Transcript Lecture 4
MGT 821/ECON 873
Credit Risk and Credit Derivatives
1
Part 1:
Credit Risk
2
Credit Ratings
In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, CCC, CC, and C The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B, Caa, Ca, and C Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade” 3
Historical Data
Historical data provided by rating agencies are also used to estimate the probability of default 4
Cumulative Ave Default Rates (%)
(1970-2006, Moody’s) Aaa Aa A Baa Ba B Caa-C 1 0.000 0.008 2 0.000 0.019 3 0.000 0.042 4 0.026 0.106 5 0.099 0.177 7 0.251 0.343 10 0.521 0.522 0.021 0.181 1.205 0.095 0.506 3.219 0.220 0.930 5.568 0.344 1.434 7.958 0.472 1.938 0.759 2.959 1.287 4.637 10.215 14.005 19.118 5.236 11.296 17.043 22.054 26.794 34.771 43.343 19.476 30.494 39.717 46.904 52.622 59.938 69.178 5
Interpretation
The table shows the probability of default for companies starting with a particular credit rating A company with an initial credit rating of Baa has a probability of 0.181% of defaulting by the end of the first year, 0.506% by the end of the second year, and so on 6
Do Default Probabilities Increase with Time?
For a company that starts with a good credit rating default probabilities tend to increase with time For a company that starts with a poor credit rating default probabilities tend to decrease with time 7
Default Intensities vs Unconditional Default Probabilities
The default intensity (also called hazard rate) is the probability of default for a certain time period conditional on no earlier default The unconditional default probability is the probability of default for a certain time period as seen at time zero What are the default intensities and unconditional default probabilities for a Caa rate company in the third year?
8
Default Intensity (Hazard Rate)
The default intensity (hazard rate) that is usually quoted is an instantaneous If
V
(
t
) is the probability of a company surviving to time
t V
(
t
t
)
V
(
t
) (
t
)
V
(
t
) This leads to
V
(
t
)
e
0
t
(
t
)
dt
The cumulative probabilit y of default by time
t
is
Q
(
t
) 1
e
(
t
)
t
9
Recovery Rate
The recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face value 10
Recovery Rates
(Moody’s: 1982 to 2006) Class Senior Secured Senior Unsecured Senior Subordinated Subordinated Junior Subordinated Mean(%) 54.44 38.39 32.85 31.61 24.47 11
Estimating Default Probabilities
Alternatives: Use Bond Prices Use CDS spreads Use Historical Data Use Merton’s Model 12
Using Bond
Average default intensity over life of bond is approximately 1
s R
where
s
is the spread of the bond’s yield over the risk-free rate and
R
is the recovery rate 13
More Exact Calculation
Assume that a five year corporate bond pays a coupon of 6% per annum (semiannually). The yield is 7% with continuous compounding and the yield on a similar risk free bond is 5% (with continuous compounding) Price of risk-free bond is 104.09; price of corporate bond is 95.34; expected loss from defaults is 8.75
Suppose that the probability of default is
Q
per year and that defaults always happen half way through a year (immediately before a coupon payment. 14
Calculations
Time (yrs) 0.5
Def Prob
Q
Recovery Amount 40 Risk-free Value 106.73
1.5
2.5
Q Q
40 40 105.97
105.17
3.5
4.5
Total
Q Q
40 40 104.34
103.46
LGD 66.73
65.97
65.17
64.34
63.46
Discount Factor 0.9753
0.9277
0.8825
0.8395
0.7985
PV of Exp Loss 65.08
Q
61.20
Q
57.52
Q
54.01
Q
50.67
Q
288.48
Q
15
Calculations
We set 288.48
Q
= 8.75 to get
Q
= 3.03% This analysis can be extended to allow defaults to take place more frequently With several bonds we can use more parameters to describe the default probability distribution 16
The Risk-Free Rate
The risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero rate (or sometimes 10 bps below the LIBOR/swap rate) To get direct estimates of the spread of bond yields over swap rates we can look at asset swaps 17
Real World vs Risk-Neutral Default Probabilities
