60 80 100 0.0 0.5 1.0 1.5 (c) Entity−specific factors x2,t (e) entity 1 entity 2 entity 3 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 (d) Credit−event variables δt (e) entity 1 entity 2 entity 3 20 40 60 80 100 20 40 60 80 (e) 5−year CDS premiums entity 1 entity 2 entity 3 20 40 60 80 100 20 40 60 (f) 10−year CDS premiums entity 1 entity 2 entity 3

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Thank you!

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices back

Definition The non-negative r.v. X ∼ γ0(λ, µ), λ > 0 and µ > 0, if X | Z ∼ γZ(µ) with Z ∼ P(λ). Two-step simulation:

1

Z ∼ P(λ) = ⇒ Z(ω) ∈ {0, 1, 2, . . .} and P(Z = 0) = exp(−λ).

2

X|Z ∼ γZ(µ).

0.00 0.05 0.10 0.15 100 200 300 400 500

Periods

Simulation of an ARGo process

• P(R=0) = 0.6

0.00 0.25 0.50 0.75 1.00 0.00 0.05 0.10 0.15

Values Probability

Cumulative distribution function

[Simulation from Monfort, Pegoraro, Renne and Roussellet (2016)]

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

A mixture distribution In other words, X ∼ γ0(λ, µ) if its (complicated) p.d.f. is:

fX(x ; λ, µ) =

+ ∞

• z=1

exp(−x/µ) xz−1 (z − 1)! µz × exp(−λ)λz z !

• 1{x>0} + exp(−λ)1{x=0}

However, simple Laplace transform: ϕX(u ; λ, µ) := E [exp(uX)] = exp

• λ

uµ (1 − uµ)

• for

u < 1 µ . = ⇒ Exponential-affine in λ.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Introducing dynamics: the ARG0 process Main goal: Build a dynamic affine process with zero point mass. Definition (Xt) is a ARG0(α, β, µ) if (Xt+1|Xt) is Gamma-zero distributed: (Xt+1|Xt) ∼ γ0(α + βXt, µ) for α ≥ 0, µ > 0, β > 0 . Again, simple conditional LT, exponential-affine in Xt: ϕX,t(u ; α, β, µ) := Et [exp(uXt+1)] = exp

• uµ

1 − uµ(α + β Xt)

• ,

for u < 1 µ .

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Introducing dynamics: the ARG0 process

0.00 0.05 0.10 0.15 100 200 300 400 500

Periods

Simulation of an ARGo process

• P(R=0) = 0.6

0.00 0.25 0.50 0.75 1.00 0.00 0.05 0.10 0.15

Values Probability

Cumulative distribution function

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Interesting features and properties Key properties: Non-negative process. Affine process: the conditional Laplace transform is exp-affine. ϕX,t(u ; α, β, µ) := Et [exp(uXt+1)] = exp [a(u)Xt + b(u)] Staying at zero with probability: P(Xt+1 = 0 | Xt = 0) = exp(−α) = 0.

α = 0 = ⇒ zero is not absorbing. in our multivariate yield curve model this probability will be time-varying, function of all date-t factors;

Closed-form moments (affine conditional cumulants).

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign Credit Risk

Calibrated Parameter

back

• We set the preference parameters as in the baseline case of Bansal and Yaron

(2004): δ∗ = 0.998, ψ∗ = 1.5 and γ∗ = 7. In order to facilitate the estimation:

• 1. We set µ(δ) = 0.6, making our model consistent with the 1983-2015 average of

sovereign-default recovery rates.

• 2. We assume that:

θc = [0, uy ω′, uδ ω′]′ , where ω is an E-dimensional vector giving the shares of the countries’ GDPs in the total GDP of this reduced euro area (i.e.

i ωi = 1). Therefore, θc depends

• n two parameters only, that are uy and uδ (calibrated at uδ = −0.2

detail ). 30 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign Credit Risk

Calibrated Parameter

• 3. α(y) (common factor dynamics) and α(1) (first entity-specific factor dynamics)

are taken to be such that the marginal means of all components of w∗

t are equal

to one.

• 4. We assume that the specification of the country-specific factors x(e)

t

are the same across member states.

• 5. We assume that, for each member state, the standard deviation of the

measurement errors ηcds,t are the same across maturities.

