slides_Qgroup.dviDarrell Duffie, Stanford University Q Group, October, 2005
Based on collaboration with:
• How much are investors in corporate debt paid for taking
default risk, above their expected default loss?
• Our analysis is based on Moodys KMV estimates of default
probabilities and CIBC data on default swap (CDS) prices.
• The default risk premium is bigger, per dollar of expected
default loss, for high-quality firms.
• The default risk premium, at fixed credit quality, was
dramatically reduced from mid-2002 to the end of 2003,
especially in the broadcasting and entertainment sector.
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2
4
6
8
10
12
14
16
18
Figure 1: Estimated actual and risk-neutral 1-year default probabilities for
Royal Caribbean Cruises.
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14
12
10
8
6
4
2
0
Aaa Aa1 Aa2 Aa3 A1 A2 Baa1 Baa2 Baa3 Ba1 Ba2 Ba3 B1 B2 B3
0.00 0.00 0.02 0.13 0.10 0.48
0.70 0.67
Figure 2: Default Rate by Moody’s Modified Credit Rating.
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Moody’s, 2004.
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• Asset value and volatility are computed jointly from a modified
Black-Scholes options pricing model, treating equity as a call
on assets struck at liabilities.
• The liability default boundary point is measured as short-term
debt plus a fraction (half) of long-term debt.
• The “distance to default” is the number of standard deviations
by which the expected asset value exceeds the default point.
• This firm’s current EDF is the fraction of those firms in
previous years with the same distance to default that actually
did default within one year.
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−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.01
0
0.01
0.02
0.03
0.04
0.05
cy of
d ef
au lt
w it
h in
on e
ye ar
Figure 4: The dependence of empirical default frequency on distance to default.
(Source: Duffie, Saita, Wang (2005).
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+
+
+
+ +
+ +
+ +
+ + +
+ + + + + +
+
+
30 40 50 60 70 80 90 100
Figure 5: Distribution of senior unsecured recovery rates, 1982 - 2002. Source:
Moody’s Default and Recovery Report (2003).
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e
Figure 6: Time variation in average recovery rates, 1982 - 2003. Source:
Moody’s.
Default rate
R ec
o v
er y
r at
£
Figure 7: Correlation of Speculative Grade Default and Recovery Rates.
Source: Moodys Default and Recovery Report (2004).
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Figure 8: Default swap: buyer of protection pays the CDS rate U
quarterly, and at the default time τ delivers bond worth Y (τ) in
exchange for notional (100).
N u m
p ro
v id
er s
ers
Figure 9: Distribution of CDS quote providers by number of quotes provided.
Data source: CIBC.
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Aaa Aa A Baa Ba B Caa Ca C Unrated
Figure 10: Distribution of firms by median credit rating during the sample
period. Sources: CIBC and Moody’s.
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0 200 400 600 800 1000 1200 1400 1600 1800 2000 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
C D
ts )
Figure 11: Scatter plot of EDF and CDS observations and OLS fitted relation-
ship. Source: CIBC (CDS) and Moody’s KMV (EDF).
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−7
−6
−5
−4
−3
−2
−1
0
L o g a ri
th m
)
Figure 12: Scatter plot of EDF and CDS observations, logarithmic, and OLS
fitted relationship. Source: CIBC (CDS) and Moody’s KMV (EDF).
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log CDSi = 1.45 + 0.76 log EDFi + ∑
βjDmonth, sector(i) + zi,
• R2 = 74.4%.
• One-sigma confidence band for a given CDS rate places it
between 59% and 169% of the fitted rate.
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M ea
n r
ec o
v er
y r
at e
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• The probability of survival for t years is p(t) = E (
e− R
p∗(t) = E∗
12 years of monthly observations of 1-year EDFs by maximum
likelihood.
• Risk-neutral intensity:
where ut is an independent gaussian auto-regressive process.
• Fit a, b, and dynamic parameters from 1-year and 5-year CDS.
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6
8
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1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
)
Figure 16: Estimated default risk premia, λ∗/λ, for Royal Caribbean Cruises.
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Jul02 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04 Sep04 Dec04 0
1
2
3
4
5
6
7
8
date
u lt
p ro
Figure 17: Median default risk premia, broadcasting-entertainment.
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