Credit Diversification.
Central Limit Theorem:
Let X1, X2, X3, ……, Xn be independent random variables that are identically distributed
(i.e., all have the same probability function in the discrete case or density function in the
n
continuous case) and have finite mean, , and variance, 2. Then lim S n   X n is
n 
i 1
asymptotically normal distributed.
PDF and CDF for sum of dots on a single fair die
Cumulative Frequency
Relative Frequency
1.2
0.18
0.16
1
0.14
0.8
0.12
0.1
0.6
0.08
0.06
0.4
0.04
0.2
0.02
0
0
0
2
4
6
8
0
1
2
4
6
8
PDF and CDF for sum of dots on 2 fair die
Relative Frequency
Cumulative Frequency
0.18
1.2
0.16
1
0.14
0.12
0.8
0.1
0.6
0.08
0.06
0.4
0.04
0.2
0.02
0
0
0
5
10
15
0
5
10
15
PDF and CDF for sum of dots on 3 fair die
Relative Frequency
Cumulative Frequency
0.14
1.2
0.12
1
0.1
0.8
0.08
0.6
0.06
0.4
0.04
0.2
0.02
0
0
0
5
10
15
0
20
5
10
15
A portfolio’s credit loss (CL) is the sum of each component’s credit loss. If the credit
loss of each individual credit risk is distributed binomial and credit events are
independent for components of the portfolio, the resulting distribution of portfolio credit
loss will be asymptotically normal distributed. i.e. as enough independent credit risks of
the same type (PD) are combined together the resulting distribution of portfolio credit
loss will be asymptotically normal distributed.
2
20
PDF and CDF for credit losses from 10 uncorrelated risks, CE=$100,000, PD = 0.10
Credit Loss pdf 10 uncorrelated credit risks,
CE=$100,000
Cumulative
1.2
0.45
1
0.4
0.35
0.8
0.3
0.25
0.6
0.2
0.4
0.15
0.1
0.2
0.05
$1 $0
00
,
$2 000
00
,0
$3 00
00
,
$4 000
00
,
$5 000
00
,0
$6 00
00
,
$7 000
00
,0
$8 00
00
,
$9 000
00
,0
00
L
C
C
$1 $0
00
,
$2 000
00
,
$3 000
00
,
$4 000
00
,
$5 000
00
,
$6 000
00
,
$7 000
00
,
$8 000
00
,
$9 000
00
,0
00
0
L
0
PDF and CDF for credit losses from 25 uncorrelated risks, CE=$40,000, PD = 0.10
Credit Loss pdf 25 uncorrelated credit risks,
CE=$40,000
Cumulative
1.2
0.3
1
0.25
0.8
0.2
0.6
0.15
$920,000
$840,000
$760,000
$680,000
$600,000
$520,000
$440,000
$360,000
C
L
$8
0,
00
0
$2
00
,0
00
$3
20
,0
00
$4
40
,0
00
$5
60
,0
00
$6
80
,0
00
$8
00
,0
00
$9
20
,0
00
3
$280,000
0
$200,000
0
$120,000
0.2
CL
0.05
$40,000
0.4
0.1
PDF and CDF for credit losses from 50 uncorrelated risks, CE=$20,000, PD = 0.10
Credit Loss pdf 50 uncorrelated credit risks,
CE=$20,000
Cumulative
1.2
0.2
0.18
1
0.16
0.8
0.14
0.12
0.6
0.1
0.08
0.4
0.06
0.04
0.2
0.02
0
$6 CL
0
$1 ,00
40 0
$2 ,00
20 0
$3 ,00
00 0
$3 ,00
80 0
$4 ,00
60 0
$5 ,00
40 0
$6 ,00
20 0
$7 ,00
00 0
$7 ,00
80 0
$8 ,00
60 0
$9 ,00
40 0
,0
00
C
$6 L
0,
0
$1 00
40
,
$2 000
20
,
$3 000
00
,
$3 000
80
,
$4 000
60
,
$5 000
40
,
$6 000
20
,
$7 000
00
,
$7 000
80
,
$8 000
60
,
$9 000
40
,0
00
0
PDF and CDF for credit losses from 200 uncorrelated risks, CE=$5,000, PD = 0.10
Credit Loss pdf 200 uncorrelated credit risks,
CE=$5,000
Cumulative
1.2
0.1
0.09
1
0.08
0.07
0.8
0.06
0.6
0.05
0.04
0.4
0.03
0.02
0.2
0.01
0
The greater the number of independent credit risks pooled together in a portfolio the
closer the distribution of credit losses is to the normal distribution. This is a direct result
4
$975,000
$905,000
$835,000
$765,000
$695,000
$625,000
$555,000
$485,000
$415,000
$345,000
$275,000
$205,000
$135,000
$65,000
CL
$970,000
$895,000
$820,000
$745,000
$670,000
$595,000
$520,000
$445,000
$370,000
$295,000
$220,000
$70,000
$145,000
CL
0
of the normal distribution. As the probability of default on any one credit risk approaches
PD = 1/2 , the convergence to the normal distribution is more rapid.
