Title Here (Times New Roman 42pt)Sudheer Chava GARP Atlanta June 2014 16 / 28. CDS Based Estimation: Bootstrapping General form of premium leg: PremT t = S T t RPV T t Where: RPVT

Credit Risk

Dr. Sudheer Chava

Professor of Finance

Director, Quantitative and Computational

Finance

Georgia Tech, Ernest Scheller Jr. College

of Business

June 2014

2

The views expressed in the following material are the

author’s and do not necessarily represent the views of

the Global Association of Risk Professionals (GARP),

its Membership or its Management.

Information for Credit Risk Evaluation

Multiple Sources of information

Credit Rating Agencies!

Accounting Information

Stock Prices

Credit Default Swap prices

Bond Markets

Sudheer Chava GARP Atlanta June 2014 2 / 28

Methods for Credit Risk Evaluation

Multiple methods to evaluate credit risk.

If available directly use Credit Rating Agencies ratings

If not rated, compute synthetic credit ratings based on Credit RatingAgencies ratings

Independent Internal credit score models

Implied default probabilities from market prices (bond market, stockmarket, credit default swap market)

Statistical Models

Risk-Neutral vs Physical default probabilities


Credit Ratings

If credit ratings are not available

can try to replicate credit ratings

many methods

Comparables (similar to matrix pricing in bond market)

Statistical Models

Steps involved

identify plausible factors used by credit rating agencies (reverseengineer)

project the firm’s characteristics onto the rated universe

calculate a synthetic credit rating


Statistical Models for Predicting Defaults

Static Models

Linear Discriminant Analysis (DA) (eg: Altman’s Z-score)

Logistic Regression and Probit Models (eg: Static models)

Hazard Models (eg: Chava and Jarrow (2004), Chava, Stefanescu andTurnbull (2012))


Statistical Models: Steps

Steps involved in implementing statistical models of default

1 Define default

2 Decide on the sample selection criteria

3 Decide on a set of explanatory variables (eg. accounting data) thatmay have an impact on the credit risk of the firm

4 Identify the default status of all the firms in the sample

5 Gather data on the explanatory variables for all sample firms

6 Run the statistical model (eg. DA, logistic model or hazard model)

7 Evaluate the in-sample performance of the model

8 Evaluate the out-of-sample performance of the model


Equity Based Estimation: Intuition

Structural Model of Default: Debt in Merton model can be decomposedinto

A risk-free security with the same face value, F ∗ and the samematurity T as the risky debt in the firm’s capital structure

A put option on the firm’s assets struck at the face value of debt.

The lender or purchaser of the firm’s debt implicitly writes this put optionto the firm’s shareholders, who can put the firm’s assets back to the debtholder in case V < F . This option is similar to a credit default swap (CDS)

D = e−rTF ∗ − CDS

E + e−rTF ∗ = CDS + V


Distance to Default model

Based on the Black-Scholes formula, value of the equity is

E = VN (d1)− e−rTFN (d2)

where

E is the market value of the firm’s equity,

F is the face value of the firm’s debt,

r is the instantaneous risk-free rate,

N (.) is the cumulative standard normal distribution function,

d1 =log(V /F ) + (r + σ2V /2)T

σV√T

d2 = d1 − σV√T



In this model, the second equation, using an application of Ito’s lemma

and the fact that∂E

∂V= N (d1), links the volatility of the firm value and

the volatility of the equity.

σE =V

EN (d1)σV



The unknowns in these two equations are

the firm value V and

the asset volatility σV .

The known quantities are

equity value E ,

face value of debt or the default boundary F ,

risk-free interest rate r ,

time to maturity T .


Distance to Default Computation

Once we compute V , σV , the probability of first passage time to thedefault boundary is given by

EDF

EDF = N (−DD) where DD is the distance to default and is defined as

DD ≡log(V /F ) + (µ− σ2V /2)T

σV√T

V is the total value of the firm;

F is a face value of firm’s debt;

µ is the expected rate of return on the firm’s assets;

σV is the volatility of the firm value, and

T is the time horizon that is set to one year.Sudheer Chava GARP Atlanta June 2014 11 / 28

CDS Based Estimation: Intuition

1-period CDS contract, Notional Amount N = 1, probability ofdefault p, recovery rate R

