Title Here (Times New Roman 42pt)Sudheer Chava GARP Atlanta June 2014 16 / 28. CDS Based Estimation:...
Transcript of Title Here (Times New Roman 42pt)Sudheer Chava GARP Atlanta June 2014 16 / 28. CDS Based Estimation:...
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
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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
Sudheer Chava GARP Atlanta June 2014 4 / 28
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))
Sudheer Chava GARP Atlanta June 2014 5 / 28
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
Sudheer Chava GARP Atlanta June 2014 6 / 28
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
Sudheer Chava GARP Atlanta June 2014 7 / 28
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
Sudheer Chava GARP Atlanta June 2014 8 / 28
Distance to Default model
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
Sudheer Chava GARP Atlanta June 2014 9 / 28
Distance to Default model
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 .
Sudheer Chava GARP Atlanta June 2014 10 / 28
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)
Sudheer Chava GARP Atlanta June 2014 12 / 28
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
Sudheer Chava GARP Atlanta June 2014 13 / 28
CDS Based Estimation: Simple Model 2
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
Sudheer Chava GARP Atlanta June 2014 14 / 28
CDS Based Estimation: Simple Model 2
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
1.60%
1.80%
0.00 1.00 2.00 3.00 4.00 5.00
De
fau
lt p
rob
ab
ilit
y
Time in years
Default Probability: Polynomial Method
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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
Sudheer Chava GARP Atlanta June 2014 16 / 28
CDS Based Estimation: Bootstrapping
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
Sudheer Chava GARP Atlanta June 2014 17 / 28
CDS Based Estimation: Bootstrapping
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
Sudheer Chava GARP Atlanta June 2014 18 / 28
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
Sudheer Chava GARP Atlanta June 2014 19 / 28
CDS Based Estimation: Bootstrapping
0.00%
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2.00%
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4.50%
5.00%
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
De
fau
lt p
rob
ab
ilit
y
Time in years
Default Probability: Bootstrapping Method
Sudheer Chava GARP Atlanta June 2014 20 / 28
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.
Sudheer Chava GARP Atlanta June 2014 21 / 28
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
Sudheer Chava GARP Atlanta June 2014 22 / 28
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
Sudheer Chava GARP Atlanta June 2014 23 / 28
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
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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
Sudheer Chava GARP Atlanta June 2014 25 / 28
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
Sudheer Chava GARP Atlanta June 2014 26 / 28
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
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CDS Implied Rating Scale vs CRA Rating Scale
Are Credit Ratings Stil Relevant? Chava, Ganduri and Ornthanlai (2013)
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Sudheer Chava GARP Atlanta June 2014 28 / 28