15.433 INVESTMENTS
Class 17: The Credit Market
Part 1: Modeling Default Risk
Spring 2003
The Corporate Bond Market
25
20
15
10
5
0
Mortgage Rates (Home Loan Mortgage Corporation)
FED Fund
Apr-01
Apr-99
Apr-97
Apr-95
Apr-93
Apr-91
Apr-87
Apr-85
Apr-83
Apr-81
Apr-79
Apr-77
Apr-75
Apr-73
Apr-71
-10
Apr-89
-5
Spread
Figure 1: Mortgage and FED rates, Source : www.federalreserve.gov/releses/hr
AAA
BAA
Spread
Figure 2: Corporate rating spreads, Source : www.federalreserve.gov/releses/hr
Jan-99
Jan-94
Jan-89
Jan-84
Jan-79
Jan-74
Jan-69
Jan-59
Jan-54
Jan-49
Jan-44
Jan-39
Jan-34
Jan-29
Jan-24
Jan-19
-
Jan-64
20
18
16
14
12
10
8
6
4
2
16%
14%
12%
10%
8%
6%
4%
1 Year Maturity
30 Years Maturity
Figure 3: Corporate rating spreads, Source : Moody’s
100%
90%
80%
70%
60%
50%
40%
30%
20%
BBB
AAA
AA-
Figure 4: Corporate rating migration for industry-sector, Source : Standard Poor’s.
D
CCC
B­
B
B+
BB­
BB
BB+
BBB­
BBB
BBB+
A­
A
A+
AA-
AA
AAA
0%
AA+
10%
Caa
B3
B2
B1
Ba3
Ba2
Ba1
Baa3
Baa2
Baa1
A3
A2
A1
Aa3
Aa2
Aa1
0%
Aaa
2%
Bond Valuation with Default Risk
100
Cashflow Conditioning on Survival
c/2
0.5
1
1.5
2
2.5
3
3.5
4
τi
Non-Default
Default
RandomDefaultTime τ%
Figure 5: Chart cash flow Conditioning on survival.
Assuming no default risk,
P0 =
8
�
er·ti + 100 · er·4
i=1
How does the default risk affect the bond price?
(1)
Modelling Default Risk
Modelling default risk is central to the pricing and hedging of credit sen­
sitive instruments.
Two approaches to modelling default risk:
• Structural approach, ”first-passage”: default happens when the to­
tal asset value of the firm falls below a threshold value (for example,
the firm’s book liability) for the first time.
• Reduced-form, ”intensity-based”: the random default time τ� is gov­
erned by an intensity process λ.
For pricing purpose, the reduced-form approach is adequate, and will be
the focus of this class.
Modelling Random Default Times
The probability of survival up to time t:
(2)
P rob(τ� ≥ t)
The probability of default? before time t:
P rob (τ� < 0) = 1 − P rob (τ� ≥ t)
(3)
We assume that T� is exponentially distributed with constant default
intensity λ:
1
Survival Probability:
(
)
Prob τ% ≥ t = e- λt
0
0
100
t (year)
Figure 6: Survival Probability.
Default Probability and Credit Quality
One-Year default probability = 1 − eλ
Default intensity λ =?
D
CCC
B­
B
B+
BB­
BB
BB+
BBB­
BBB
BBB+
AA
A+
AAAA
AA+
AAA
0%
10%
AAA
BB
20%
AA+
BB­
Figure 7: Survival Probability.
AA
B+
30%
AAB
40%
50%
A+
B­
A
CCC
60%
A­
D
70%
BBB+
80%
BBB
90%
BBB­
100%
BB+
AAA
AA+
AA
AAA+
A
ABBB+
BBB
BBBBB+
BB
BBB+
B
BCCC
D
AAA
91.95%
2.31%
0.62%
0.00%
0.00%
0.08%
0.14%
0.00%
0.07%
0.05%
0.17%
0.00%
0.00%
0.00%
0.00%
0.00%
0.19%
0.00%
AA+
4.11%
84.71%
1.36%
0.15%
0.03%
0.06%
0.04%
0.00%
0.03%
0.00%
0.00%
0.00%
0.00%
0.03%
0.00%
0.00%
0.00%
0.00%
AA
2.86%
8.75%
85.42%
3.44%
0.83%
0.49%
0.11%
0.08%
0.07%
0.11%
0.00%
0.12%
0.00%
0.00%
0.07%
0.00%
0.00%
0.00%
AA0.48%
2.88%
7.24%
83.67%
4.47%
0.66%
0.35%
0.13%
0.17%
0.21%
0.08%
0.06%
0.05%
0.10%
0.00%
0.00%
0.00%
0.00%
A+
0.16%
0.19%
2.60%
8.61%
82.27%
5.25%
1.13%
0.59%
0.45%
0.11%
0.08%
0.06%
0.09%
0.00%
0.00%
0.18%
0.19%
0.00%
A
0.20%
0.48%
1.49%
3.02%
8.08%
82.50%
8.58%
2.26%
0.93%
0.69%
0.51%
0.37%
0.05%
0.03%
0.14%
0.00%
0.00%
0.00%
A0.12%
0.10%
0.25%
0.50%
2.75%
5.44%
77.39%
8.32%
2.24%
0.59%
0.34%
0.18%
0.28%
0.23%
0.21%
0.00%
0.19%
0.00%
BBB+
0.04%
0.00%
0.50%
0.23%
0.46%
3.18%
7.21%
75.24%
7.83%
2.67%
0.67%
0.31%
0.33%
0.10%
0.00%
0.36%
0.19%
0.00%
Figure 8: Survival Probability, Migration table, Source: RiskMetrics T M .
