1. Credit Risk
Yildiray Yildirim
Martin J. Whitman School of Management
Syracuse University
yildiray@syr.edu
1
2. Objective of the class
The course is designed to familiarize students with quantitative models
dealing with the default risk. In particular, we introduce the review of
credit risk models:
statistical models
structural models
reduced form models
We also study credit derivative market, related securities such as
credit default swap,
credit linked note,
asset swaps,
credit spread options
basket default swap, and
cmbs and calculating subordination levels.
2
3. Cash Flows
Trustee
Servicer
Oversees Pool
(Master Svcr,
Sub-Svcrs)
Collects CF
Special
Sevicer
Deals with
defaults,
workouts
3
4. Books
Required
David Lando, Credit Risk Modeling: Theory and Applications, Princeton
University Press, 2004
Gunter Loeffler, Peter N. Posch, Credit Risk Modeling using Excel and
VBA , ISBN: 978-0-470-03157-5, 2007
Supplementary
Damiano Brigo and Fabio Mercurio, Interest Rate Models – Theory and
Practice with simile, inflation and credit, Springer, 2006
Satyajit Das, (2005) Credit Derivatives : CDOs and Structured Credit
Products (Wiley Finance), ISBN: 0470821590
Related academic articles related to new developments on the topic.
4
5. Outline
1 Introduction to Credit Risk
2 Statistical Techniques for Analyzing Default
3 Structural Modelling of Credit Risk
4 Intensity – Based Modelling of Credit Risk
5 Credit Derivatives
5
6. Market risk: unexpected changed in market prices or rates
Liquidity risk: the risk of increased costs, or inability, to adjust
financial positions, or lost of access to credit
Operational risk: fraud, system failures, trading errors (such as
deal mis-pricing)
Credit risk: is the risk that the value of financial asset or
portfolio changes due to changes of the credit quality of issuers
(such as credit rating change, restructuring, failure to pay,
bankruptcy)
6
7. Interest rate risk is isolated via interest rate swaps
Exchange rate risk via foreign exchange derivatives
Credit risk via credit derivatives
Credit derivative transfer the credit risk contained in a loan from
the protection buyer to the protection seller without effecting the
ownership of the underlying asset. Essentially, it is a security with a
payoff linked to credit related event.
These risks can separately sold to those willing to bear them:
US corporate want to hedge political risks (wars, labor strikes,
… ) for its investment in Mexico. WHAT DO TO DO?
7
8. Solution: Buy credit default swap reference a USD-denominated
Mexican bond and write down all the concerned events as credit
events (beside the default of the bond).
Using financial/credit instruments to provide protection against default
risk is not new.
Letters of credit or bank guarantees have been applied for some
time, also securitization is a commonly used tool.
However, credit derivatives show a number of differences:
1. Their construction is similar to that of financial derivatives, trading takes
place separately from the underlying asset.
2. Credit derivatives are regularly traded. This guarantees a regular marking
to market of the relevant positions.
3. Trading takes place via standardized contracts prepared by the
International Swaps and Derivatives Association (ISDA).
8
9. Sources of Credit Risk for Financial Institution
Potential defaults by:
Borrowers
Counterparties in derivatives transactions
(Corporate and Sovereign) Bond issuers
Banks are motivated to measure and manage Credit Risk.
Regulators require banks to keep capital reflecting the credit risk
they bear (Basel II).
9
10. Two determinants of credit risk:
Probability of default
Loss given default / Recovery rate
Consequence:
Cost of borrowing > Risk-free rate
Spread = Cost of borrowing – Risk-free rate
(usually expressed in basis points, 100bp=1%)
Function of a rating
Internal (for loans)
External: rating agencies (for bonds)
10
11. Spreads of investment grade zero-coupon bonds: features
Baa/BBB
Spread
• Spread over treasury zero-curve increases
over
as rating declines
Treasur A/A
ies
• increases with maturity
Aa/AA • Spread tends to increase faster with
maturity for low credit ratings than for
high credit ratings.
Aaa/AAA
Maturity
11
12. Difference in yield between Moody's Baa-rated corporate debt and constant-maturity
10-year Treasuries based on monthly averages, with NBER recessions indicated as
shaded regions.
Data sources: http://research.stlouisfed.org
12
13. Moody‘s, S&P, and Fitch are the most respected private and
public debt rating agencies.
A credit rating is “an opinion on the future ability and legal
obligation of an issuer to make timely payments of principal and
interest on a specific fixed income security”- Moody’s
Ratings reflect primarily default probabilities (PD).
13
14. Letter grades to reflect safety of bond issue
Very High High Speculative Very Poor
Quality Quality
S&P AAA AA A BBB BB B CCC D
Moody’s Aaa Aa A Baa Ba B Caa C
Investment-grades Speculative-grades
14
15. Rating process includes quantitative and qualitative.
Quantitative analysis is mainly based on the firm’s financial
reports. (now it includes model based approaches such as
structural and intensity based)
Qualitative analysis is concerned with management quality,
reviews the firm’s competitive situation as well as an
assessment of expected growth within the firm’s industry plus
the vulnerability to technological changes, regulatory changes,
labor relations, etc.
15
16. Credit risk is the largest element of risk in the books of most
banks
Typical elements of (individual) credit risk:
Default probability
Recovery rate, Loss given default
16
17. Why credit rating and what is it ?
Typical situation for a bank:
someone applies for a loan (a company, a person, a state, ...)
the bank has to decide to grant the loan or not
then
If the bank is too restrictive, it will loose a lot of business
If the bank is not carefully checking the risk of default (i.e. the credit
is not or only partly repaid) related to the customer, then it will loose
too much money
Therefore,
Every bank needs a systematic way to judge the quality of the
customer with respect to the ability to pay the credit back (Credit
rating)
17
18. So far:
For each credit given a bank has to deposit 8% of the sum as a safety
loading against the default of the credit independent on the reliability and
the quality of the (private) debtor.
Good and bad credits are treated totally similar
Desire of banks: reduction of the safety deposit
US-approach: Companies should be rated by rating agencies (typically
US companies)
EU-approach: Banks are allowed to use internal models, but have to
prove that they are based on statistical and mathematical methods (and
satisfy some quality requirements ...) which have to be validated and
backtested regularly.
Consequence of EU-approach (Basel II):
Banks have to set up internal models (or have to stick to the standard
approach or can rely on rating agencies)
Models have to be documented, and regularly backtested.
18
19. Rating Process
Assign analytical team Rating Committee
Request Rating Meet issuer Issue Rating
Meeting
Conduct basic research
Appeals
Surveilance
Process
19
20. Credit risk literature has been essentially developed in two
direction mathematically:
Structural Models: (Merton’ 74)
First passage time approach: Black and Cox’ 76, Longstaff and Schwartz’
95, Leland’ 94, Leland and Toft’ 96,
The first passage time approach extends the original Merton model by
accounting for the observed feature that default may occur not only at the
debt’s maturity, but also prior to this date. Default happens when the
underlying process hits a barrier. It can be exogenous or endogenous
w.r.t. the model.
