1 CDO Basics—What Is a CDO? Why CDOs? Types of CDOs
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the originating bank, to another, the investor. The investor can invest in
different CDO tranches. Each tranche has a different degree of credit risk.
The credit risk is distributed with a waterfall principle: If losses accumulate
and the detachment level of a tranche is breached, additional credit losses
ﬂow into the adjacent higher tranche. A CDO is typically arranged by a
special purpose vehicle (SPV), which is AAA rated to minimize counterparty risk.
5.1.2 Why CDOs?
There are three main parties in a CDO:
1. The originator (or protection buyer), who transfers the credit risk.
2. The investor, who assumes the credit risk.
3. The special purpose vehicle (SPV), which manages the CDO.
The motivation for the originator is naturally to transfer the credit risk,
which improves his credit rating, frees credit lines, reduces regulatory capital,
and lowers funding cost. The motivation for the investor is to receive high
yields. The motivation for the SPV is fee income.
CDOs include several sound ﬁnancial properties:
■
■
■
Diversiﬁcation. Since typically 125 assets are in a CDO, a skilled
originator will choose assets with a low correlation to achieve high
diversiﬁcation beneﬁts (see Chapter 1, section 1.3.1, “Investments and
Correlation”).
Subordination. This means that mezzanine and higher tranches are
protected by lower tranches, since lower tranches absorb default losses
from the underlying basket of credits ﬁrst.
Overcollateralization. Typically the assets in a CDO have a higher value
than the liabilities that the SPV owes to the investors. This overcollateralization adds an additional element of protection for investors.
The drawback of CDOs lies in their relative pricing complexity. We have
to ﬁnd the default probability function with respect to time of 125 assets
for the duration of the CDO, which can be up to 10 years. This alone is
difﬁcult to estimate. Furthermore, we have to correlate the default functions
of the 125 assets! This is where the copula function comes in.
First let’s have a look at where the CDO market is today.
From Table 5.1 we observe that the CDO market is recovering nicely
since 2009; however, the CDO issuance is far below the record 2006
levels.
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TABLE 5.1
103
Global CDO Issuance in USD Millions
Total CDO Issuance
(in USD millions)
Year
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
until 7/15/2013
86,629.8
157,820.7
251,265.3
520,644.6
481,600.7
61,886.8
4,336.0
8,665.1
31,131.3
45,399.8
44,403.0
Source: www.Sifma.com.
5.1.3 Types of CDOs
There are three main types of CDOs, which are displayed in Figure 5.1.
In a cash CDO, the originating bank sells assets to the SPV, which then
creates tranches. Each tranche is exposed to a certain degree of default risk.
The ﬁrst losses from asset defaults ﬂow into the equity tranche. Further losses
ﬂow into the next higher mezzanine tranche, and so on. Figure 5.2 shows the
cash ﬂows of a typical cash CDO.
In a synthetic CDO, assets are not sold from the originating bank to the
SPV, but the SPV assumes the credit risk via selling credit default swaps
(CDSs). The SPV receives the CDS spreads from the originating bank and the
cash from the investor, and invests these cash ﬂows into risk-free assets. A
synthetic CDO is displayed in Figure 5.3.
CDOs
Cash CDOs
FIGURE 5.1 Main Types of CDOs
Synthetic
CDOs
Unfunded
CDOs
(iTraxx, CDX)
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Senior Tranche
(exposed to next
88% of defaults)
Originating
Bank
Initial Cash
Investment
Cash
Mezzanine Tranche
(exposed to next
2.25% of defaults)
SPV
Assets
Tranche Spread
Mezzanine Tranche
(exposed to next
2.75% of defaults)
Equity Tranche
(exposed to first
7% of defaults)
Investor
(sells protection)
FIGURE 5.2 A Cash CDO
A third type of CDOs are unfunded CDOs such as the family of CDX
indexes or the iTraxx indexes, also called credit default swap indexes. The
most popular CDX index is the CDX.NA.IG, which references 125 investment grade CDSs in North America. The most popular iTraxx index is the
iTraxx Europe, which references 125 investment grade CDSs in Europe.
Importantly, the CDX and iTraxx indexes are unfunded; therefore no initial
principal amount is exchanged between the buyer (investor) and the seller.
