The pricing of unexpected
credit losses
Jeff Amato and Eli Remolona
Second Credit Risk Conference
Recent Advances in Credit Risk Research
May 26-27, 2005, London
Policy question: Are corporate spreads too tight?
US corporate spreads
Option-adjusted, in basis points
BBB-rated 350
A-rated
300
250
200
150
100
50
Jan 02 May 02 Sep 02 Jan 03 May 03 Sep 03 Jan 04 May 04 Sep 04 Jan 05
Source: Merrill Lynch
λt
What theory says
Jarrow, Lando and Yu (2003) derive propositions of
conditional diversification. Proposition 3.2 goes something
like this:
Assume economy consists of money market account
with short rate r(t) and infinite collection of traded
securities in which (a) default risk is conditionally
diversifiable and (b) defaults are conditionally
independent under the pricing measure Q. Then the Q-
intensity is equal to the P-intensity for all i.
In English: If you can diversify in the sense of JLY, spreads
should be equal to expected losses from default.
The facts
The credit spread puzzle in US
corporates
(Jan 1997-Jul 2004, 5-year duration)
OAS Expected Spread
spread loss Ratio
AAA/Aaa 70.1 0.1 625.4
AA/Aa 80.7 0.9 55.4
A/A 108.1 6.2 13.2
BBB/Baa 181.5 40.1 4.1
BB/Ba 338.5 147.9 2.2
The spread puzzle and the
pricing of credit risk
The credit spread puzzle
Possible explanations
How diversified are credit portfolios?
Evidence from CDOs
Measuring credit risk
Pricing credit risk
Conclusion
Concepts and assumptions
Physical processes
Doubly stochastic model for marginal default
probabilities
Physical intensities for each issuer: λi
Constant LGD
Risk-neutral processes
Risk-neutral intensities: λiQ
The spread puzzle in terms of ratio
of risk neutral to physical
Amato and Remolona (2004):
λQ/ λ = 55 for Aa; = 4 for Baa
Ratio increases with credit quality
Other papers:
Driessen (2003): λQ/ λ = 2 to 6
Bernt, Douglas, Duffie, Ferguson and Schranz
(2004): λQ/ λ exceeds 1 and varies
substantially over time
Possible explanations
Elton, Gruber, Agrawal and Mann (2001) look at levels of
US credit spreads:
Taxes explain 28% to 73% depending on maturity and
credit rating – hence most of Aaa spread but little of
Baa spread
“Systematic risk” explains 19% to 41%.
Collin-Dufresne, Goldstein and Martin (2001) look at
changes in spread but find no macro or financial variables
to explain them
Huang and Huang (2002) look at the five most popular
structural models of credit risk. None can explain spread.
How about liquidity?
Puzzle remains even for benchmark bonds
Finance company spreads on August 6, 2003
as quoted by Deutsche Bank
Size S&P Bid
Issuer Coupon Maturity
($bn) rating spread
Ford Motor Jan
6.500 4.00 BBB 225
Credit 2007
Mar
GE Capital 5.375 2.00 AAA 15
2007
Apr
CIT 7.375 1.25 A 50
2007
So why are spreads so wide?
Conditional diversification of Jarrow, Lando and Yu:
Diversifiability implies risk of individual defaults
not priced
Risk premia driven solely by systematic risk
But what if investors not really able to diversify?
Then idiosyncratic risk – unexpected losses in
portfolio -- will be priced
Default correlations only make things worse
In case of CDOs, large subordination needed for
unexpected losses
How diversified are actual corporate
bond portfolios?