Expected payout of protection seller: L = p(1− R)

CDS spread: S = p(1−R)1+r

Implied Probability of Default: p = S(1+r)(1−R)


CDS Based Estimation: Simple Model 1

CDS pricing equation: PremT = ProtT

Simple Model Assumptions: Constant default intensity λ, Constant knownrecovery rate R, Ignoring accrued premium between 2 payments

ProtT = (1− R)f .T∑i=1

exp(−ri ti )(Q(τ ≥ ti−1)− Q(τ ≥ ti ))

PremT =f .T∑i=1

exp(−ri ti )s

mQ(τ ≥ ti )

Survival probability Q(τ ≥ tn) = exp(−λtn)

Solve for λ based on (liquid) CDS spreads observed in the markets



CDS pricing equation: PremT = ProtT

Simple Model Assumptions: Polynomial form for default intensity λ,Constant known recovery rate R, Take into account accrued premium

PremT =f .T∑i=1

exp(−ri ti )s

mQ(τ ≥ ti )

ProtT =f .T∑i=1

L(i) exp(−ri ti )(Q(τ ≥ ti−1)− Q(τ ≥ ti ))

Default occurs at ti then: L(i) = (1− R)− s

m

(ti − tlast coupon)

∆tcoupon

Where: λ(t) = a + bt + ct2; Q(τ ≥ tn) = exp(−∑n

i=1 λ(ti )∆ti )

Solve for a, b, and c based on (liquid) CDS spreads (1-yr, 3-yr, 5-yr CDSspreads) observed in the markets



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fau

lt p

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ab

ilit

y

Time in years

Default Probability: Polynomial Method


CDS Based Estimation: Bootstrapping

General form of protection leg:

ProtTt = (1− R)

∫ t+T

t

EQt

[λu exp

(−∫ u

t

(rs + λs)ds

)]du

Assume: Deterministic interest rates and default intensities. Let t = 0 then:

ProtT = (1− R)

∫ T

0

e−∫ t0(λs+rs ) dsλt dt

For Bootstrapping procedure: Let λ and r be piecewise continuous functionsand constant between two CDS tenors that trade in the market.

Then: ProtT = (1− R)

N∑j=1

(λj

λj + rj

)S(Tj−1)D(Tj−1)

[1− e−(λj+rj )h

]Where: S(Tj) = e−

∑jk=1 λjh, D(Tj) = e−

∑jk=1 rjh, and

Tjε{T 1, T

3,T 5,T 7,T 10}

λj , rj are default intensities and forward rates between Tj−1 and Tj ,h = Tj − Tj−1 and N is the N th CDS of tenor T



General form of premium leg: PremTt = ST

t RPV Tt

Where:

RPV Tt =

N∑n=1

δ(tn−1, tn)EQt

[exp

(−∫ tn

t

(rs + λs)ds

)]+

N∑n=1

∫ tn

tn−1

δ(tn−1, u)EQt

[λu exp

(−∫ u

t

(rs + λs)ds

)]du

let δ(tn−1, tn) is the day count fraction between two consecutive premiumpayment dates, Frequency of premium payment: m per year, t = 0,N = m × T then:

RPV T =N∑

n=1

1

me−

∫ tn0

(λs+rs ) ds +N∑

n=1

∫ tn

tn−1

(t − tn−1)e−∫ t0(λs+rs ) ds λt dt



Solving we get:

Payment Term =N∑

n=1

1

mS(tn)D(tn)

Accrued Term =N∑

n=1

λnS(tn−1)D(tn−1)

∫ tn

tn−1

e−(λn+rn)(t−tn−1)(t − tn−1) dt

SoRPV T =N∑

n=1

1

mS(tn)D(tn)+

N∑n=1

1

m

λn(λn + rn)

S(tn−1)D(tn−1)(

1− e−(λn+rn).1m

)

Where: CDST =ProtT

RPV T


Bootstrapping Forward Curve of Interest Rates

Bootstrapping Interest Rates data: interest rate swaps with maturities of1,2,3,4,5,7 and 10 years and US Libor money market deposits withmaturities 1,3,6,9 months.