Hence the trading of the CDX and iTraxx indexes is similar to buying and
selling futures contracts. The cash ﬂows of an unfunded, tranched CDO are
displayed in Figure 5.4.
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105
Super-Senior Tranche
(exposed to next
87.5% of defaults)
Risk-Free
Asset
Seller
Initial Cash Investment
+ CDS Spread
Initial Cash
Investment
CDS Spread
Protection
Buyer
Payment in Case
of Default
Senior Tranche
(exposed to next
3.75% of defaults)
Coupons
SPV
Tranche Spread
Mezzanine Tranche
(exposed to next
2.25% of defaults)
Mezzanine Tranche
(exposed to next
2.75% of defaults)
Equity Tranche
(exposed to first
3.75% of defaults)
Investor
(sells protection)
FIGURE 5.3 A Synthetic CDO
5.2 VALUING CDOs
There are three main input factors when valuing a CDO:
1. The default probability of each of the 125 assets.
2. The default correlation between the 125 assets in the portfolio.
3. The recovery rate in case of default.
Let’s discuss brieﬂy how to derive the default probability function before
we concentrate on the most signiﬁcant element, the default correlation.
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Super-Senior Tranche
(exposed to
12% to 22% of defaults)
Nonfunded CDOs
(e.g., CDX, iTraxx)
Senior Tranche
(exposed to
9% to 12% of defaults)
Payout in Case
of Default
Mezzanine Tranche
(exposed to
6% to 9% of defaults)
Tranches as
European
iTraxx
Buyer
(buys protection)
Spread
Mezzanine Tranche
(exposed to
3% to 6% of defaults)
Junior Tranche
(exposed to
first 3% of defaults)
Seller
(sells protection)
FIGURE 5.4 A Tranched, Nonfunded CDO Such as the iTraxx
5.2.1 Deriving the Default Probability for
Each Asset in a CDO
Most investment banks, hedge funds, and SPVs use an extension of the
seminal Merton 1974 model to derive the default probability for each asset in
a CDO. Let’s calculate this default probability.
In 1973, Fischer Black and Myron Scholes, and separately Robert
Merton, created their famous Black-Scholes-Merton (BSM) option pricing
model. The well-known equation for a call is
C = S0 N(d1 ) − Ke−rT N(d2 )
(5.1)
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107
S0
s2
ln
T
+
pﬃﬃﬃﬃ
Ke−rT
2
pﬃﬃﬃﬃ
and d2 = d1 − s T
where d1 =
s T
C: call price
S0: current stock price
N: cumulative standard normal distribution
K: strike price
r: continuously compounded risk-free interest rate
T: option maturity, measured in years
s: implied volatility of S
One year later, in 1974, Robert Merton transferred the option framework of equation (5.1) to corporate ﬁnance. He applied the equation
Equity = Assets – Liabilities, and argued that the equity value of a company
has similar properties as a call: If the asset value of a company increases,
equity increases with unlimited upside potential. In addition, the value of
equity is asymmetric, since it can only go to zero. This is the case when the
asset value drops below the debt value, which is the case of default. With this
rationale, Merton derived
E = V 0 N(d1 ) − D e−rT N(d2 )
(5.2)
V0
s2
+
ln
T
pﬃﬃﬃﬃ
De−rT
2
pﬃﬃﬃﬃ
with d1 =
and d2 = d1 − s T
s T
where
V0: current asset value of the company
D: debt of the company
s: implied volatility of V
T: time to maturity of debt D
Other variables are deﬁned as in equation (5.1).
Note that equations (5.1) and (5.2) are mathematically identical. Just the
variables are redeﬁned.
The asymmetric payoff of equity implies, as is the case with a call, that
there is time value of equity, as seen in Figure 5.5.
Figure 5.5 outlines the relationship between a company’s equity value
and its asset value at a certain point in time before debt maturity. If we assume
that the asset value grows with a certain rate r, we derive the probability of
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Value of
equity E
E
Time value
Intrinsic value
max(V – D, 0)
D
Asset
value V
FIGURE 5.5 Equity Value with Respect to Asset Value in the Merton 1974 Model
default as the probability of the asset value being smaller than the value of
debt at debt maturity T, as seen in Figure 5.6.