• 6. The standard deviation of the measurement error ηc,t is set to 0.20%. This

standard deviation corresponds to the one of the cycle component of per capita real consumption resulting from an application of the Hodrick-Prescott filter with a smoothing parameter of 10.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Adjust. Value St.dev. Adjust. Value St.dev. α(y) (×102) 1.969 0.131 σ1 (×102) 8.002 0.516 α(e) (×102) 1.396 0.080 σ2 (×102) 4.183 0.270 σ3 (×102) 24.937 1.609 β(y)

y

0.980 0.001 σ4 (×102) 25.065 1.617 β(e)

x

0.981 0.001 c0 0.501 0.003 β(e)

y

(×102) 0.500 0.020 µc (×105) 83.770 25.001 β(δ)

1,x

(×104) 0.402 0.130 β(δ)

2,x

(×104) 0.209 0.076 γ 7.000 − β(δ)

3,x

(×104) 0.610 0.155 ψ 1.500 − β(δ)

4,x

(×104) 0.833 0.230 δ 0.998 − uy −1.062 0.306 uδ −0.200 −

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices back

Standard asset-pricing results Absence of arbitrage opp. ⇒ ∃ stochastic discount factor Mt,t+1 > 0. Risk-free short-term interest rate: rt = − log E(Mt,t+1|wt). Risk-neutral measure Q (equivalent to physical measure P): dQ/dP = Mt,t+1 exp(rt). If Pt is the date-t price of an asset, then: Pt = EP [Mt,t+1Pt+1] = EQ [exp(−rt)Pt+1] . Economic implication of Assumption S.3 From an economic point of view, the stochastic discount factor (SDF) characterizes "bad/good" states of the world. In standard macro equilibrium models, the SDF tends to be higher when consumption is low ("bad" state of the world). If "specific defaults" have an effect on aggregate consumption, the SDF should depend on them. Examples of "specific defaults": government, large firms. ⇒ Not the case under Assumption S.3.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Credit Event risk pricing

back

If θδ = 0, then credit event risks are priced sources of risk [see Gourieroux, Monfort and Renne (2014)]. δ(e)

t

has a stochastic Q-intensity proportional to the historical one: λQ

e,t

=

• α(δ)

e

+ β(δ)

e,y yt +

β(δ)

e,δ δt−1 =

• 1

1 − θe,δ µ(δ)

e

• λP

e,t .

λQ

e,t = λP e,t if θe,δ = 0, i.e. if e’s credit event not priced. 34 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Defaultable Bond Pricing

Recovery of Market Value (RMV)

back

Proposition – Recovery of Market Value (RMV) convention Recovery Value at date t + i: V(e)

t+i,h−i = EQ

• exp
• −

h−1

• j=i
• rt+j + δ(e)

t+j+1

• wt+i
• .

(8) Under our Assumptions, price Be(t, h) at date t < τ (e): Be(t, h) = V(e)

t,h,

(9) where V(e)

t,h = exp

(Ah − ξ1)′ wt + (Bh − h ξ0) . (10)

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Defaultable Bond Pricing

Recovery of Market Value (RMV)

Equation (9) is a key result of the paper. It reads: Be(t, h) = EQ

• exp
• −
• h
• i=1

rt+i−1 + δ(e)

t+i

• wt
• .

(11) ⇒ Familiar no-arbitrage bond pricing formula based on a default-adjusted short rate (rt+i−1 + δ(e)

t+i) where

the credit events are priced sources of risk, the no-jump condition is relaxed, contagion is allowed and the recovery rate is stochastic. Generalization of the Duffie and Singleton (1999) setting.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Defaultable Bond Pricing

Recovery of Face Value (RFV)

back

Lemma Let Z1 be a random variable valued in Rd (d ≥ 1) and Z2 be a random variable valued in R+ = [0, +∞). Then, we have: E exp(u′

1Z1) 1{Z2=0}

• =

lim

u2→−∞E

exp(u′

1Z1 + u2 Z2)

. (12) RFV: Brennan and Schwartz (1980) and Duffie (1998); recovery payment = RR(e)

t+i, i.e. V(e) t+i,h−i ≡ 1.

See also Chen and Filipovic (2007).

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Proposition 4 – Recovery of Face Value (RFV) convention Price at date t < τ (e) of a ZCB issued by entity e and maturing in h periods: Be(t, h) =

h

• i=1
• ΛQ

(1,t,i) − ΛQ (2,t,i)

• + ΛQ

(3,t,h) ,

(13) where: ΛQ

(1,t,i)

:= lim

u→−∞ΨQ (t,i)(ae, u

eδ − ξ1, −aw,e) ΛQ

(2,t,i)

:= lim

u→−∞ΨQ (t,i)(ae, u

eδ − ξ1, u eδ − aw,e) ΛQ

(3,t,i)

:= lim

u→−∞ΨQ (t,i)(0, u

eδ − ξ1, u eδ) (14) with u ∈ R and where [denoting ϕQ

w,t,i (u2, . . . , u2, u1) = ϕQ w,t,i (u2, u1)]:

ΨQ

(t,i)(κ, u1, u2)

:= exp −iξ0 + κ + u′

2wt

• ϕQ

w,t,i(u2, u1)

(15)

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Multi Currency Credit Default Swap Pricing

back

We consider a CDS whose notional is equal to one unit of the foreign currency (i.e. to exp(st) units of the domestic currency). Fixed Leg:

• if entity e has not defaulted at date t + i (≤ t + h), the cash flow on this date,

expressed in the domestic currency, is: S(e)f

t,t+h exp(st+i).

