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that have exposures to spread risk. As an example, consider an institution
that is holding a debt instrument of a given corporation. If the institution
wants to reduce its exposure to the credit risk associated with that corpora-
tion, it could buy a put option on the debt instrument it holds. Should the

4
The relationship between option prices and volatility is discussed brie¬‚y in the
following section and in Part III of this book.
8.3 Valuation Considerations 95

credit quality of the corporation deteriorate, which would result in a decline
in the market value of the debt instruments issued by the corporation, the
institution could exercise its put option to o¬set the associated losses in its
credit portfolio. In this regard, the put option is akin to buying protection
in a credit default swap (see Chapter 6). The credit default swap gives the
protection seller the right to put (sell) an underlying asset to the protection
buyer for its par value upon default by the reference entity. There are some
di¬erences, however. For instance, the put option allows the contract to be
exercised even if no default event takes place, and its premium is typically
paid upfront.
Lastly, prospective borrowers may want to buy spread options with the
purpose of capping their future borrowing costs. For instance, a corporation
may buy an option to obtain a two-year loan from a bank at a spread of
LIBOR plus, say, 150 basis points in one-year™s time. This is essentially an
option to sell a two-year ¬‚oater to the bank in one year, where the strike
spread is set at 150 basis points. If the corporation™s funding costs for a
two-year loan have risen above 150 basis points by the exercise date of the
option, the corporation exercises its right under the option, otherwise it
lets the option expire unexercised.


8.3 Valuation Considerations
We have been able to rely on the static replication approach to price the
various credit derivatives examined thus far in this part of the book. For
instance, under the assumption of liquid and frictionless markets, all we
needed to do to, say, determine the arbitrage-free value of a credit default
swap premium was to examine the costs of setting up a portfolio consist-
ing of a long position in a risky ¬‚oater and a short position in a riskless
¬‚oater. Moreover, once that portfolio was set up, we could essentially rest
assured that a short position in the portfolio would exactly o¬set any gains
and losses associated with the credit default swap. (Again, as we saw in
Chapter 1, that is what makes this a static replication.) With bond and
spread options, we encounter our ¬rst example in this part of the book
where static replication on the basis of simple cash instruments does not
work. It can be shown that a replicating portfolio does exist, but that
portfolio needs to be rebalanced dynamically, on a continuous basis, so it
can truly replicate the option.5 For now, however, we will limit ourselves to
discussing brie¬‚y the main determinants of the market price of spread and
bond options. We will revisit the valuation of these derivatives in greater
depth later, in Chapter 18.

5
See, e.g., Baxter and Rennie (2001)[6] and Bjork (1998)[7] for a discussion of
replicating portfolios for standard call and put options.
96 8. Spread and Bond Options

Those familiar with call and put options written on stocks should have
no di¬culty understanding the main factors that determine the prices of
spread and bond options. As an illustration, we will examine the case of a
call option on a bond, but the basic points made here are also applicable
to put options on bonds and to puts and calls on spreads, as we shall see in
Chapter 18. The payo¬ V (T ) of a call option on a bond can be written as

V (T ) = Max[B(T ) ’ K, 0] (8.1)

where T is the exercise date of the option, B(T ) is the market price of the
underlying bond at that date, and K is the strike price. In words, if the
price of the bond happens to be above the strike price at time T , the option
holder will exercise the option, buying the bond for the strike price K and
selling it in the open market for its market price B(T ) for an immediate
pro¬t of B(T ) ’ K. In contrast, if the market value of the bond happens
to be below the strike price, the option has no value to its holder, i.e.,
V (T ) = 0 when B(T ) < K.
Equation (8.1) shows that, from the perspective of the buyer of a call
option, the further the bond price rises above the strike price written into
the option the better. Thus, other things being equal, a call option with
a lower strike price should be more expensive than an otherwise identical
option of a higher strike price. The former presents an easier “hurdle” for
the bond price and thus a potentially higher payo¬ to the option holder.
For a put option on a bond, the reverse is true. Puts with lower strike
prices are more valuable than comparable puts with higher strike prices.
The volatility of the bond price is a key determinant of the option price.
Higher volatility increases the chances that B(T ) in equation (8.1) will end
up above K so that the option will expire “in the money.” Thus options
written on bonds whose prices are subject to greater volatility are more
expensive than otherwise comparable (same maturity and strike and initial
bond prices) options written on bonds with relatively stable prices.