The default probabilities backed out of bond prices or credit default swap spreads are risk neutral default probabilities The default probabilities backed out of historical data are real-world default probabilities 18
A Comparison
Calculate 7-year default intensities from the Moody’s data (These are real world default probabilities) Use Merrill Lynch data to estimate average 7-year default intensities from bond prices (these are risk-neutral default intensities) Assume a risk-free rate equal to the 7-year swap rate minus 10 basis point 19
Real World vs Risk Neutral Default Probabilities, 7 year averages
Rating Aaa Aa A Baa Ba B Caa-C Real-world default probability per yr (% per annum) 0.04 0.05 0.11 0.43 2.16 6.10 13.07 Risk-neutral default probability per yr (% per year) 0.60 0.74 1.16 2.13 4.67 7.97 18.16 Ratio 16.7 14.6 10.5 5.0 2.2 1.3 1.4 Difference 0.56 0.68 1.04 1.71 2.54 1.98 5.50 20
Risk Premiums Earned By Bond Traders
Rating Aaa Aa A Baa Ba B Caa Bond Yield Spread over Treasuries (bps) 78 87 112 170 323 521 1132 Spread of risk-free rate used by market over Treasuries (bps) 42 42 42 42 42 42 42 Spread to compensate for default rate in the real world (bps) 2 4 7 26 129 366 784 Extra Risk Premium (bps) 34 42 63 102 151 112 305 21
Possible Reasons for These Results
Corporate bonds are relatively illiquid The subjective default probabilities of bond traders may be much higher than the estimates from Moody’s historical data Bonds do not default independently of each other. This leads to systematic risk that cannot be diversified away.
Bond returns are highly skewed with limited upside. The non-systematic risk is difficult to diversify away and may be priced by the market 22
Which World Should We Use?
We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of default We should use real world estimates for calculating credit VaR and scenario analysis 23
Merton’s Model
Merton’s model regards the equity as an option on the assets of the firm In a simple situation the equity value is max(
V T
−
D
, 0) where
V T
is the value of the firm and debt repayment required
D
is the 24
Equity vs. Assets
An option pricing model enables the value of the firm’s equity today,
E
0 , to be related to the value of its assets today,
V
0 , and the volatility of its assets, s
V E
0 ( 1 )
De
rT
2 ) where
d
1 ln (
V
0
D
) (
r
s 2
V
s
V T
2 )
T
;
d
2
d
1 s
V T
25
Volatilities
s
E E
0
E
V
s
V V
0 ( 1 ) s
V V
0 This equation together with the option pricing relationship enables
V
0 be determined from
E
0 and s
E
and s
V
to 26
Example
A company’s equity is $3 million and the volatility of the equity is 80% The risk-free rate is 5%, the debt is $10 million and time to debt maturity is 1 year Solving the two equations yields
V
0 =12.40 and s
v
=21.23% 27
Example continued
The probability of default is
N
(-
d
2 ) or 12.7% The market value of the debt is 9.40
The present value of the promised payment is 9.51
The expected loss is about 1.2% The recovery rate is 91% 28
Estimating volatility and asset value
Iteration MLE 29
The Implementation of Merton’s Model
Choose time horizon Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote it by
D
Use Merton’s model to calculate a theoretical probability of default Use historical data or bond data to develop a one-to-one mapping of theoretical probability into either real-world or risk-neutral probability of default.
30
Credit Risk in Derivatives Transactions
Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability 31
General Result
Assume that default probability is independent of the value of the derivative
f
Consider times
t
1 ,
t
2 ,…
t n q i
and default probability is at time
t i
. The value of the contract at time
t i
is
i
and the recovery rate is
R
The loss from defaults at time
t i
is
q i
(1-
R
)
E
[ max (
f i
, 0 )]. Defining
u i
=
q i
( 1 -
R
) and
v i
as the value of a derivative that provides a payoff of max(
f i
, 0 ) at time
t i
, the cost of defaults is
n i
1
u i v i
32
If Contract Is Always a Liability
f
0 *
f
0
e
(
y
*
y
)
T
where derivative provides a payoff at time
T
.