Chp. 19 – Measuring Actuarial Default Risk
Credit event – broader than the definition of default event
ISDA – International Swaps and Derivatives Dealers Association
http://www.isda.org/index.html
The International Swaps and Derivatives Association is the global trade association representing
participants in the privately negotiated derivatives industry, a business covering swaps and
options across all asset classes (interest rate, currency, commodity and energy, credit and
equity). ISDA was chartered in 1985, and today numbers over 625 member institutions from 47
countries on six continents. These members include most of the world's major institutions who
deal in, as well as leading end-users of, privately negotiated derivatives. The membership
includes associated service providers and consultants.
Since its inception, ISDA has pioneered efforts to identify and reduce the sources of risk in the
derivatives and risk management business. Among its most notable accomplishments are:
developing the ISDA Master Agreement; publishing a wide range of related documentation
materials and instruments covering a variety of transaction types; producing legal opinions on the
enforceability of netting and collateral arrangements (available only to ISDA members); securing
recognition of the risk-reducing effects of netting in determining capital requirements; promoting
sound risk management practices, and advancing the understanding and treatment of derivatives
and risk management from public policy and regulatory capital perspectives.
Bankruptcy
Failure to pay
Obligation/cross default
Obligation/cross acceleration
Repudiation/moratorium
Restructuring
Downgrade
Currency inconvertibility
Governmental action
The definition of a credit event is now doubly important given the increased importance
of credit derivatives, i.e. contracts which have payoffs conditional on the occurrence of a
credit event.
Standard and Poor’s and Moody’s are the largest (most rated companies/instruments)
domestic credit rating agencies.
Investment grade instruments are rated BBB or greater by Standard and Poor’s and Baa
or greater by Moody’s.
5
Credit Rating – an opinion of the future ability, legal obligation, and willingness of a
bond issuer or other obligor to make full and timely payments on principal and interest
due to investors.
Investment grade:
Highest grade
High grade
Upper medium grade
Medium grade
Speculative grade:
Lower medium grade
Speculative
Poor standing
Highly speculative
Lowest quality, no interest
In default
Modifiers


Standard & Poor’s
Moody’s Services
AAA
AA
A
BBB
Aaa
Aa
A
Baa
BB
B
CCC
CC
C
D
A+, A, A-
Ba
B
Caa
Ca
C
A1, A2, A3
High grade companies are less reliant on debt in their capital structure, i.e. smaller
total debt/equity.
High grade companies have larger cash flow per dollar of interest expense.
Cumulative default probabilities, the proportion of issuers that default within time period,
are reported by credit rating agencies.
The marginal default probability, the likelihood an issuer of a given credit quality will
default in a specific period is extracted from reported cumulative default probabilities.
c = cumulative default probability
d = marginal default probability
(1- cn) = (1 – cn-1)*(1 – dn)
(1  cn )
(1  cn 1 )
If the annual default rate is d, the monthly default rate, assuming it is constant, is
dn  1 
(1  d M )12  (1  d )
d M  1  (1  d )
1
12
6
Standard and Poor's cumulative default rates by credit rating 12/31/80 - 12/31/2002
Columns years 1-15
Standard and Poor's Credit Week January 29, 2003 pg. 17
Cumulative
AAA
1
2
0
0