8.4 Variations on Basic Structures
There are several variants of the basic spread and bond options described
so far in this chapter. For instance, an even simpler spread option than the
ones described above is one written directly on a ¬‚oating-rate note, instead
of on an asset swap. Alternatively the underlying in a spread option may
well be a credit default swap. Indeed, we mentioned credit default swaptions
at the end of Chapter 6 when we discussed variations on the vanilla credit
default swap agreement.
For the sake of simplicity, this chapter has focused on European options,
or those with a single, ¬xed exercise date. Spread and bond options are
8.4 Variations on Basic Structures 97

also negotiated with “American” and “Bermudan” exercise structures.
American-style spread and bond options allow the option holder to exer-
cise the option at any time during the life of the option (some contracts
may stipulate an initial “no-exercise” period, e.g., a ¬ve-year option that
becomes exercisable any time starting in one year). Bermudan-style options
give the option buyer the right to choose one of several ¬xed exercise dates.
For instance, the option may be exercised on any one of the coupon dates
of the underlying bond.
There are many other variations in spread and bond option contracts as
these tend to be less standardized than, say, credit default swap agreements.
For instance, while many options are physically settled, others are settled
in cash, and some spread options may be written in terms of spreads over
a given US Treasury yield, rather than short-term LIBOR.
9
Basket Default Swaps




Unlike the basic forms of the contracts discussed in Chapters 4 through
8, basket default swaps are credit derivatives written on a “basket” or
portfolio of assets issued by more than one reference entity. In particular,
a payment by the protection seller in a basket swap can be triggered by a
default of any one of the entities represented in the basket, provided that
default meets the requirements speci¬ed in the contract.
We brie¬‚y encountered a common variety of a basket default swap”the
¬rst-to-default basket”in Chapter 1 when we introduced di¬erent types of
credit derivatives. We will now take a closer look at this instrument, this
time to illustrate the basic structure of basket default swaps. We also use
this chapter to highlight the importance of default correlation in the pricing
of multi-name credit derivatives, an issue we will examine in greater detail
in Part IV of this book.



9.1 How Does It Work?
Let us look at a particular example. Consider an institution that wants to
hedge its exposure to ¬ve di¬erent reference entities. The institution enters
into a ¬rst-to-default (FTD) basket contract with a derivatives dealer where
the reference basket is composed of those ¬ve entities. The institution,
which is the protection buyer in the contract, agrees to make periodic
premium payments to the dealer (the protection seller), much like in a
single-name credit default swap. In return, the dealer commits to making
100 9. Basket Default Swaps




FIGURE 9.1. Diagram of a First-to-default Basket


a payment to the institution (the protection buyer) if and when any one
of the entities included in the reference basket defaults during the life of
the contract. The catch is that the payment will be made for only the
¬rst default. The FTD basket contract is terminated after the protection
coverage regarding the ¬rst default in the basket is settled.
Figure 9.1 illustrates the basics of an FTD basket. In this example, we
assume that the notional amount of debt covered by the contract is $10
million for each of the ¬ve entities in the basket reference. Let us say that
the FTD basket premium is quoted as 200 basis points per year and that
recovery rates are zero. That means that the annual payment made by the
protection buyer would be
200
— $10 million = $200,000
1002
or, typically, $50,000 per quarter. Why was the premium payment based
on only $10 million, rather than on $50 million, which is the sum of the
notional amounts covered for each reference entity? Because the protection
seller will cover only the ¬rst default, and the notional amount associated
with each individual default is only $10 million.
If none of the reference entities defaults during the life of the FTD bas-
ket, the protection seller simply keeps on collecting the premium payments
made by the protection buyer. What happens when the ¬rst default takes
place? Similar to a single-name credit default swap, which we discussed in
Chapter 6, the protection seller pays the buyer the di¬erence between the
face value of the defaulted debt and its recovery value. The contract can
be either cash or physically settled.
Carrying on with the example in Figure 9.1, suppose reference entity #3
defaults and the contract calls for physical settlement. The protection buyer
9.3 Valuation Considerations 101

will deliver to the protection seller $10 million worth of par value of eligible
debt instruments issued by reference entity #3. These instruments are
now worth only their recovery value, but the protection seller will pay par
for them. The FTD basket swap terminates once the protection payment
related to the ¬rst default is made. The protection seller has no further
exposure to the remaining names in the reference basket.



9.2 Common Uses
Protection buyers ¬nd basket swaps attractive because they tend to be
less expensive than buying protection on each name in the basket sepa-
rately through single-name credit default swaps. Of course, the lower cost
of protection in, say, the FTD basket swap stems from the fact that the
protection seller is only really protected from the ¬rst default. Still, in the
case of the FTD basket, protection buyers ¬nd some comfort in the fact
that it will take more than one default in their portfolio before they actually
experience a loss. In market parlance, the credit risk associated with the
“¬rst loss” in the reference basket has been transferred to the protection
seller.
From the perspective of investors (protection sellers), basket swaps pro-
vide an opportunity for yield enhancement with a limited downside risk.
For instance, we will see below that the FTD basket premium is typically
above the credit default swap premiums of any one individual entity refer-
enced in the basket. This is because the protection seller is exposed to the
credit risk of all names in the reference basket. As a result, a protection
seller in an FTD basket composed of, say, A-rated names, could conceiv-
ably earn a premium that would be typical of, say, a credit default swap
written on a single BBB-rated name, without actually having to expose
itself to a BBB-rated entity.
We mentioned above that the protection seller has a limited downside
risk. Why? Because the seller will ultimately be liable to cover at most one
default among the names included in the basket. The protection seller can
thus substantially leverage its credit exposure: Going back to the example
in Figure 9.1, the protection seller was exposed to assets totaling $50 million
in notional amount, but the most it could lose was $10 million.