f
0 * is the actual value of the derivative and
f
0 is the default free value.
y
* is the yield on zero coupon bonds issued by the seller of the derivative and
y
is the risk free yield for this maturity 33
Credit Risk Mitigation
Netting Collateralization Downgrade triggers 34
Default Correlation
The credit default correlation between two companies is a measure of their tendency to default at about the same time Default correlation is important in risk management when analyzing the benefits of credit risk diversification It is also important in the valuation of some credit derivatives, eg a first-to-default CDS and CDO tranches. 35
Measurement
There is no generally accepted measure of default correlation Default correlation is a more complex phenomenon than the correlation between two random variables 36
Binomial Correlation Measure
One common default correlation measure, between companies
i
and
j
is the correlation between A variable that equals 1 if company
i
between time 0 and time
T
defaults and zero otherwise A variable that equals 1 if company
j
between time 0 and time
T
defaults and zero otherwise The value of this measure depends on
T
. Usually it increases at
T
increases.
37
Binomial Correlation
continued Denote
Q i
(
T
) as the probability that company
A
will default between time zero and time
T
, and
P ij
(
T
) that both
i
and
j
as the probability will default. The default correlation measure is
ij
(
T
)
P ij
(
T
)
Q i
(
T
)
Q j
(
T
) [
Q i
(
T
)
Q i
(
T
) 2 ][
Q j
(
T
)
Q j
(
T
) 2 ] 38
Survival Time Correlation
Define
t i
and
Q i
(
t i
) as the time to default for company
i
as the probability distribution for
t i
The default correlation between companies
i
and
j
can be defined as the correlation between
t i
and
t j
But this does not uniquely define the joint probability distribution of default times 39
Gaussian Copula Model
Define a one-to-one correspondence between the time to default,
t i
, of company
i
and a variable
x i
by
Q i
(
t i
) =
N
(
x i
) or
x i
=
N
-1 [
Q
(
t i
)] where
N
is the cumulative normal distribution function. This is a “percentile to percentile” transformation. The
p
percentile point of the
Q i p
percentile point of the
x
i normal distribution distribution is transformed to the distribution.
x i
has a standard We assume that the
x i
are multivariate normal. The default correlation measure, r
ij
between companies
i
and
j
is the correlation between
x i
and
x j
40
Binomial vs Gaussian Copula Measures
The measures can be calculated from each other
P ij
(
T
)
M
[
x i
,
x j
; r
ij
] so that
ij
(
T
)
M
[
x i
,
x j
; r
ij
]
Q i
(
T
)
Q j
(
T
) [
Q i
(
T
)
Q i
(
T
) 2 ][
Q j
(
T
)
Q j
(
T
) 2 ] where
M
is the cumulative probabilit y distributi on bivariate function normal 41
Comparison
The correlation number depends on the correlation metric used Suppose
T
of r
ij
= 1,
Q i
(
T
ij
(
T
) equal to 0.024.