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0003 0.0006 0.001 0.0017 0.0025 0.0038 0.0043 0.0048 0.0048 0.0048 0.0048 0.0056 0.0067
AA
0.0001 0.0003 0.0008 0.0016 0.0027 0.0039 0.0053 0.0065 0.0075 0.0085 0.0095 0.0106 0.0115 0.0122 0.013
A
0.0005 0.0015 0.0028 0.0044 0.0062 0.0081 0.0103 0.0125 0.0152 0.0182 0.0206 0.0226 0.0243 0.0261 0.0288
BBB
0.0037 0.0094 0.0152 0.0234 0.032 0.0402 0.0474 0.054 0.0599 0.0668 0.074 0.0797 0.0855 0.091 0.0977
BB
0.0138 0.0407 0.0716 0.0996 0.1234 0.1465 0.1646 0.1802 0.196 0.2082 0.2198 0.2279 0.2358 0.2399 0.2451
B
0.062 0.1327 0.1907 0.2345 0.2659 0.2908 0.3141 0.3327 0.3458 0.3587 0.3698 0.3797 0.3895 0.3996 0.4109
CCC
0.2787 0.3602 0.4179 0.4626 0.5046 0.5217 0.536 0.5436 0.5616 0.5721 0.5815 0.5895 0.5959 0.607 0.607
Investment 0.0013 0.0034 0.0057 0.0087 0.012 0.0152 0.0183 0.0213 0.0241 0.0272 0.0302 0.0326 0.035 0.0373 0.0403
Speculative 0.0517 0.1027 0.1481 0.1846 0.2131 0.2367 0.2571 0.2736 0.2883 0.3007 0.312 0.3209 0.3297 0.3372 0.3452
All
0.0167 0.0336 0.0486 0.0612 0.0714 0.0802 0.088 0.0947 0.1007 0.1064 0.1117 0.116 0.1202 0.124 0.1285
Standard and Poor's marginal default rates by credit rating 12/31/80 - 12/31/2002
Columns years 1-15
Standard and Poor's Credit Week January 29, 2003 pg. 17
Marginal
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
AAA
0.0000 0.0000 0.0003 0.0003 0.0004 0.0007 0.0008 0.0013 0.0005 0.0005 0.0000 0.0000 0.0000 0.0008 0.0011
AA
0.0001 0.0002 0.0005 0.0008 0.0011 0.0012 0.0014 0.0012 0.0010 0.0010 0.0010 0.0011 0.0009 0.0007 0.0008
A
0.0005 0.0010 0.0013 0.0016 0.0018 0.0019 0.0022 0.0022 0.0027 0.0030 0.0024 0.0020 0.0017 0.0018 0.0028
BBB
0.0037 0.0057 0.0059 0.0083 0.0088 0.0085 0.0075 0.0069 0.0062 0.0073 0.0077 0.0062 0.0063 0.0060 0.0074
BB
0.0138 0.0273 0.0322 0.0302 0.0264 0.0264 0.0212 0.0187 0.0193 0.0152 0.0147 0.0104 0.0102 0.0054 0.0068
B
0.0620 0.0754 0.0669 0.0541 0.0410 0.0339 0.0329 0.0271 0.0196 0.0197 0.0173 0.0157 0.0158 0.0165 0.0188
CCC
0.2787 0.1130 0.0902 0.0768 0.0782 0.0345 0.0299 0.0164 0.0394 0.0240 0.0220 0.0191 0.0156 0.0275 0.0000
Investment 0.0013 0.0021 0.0023 0.0030 0.0033 0.0032 0.0031 0.0031 0.0029 0.0032 0.0031 0.0025 0.0025 0.0024 0.0031
Speculative 0.0517 0.0538 0.0506 0.0428 0.0350 0.0300 0.0267 0.0222 0.0202 0.0174 0.0162 0.0129 0.0130 0.0112 0.0121
All
0.0167 0.0172 0.0155 0.0132 0.0109 0.0095 0.0085 0.0073 0.0066 0.0063 0.0059 0.0048 0.0048 0.0043 0.0051
Recovery rates vary considerably with the seniority (position in the defaulting
counterparty’s capital structure) of the liability. Credit rating agencies measure recovery
rates using the value of debt right after default. This is viewed as the market’s best
estimate of the future recovery and takes into account the value of the firm’s assets, the
estimated cost of the bankruptcy process, discounted to the present.
7
Moody’s Recovery Rates for US Corporate Debt 1970-1999
Min
1st Qu. Median Mean 3rd Qu. Max
Senior/Secured
15.00
60.00
75.00
69.91 88.00
98.00
Senority/Security
Bank Loans
Equipment Trust
Bonds
Senior/Secured
Bonds
Senior/Unsecured
Bonds
Senior/Subordinated
Bonds
Subordinated Bonds
Junior/Subordinated
Bonds
Preferred Stocks
All
Stdev
23.47
8.00
26.25
70.63
59.96
85.00
103.00
31.08
7.50
31.00
53.00
52.31
65.25
125.00
25.15
0.50
30.75
48.00
48.84
67.00
122.60
25.01
0.50
21.34
35.50
39.46
53.47
123.00
24.59
1.00
3.63
19.62
11.38
30.00
16.25
33.17
19.69
42.94
24.00
99.13
50.00
20.78
13.85
0.05
0.05
5.03
21.00
9.13
38.00
11.06
42.11
12.91
61.22
49.50
125.00
9.09
26.53
Derivative claims generally rank equally with senior unsecured debt.
8