Reduced Form Models: (Duffie, Singleton' 94), (Jarrow, Turnbull' 95)
Combination: (Duffie, Lando' 01), (Cetin, Jarrow, Protter, Yildirim’ 04)
20
21. Estimation:
Altman' 68 and Zmijewski' 84 → Static Models
Shumway' 01,
→ Hazard Models
Chave and Jarrow’ 02,
Yildirim’ 07
→ Estimation based on reduced
Duffee' 99 and
Janosi, Jarrow, Yildirim' 01 form model on debt prices
Delianedis and Geske’ 98 and → Estimation based on equity prices
Janosi, Jarrow, Yildirim' 01
21
22. Outline
1 Introduction to Credit Risk
2 Statistical Techniques for Analyzing Default
3 Structural Modelling of Credit Risk
4 Intensity – Based Modelling of Credit Risk
5 Credit Derivatives
22
23. Estimating Credit Scores with Logit
Typically, several factors can affect a borrower’s default
probability.
Salary, occupation, age and other characteristics of the loan
applicant;
When dealing with corporate clients: firm’s leverage, profitability or
cash flows, etc…
A scoring model specifies how to combine the different pieces of
information in order to get an accurate assessment of default
probability.
Standard scoring models take the most straightforward
approach by linearly combining those factors.
23
24. Let x denote the factors and β the weights (or coefficients) attached to
them
scorei 1x i 1 2x i 2 .. k x ik X
Assume y is an indicator function with y=1 shows the firm default and
y=0 shows the firm is not in default.
The scoring model should predict a high default probability for those
observations that defaulted and a low default probability for those that
did not defaulted.
Therefore, we need to link scores to default probabilities (λ =PD). This can
be done by representing PD as a function (F) of scores:
Pr ob(Defaulti ) F (Scorei )
24
25. Like PD, the function F should be from 0 to 1. A distribution often
considered for this purpose is the logistic distribution:
1
logistic distribution(Scorei )
1 exp( X )
We can also derive the PD using odds ratios:
25
26. being the probability of an event the model is
log( i ) 1x i 1 2x i 2 ....k x ik
1 i
Brief derivation of the event probability is as follows:.
L log X
1
1
e (X )
e (X ) 1,
1
( X )
1 e
26
27. To calculate the estimate from above equation, we use maximum likelihood estimation
(MLE). Let
1 occurence of event
y 0 non-occurence of event
and P(y=1) = ; unknown
individual Probability
y
1 1
1
2 0
3 1
Joint probability (L) = (1 ) = 2 (1 )
→ The value that maximizes 2 (1 ) is 2/3. Therefore, MLE 2 / 3.
Now, let the sample size be n . The joint distribution of observing the data is
n
L i (1 )
1yi
y
i 1
27
30. Now, you have to pick the covariates which you think will best
represents the default probabilities of firm.
Altman Z scores and ZETA models (Static Model) are the early
credit risk models based on only scores:
Z-score (probability of default), developed in 1968, is a function of:
x1: Working capital/total assets ratio captures the ST liquidity of the
firm
x2: Retained earnings/assets captures historical profitability
x3: EBIT/Assets ratio captures current profitability
x4: Market Value of Equity/ Total liability is market based measure of
leverage
x5: Sales/Total Assets is a proxy for competitive situation of the firm
Z 1.2x 1 1.4x 2 3.3x 3 0.6x 4 0.9x 5
30
31. ZETA model, 1977:
x1: returns on assets
x2: stability of earnings
x3: debt service
x4: cumulative profitability
x5: liquidity
x6: capitalization
x7: size
Even though coefficients are not specified (because ZETA company
didn’t make it available), you can estimate the coefficients yourself.
What’s the problem with static models?
Based on historical accounting ratios, not market values (with
exception of market to book ratio).
To find PD from Altman scores, you need to transform scores to
PD using the logistic distribution.
31
32. A credit rating system uses a limited number of rating grades to
rank borrowers/firms according to their default probability.
Rating assignments can be based on a qualitative process or on
a default probabilities estimated with a scoring model, or other
models we will discuss later in the class.
To translate PD estimates into ratings, one defines a set of
rating grade boundaries, e.g. rules that borrowers are assigned
to grade AAA if their PD < 0.02%, to grade AA if their PD is
between 0.02% and 0.05% and so on.
32
33. If the ratings are already know, but you want to find out the cutoff for
PD for each rating group, you can do the followings:
Calculate PD for particular rating class and average expected PD of all firms within
the rating class by running ordered logistic regression. For example:
PD using Logit model
Rating class 1 1.75%
Rating class 2 2.43%
Rating class 3 3.07%
Rating class 4 3.61%
Rating class 5 4.20%
Rating class 6 4.61%
Rating class 7 5.05%
Rating class 8 5.89%
Rating class 9 10.12%
33
34. A possible building blocks of the static model is as follow:
From economic reasoning, compile a set of variables we believe to capture
factors might be relevant for default prediction: EBIT/TA, Net Income/Equity,
etc. for each firm.
Examine the univariate distribution of these variables (skewness,
kurtosis,…) and their univariate relationship to default rates to determine
whether there is a need to treat outliers (excess kurtosis) .
Based on the previous two steps, run logistic regression and check pseuda-
R squares. It measures whether we correctly predicted the defaults.
Calculate the default probability of each firm based on the estimated
coefficients and observed values for the firms.
Look up where each firm falls in our rating matrix to assign a tentative letter
grade.
34
35. Rating Process
Assign analytical team Rating Committee
Request Rating Meet issuer Issue Rating
Meeting
Conduct basic research
Appeals
Surveilance
Process
35
37. Above analysis is based on qualified variables using standard
model. On the other hand, we also have some common sense
variables, such management strength.
Common Sense:
Look at management, firm’s balance sheet, etc. to obtain a
subjective rating.
Rate: excellent, good, average, below average, poor.
Add/subtract one credit notch for movements above/below average.
Example, excellent moves a firm from B to B+ to A-. Two credit
notches
37
38. Quantitative adjustment:
Using common sense, rate firm as above (excellent, good, average, below
average, poor).
Let excellent = 4, good = 3, average = 2, below average = 1, poor = 0.
We want to adjust the probability of default for the firm based on this
subjective rating.
PD(adjusted) = PD(quantitative) - alpha* (common sense rating time t).
To determine alpha, we need some experience with our common sense
rating. The experience will provide us with historical data.
38
39. Two methods given more data:
Method 1: add our common sense variable to the right hand side
of the logistic regression as an additional covariate.
Method 2: Run a regression on the change in a key financial ratio
already included as a covariate in the quantitative logistic
regression versus the common sense rating.
Example,
(change in assets/liabilities time t) = beta*(common sense rating time t).
Then,
alpha = (coefficient of assets/liabilities time t in the logistic regression)* beta.
Method 1 is my preferred approach.