) =
Q j
(
T
) = 0.01, a value equal to 0.2 corresponds to a value of In general
ij
(
T
) < r
ij
and
ij
(
T
) is an increasing function of
T
42
Example of Use of Gaussian Copula
Suppose that we wish to simulate the defaults for
n
companies . For each company the cumulative probabilities of default during the next 1, 2, 3, 4, and 5 years are 1%, 3%, 6%, 10%, and 15%, respectively 43
Use of Gaussian Copula
continued We sample from a multivariate normal distribution to get the
x i
Critical values of
x i
are
N
-1 (0.01) = -2.33,
N
-1 (0.03) = -1.88,
N
-1 (0.06) = -1.55,
N
-1 (0.10) = -1.28,
N
-1 (0.15) = -1.04
44
Use of Gaussian Copula
continued When sample for a company is less than -2.33, the company defaults in the first year When sample is between -2.33 and -1.88, the company defaults in the second year When sample is between -1.88 and -1.55, the company defaults in the third year When sample is between -1,55 and -1.28, the company defaults in the fourth year When sample is between -1.28 and -1.04, the company defaults during the fifth year When sample is greater than -1.04, there is no default during the first five years 45
A One-Factor Model for the Correlation Structure
x i
a i F
1
a i
2
Z i
The correlation between
x i
and
x j
is
a i a j
The
i
th company defaults by time
x i
<
N
-1 [
Q
i (
T
)] or
Z i
N
1 [
Q i
(
T
)
a i F
] 1
a i
2
T
when Conditional on
F
the probability of this is
Q i
(
T F
)
N
N
1
Q i
(
T
1 )
a i
2
a i F
46
Credit VaR
Can be defined analogously to Market Risk VaR A
T
-year credit VaR with an
X
% confidence is the loss level that we are
X
% confident will not be exceeded over
T
years 47
Calculation from a Factor-Based Gaussian Copula Model
Consider a large portfolio of loans, each of which has a probability of
Q
(
T
) of defaulting by time
T
. Suppose that all pairwise copula correlations are r so that all
a i
’s are r We are
X
= −
N
-1 (
X
) % certain that
F
is less than
N
-1 (1 −
X
) It follows that the VaR is
V
(
X
,
T
)
N N
1
Q
(
T
) 1 r r
N
1 (
X
) 48
Basel II
The internal ratings based approach uses the Gaussian copula model to calculate the 99.9% worst case default rate for a portfolio This is multiplied by the loss given default (=1−Rec Rate), the expected exposure at default, and a maturity adjustment to give the capital required 49
CreditMetrics
Calculates credit VaR by considering possible rating transitions A Gaussian copula model is used to define the correlation between the ratings transitions of different companies 50
Credit Derivatives
51
Credit Default Swaps
A huge market with over $40 trillion of notional principal Buyer of the instrument acquires protection from the seller against a default by a particular company or country (the reference entity) Example: Buyer pays a premium of 90 bps per year for $100 million of 5-year protection against company X Premium is known as the
credit default spread
. It is paid for life of contract or until default If there is a default, the buyer has the right to sell bonds with a face value of $100 million issued by company X for $100 million (Several bonds are typically deliverable) 52
CDS Structure
Default Protection Buyer, A 90 bps per year Payoff if there is a default by reference entity=100(1 R ) Default Protection Seller, B Recovery rate,
R
, is the ratio of the value of the bond issued by reference entity immediately after default to the face value of the bond 53
Other Details
Payments are usually made quarterly in arrears In the event of default there is a final accrual payment by the buyer Settlement can be specified as delivery of the bonds or in cash Suppose payments are made quarterly in the example just considered. What are the cash flows if there is a default after 3 years and 1 month and recovery rate is 40%?
54
Attractions of the CDS Market
Allows credit risks to be traded in the same way as market risks Can be used to transfer credit risks to a third party Can be used to diversify credit risks 55
Using a CDS to Hedge a Bond
Portfolio consisting of a 5-year par yield corporate bond that provides a yield of 6% and a long position in a 5-year CDS costing 100 basis points per year is (approximately) a long position in a riskless instrument paying 5% per year 56
Valuation Example
Cconditional on no earlier default a reference entity has a (risk-neutral) probability of default of 2% in each of the next 5 years. (This is a default intensity) Assume payments are made annually in arrears, that defaults always happen half way through a year, and that the expected recovery rate is 40% Suppose that the breakeven CDS rate is
s
per dollar of notional principal 57
Unconditional Default and Survival Probabilities
Time (years) 1 2 3 4 5 Default Probability 0.0200
0.0196
0.0192
0.0188
0.0184
Survival Probability 0.9800
0.9604
0.9412
0.9224
0.9039
58
Calculation of PV of Payments
(Principal=$1) Time (yrs) 1 2 3 4 5 Total Survival Prob 0.9800
0.9604
0.9412
0.9224
0.9039
Expected Paymt 0.9800
s
0.9604
s
0.9412
s
0.9224
s
0.9039
s
Discount Factor 0.9512
0.9048
0.8607
0.8187
0.7788
PV of Exp Pmt 0.9322
s
0.8690
s
0.8101
s
0.7552
s
0.7040
s
4.0704
s
59
Present Value of Expected Payoff ( Principal = $1) Time (yrs) Default Probab.