• PV of the fixed-leg payments, expressed in the domestic currency:

S(e)f

t,t+h

h

i=1 EQ

• exp
• st+i − i

j=1 rt+j−1

• 1

{δ(e)′

t:t+i 1=0}

• wt
• ,

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Multi Currency Credit Default Swap Pricing

Floating Leg:

• under the RFV convention, the protection seller will make a payment of

(1 − RR(e)

t+i) exp(st+i) (Loss-Given-Default) at date t + i in case of default over

the time interval ]t + i − 1, t + i ].

• PV of the floating leg expressed in the domestic currency:

h

i=1 EQ

exp

• st+i − i

j=1 rt+j−1

• (1 − RR(e)

t+i)

• 1

{δ(e)′

t:t+i−11=0} − 1

{δ(e)′

t:t+i 1=0}

• wt
• ,
• We assume RR(e)

t

= exp −ae − a′

w,e wt

• and st = χ + u′

swt. 40 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Proposition – Price of a multi-currency CDS S(e)f

t,t+h =

h

i=1 ΛQ (t,i)

h

i=1 limu→−∞ ΨQ (t,i) (χ, u˜

eδ − ξ1, u˜ eδ + us) , (16) where: ΛQ

(t,i)

:= limu→−∞

• ΨQ

(t,i)

• χ, u

eδ − ξ1, us

• − ΨQ

(t,i)

• χ, u

eδ − ξ1, u eδ + us

• −ΨQ

(t,i)

• χ − ae, u

eδ − ξ1, us − aw

• +ΨQ

(t,i)

• χ − ae, u

eδ − ξ1, u eδ + us − aw

• ,

(17) and where ΨQ

(t,i) (κ, u2, u1) is given in Proposition 4. 41 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment economy

back

Representative agent with Epstein-Zin preferences: Ct is her consumption at date t, and her utility is given by the following recursion: Ut =

• (1 − δ∗)C1−ρ∗

t

+ δ∗E(U1−γ∗

t+1

| wt)

1−ρ∗ 1−γ∗

• 1

1−ρ∗

γ∗ = coefficient of relative risk aversion, ψ∗ = 1/ρ∗ = elasticity of intertemporal substitution (EIS) and δ∗ = rate of time preference.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign defaults and exchange rates

back

Equation (7): credit risk affects the exchange rate through the credit event variables only. What if we allow w∗

t to have a direct impact on the exchange rate?

Alternative specification for the exchange rate: st = χ + y3,t + us,y y1,t + u′

s,x xt ,

(18) where y3,t replaces y2,t in the vector w∗

t .

Approximately same fit as before. However, unreasonable implications: Estimated effect of (y1,t, x

′

t ) on st is too strong. For instance, during the

summer of 2011:

• The euro depreciated by about 10% while,
• according to the estimated us,y and us,x, the increase in the components of

(y1,t, x

′

t ) alone would have implied a 50% depreciation of the euro.

• To compensate, sharp decrease in y3,t. Not consistent with the assumption that

y3,t is independent from (y1,t, x

′

t ).

It is the relationship between st and δ per se, and less between st and the conditional default probabilities – driven by (y1,t, x

′

t ) – that is key to explain the

fluctuation of quanto CDS.

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Sovereign credit risk pricing in an EZ endowment economy

RR and Effect on Consumption

back Panel (a): Distribution of δ

δ Density 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 −40 −30 −20 −10 10 20

Panel (b): δ versus ∆c

δ ∆c

RUS 1998 PAK 1999 ECU 1999 UKR 2000 ARG 2001 MOL 2002 URU 2003 NIC 2003 RDO 2005 BEL 2006 ECU 2008 JAM 2010 GRC 2012 GRC 2012 BEL 2012 CYP 2013 JAM 2013

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

Zero-coupon bond (default case)

back

Figure: At t, investor I buys a bond issued by e, maturity date t + h

timeline Debit Credit t Be(t, h) τ (e) = t + i < t + h RR(e)

t+i × V(e) t+i,h−i

Recovery payoff:

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Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

CDS Pricing (payoffs denominated in domestic currency)

back

Figure: At t, purchase of credit protection, maturity date t + h

timeline Debit (fixed leg) Credit (floating leg) t S(e)

t,t+h

S(e)

t,t+h

S(e)

t,t+h

S(e)

t,t+h

τ (e) ≤ t + h 1 − RR(e)

τ(e) 46 / 24

Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices

CDS Pricing (payoffs denominated in foreign currency)

back

Figure: At t, purchase of credit protection, maturity date t + h

timeline Debit (fixed leg) Credit (floating leg) t S(e)f

t,t+h

FXt+1× S(e)f

t,t+h

FXt+2× S(e)f

t,t+h

FXt+3× S(e)f

t,t+h

FXt+4× τ (e) ≤ t + h

• 1 − RR(e)

τ(e)

• FXτ(e)×

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