39
40. Discriminate analysis (DA)
Besides checking the significances of coefficients in your statistical
analyses, you may also need to check the power of the estimates, or in
another words, which model delivers acceptable discriminatory power
between the defaulting and non-defaulting obligor.
The basic assumption in DA is that we have two populations which are
normally distributed with different means.
In logistic regression, we have certain firm characteristics which influence
the probability of default. Given the characteristics, nonsystematic variation
determines whether the firm actually defaults or not.
In DA, the firms which default are given, but the firm characteristics are then
a product of nonsystematic variation.
Cumulative Accuracy Profile (CAP), Receiver Operating Characteristic
(ROC), Bayesian error rate, conditional
Information Entropy Ratio (CIER), Kendall’s tau and Somers’ D, Brier
score are some statistical techniques one can use.
Among those methodologies, the most popular ones are Cumulative
Accuracy Profile (CAP) and Receiver Operating Characteristic (ROC).
40
41. ROC:
ROC depends on the distributions of rating scores for defaulting and non-
defaulting debtors.
For a perfect rating model the left distribution and the right distribution in below
figure would be separate.
For real rating systems, perfect discrimination in general is not possible.
Distributions will overlap as illustrated in figure (from BCBS working paper) .
41
42. Assume one has to use the rating scores to decide which debtors will
survive during the next period and which debtors will default.
One possibility would be to introduce a cut-off value C, then each
debtor with a rating score lower than C is classed as a potential
defaulter, and each debtor with a rating score higher than C is classed
as a non-defaulter.
If the rating score is below the cut-off value C and the debtor
subsequently defaults, the decision was correct. Otherwise the
decision-maker wrongly classified a non-defaulter as a defaulter.
If the rating score is above the cut-off value and the debtor does not
default, the classification was correct. Otherwise a defaulter was
incorrectly assigned to the non-defaulters’ group.
42
43. Then one can define a hit rate HR(C) and false alarm rate
FAR(C) as:
H (C )
HR(C )
ND
H (C ) : is the number of defaulters predicted
correctly with cut-off value C
N D : is the total number of defaulters in the sample
F (C )
FAR(C )
N ND
F (C ) : is the number of false alarm
N ND : is the total number of non-defaulters in the sample
43
44. To construct the ROC curve, the quantities HR(C) and FAR(C) are computed for
all possible cut-off values of C that are contained in the range of the rating
scores.
The ROC curve is a plot of HR(C) versus FAR(C), illustrated in the figure.
The accuracy of a rating model’s performance increases the steeper the ROC
curve is at the left end, and the closer the ROC curve’s position is to the point
(0,1). Similarly, the larger the area under the ROC curve, the better the model.
The area A is 0.5 for a random model without discriminative power and it is 1.0
for a perfect model. In practice, it is between 0.5 and 1.0 for any reasonable
rating model.
44
45. Transition Probability
We already introduced the way to translate default probability
estimates into ratings based on defined set of rating grade
boundaries.
We will now introduce methods for answering questions such as
With what probability will the credit risk rating of a borrower
decrease by a given degree?
This means we will show how to estimate probabilities of rating
transition (transition matrix).
45
46. Consider a rating system with two rating classes A and B, and a default
category D. The transition matrix for this rating system is:
A B D(efault)
A Probability of Probability of Probability of
staying in A migrating from A default from A
to B
B Probability of Probability of Probability of
migrating from B staying in B default from B
to A
Transition matrices serve as an input to many credit risk analyses. They
are usually estimated from observed historical rating transitions in two
ways:
Cohort approach
Hazard approach
For a rating system based on a quantitative model, one could try to
derive transition probabilities within the model.
Markov chain is the critical part of the transition matrices.
46
47. Example:
One-Year Ratings Transition Matrix from 1981-2000 Source: Standard & Poor's.
probability of migrating to rating by year end (%)
original
AAA AA A BBB BB B CCC Default
rating
AAA 93.66 5.83 0.4 0.08 0.03 0 0 0.00
AA 0.66 91.72 6.94 0.49 0.06 0.09 0.02 0.02
A 0.07 2.25 91.76 5.19 0.49 0.2 0.01 0.03
BBB 0.03 0.25 4.83 89.26 4.44 0.81 0.16 0.22
BB 0.03 0.07 0.44 6.67 83.31 7.47 1.05 0.96
B 0 0.1 0.33 0.46 5.77 84.19 3.87 5.28
CCC 0.16 0 0.31 0.93 2 10.74 63.96 21.90
Default 0 0 0 0 0 0 0 100.00
For example, based upon the matrix, a BBB-rated bond has a 4.44%
probability of being downgraded to a BB-rating by the end of one year.
To use a ratings transition matrix as a default model, we simply take the
default probabilities indicated in the last column and ascribe them to
bonds of the corresponding credit ratings. For example, with this
approach, we would ascribe an A-rated bond a 0.03% probability of
default within one year.
47
48. Default probability in two years:
A B D
A PAA PAB PAD
B PBA PBB PBD A
D - - 1.0
Default in one year: P1= PAD
Default in two years: P2= PAA x PAD + PAB x PBD
48
49. Cohort approach
It is a traditional technique estimates transition probabilities
through historical transition frequencies.
It doesn’t make full use of the available data. The estimates are
not affected by the timing and sequencing of transitions within a
year.
49
50. Example for cohort analyses:
AAA AA A BBB BB B CCC/C Df TOTAL
AAA 92 6 0 0 0 0 0 0 98
AA 1 393 15 1 0 0 0 0 410
A 0 17 114 35 1 0 0 0 167
BBB 0 1 33 1331 27 2 0 0 1394
BB 1 0 1 41 797 53 2 4 899
B 0 0 0 1 57 653 19 13 743
CCC/C 0 0 1 0 1 21 75 19 117
D 0 0 0 0 0 0 0 1 1
AAA AA A BBB BB B CCC/C D
AAA 93.9% 6.1% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
AA 0.2% 95.9% 3.7% 0.2% 0.0% 0.0% 0.0% 0.0%
A 0.0% 10.2% 68.3% 21.0% 0.6% 0.0% 0.0% 0.0%
BBB 0.0% 0.1% 2.4% 95.5% 1.9% 0.1% 0.0% 0.0%
BB 0.1% 0.0% 0.1% 4.6% 88.7% 5.9% 0.2% 0.4%
B 0.0% 0.0% 0.0% 0.1% 7.7% 87.9% 2.6% 1.7%
CCC/C 0.0% 0.0% 0.9% 0.0% 0.9% 17.9% 64.1% 16.2%
D 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%
50
51. We can now formulate the above example:
ni (t ) is the number of firms in state i at date t
nij (t ) is the number of firms which went from i at date t-1 to j at date t.
T 1
N i (T ) ni (t ) is the total number of firm exposures recorded at the
t 0
beginning of the transition periods.
T 1
N ij (T ) nij (t ) is the total number of transitions observed from
t 0
i to j over the entire period.