Corporate bond funds
Largest = 385 names (Vanguard)
Top five: 658 unique names
Cash arbitrage CDOs
100-150 issuers in collateral pools
Diversity scores: 45-60 names
Synthetic CDOs: 150-250 names
Traded CDS indices
DJ CDX: 125 names
DJ iTraxx Europe: 125 names
Implications of “small” portfolio
Consider hypothetical portfolio:
$1 million divided evenly among 100 Baa-
rated names with independent defaults
5-year horizon; Prob(default in 5 yrs) = 3.4%
Losses calculated using Binomial
distribution
Portfolio return distribution highly skewed
Actual loss can be much larger than EL
EL = $17000, but Prob(loss > 3 x EL) > 0.2%
Subordination in cash arbitrage CDOs
Investment High yield
grade collateral
collateral
Tranche Size % of Size % of
$mn deal $mn deal
Senior 383 81.2 240 66.6
Mezzanine 53 11.2 69 19.3
Equity 36 7.6 50 14.0
Subordination 89 18.8 119 33.4
Total 472 100 359 100
Subordination ratios so high because of lack of
diversification
Measuring risk in a credit portfolio
Most common measure is volatility – easy to calculate but
bad for asymmetric return distributions
Lower partial moments better at capturing downside risk --
but complicated
Expected shortfall measures mean loss beyond a
threshold
Measure of choice for risk managers is Value-at risk (VaR)
Arbitrary choice of confidence level
Not coherent – violates subadditivity
Nonetheless a form of VaR gaining ground in credit
markets -- especially for CDOs and CDS index tranches
The risk measure in CDO subordination
Rating agencies calculate required subordination in
different ways but essentially the same
To oversimplify, for collateral pool of N names (same
default probability):
CDO manager sets k so probability of kth default is
equal to specified survival probability of senior tranches
k ∗ (N , λi ) ≡ min k s.t. F ( N , k , λi ) ≥ 1 − prob( Aaa default )
Use physical intensities
k*/N is subordination ratio – a measure of risk
CDO subordination: k*/N and diversification
Lessons from arbitrage CDOs
Since the more diversified the collateral pool the smaller
the subordination, CDO managers have strong incentive
to diversify
Yet actual CDOs far from fully diversified
Subordination large due to idiosyncratic risk – unexpected
losses in smallish portfolios
Subordination structure uses k*/N as measure of risk
Just a VaR calculation, but confidence level set at Aaa
survival probability
VaRα ( N , λi ) ≡ inf {k ∈ Z : F (k , N , λi ) ≥ α }
Measuring risk in small credit portfolios
Set VaR confidence level at 99.999%
Confidence level no longer arbitrary
Now coherent for non-Aaa portfolios
Measure risk as ratio of this VaR to size of portfolio
-- like k*/N
Captures downside risk of unexpected losses in
small portfolios
Risk declines as portfolio gets bigger
Also accounts for default correlations
Mapping the physical into the
risk neutral
Go from physical to risk-neutral intensities
Mapping involves risk and price of risk:
Risk measure to account for available scope for
diversification and systematic risk
Market price of risk common across securities
Measure of risk is:
ω Aaa ( N , λi , ρ ) ≡ VaR Aaa (N , λi , ρ ) / N
Price risk relative to Aaa risk
Linear pricing conjecture
Specify expected excess returns = spreads – EL
Express as differential over Aaa excess returns
Conjecture that this is linear in risk
~
λiQ = expected excess return
~ ~ ~ ~
Q
λiQ − λ Q = (λ M − λ Q )ω Aaa ( N , λi , ρ )
Aaa Aaa
In equilibrium, market price of risk is same across bonds
~Q ~Q
~Q ~Q λi − λ Aaa
J ≡ λM − λ Aaa ≡
ω Aaa
Can we match the cross-sectional
stylized facts?
Calculate risk measure
Different credit ratings
Different portfolio sizes
Different correlations
For default correlations, use Hull-White copula
approach
Ratio of expected excess returns to risk should be
same across ratings
So in plot of excess returns against risk, observations
should lie on straight line drawn from origin
Excess returns and risk
Asset correlation = 0
Excess returns and risk:
Asset correlation = 0.3
Systematic versus idiosyncratic risk
systematic risk
idiosyncratic risk
Implications of pricing conjecture
Scope for diversification limited to about 100 names
Perceived average asset correlation is about 0.3
For Baa bonds:
idiosyncratic risk ≈ one fourth of priced credit risk
systematic risk ≈ three fourths
Market price of risk -- roughly expected excess return
on subordinate tranches -- around 150 basis points in
our sample. Compare to:
70 basis points for Baa risk
120 basis points for Ba risk
Conclusions
Credit risk diversification requires very large portfolio
because of skewness in return distribution
Corporate bond funds, traded CDS indices and arbitrage
CDOs suggest actual portfolios not so large
Risk of unexpected credit losses will be significant
To account for this risk, CDO subordination structures
rely on VaR at Aaa confidence level
Measured this way, this risk can explain credit spreads,
and relationship may even be linear