In Figure 5.6, using the terminology of Moody’s KMV, EDF is the
expected default frequency (i.e., the default probability), and DD is the
distance to default, which is a representation of the risk-neutral d2 of equation
(5.2). DD is the difference between the expected asset value and debt value at
debt maturity T. There is an inverse relationship between EDF and DD.
Importantly, in equation (5.1) the probability of exercising a call option
at option maturity T is Prob(ST > K) = N(d2). The probability of not
exercising the call option is 1 − N(d2) = N(−d2). In analogy, the probability
of the asset value V being smaller than the debt value D at time T, which
means default at T, follows from the Merton 1974 model of equation (5.2) as
N(−d2). Hence, the default probability in the Merton model is derived
conveniently with a closed form solution as N(−d2).1
The ingenious Merton 1974 model outlines the principles of a company’s
default using structural properties such as asset and debt. The main limitations of the model are that only one form of debt D is modeled and that
default can occur only at debt maturity T. Naturally, numerous extensions of
the model have been created to bring the model in line with the complexities
of reality. In particular:
■
The ﬁrst passage time models of Black and Cox (1976); Kim,
Ramaswamy, and Sundaresan (1993); Longstaff and Schwartz (1995);
1. For an analysis of this property, see Meissner (2007).
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Asset value V
109
Cumulative
asset return
distribution N
r
DD = d2
N(–d2) = EDF
Debt D
T
Time
FIGURE 5.6 Default Probability EDF = N(−d2) in the Merton Model for Asset Value
V < Debt D
■
■
■
and Briys and de Varenne (1997) evaluate the default probability before
debt maturity T by introducing an exogenous, continuous default barrier. Once the asset value falls below the barrier, default occurs. Hence
the ﬁrst time passage models effectively turn the European-style model of
equation (5.2) into an American-style model.
The asset return distribution at debt maturity T does not grow with the
risk-free rate r and is not assumed normally distributed (see Figure 5.6).
Instead a real-world historical asset growth rate and asset distribution is
applied. For example, Moody’s KMV database contains 30 years of
information on over 6,000 public and 150,000 private company default
events.
The debt value is not considered constant as in Figure 5.6. Instead,
empirical data is used to project a realistic increase or decrease in debt.
Other default criteria besides asset and debt value are taken into
consideration, such as liquidity risk and systemic risk, as well as
company-speciﬁc data (product line, competition, quality of management, etc.).
The Merton model, which we just discussed, is called a structural
approach, since it uses the capital structure of the entity as inputs to derive
the default probability. A different way to determine the default probability
of an entity is the reduced form approach. Here market prices such as bond
prices or credit default swap prices are the inputs to derive the default
probabilities; see Jarrow and Turnbull (1995); Jarrow, Lando, and Turnbull
(1997); and Dufﬁe and Singleton (1999). The approach is called reduced form
since it does not apply the capital structure of an entity as inputs.
Let’s now discuss the critical aspect of the Gaussian copula with respect
to valuing CDOs.
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5.2.2 Deriving the Default Correlation of the
Assets in a CDO
In the previous section, we derived the individual default probability l of
each asset i, li, in the CDO. The probability of default of an asset li is now
mapped via
N −1 (li )
(5.3)
where N−1 is the inverse of a standard normal distribution (=normsinv(li)
in Excel, norminv(li) in MATLAB). Equation (5.3) maps the default probabilities to a standard normal distribution. For example, if li = 5%, then
N−1(0.05) = −1.645, which is the x-axis value of the 5th percentile of a
standard normal distribution.2 This procedure allows a comparison of the
default probabilities with samples from an n-variate normal asset distribution Mn.
We will now determine the default threshold. This is the value that, when
breached, will constitute default of the entity or asset in question. To derive
the threshold, typically the popular Gaussian copula model is applied.
We slightly rewrite the right side of equation (4.12) and derive the default
threshold as
Mn N −1 (u1 ); . . . ; N −1 (un ); rM
(5.4)
Mn is the n-variate Gaussian distribution, N−1 is again the inverse of a
standard normal distribution, and ux is a uniform random vector ux e [0, 1];
=rand() in Excel/VBA or randn() in MATLAB. rM is the asset correlation
matrix. An example of an asset correlation matrix is shown in Table 5.2.