Then,
N ij (T )
Pij
N i (T )
51
52. Multilevel mixture model (Yildirim’07):
P (t t t | t,Y 1)
(t; x ) lim t 0
country k
t
p(z ) P (Y 1; z )
industry type j
firm i
1- pijk (z)
pijk(z)
Yijk=1 Yijk =0
(eventually default) (never default) (δijk =0)
λijk (t;x) 1-λijk(t;x)
failed right censored
(δijk=1) (δijk=0)
52
53. Unconditional probability density is given for those experiencing the default by
P ( 1; x ; z ) p(z )f (t; x | Y 1) ,
and for those being a long-term survivors, unconditional probability density is given by
P ( 0; x ; z ) S (t; x ; z ) P (Y 0; z ) P (Y 1; z )P (t ; x | Y 1)
(1 p(z )) p(z ) 1 F (t ; x | Y 1)
(1 p(z )) p(z )S (t ; x | Y 1).
53
54. We can now write the likelihood function of the mixture of long-term survival and
eventual default of i th loan at time t for a given property type and region as:
(1i )
Li (, ; x , z ) [ p(z i )f (t; x i | Y 1)] i [1 p(z i ) p(z i )S (t; x i | Y 1)] ,
where f (t; x i | Y 1) i (t; x i )S (t; x i | Y 1) , and and are the vectors of
estimated parameters. Given yi , the complete likelihood function of i th loan can be
written as:
y (1yi ) y (1i )
Li (, ; x , z ) [{p(z i )i (t; x i )S (t; x i )} i ] i [{1 p(z i )} {p(z i )S (t; x i )} i ]
After rearranging the terms we can write the following:
(1yi ) (yi i )
y y
Li (, ; x , z ) p(z i ) i {1 p(z i )} i (t; x i ) i {1 i (t; x i )} S (t; x i ) i .
54
55. We can now write the complete data likelihood function:
I kj
Jk
N
L(, ; x , z | u, v, w ) p(z ijk ) ijk {1 p(z ijk )}
(1yijk )
y
k 1 j 1 i 1
(yijk ijk ) y
ijk (t ; x ijk ) ijk {1 ijk (t; x ijk )} S (t; x ijk ) ijk ,
where some of the covariates are time series. EM algorithm is used to find the
parameter estimates and the correlations.
'
' zijk uijk v jk wk ' x ijk (t )uijk v jk wk
e e
pijk (z ) and ijk (t; x )
' zijk uijk v jk wk '
' x ijk (t )uijk v jk wk
1 e 1 e
55
56. Outline
1 Introduction to Credit Risk
2 Statistical Techniques for Analyzing Default
3 Structural Modelling of Credit Risk
4 Intensity – Based Modelling of Credit Risk
5 Credit Derivatives
56
57. 2 approaches:
Structural models (Black Scholes, Merton, Black & Cox, Leland..)
Utilize option theory
Diffusion process for the evolution of the firm value
Better at explaining than forecasting
Reduced form models (Jarrow, Lando & Turnbull, Duffie Singleton)
Assume Poisson process for probability default
Use observe credit spreads to calibrate the parameters
Better for forecasting than explaining
57
58. Merton Model
Consider a firm with market value V, and financed by equity and
a zero coupon bond with face value K and maturity date T.
Market is frictionless (no taxes, transaction costs)
Continuous trading
r>0 and constant
We want to price bonds issued by a firm whose market value
follows a geometric Brownian motion where W is a standard
Brownian motion under P measure. Then, we have a closed
form solution of the firm value as
58
59. Default Time
The firm defaults at the maturity if the assets are not sufficient to fully pay
off the bond holders.
τ be the default time, we have
Let
59
61. Payoff at Maturity
We have the following payoffs at the maturity T:
Therefore, we can define bond and equity value as
61
62. Pay-off
Equity
holders
Bond holders
L
L
0 asset value AT
62
63. Equity Value
Payoff of E(T) is equal to a call option on the value of firm’s
asset with a strike K and maturity T.
Under the structure we have, the equity value at time 0 is given
by the Black-Scholes call option formula:
63
64. Bond Value
Bond value at T is
meaning that it is equal to loan amount K minus a put option on
the asset value of the firm with strike of bond’s face value K and
maturity T.
Now, we can write the bond value at time 0 as
Using the put-call parity for European option on non-dividend
stock, we can write the bond value as:
64
67. First Passage Time Approach
We now consider the basic extension of the Merton Model.
Based on (Black-Cox’ 76).
The idea is to let defaults occur prior to the maturity of the bond.
In mathematical terms, default will happen when the level of the
asset value hits a lower boundary, modeled as a deterministic
function of time.
If one is looking for closed form solutions, one needs to study
BM hitting to a linear boundary.
Suppose that the default take place the first time the asset value
falls to some predetermined threshold level:
67
69. Equity Valuation
If the firm value falls below the barrier at some point during the
bond’s term, the firm defaults.
Then firm stops operating, bond investors take over its assets
and equity investors receive nothing.
Equity position is equivalent to a European down-and-call option
and we have a closed form solution for it: (a messy one)
69
70. Bond Value
Consider a zero coupon bond which has R recovery in the event
of a default. Then,
70
72. Implementation in Practice
A key problem in the practical application of option-based
techniques is the fact that we rarely can observe the asset
value, and volatility of the firm.
However, for traded firms we can observe the value of equity
and its standard deviation.
As we have seen before, we can write equity as a contingent
claim on the value of the firm’s assets. By inverting this
equation, we can back out the asset value and the asset
volatility.
(We can assume asset and equity volatility equal to each other if
leverage is small and doesn’t change much over time)
72
78. Francois and Morellec (2002):
models liquidation using excursion theory. Liquidation occurs when the
value of its assets goes below the distress threshold and remains below that
level for an extended time. If the value of firm goes above the barrier, then
distress clock is reset to zero.
Moraux (2002):
uses the excursion but assumes the liquidation will be triggered cumulative
excursion time is under the exogenously set barrier.
But again, the process can stay some constant units of time below the
barrier, but we could still have small jumps below the barrier. these
models don’t account how far the firm value can decrease during the
excursion. They only look how much time the value stays under the
barrier.
In this new model, I define default in terms of the area of the excursion
to overcome this obstacle.
Now, the default happens first time the area of excursion is above a
specified level.
78
80. New Structural Model based on Yildirim’ 06
We consider a continuous trading economy with a money market account
where default free zero-coupon bonds are traded. In this economy there is
a risky firm with debt outstanding in the form of zero-coupon bonds.
Let V be the value of the firm, normalized by the value of the money
market account,
With x>0, , and B is a standard Brownian motion.
80
81. New Default Definition
Anderson and Sundaresan’ 96, Mella-Barral and Perraudin (1997),
Mella-Barral(1999), Fan and Sundaresan (2000) and Acharya,
Huang, Subrahmanyam and Sundaram (2002) clarified the
distinction between the default time and the liquidation time.