16%
14%
12%
10%
8%
6%
4%
2%
0%
Aaa Aa1 Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3 Ba1 Ba2 Ba3 B1
B2
Figure 9: One-Year Default Rates by Modified Ratings, 1983-1995, Source: Moodys (1996).
B3
BBB
0.04%
0.38%
0.22%
0.15%
0.40%
1.11%
3.00%
8.36%
77.76%
9.46%
4.21%
1.59%
0.52%
0.13%
0.14%
0.00%
0.56%
0.00%
Pricing A Defaultable Bond
For simplicity, let’s first assume that the riskfree interest rate r is a
constant. Consider a τ -year zero-coupon bond issued by a firm with
default intensity λ:
P0 = $100 · e−r·τ · P rob(τ� ≥ τ )
(4)
P0 = $100 · e−r·τ · e−λ·τ
(5)
P0 = $100 · e−(r+λ)·τ
(6)
where we assume that conditioning on a default, the recovery value of
the bond is 0 (we have also assumed risk-neutral pricing).
The yield on the defaultable bond is r + λ, resulting in a credit spread
of λ.
Time Variation of Default Probability
10
8
6
4
2
0
-2
GDP Growth / Chain QQQ%
Linear (GDP Growth / Chain QQQ%)
Figure 10: Chart Annual GDP Growth Rate, source: Bureau of Economic Analysis Stochastic
Dec-01
Dec-00
Dec-99
Dec-98
Dec-97
Dec-96
Dec-95
Dec-94
Dec-93
Dec-92
Dec-91
Dec-90
Dec-89
Dec-88
Dec-87
Dec-86
-4
Default Intensity
In general, the credit quality of a firm changes over time.
A more realistic model is to treat the arrival intensity as a random
process.
Suppose that intensities are updated with new information at the begin­
ning of each year, and are constant during the year. Then the probability
of survival for t years is
For example,
�
�
E
e−λ0 +λ1 +···+λt−1
(7)
�
�
¯ − λt + εt+1
λt+1 − λt = k λ
(8)
Can you calculate the probability of survival for τ years? What is the
price of a τ -year zero-coupon bond? What if the riskfree interest rate is
also stochastic?
Example: A portfolio consists of two long assets $100 each. The prob­
ability of default over the next year is 10% for the first asset, 20% for
the second asset, and the joint probability of default is 3%. What is
the expected loss on this portfolio due to credit risk over the next year
assuming 40% recovery rate for both assets.
Probabilities:
0.1 · (1 − 0.2)
−
def ault probability of A
(9)
0.2 · (1 − 0.1)
−
def ault probability of B
(10)
0.03
−
joint def ault probability
(11)
Expected losses:
0.1 · (1 − 0.2) · 100 · (1 − 0.4) = 4.8
(12)
0.2 · (1 − 0.1) · 100 · (1 − 0.4) = 10.8
(13)
0.03 · 200 · (1 − 0.4) = 3.6
4.8 + 10.8 + 3.6 = $19.2 mio.
(14)
(15)
Example: Assume a 1-year US Treasury yield is 5.5% and a Eurodollar
deposit rate is 6%. What is the probability of the Eurodollar deposit to
default assuming zero recovery rate)?
1
1−π
=
1.06
1.055
π = 0.5%
(16)
(17)
Example: Assume a 1-year US Treasury yield is 5.5% and a and a
default probability of a one year CP is 1%. What should be the yield on
the CP assuming 50% recovery rate?
1
1−π
0.5π
=
+
1+x
1.055 1.055
= 6%
(18)
(19)
Some Practitioner’s Credit Risk Model
RiskMetrics: CreditM etricsT M
http://riskmetrics.com/research
Credit Suisse Financial Products: CreditRisk+
http://www.csfb.com/creditrisk
KMV Corporation / CreditM onitor T M
http://www.kmv.com
Focus:
BKM Chapter 14
• p. 415-422 (definitions of instruments, innovation in the bond mar­
ket)
• p. 434-441 (determinants of bond safety, bond indentures)
Style of potential questions: Concept check questions, p. 448 ff. ques­
tion 31
Questions for Next Class
Please read:
• Reyfman,
• Toft (2001), and
• Altman, Caouette, Narayanan (1998).