The Price for Bearing Default Risk Darrell Duffie, Stanford - - PowerPoint PPT Presentation

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.

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Dec01 Jul02 Jan03 Aug03 Feb04 Sep04 Mar05 2 4 6 8 10 12 14 16 18 20 risk−neutral actual Default probability (percent)

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 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 2.23 3.58 7.98 12.16 0.00 0.00 0.07 Default Rate (percent)

Figure 2: Default Rate by Moody’s Modified Credit Rating.

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70% 60% 50% 40% 30% 20% 10% 0%

Investment-Grade Ba Caa B

Upgraded Unchanged Downgraded Last rating change:

3-year default rate

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

• n 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 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.01 0.01 0.02 0.03 0.04 0.05 Distance to default Frequency of default within one year

Figure 4: The dependence of empirical default frequency on distance to default.

(Source: Duffie, Saita, Wang (2005).

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350 300 250 200 150 100 50 10 20 Number of Observations Post Default Prices in US Dollars

+ + + + + + + + + + + + + + + + + + + +

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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0% 10% 20% 30% 40% 50% 60%

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Value-Weighted Mean Issuer-Weighted Mean Long-Term Issuer Mean Year Recovery rate

Figure 6:

Time variation in average recovery rates, 1982 - 2003. Source: Moody’s.

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60% 50% 40% 30% 20% 10% 0% 0.0% 0.5% 1.5% 2.5% 3.5% 1.0% 2.0% 3.0% 4.0%

Default rate Recovery rate

1994 1998 1982 1992 1987 1986 1988 2003 1999 1989 1990 2002 1991 2000 2001 1984 1997 1996 Recovery rate = 50.3 - 6.3 Default rate

2

R = 0.60

£

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).

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2 4 6 8 10 12 14 16 200 200-400 400-800 800-1,600 1,600-3,200 3,200-6,400 6,400-12,800 12,800-25,600 25,600 Number of quote providers < >

Figure 9: Distribution of CDS quote providers by number of quotes provided.

Data source: CIBC.

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5 10 15 20 25 30 35 40 45 50 55 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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200 400 600 800 1000 1200 1400 1600 1800 2000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Moody’s KMV 5-year EDF (basis points) CDS 5-year rate (mid-quote, basis points)

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 −8 −7 −6 −5 −4 −3 −2 −1

Logarithm of Moody’s KMV 5-year EDF Logarithm of CDS 5-year rate (mid-quote)

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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CDS versus EDF (5-year)

For 33,912 paired daily median observations over 2000-2004: log CDSi = 1.45 + 0.76 log EDFi + ˆ βjDmonth, 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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0.00 0.50 1.00 1.50 2.00 2.50 3.00 D e c

• F

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A p r

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D e c

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F e b

• 3

A p r

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Month Time effect on risk premium Oil and gas Broadcasting Healthcare

Figure 13: Monthly dummy multipliers in CDS-to-EDF fit.

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10 20 30 40 50 60 Healthcare Media, Broadcasting and Cable Oil and Oil Services Utility-Gas Mean recovery rate

Figure 14: Sectoral recovery differences.

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Default Intensity

• λt is the conditional mean arrival rate of default.
• The probability of survival for t years is p(t) = E
• e−

R t

0 λ(s) ds

.

• The risk-neutral probability of survival for t years is

p∗(t) = E∗ e−

R t

0 λ∗(s) ds

.

• p∗(t) < p(t) because

– λ∗

t > λt.

– E∗(λ∗

t ) > E(λ∗ t ). Stanford University, 2005

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.

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Dec01 Jul02 Jan03 Aug03 Feb04 Sep04 Mar05 2 4 6 8 10 12 14 16 18 20 risk−neutral actual

date default intensity (%)

Figure 15: Implied default intensities for Royal Caribbean Cruises.

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Dec01 Jul02 Jan03 Aug03 Feb04 Sep04 Mar05 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

λ∗(t)/λ(t)

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 1 2 3 4 5 6 7 8 9 instantaneous 1 year 5 year

date risk-neutral-to-actual default probability

Figure 17: Median default risk premia, broadcasting-entertainment.

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