Gilson, Kose and Lang’90 show that almost half of the companies
in financial distress avoid liquidation through out-of-court debt
restructuring.
We define default times as the times the value of the firm’s asset
reaches the default threshold.
.
81
82. Let define the time of insolvency as:
Mathematically, it is the time where the firm value stays under the
default threshold, b, and the cumulative sum (this won’t let us to set
the ‘distress clock’ to zero, so it keeps in mind the history of the
financial distress) of the values are above another exogenous
barrier
We can calculate the area of the cumulative excursion below a
barrier b as:
82
84. Valuation of a Risky Zero-coupon Bond
Let denote the price process of a risk zero coupon
bond issued by this firm that pays $1 at time T if no default
occurs prior to that date.
Then, under the no arbitrage assumption, S is given by
We will assume that interest rates are deterministic, and we
have constant recovery rate in the case of default.
84
87. Using our model, we can easily calculate the expected loss
This paper models the default different then the occupation time
and the excursion models. In these models, we have the
problem of triggering default whenever the process below the
barrier, and we don’t account the fact it could be below the
barrier but fluctuates very close to the barrier.
We separate default from insolvency.
87
88. Option Pricing Approach - KMV
Firms within the same rating class have the same default rate
The actual default rate (migration probabilities) are equal to the
historical default rate
A firm is in default when it cannot pay its promised payments.
This happens when the firm’s asset value falls below a threshold
level.
88
89. On the KMV approach
We argue that the equity value is an equilibrium price, reflecting
the information of analysts and investors, and as such it is the
best estimate for the asset price.
Note that if the firm asset value and PD satisfies
89
91. DD “Distance to Default” is another way of stating the default
probability
An actual test of whether this is a good model for default would
then look at historically how well DD predicted defaults.
Utilize the database of historical defaults to calculate empirical
PD (called “Expected Default Frequencies”-EDF)
91
92. KMV: maps historical DDs to actual defaults for a given risk horizon
In practice there are problems with KMV:
Static model (assumes leverage is unchanged):
KMV approach doesn’t take dynamics of borrowers’ financial decisions into
account
In reality firms may issue additional debt or reduce debt before the risk horizon
Collin-Dufresne and Goldstein (2001) model leverage changes
Does not distinguish between different types of debt – seniority, collateral,
covenants, convertibility. Leland (1994), Anderson, Sundaresan and Tychon
(1996) and Mella-Barral and Perraudin (1997) consider debt renegotiations
and other frictions.
Asset value and volatility is not observable
In practice, EDF doesn’t converge to zero when DD gets larger.
92
93. A commercial implementation of the Merton model is the EDF
measure of Moody’s KMV.
It uses modified Black-Scholes-Merton model that allows
different type of liabilities.
Default is triggered if the asset value falls below the sum of
short term debt plus a fraction of long term debt. This rule is
derived from an analysis of historical defaults.
Distance to default that comes out of the model is transformed
into default probabilities by calibrating it to historical default
rates.
93
94. Implementing Merton Model with one year horizon
Et At (d1 ) Le r (T t )(d2 )
1
In(At / L) (r 2 )(T t )
2
d1 ; d2 d1 (T t )
(T t )
•We will have 260 equations and 260 unknowns
E L e rt (T t )(d )
t 2
At
t
•We will calculate As
(d1 )
E L e rt 1 (T (t 1))(d ) •Sigma is also not known, but we will have one
t 1 2
At 1
t 1 to calculate of all these equations estimated
(d1 ) from time series As.
.
.
.
E (d2 )
r (T (t 260))
Lt 260e t 260
t 260
At 260
(d1 )
94
95. iteration 0 : Set starting values At a for each a 0,1,..260. A sensible choice
is to set At a equal to the sum of market value of equity Et a and the book
value of liabilities Lt a . Set equal to the standard deviation of the log asset
returns computed with the At a .
iteration k : Insert At a and from the previous iteration into BS model to calculate
d1 and d2 . Input these into asset value equation to calculate the new At a . Again
use the At a to compuate the asset volatility.
We go until the procedure converges. If the sum of squared differences between
consecutive asset values is below some small number, say 1010 , we stop.
We will now implement this procedure for Enron, 3 months before its default
(data from 8/31/00 to 8/31/01) in December 2001. First, we will find asset value,
and asset volatility. Second, we will find , that is log of exected asset return.
And finally, we will calculate default probability.
95
101. We can finally calculate the default probability 3 months before
December 2001.
101
102. Outline
1 Introduction to Credit Risk
2 Statistical Techniques for Analyzing Default
3 Structural Modelling of Credit Risk
4 Intensity – Based Modelling of Credit Risk
5 Credit Derivatives
102
103. Intensity based models
The structural approach is based on solid economic arguments; it
models default in terms of fundamental firm value.
If the firm value has no jumps, this implies that the default event is not a
total surprise. There are pre-default events which announces the
default of a firm. We say default is predictable (predictable - accessible
stopping time).
The intensity based model is more ad-hoc (reduced form model) in the
sense that one can not formulate economic argument why a firm
default; one rather takes the default event and its stochastic structure
as exogenously given.
Instead of asking why the firm defaults, the intensity model is calibrated
from market prices. It is the most elegant way of bridging the gap
between credit scoring or default prediction models and the models for
pricing default risk.
103
104. If we want to incorporate into our pricing models not only the
firm’s asset value but other relevant predictors of default, and
turn this into a pricing model, we need to understand the
dynamic evolution of the covariates, and how they influence
default probabilities.
Natural way of doing this is intensity-based models.
Difference between defaultable bond pricing and treasury bond
pricing.
We model default as some unpredictable Poisson like event
(e.g. default comes from a surprise like processes-
unpredictable-inaccessible stopping time). Reduced form
models are tractable and have better empirical performance.
104
115. Simple Binomial Reduced Form Model
$δt if default
λt
v(0,t)
1-λt
$1 if no default
rt t 1t [ rt t 1t ]
1 t e e
rt
v(0, t ) e e
tt
c
1 c for small c
Note that e
115
116. Expected Losses (EL) = PD x LGD
Identification problem: cannot disentangle PD from LGD.
Intensity-based models specify stochastic functional form for
PD.
Jarrow & Turnbull (1995): Fixed LGD, exponentially distributed
default process.
Das & Tufano (1995): LGD proportional to bond values.
Jarrow, Lando & Turnbull (1997): LGD proportional to debt
obligations.
Duffie & Singleton (1999), Jarrow, Janosi and Yildirim (2002), :
LGD and PD functions of economic conditions.
Jarrow, Janosi and Yildirim (2003): LGD determined using equity
prices.
116
117. Calibration
We calibrate the model directly from market prices of various
credit sensitive securities.
One often uses liquid debt prices or credit default swap spreads,
although Janosi, Jarrow and Yildirim (2003) uses equity as well.