We now look at a certain time frame t and derive the mapped default
probability of asset i at time t, N−1(li,t), following equation (5.3). We also
derive Mn in equation (5.4) for a certain time t, Mn,t, and then derive a sample
Mn;t ( ? ) using Cholesky decomposition, which was explained in Appendix 4A
of Chapter 4. If the mapped individual default probability N−1(li,t) is
bigger than the threshold sample Mn;t ( ? ), default of asset i occurs and vice
versa. Formally:
É
(5.5)
ti;t = 1È −1
N (li;t ) > Mn;t ( ? )
In equation (5.5), 1 is an indicator variable. That is, 1 assumes the value 1
if N −1 (li;t ) > Mn;t ( ? ) and zero otherwise. We now perform Monte Carlo
simulations; that is, we derive multiple results (e.g., 100,000) of equation
(5.5) and average those results. This gives us a certain probability of default of
2. See Chapter 4, section 4.3 for details of copula mapping.
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TABLE 5.2
111
(Fictitious) Asset Correlation Matrix Underlying Figure 5.7
Asset Correlation Matrix
1
0.15
0.15
0.15
0.15
0.15
0.05
0.05
0.05
0.05
0.15
1
0.15
0.15
0.15
0.05
0.05
0.05
0.05
0.05
0.15
0.15
1
0.15
0.15
0.15
0.15
0.05
0.05
0.05
0.15
0.15
0.15
1
0.15
0.15
0.05
0.05
0.05
0.05
0.15
0.15
0.15
0.15
1
0.15
0.15
0.05
0.05
0.05
0.15
0.05
0.15
0.15
0.15
1
0.15
0.05
0.05
0.05
0.05
0.05
0.15
0.05
0.15
0.15
1
0.15
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.15
1
0.15
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.15
1
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
1
asset i at time t. For example, if the average result of equation (5.5) for a
certain asset i in the CDO is 0.1, then the default probability of this asset is
10% at time t. We apply equation (5.5) for all n assets in the CDO. This gives
us the correlated default distribution of all assets in the CDO. The defaults in
the distribution are correlated since the threshold Mn;t ( ? ) includes the
correlation of the defaults via the correlation matrix rM. Figure 5.7 shows
a possible default distribution generated by the Gaussian copula model.
In Figure 5.7, the defaults are put into 10% bins. We observe that there
is approximately a 19% probability that 10% of the assets default,
CDO Total Default Distribution
30%
25%
Probability
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20%
15%
10%
5%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90% 100%
Default
FIGURE 5.7 A Loss Distribution of the Gaussian Copula Model
Inputs are: 10 assets, default probability of every asset 5%, recovery rate 5%,
correlations as in Table 5.2. See the model “CDO Gauss educational.xlsm” at www
.wiley.com/go/correlationriskmodeling, under “Chapter 5,” for the generation of the
loss distribution.
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Defaults
SuperSenior
Senior
Mezzanine 2
Mezzanine 1
Equity
Tranche
Probability
FIGURE 5.8 Mapping of the Default Distribution to Tranches
approximately a 26% probability that 20% of the assets default, and so forth.
We see that the loss distribution is somewhat lognormal; however, other
simulations display other shapes.
We now map the default distribution to the tranches of the CDO.
Assuming a continuous default distribution, the mapping is shown in
Figure 5.8.
Figure 5.8 gives us the correlated default probability of each tranche. The
tranche spread s, which is effectively a coupon that the tranche investor
receives (see Figures 5.2 to 5.4) is directly related to the default probability l
via equation (5.6):
s ≈ l(1 − R)
(5.6)
where R is the recovery rate.
Equation (5.6) is also called the “credit triangle,” since three parameters
are involved and two parameters are necessary to derive the remaining third.
If the recovery rate is already included in the loss distribution, we have s ≈ l.
This relationship is intuitive since the default probability l is the risk that the
investors take, and they should be compensated for this risk by receiving a
similar amount, the spread s. The relationship s ≈ l (1 – R) was formally
derived by Lando (1998) with R = 0 and by Dufﬁe and Singleton (1999) with
R ≠ 0.
Once we have derived the correlated default probability distribution l,
we can derive the loss distribution L via
L = EAD l (1 − R)
(5.7)