Rec. Expected Rate Payoff Discount Factor PV of Exp. Payoff 0.5
1.5
2.5
3.5
4.5
Total 0.0200
0.0196
0.0192
0.0188
0.0184
0.4
0.4
0.4
0.4
0.4
0.0120
0.0118
0.0115
0.0113
0.0111
0.9753
0.9277
0.8825
0.8395
0.7985
0.0117
0.0109
0.0102
0.0095
0.0088
0.0511
60
PV of Accrual Payment Made in Event of a Default. ( Principal = $1) PV of Pmt Time 0.5
1.5
2.5
3.5
4.5
Total Default Prob 0.0200
0.0196
0.0192
0.0188
0.0184
Expected Accr Pmt 0.0100
s
0.0098
s
0.0096
s
0.0094
s
0.0092
s
Disc Factor 0.9753
0.9277
0.8825
0.8395
0.7985
0.0097
s
0.0091
s
0.0085
s
0.0079
s
0.0074
s
0.0426
s
61
Putting it all together
PV of expected payments is 4.0704
s
+0.0426
s
= 4.1130
s
The breakeven CDS spread is given by 4.1130
s
= 0.0511 or
s
= 0.0124 (124 bps) The value of a swap negotiated some time ago with a CDS spread of 150bps would be 4.1130
×0.0150−0.0511 or 0.0106 times the principal.
62
Implying Default Probabilities from CDS spreads
Suppose that the mid market spread for a 5 year newly issued CDS is 100bps per year We can reverse engineer our calculations to conclude that the default intensity is 1.61% per year.
If probabilities are implied from CDS spreads and then used to value another CDS the result is not sensitive to the recovery rate providing the same recovery rate is used throughout 63
Other Credit Derivatives
Binary CDS First-to-default Basket CDS Total return swap Credit default option Collateralized debt obligation 64
Binary CDS
The payoff in the event of default is a fixed cash amount In our example the PV of the expected payoff for a binary swap is 0.0852 and the breakeven binary CDS spread is 207 bps 65
Credit Indices
CDX NA IG is a portfolio of 125 investment grade companies in North America itraxx Europe is a portfolio of 125 European investment grade names The portfolios are updated on March 20 and Sept 20 each year The index can be thought of as the cost per name of buying protection against all 125 names The way the index is traded is more complicated (See Example 23.1, page 534) 66
CDS Forwards and Options
Example: European option to buy 5 year protection on Ford for 280 bps starting in one year. If Ford defaults during the one-year life of the option, the option is knocked out Depends on the volatility of CDS spreads 67
Basket CDS
Similar to a regular CDS except that several reference entities are specified In a first to default swap there is a payoff when the first entity defaults Second, third, and
n
th to default deals are defined similarly Why does pricing depends on default correlation?
68
Total Return Swap
Agreement to exchange total return on a portfolio of assets for LIBOR plus a spread At the end there is a payment reflecting the change in value of the assets Usually used as financing tools by companies that want an investment in the assets Total Return Payer Total Return on Assets LIBOR plus 25bps Total Return Receiver 69
Asset Backed Securities
Security created from a portfolio of loans, bonds, credit card receivables, mortgages, auto loans, aircraft leases, music royalties, etc Usually the income from the assets is tranched A “waterfall” defines how income is first used to pay the promised return to the senior tranche, then to the next most senior tranche, and so on.