Now, we will look at the empirical studies for RFM (by Janosi,
Jarrow and Yildirim (2002) and (2003))
Estimating Expected Losses and Liquidity Discounts
Implicit in Debt Prices
Estimating Default Probabilities Implicit in Equity Prices
117
118. Estimating Expected Losses and Liquidity Discounts
Implicit in Debt Prices (Janosi, Jarrow, Yildirim’ 02)
Introduction
Model Structure
Data
Estimation
Conclusion
118
121. Introduction
Comprehensive empirical implementation of a reduced form credit risk model
that incorporates both liquidity risk and correlated defaults
Liquidity discount is modeled as a convenience yield
Correlated defaults arise due to the fact that a firm’s default intensities depend
on common macro-factors:
spot rate of interest
equity market index
A linear default intensity process and Gaussian default-free interest rates
Time period covered: May 1991 – March 1997
Monthly bond prices on 20 different firm’s debt issues across various industry
groupings
Five different liquidity premium models
Best performing liquidity premium model: constant liquidity version of the affine
model
121
122. Model
Frictionless markets with no arbitrage opportunities.
Traded are:
Default-free zero-coupon bonds and
Risky (defaultable) zero-coupon bonds of all maturities
Default-free Debt:
P(t,T): time t price of a default-free dolar paid at T.
f(t,T): default-free forward rate
The spot rate is given by r(t)=f(t,t).
122
123. Risky Debt:
Consider a firm issuing risky debt to finance its operations
v(t,T): represent the time t price of a promised dollar to be paid by
this firm at time T
τ : represents the first that this firm defaults
N(t) : denote the point process indicating whether or not default
has occurred prior to time t.
λ(t) : represents its random intensity process
λ(t)∆ : approximate probability of default over the time interval
[t, t+∆]
δ(τ) : fractional recovery rate, if default occurs
123
124. Risk Neutral Valuation:
Under the assumption of no arbitrage, standard arbitrage pricing
theory implies that there exists an equivalent probability Q such
that the present values of the zero-coupon bonds are computed
by discounting at the spot rate of interest and then taking an
expectation with respect to Q. That is,
124
125. Coupon Bonds:
Coupon-bearing government and corporate debt
The price of the default free coupon bond
n
B(t,T ) C p(t,t )
j 1 tj j
The price of a risky coupon-bearing bon
n
B (t,T ) C v(t,t )
j 1 tj j
125
127. (Spot Rate Evaluation)
Single factor model with deterministic volatilities
Gaussian, the extended Vasicek Model
(Market Index Evaluation)
Geometric Brownian motion with drift r(t) and volatility a m(t)
Z(t): a measure of the cumulative excess return per unit of risk
(above the spot rate of interest) on the equity market index.
j: Correlation coefficient between the return on the market index
and changes in the spot rate.
Given the evolutions of the state variables, we next need to
specify their relationship to the bankruptcy parameters, the
recovery rate and the liquidity discount.
127
128. (Expected Loss: A Function of the Spot Rate and the Market
Index)
(1 (t ))(t ) ma xa a r(t ) a Z (t ),0]
[
01 2
(t ) where
a ,a ,a , are constants.
012
Given these expressions, it is shown that the default free zero-
coupon and the risky zero-coupon bond’s price can be written
as:
128
130. (Liquidity Discount)
Substitution of the above expression into the risky coupon bond
price formula completes the empirical specification of the
reduced form credit risk model.
130
131. Description of Data
University of Houston’s Fixed Income Data Base - monthly bid
prices (Lehman Brothers)
U.S. Treasury securities: (all outstanding bills, notes and bonds
included)
Over 2 million entries
Filtered the data
Used remaining 29,100 entries
Corporate bond prices:
Excluded issues contained embedded options
The time period covered: May 1991 – March 1997.
Twenty different firms chosen to stratify various industry groupings:
financial, food and beverages, petroleum, airlines, utilities, department
stores, and technology (Table 1)
For the equity market index, we used the S&P 500 index with
daily observations obtained from CRSP.
131
132. Ticker SIC First Date Last Date Number Moodies S&P
Symbol Code used in the used in the of
Estimation Estimation Bonds
Financials
SECURITY PACIFIC CORP spc 6021 12/31/1991 07/31/1994 7 A3 A
FLEET FINANCIAL GROUP flt 6021 12/31/1991 10/31/1996 3 Baa2 BBB+
BANKERS TRUST NY bt 6022 01/31/1994 04/30/1994 3 A1 AA
MERRILL LYNCH & CO mer 6211 12/31/1991 03/31/1997 14 A2 A
Food & Beverages
PEPSICO INC pep 2086 12/31/1991 03/31/1997 8 A1 A
COCA - COLA cce 2086 12/31/1991 06/30/1994 3 A2 AA-
ENTERPRISES INC
Airlines
AMR CORPORATION amr 4512 02/29/1992 08/31/1994 2 Baa1 BBB+
SOUTHWEST AIRLINES luv 4512 05/31/1992 03/31/1997 3 Baa1 A-
CO
Utilities
CAROLINA POWER + cpl 4911 08/31/1992 01/31/1993 3 A2 A
LIGHT
TEXAS UTILITIES ELE CO txu 4911 04/30/1994 03/31/1997 4 Baa2 BBB
Petroleum
MOBIL CORP mob 2911 12/31/1991 02/29/1996 3 Aa2 AA
UNION OIL OF ucl 2911 12/31/1991 03/31/1997 6 Baa1 BBB
CALIFORNIA
SHELL OIL CO suo 2911 03/31/1992 02/28/1995 5 Aaa AAA
Department Stores
SEARS ROEBUCK + CO s 5311 12/31/1991 08/31/1996 7 A2 A
DAYTON HUDSON CORP dh 5311 04/30/1993 03/31/1997 2 A3 A
WAL-MART STORES, INC wmt 5331 12/31/1991 03/31/1997 3 Aa3 AA
Technology
EASTMAN KODAK ek 3861 01/31/1992 09/30/1994 3 A2 A-
COMPANY
XEROX CORP xrx 3861 12/31/1991 03/31/1997 4 A2 A
TEXAS INSTRUMENTS txn 3674 10/31/1992 03/31/1997 3 A3 A
INTL BUSINESS ibm 3570 01/31/1994 03/31/1997 3 A1 AA-
MACHINES
Table 1: Details of the Firms Included in the Empirical Investigation.
Ticker Symbol is the firm’s ticker symbol. SIC is the Standard Industry Code. Number of Bonds is the
number of the firm’s different senior debt issues outstanding on the first date used in the estimation.
Moodies refers to Moodies’ debt rating for the company’s senior debt on the first date used in the
estimation. S&P refers to S&P’s debt rating for the company’s debt on the first date used in the estimation.