Let swap rate s, frequency of swap payments m (usually quarterly), maturityof swap T , Number of payments N = m × T

Swaps are priced such that s is the coupon payment on a bond trading atpar with coupon payment frequency m and maturity T

Pricing equation: 1 =N∑

n=1

s

me−

∫ tn0

rt dt + 1.e−∫ tN0 rt dt

Bootstrap forward curve using above equation and assuming forward rate rtto be a piecewise linear continuous function and constant between any 2swap payments



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De

fau

lt p

rob

ab

ilit

y

Time in years

Default Probability: Bootstrapping Method


Bond Based Estimation: Intuition

1-period Bond contract, Principal Amount N = 1, probability ofdefault p, recovery rate R

Bond price: B = (1−p)+p(R)1+r

Implied Probability of Default: p = 1−(1+r)B(1−R)

All methods above used to estimate default probabilites from CDScan be used for Bonds as well.


Bond Based Estimation: Exponential Spline

Applied widely in the industry

Bond Price:

B =N∑i=1

exp(−ri ti )CF tot(ti )Q(τ ≥ ti ) +N∑i=1

exp(−ri ti )(1.Rprincipal + CF int(ti ).Rinterest)(Q(τ ≥ ti−1)− Q(τ ≥ ti ))

Let Q(τ ≥ t) =3∑

k=1

β3e−kαt

Survival probability at t=0 is 1 implies3∑

k=1

βi = 1

Decay parameter α interpreted as long-maturity asymptotic limit ofhazard rate


Equity, CDS Based Estimation: Fair-value CDS spreads

Risky Debt = Default-free Debt - Expected Loss Value

B = Fe−rT − ELV

Put option = Expected Loss Value(ELV):ELV = Be−rTN (−d2)− VN (−d1)

Fair Value Credit Spread:

S = y − r =log(F/D)

T− r = − 1

Tlog(1− ELV

Be−rt)

Government subsidy = Equity implied FVCDS - market CDS


CDS Implied Ratings: Intuition

Estimate CDS boundaries separating two adjacent (closest) ratinggroups in a non-parametric manner

Misclassifications: CDS spreads of bonds with higher rating is higherthan CDS spreads of bonds with lower rating

Estimation of CDS boundary: minimize a penalty function with theobjective of reducing the number of such misclassifications

Example: Minimize penalty function F to estimate boundary betweenAA and A rating categories is defined by:

F (bAA−A) = 1m

∑mi=1[max(si,AA−bAA−A, 0)]2+ 1

n

∑nj=1[max(bAA−A−sj,A, 0)]2


CDS Implied Ratings Construction

F (bAA−A) = 1m

∑mi=1[max(si,AA−bAA−A, 0)]2+ 1

n

∑nj=1[max(bAA−A−sj,A, 0)]2

Where:

si,AA is the CDS spread of AA-rated firm i

sj,A is the CDS spread of A-rated firm j

m is number of firms in the AA rating class

n is number of firms in the A rating class

The penalty function for estimating boundaries between otheradjacent rating classes are defined similarly


CDS Implied Rating Model

F =∑r

nr∑i

(si−b+r )2

nrif si > b+r

(b−r −si )2

nrif si < b−r

else 0

Minimize F where:

i iterates over all spread observations

r iterates over all rating categories

si is the i th spread, which will be in some r for all cases

b+r is the upper spread boundary for rating category r

b−r is the lower spread boundary for rating category r

nr number of spreads in rating category r

Note: b+r = b−r+1 The upper bound of a category is the lower bound forthe next higher category


CDS Implied Rating Model

The Fitch CDS-IR model first penalizes spreads that are above therating boundary for its rating category (i.e si > b+r )

A symmetrical penalty is assessed for spreads that are below theapplicable rating boundary (i.e si < b−r )

The penalty rises with the square of the distance making this functiondifferentiable in all cases

It is summed across all rating categories and across all CDS spreadswithin each category

With such a specification, the boundaries would cannot cross whichcan sometimes be a problem when the penalty function is minimizedindividually


CDS Implied Rating Scale vs CRA Rating Scale

Are Credit Ratings Stil Relevant? Chava, Ganduri and Ornthanlai (2013)

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Documents

Title Here (Times New Roman 42pt)Sudheer Chava GARP Atlanta June 2014 16 / 28. CDS Based Estimation: Bootstrapping General form of premium leg: PremT t = S T t RPV T t Where: RPVT