The Price for Bearing Default Risk
Darrell Duffie, Stanford University
Q Group, October, 2005
Based on collaboration with:
Antje Berndt
Rohan Douglas
Mark Ferguson
David Schranz
Stanford University, 2005
Main Objective
• 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.
Stanford University, 2005
20
risk−neutral
actual
18
16
Default probability (percent)
14
12
10
8
6
4
2
0
Dec01
Jul02
Jan03
Aug03
Feb04
Figure 1:
Sep04
Mar05
Estimated actual and risk-neutral 1-year default probabilities for
Royal Caribbean Cruises.
Stanford University, 2005
14
12.16
12
Default Rate (percent)
10
7.98
8
6
3.58
4
2.23
2
0
0.00
0.00
0.00
0.07
0.00
0.02
0.13
Aaa
Aa1
Aa2
Aa3
A1
A2
Baa1
0.10
Baa2
0.48
Baa3
0.70
0.67
Ba1
Ba2
Ba3
B1
B2
B3
Figure 2: Default Rate by Moody’s Modified Credit Rating.
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70%
60%
3-year default rate
50%
Last rating change:
Upgraded
Unchanged
Downgraded
40%
30%
20%
10%
0%
Investment-Grade
Ba
B
Caa
Figure 3: Upgrade-downgrade momentum (1996-2003 data). Source:
Moody’s, 2004.
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Moody’s KMV Estimated Default Frequency
• 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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0.05
Frequency of default within one year
0.04
0.03
0.02
0.01
0
−0.01
−0.5
0
0.5
1
1.5
2
2.5
3
Distance to default
3.5
Figure 4:
4
4.5
5
The dependence of empirical default frequency on distance to default.
(Source: Duffie, Saita, Wang (2005).
Stanford University, 2005
+
350
+
+
300
+
Number of Observations
+
250
+
+
200
+
+
150
+ +
100
+
+ +
+ + +
50
+
+
+
10
20
30
40
50
60
70
80
90
100
Post Default Prices in US Dollars
Figure 5:
Distribution of senior unsecured recovery rates, 1982 - 2002. Source:
Moody’s Default and Recovery Report (2003).
Stanford University, 2005
60%
50%
Recovery rate
40%
30%
Value-Weighted Mean
Issuer-Weighted Mean
Long-Term Issuer Mean
20%
10%
0%
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
Year
Figure 6:
Time variation in average recovery rates, 1982 - 2003.
Moody’s.
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Source:
60%
Recovery rate
50%
40%
1997
1987
1984
1986
1992
1994
1996
1998
2003
1988
1982
1991
1999
1989
30%
2002
1990
2000
2001
20%
10%
0%
0.0%
Recovery rate = 50.3 - 6.3£ Default rate
R2 = 0.60
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
Default rate
Figure 7:
Correlation of Speculative Grade Default and Recovery Rates.
Source: Moodys Default and Recovery Report (2004).
Stanford University, 2005
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).
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viders
16
14
Number of quote providers
12
10
8
6
4
2
0
< 200
200-400
400-800
800-1,600
1,600-3,200
3,200-6,400
Figure 9:
6,400-12,800 12,800-25,600
> 25,600
Distribution of CDS quote providers by number of quotes provided.
Data source: CIBC.
Stanford University, 2005
55
50
45
40
35
30
25
20
15
10
5
0
Aaa
Aa
A
Baa
Ba
B
Caa
Figure 10:
Ca
C
Unrated
Distribution of firms by median credit rating during the sample
period. Sources: CIBC and Moody’s.
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CDS 5-year rate (mid-quote, basis points)
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0
200
400
600
800
1000
1200
1400
1600
1800
Moody’s KMV 5-year EDF (basis points)
Figure 11:
2000
Scatter plot of EDF and CDS observations and OLS fitted relationship. Source: CIBC (CDS) and Moody’s KMV (EDF).
Stanford University, 2005
Logarithm of CDS 5-year rate (mid-quote)
0
−1
−2
−3
−4
−5
−6
−7
−8
−7
−6
−5
−4
−3
−2
Logarithm of Moody’s KMV 5-year EDF
Figure 12:
−1
Scatter plot of EDF and CDS observations, logarithmic, and OLS
fitted relationship. Source: CIBC (CDS) and Moody’s KMV (EDF).
Stanford University, 2005
CDS versus EDF (5-year)
For 33,912 paired daily median observations over 2000-2004:
X
log CDSi = 1.45 + 0.76 log EDFi +
β̂j Dmonth, sector (i) + zi ,
(0.05)
(0.02)
• Standard errors estimated for panel correlation.
• 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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Fe 1
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A 2
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Time effect on risk premium
3.00
2.50
Oil and gas
Broadcasting
Healthcare
2.00
1.50
1.00
0.50
0.00
Month
Figure 13:
Monthly dummy multipliers in CDS-to-EDF fit.
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60
Mean recovery rate
50
40
30
20
10
0
Healthcare
Media, Broadcasting
and Cable
Oil and Oil Services
Figure 14: Sectoral recovery differences.
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Utility-Gas
Default Intensity
• λt is the conditional mean arrival rate of default.
Rt
λ(s)
ds
.
• The probability of survival for t years is p(t) = E e− 0
• The risk-neutral
of survival for t years is
R tprobability
− 0 λ∗ (s) ds
∗
∗
p (t) = E e
.
• p∗ (t) < p(t) because
– λ∗t > λt .
– E ∗ (λ∗t ) > E(λ∗t ).
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Dynamic Default Intensity Models
• Actual intensity, λt log-normal with mean reversion, fitted from
12 years of monthly observations of 1-year EDFs by maximum
likelihood.
• Sector homogeneity of volatility and mean reversion.
• Risk-neutral intensity:
log λt = a + b log λt + ut ,
where ut is an independent gaussian auto-regressive process.
• Fit a, b, and dynamic parameters from 1-year and 5-year CDS.
Stanford University, 2005
20
risk−neutral
actual
default intensity (%)
18
16
14
12
10
8
6
4
2
0
Dec01
Jul02
Jan03
Aug03
Feb04
Sep04
Mar05
date
Figure 15:
Implied default intensities for Royal Caribbean Cruises.
Stanford University, 2005
5.0
4.5
λ∗ (t)/λ(t)
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
Dec01
Figure 16:
Jul02
Jan03
Aug03
Feb04
Sep04
Mar05
Estimated default risk premia, λ∗ /λ, for Royal Caribbean Cruises.
Stanford University, 2005
risk-neutral-to-actual default probability
9
instantaneous
1 year
5 year
8
7
6
5
4
3
2
1
0
Jul02
Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04 Sep04 Dec04
date
Figure 17:
Median default risk premia, broadcasting-entertainment.
Stanford University, 2005