132
133. Estimation of the State Variable Process Parameters
(A) Spot Rate Process Parameter Estimation
The inputs to the spot rate process evolution:
The forward rate curves (f(t,T) for all months, e.g. Jan 1975 –
March 1997)
The spot rate parameters (a, a r)
133
135. (Estimation of the forward rate curve) two-step procedure is utilized
(1) for a given time t,
choose (p(t,T) for all relevant T max{ : i I } )
T
t
i
2
bid
B (t,T ) B (t,T )
to minimize i i ii
iI
t
(2) fit a continuous forward rate curve to the estimated zero-coupon bond prices
(p(t,T) for all T max{ : i I } ) - Janosi' 00 (Figure 1)
T
i
• For ∆ = 1/12 (a month), the expression is:
2
( )
aT t
/ a 2 .
2
t
,T )/ P(t,T )) r(t )] e 1
var [log(P(t
rt t
t
• compute the sample variance, denoted v , using T ∈ {3 months, 6
tT
months, 1 year, 5 years, 10 years, 30 years}
estimate the parameters ( ,a )
•
rt t
135
136. (B) Market Index Parameter Estimation
Using the daily S&P 500 index price data and the 3-month T-bill
spot rate data:
The parameters of the market index process -- (a m, a)
The cumulative excess return on the market index – (Z(t))
For a given date t, e.g May 24, 1990 – March 31, 1997
Go back in time 365 days and estimate the time dependent sample
variance and correlation coefficients using the sample moments
(a mt , a t)
σ mt = vart M (t ) − M (t − ∆) 1
2
∆
M (t − ∆)
and
ϕt = corrt M (t ) − M (t − ∆) , r (t ) − r (t − ∆)
M (t − ∆)
136
137. (C) Default and Liquidity Discount Parameter Estimation
a (1 ),a (1 ),a (1 )
The default parameters are
0 0 1 1 2 2
The liquidity discount parameters are
0, 1, 2, 3
choose (a ,a ,a , , , , ) to
0t 1t 2t 0t 1t 2t 3t
2
bid
minimize B (t,T ) B (t,T )
i i
iI li li
t
s.t. a 0
0t
137
139. Figure 1a: Liquidity Discount: exp(-γ (t,T))
1.08
Model 1
Model 2
1.06
Model 3
Model 4
1.04 Model 5
1.02
1
0.98
Dec91 Jan93 Nov93 Sep94 Jul95 May96 Mar97
Figure 1b: Expected Loss: a(t)=a0+a1r(t)+a2Z(t)
0.04
Model 1
Model 2
0.03
Model 3
Model 4
0.02 Model 5
0.01
0
-0.01
Dec91 Jan93 Nov93 Sep94 Jul95 May96 Mar97
Figure 1: Time Series Estimates of Xerox’s Liquidity Discount and Expected Loss (per
unit time) from December 1991 to March 1997.
139
144. Conclusion
This paper provides an empirical investigation of a reduced form
credit risk model that includes both liquidity risk and correlated
defaults.
Based on both in- and out- of sample, the evidence supports the
importance of including a liquidity discount into a credit risk
model to capture liquidity risk.
The model fits the data quite well.
The default intensity appears to depend on the spot rate of
interest, but not a market index.
The liquidity discount appears to be firm specific, and not market
wide.
144
145. Estimating Default Probabilities Implicit in Equity Prices
(Janosi, Jarrow, Yildirim’03)
GOAL:
To estimate default probabilities using equity prices in conjunction
with a reduced form modeling approach.
RESULTS:
First, the best performing intensity model depends on the spot rate
of interest but not an equity market index.
Second, due to the large variability of equity prices, the point
estimates of the default intensities obtained are not very reliable.
Third, we find that equity prices contain a bubble component not
captured by the Fama-French (1993,1996) four-factor model for
equity’s risk premium.
Fourth, we compare the estimates of the intensity process obtained
here with those obtained using debt prices from Janosi, Jarrow,
Yildirim (2000) for the same fifteen firms over the same time period.
The hypothesis that these two intensity functions are equivalent
cannot be rejected.
145
149. To obtain an empirical formulation of above model, more
structure needs to be imposed on the stochastic nature of the
economy.
Consider an economy that is Markov in three state variables:
Spot rate of interest
The cumulative excess return on an equity index
Liquidating dividend process
149
154. Unfortunately, observing only a single value for the stock price
at each date leaves this system under determined as there are
more unknowns (L(t),λ0,λ1,λ2) than there are observables (ζ(t)).
To overcome this situation, we use Liquidation Value Evolution
in conjunction with expression above to transform ζ(t)
expression into a time series regression.
154
155. This is a generalization of the typical asset-pricing model to include a firm’s default parameters.
This expression forms the basis for our empirical estimation in the subsequent sections.
The first interpretation of expression above is that it is equivalent to a reduced form credit risk model for the firm’s equity.
155
156. Data
Firm equity data are obtained from CRSP.
For equity market index, the S&P 500 index is used.
For estimating an equity risk premium, we will employ the Fama-
French benchmark portfolios (book-to-market factor (HML),
small firm factor (SMB)), and a momentum factor (UMD). These
monthly portfolio returns were obtained from Ken French’s
webpage
U.S. Treasury securities are obtained from University Houston’s
Fixed Income Database.
The time period covered: May 1991-March 1997.
The same twenty firms as in Janosi, Jarrow, Yildirim (2002)
were initially selected for analysis.
156
168. Conclusion
First, equity prices can be used to infer a firm’s default
intensities.
Second, due to the noise present in equity prices, the point
estimates of the default intensities that are obtained are not very
precise.
Third, equity prices appear to contain a bubble component, as
proxied by the firm’s P/E ratio.
Fourth, we compare the default probabilities obtained from
equity with those obtained implicitly from debt prices using the
reduced form model contained in Janosi, Jarrow, Yildirim
(2000). We find that due to the large standard errors of the
equity price estimates, one cannot reject the hypothesis that
these default intensities are equivalent.
168
170. Outline
1 Introduction to Credit Risk
2 Statistical Techniques for Analyzing Default
3 Structural Modelling of Credit Risk
4 Intensity – Based Modelling of Credit Risk
5 Credit Derivatives
170
171. Outline
Overview of the credit derivatives market
Credit derivative valuation
Valuation of CDS
“A Simple Model for Valuing Default Swaps when both Market
and Credit Risk are Correlated”, Journal of Fixed Income, 11 (4),
March 2002 (Robert Jarrow and Yildiray Yildirim)
Valuation of Bond Insurance
Valuation of CLN
171
172. Overview of the credit derivatives market
Market risk: changes in value associated with unexpected
changes in market prices or rates
Credit risk: changes in value associated with unexpected
changes in credit quality (such as credit rating change,
restructuring, failure to pay, bankruptcy)
Liquidity risk: the risk of increased costs, or inability, to adjust
financial positions, or lost of access to credit
Operational risk: fraud, system failures, trading errors (such
as deal mispricing)
172
173. Interest rate risk is isolated via interest rate swaps
Credit risk via credit derivatives
Exchange rate risk via foreign exchange derivatives
These risks can separately sold to those willing to bear them.
From microeconomic point of view, this should result in an
increase in allocation efficiency.