70
Possible Structure
(Figure 23.3) Asset 1 Asset 2 Asset 3 Asset n Principal=$100 million SPV Tranche 1 (equity) Principal=$5 million Yield = 30% Tranche 2 (mezzanine) Principal=$20 million Yield = 10% Tranche 3 (super senior) Principal=$75 million Yield = 6% 71
The Mezzanine Tranche is Most Difficult to Sell…
Subprime Mortgage Portfolio Equity Tranche (5%) Not Rated Mezzanine Tranche (20%) BBB Super Senior Tranche (75%) AAA The mezzanine tranche is repackaged with other similar mezzanine tranches Equity Tranche (5%) Mezzanine Tranche (15%) BBB Super Senior Tranche (80%) AAA 72
The Credit Crunch
Between 2000 and 2006 mortgage lenders in the U.S. relaxed standards (liar loans, NINJAs, ARMs) Interest rates were low Demand for mortgages increased fast Mortgages were securitized using ABSs and ABS CDOs In 2007 the bubble burst House prices started decreasing. Defaults and foreclosures, increased fast.
73
Collateralized Debt Obligations
A cash CDO is an ABS where the underlying assets are corporate debt issues A synthetic CDO involves forming a similar structure with short CDS contracts on the companies In a synthetic CD0 most junior tranche bears losses first. After it has been wiped out, the second most junior tranche bears losses, and so on 74
Synthetic CDO Structure
CDS 1 CDS 2 CDS 3 Tranche 1: 5% of principal Responsible for losses between 0% and 5% Earns 1500 bps Trust Tranche 2: 10% of principal Responsible for losses between 5% and 15% Earns 200 bps CDS n Tranche 3: 10% of principal Responsible for losses between 15% and 25% Earns 40 bps Average Yield 8.5% Tranche 4: 75% of principal Responsible for losses between 25% and 75% Earns 10bps 75
Synthetic CDO Details
The bps of income is paid on the remaining tranche principal. Example: when losses have reached 7% of the principal underlying the CDSs, tranche 1 has been wiped out, tranche 2 earns the promised spread (200 basis points) on 80% of its principal 76
Single Tranche Trading
This involves trading tranches of portfolios that are unfunded Cash flows are calculated as though the tranche were funded 77
Quotes for Standard Tranches of CDX and iTraxx
Quotes are 30/360 in basis points per year except for the 0 3% tranche where the quote equals the percent of the tranche principal that must be paid upfront in addition to 500 bps per year. CDX NA IG (Mar 28, 2007): Tranche 0-3% 3-7% 7-10% Quote 26.85% 103.8
20.3
10-15% 10.3
15-30% 4.3
30-100% 2 iTraxx Europe (Mar 28, 2007) Tranche Quote 0-3% 11.25% 3-6% 57.7
6-9% 14.4
9-12% 6.4
12-22% 2.6
22-100% 1.2
78
Valuation of Synthetic CDOs and basket CDSs
A popular approach is to use a factor based Gaussian copula model to define correlations between times to default Often all pairwise correlations and all the unconditional default distributions are assumed to be the same Market likes to imply a pairwise correlations from market quotes. 79
Valuation of Synthetic CDOs and Basket CDOs
continued
Q
(
t F
)
N N
1 [
Q
(
t
)] 1 r r
F
The probability of
k
defaults from
n
names by time
t
conditional on
F
(
n
n
!
k
)!
k
!
Q
(
t
is
F
)
k
[ 1
Q
]
n
k
This enables cash flows conditional on
F
to be calculated. By integrating over
F
the unconditional distributions are obtained 80
Implied Correlations
A compound correlation is the correlation that is implied from the price of an individual tranche using the one-factor Gaussian copula model A base correlation is correlation that prices the 0 to
X
% tranche consistently with the market where
X
% is a detachment point (the end point of a standard tranche) 81
Procedure for Calculating Base Correlation
Calculate compound correlation for each tranche Calculate PV of expected loss for each tranche Sum these to get PV of expected loss for base correlation tranches Calculate correlation parameter in one-factor Gaussian copula model that is consistent with this expected loss 82
Implied Correlations for iTraxx on March 28, 2007
Tranche Compound Correlation 0-3% 18.3% 3-6% 9.3% 6-9% 14.3% 9-12% 18.2% 12-22% 24.1% Tranche Base Correlation 0-3% 18.3% 0-6% 27.3% 0-9% 34.9% 0-12% 41.4% 0-22% 58.1% 83