Credit derivative transfer the credit risk contained in a loan
from the protection buyer to the protection seller without
effecting the ownership of the underlying asset. Essentially, it is
a security with a payoff linked to credit related event.
173
174. Using financial/credit instruments to provide protection against
default risk is not new. Letters of credit or bank guarantees have
been applied for some time. However, credit derivatives show a
number of differences:
1. Their construction is similar to that of financial derivatives, trading
takes place separately from the underlying asset.
2. Credit derivatives are regularly traded. This quarantees a regular
marking to market of the relevant positions.
3. Trading takes place via standardized contracts prepared by the
International Swaps and Derivatives Association (ISDA).
174
175. CLNs and
Asset sw aps
Repacks
12%
9% Credit Spread
opt ions
Synt het ic 3%
Source: British
Securit isat ions
$tri Bankers Association
26%
2.0
1.5
1.0
0.5
0.0
Credit Def ault 96 97 98 99 00 01 02
Sw aps
Basket def ault
45%
sw aps
5%
Source: Risk, Feb 2001
175
176. Depending on the need, we have different kind of credit
derivatives. The following products are regularly used:
Total Return Swap
pays all
cashflows
based on its
asset Protection
Protection
Seller
Buyer
(risk buyer)
(risk seller)
pays
periodic
interest
payments
e.g.
Reference Security
LIBOR+20bp
• Bond/Loan
176
177. Asset Swap
The investor in the asset swap basically swaps fixed coupon
receipts into floating receipts to eliminate interest rate risk, but
kept the credit risk and earns 20 b.p. as compensation.
pays libor
Protection
Protection
Seller
Buyer
(risk buyer)
(risk seller)
e.g. pays
fixed – 20bp
Pays fixed
Pays libor
3rd party
177
178. Credit Default Swap (CDS)
Protection buyer has $50M Exxon bond (=notional amount)
He is willing to pay 120bp annually as a fraction of notional amount for 5 years
120bp*$50M=60,000.
Total Payment=$300,000 if no default
If default you recovery 55% of the notional amount $27,500,000
If 55% is set at the beginning, then you know how much you can get back, it is
fixed in case of default. This is called “Default Digital Swap=DDS”.
pays a fixed
periodic fee
Protection
Protection
Seller
Buyer
(risk buyer)
(risk seller)
pays
contingent
on credit event
Reference Security
• Bond/Loan
178
179. Basket Default Swap (BDS)
Also called first-to-default or kth-to-default basket credit default
swap. If any of the loans first defaults, then protection seller will pay
the par amount of the bond in default and collect the bond in default
for whatever recovery available to seller.
pays a fixed
periodic fee
Protection
Protection
Seller
Buyer
(risk buyer)
(risk seller)
pays
contingent
on credit event
Reference Security
• Portfolio (basket) of
assets
179
180. Credit Spread Swap (CSS)
pays a fixed
periodic fee
Protection
Protection
Buyer
Seller
(risk seller)
(risk buyer)
pays periodic
variable
rate=spread
bond over
Treasury
Reference Security
• Bond
180
181. Credit Spread Option (CSO)
A derivative on the spread between the default-risky asset and the
bank liability curve.
This product provides protection against both the credit event and
also any other changes in the spread.
EXAMPLE (Call Option)
At exercise, $ payoff = MAX { 0 , (Final spread - Strike spread) }
x notional principal x term
For buyer of spread call, gain when spread increases above strike
(in –the-money). No gain, just lose premium otherwise.
181
182. The above products are off-balance-sheet derivatives. One can
repackage them to create new tradable securities. One example
is CLN.
Credit-Linked Note (CLN)
In CDS, the protection buyer is exposed to the risk of default of the
protection seller. Note that the seller can, in principle, sell protection
without coming up with any funds. Therefore, we refer to the default
swap transactions as “unfunded” transfers of credit risk.
CLN is a “funded” alternative to this transaction. Here the
protection seller buys a bond, called CLN, from the protection buyer
and the CF on this bond is linked to the performance of a reference
issuer. If the reference issuer defaults, the payment on the CLN
owner is reduced.
182
183. pays CF linked
to CF of
Protection
Protection underlying asset
Seller
Buyer
(risk buyer)
(risk seller)
buys CLN
If reference issuer is defaulted,
payment to CLN is reduced.
Reference Security
• Bond
183
184. Collateralized Debt Obligations (CDOs)
These are the portfolio products. We have two different types of
CDOs
1. Collateralized Loan Obligations (CLOs) Senior
2. Collateralized Bond Obligations (CBOs) Aaa/AAA to
interest/ A-/A3
principal
asset sold Special
Mezzanine
Purpose
Bank
BBB+/Baa to
Vehicle
funds B-/B3
funds
(SPV)
Equity
A Special Purpose Vehicle (SPV) is setup to hold the portfolio of assets. It
issues notes with different subordination, so called tranches, then sells to
investors. The principal payment and interest income (LIBOR + spread) are
allocated to the notes according to the following rule: senior notes are paid
before mezzanine and lower rated notes. Any residual cash is paid to the equity
note
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185. Credit Derivative Valuation
Derivatives are the securities written against the underlying
asset. Below is a bond insurance, CDS:
pays a fixed
X% Protection
Protection Net coupon payment=(C- X)%
Seller
Buyer Whether defaulted or not.
(risk buyer)
(risk seller)
pays $(1000-V)
if defaulted
$V if C%
defaulted
Reference Security
Bond with notional
payment of $1000
185
186. Credit Derivative Valuation
Derivatives are the securities written against the underlying
asset.
pays a fixed
X% Protection
Protection Net coupon payment=(C- X)%
Seller
Buyer Whether defaulted or not.
(risk buyer)
(risk seller)
pays $(1000-V)
if defaulted
$V if C%
defaulted
Reference Security Payout from defaultable bond +CDS
Bond with notional
payment of $1000 Payout from investing in riskless bond
186
188. A Simple Model for Valuing Default Swaps when
both Market and Credit Risk are Correlated
(Jarrow, Yildirim’03)
CDS pricing (Theory/Estimation)
The existing literature investigating the valuation of default swaps (see
Hull and White [2000, 2001], Martin, Thompson and Browne [2000],
Wei [2001], and the survey paper by Cheng [2001]) gives the
impression that simple models for pricing default swaps are only
available when credit and market risk are statistically independent.
This paper provides a simple analytic formula for valuing default swaps
with correlated market and credit risk .
We illustrate the numerical implementation of this model by inferring the
default probability parameters implicit in default swap quotes for twenty
two companies over the time period 8/21/00 to 10/31/00.
188
189. The data used for this investigation was downloaded from
Enron’s web site. The twenty-two different firms were chosen to
stratify various industry groupings: financial, food and
beverages, petroleum, airlines, utilities, department stores, and
technology.
For comparison purposes, the standard model with statistically
independent market and credit risk (a special case of our model)
is also calibrated to this market data.
One can also easily calibrate our simple model with correlated
market and credit risk to exactly match the observed default
swap quote term structure.
189