9.3 Valuation Considerations
What determines the premium paid by the protection buyer in a credit
basket swap? The main determinants are (i) the number of entities in the
reference basket, (ii) the credit quality and expected recovery rate of each
102 9. Basket Default Swaps

basket component, and (iii) the default correlation among the reference
entities.1
Understanding the role of the ¬rst two determinants”number and credit
quality of the reference entities”is relatively straightforward. In general,
other things being equal, the larger the number of entities included in the
basket the greater the likelihood that a default event will take place and
thus the higher the premium that protection buyers will pay. Likewise, for
a given number of names in the basket, the lower the credit quality and
recovery rates of those names, the more expensive the cost of protection
will be.
Though important, the number and credit quality of the entities in the
reference basket only tell part of the story when it comes to pricing a credit
basket swap. Indeed, the role of default correlation is so important in the
pricing of basket swaps that market participants tend to characterize these
instruments mostly as default correlation products. To see why this is so
we will examine the simplest FTD basket one can imagine, one that is
composed of only two reference names, which we shall refer to as XYZ
Corp. and AZZ Bank.
Consider an investor who is exposed to both XYZ Corp. and AZZ Bank,
but who wants to use credit derivatives to at least reduce this exposure.
One option for such an investor is simply to buy protection in two sepa-
rate credit default swaps”one referencing XYZ, the other written on AZZ.
Assume that the notional amount of protection sought for each reference
entity is $1, the relevant recovery rates are 0, and the maturity of the
desired credit default swaps is one year. If we make the same simplifying
assumptions adopted in Section 6.3.3, the premium for the CDS written on
XYZ is simply given by the probability ωX that XYZ will default in one
year™s time, and the cost of buying protection against a default by AZZ is
analogously determined. Thus the cost of buying protection through two
separate single-name CDS in this case would be:

SCDSs = ωX + ωA (9.1)

What if, instead of buying protection against defaults by both XYZ and
AZZ through two separate credit default swaps, the investor were to buy
protection through an FTD basket with the same maturity as the CDS and
with a notional amount of $1 for each of the names in the basket? Note that
in this case, the protection bought and sold applies to a default by either
XYZ or AZZ, whichever comes ¬rst. Let ωX or A denote the probability
that at least one of the entities will default in one year™s time. Thus, the
protection seller will be required to make a payment of $1 to the protection

1
Default correlation is discussed further in Part IV”see also Appendix B.
9.3 Valuation Considerations 103

buyer with probability ωX or A ”no payment from the protection seller will
be due otherwise. Carrying on with the simple methodology outlined in
Chapter 6, we can write the present value of the “protection leg” of this
basket swap as

PV[protection] = PV[ωX or A — $1 + (1 ’ ωX or A ) — $0] (9.2)

where PV[.] denotes the present value of the variable in brackets.
Recall now from elementary statistics that one can write ωX or A as

ωX or A = ωX + ωA ’ ωX&A (9.3)

where ωX&A denotes the probability that both XYZ and AZZ will default
in one year™s time.2 Thus, we can rewrite (9.2) as

PV[protection] = PV[ωX + ωA ’ ωX&A ] (9.4)

Let us now value the premium leg of the basket. Let Sbasket denote
the premium paid by the protection buyer under this basket agreement.
The present value of the premium leg is simply given by

PV[premiums] = PV[Sbasket ] (9.5)

and thus, if the basket has zero market value at its inception, the present
values of the premium and protection legs must be the same, which
implies that

Sbasket = ωX + ωA ’ ωX&A (9.6)

Equation (9.6) allows us to make several important points regarding the
valuation of basket swaps. As discussed in Part IV, the probability that
both XYZ and AZZ will default in one year™s time is closely related to the
default correlation between the two reference entities. In particular, if the
one-year default correlation between the two entities is positive and high,
ωX&A will be a larger number.3

2
This basic result is illustrated graphically in Chapter 19.
3
The generalization to the case of nonzero recovery is straightforward. The reader
can verify that the expression for the FTD premium becomes:

Sbasket = (ωX + ωA ’ ωX&A )(1 ’ X)

where X is the recovery rate associated with the reference entities.
104 9. Basket Default Swaps

9.3.1 A First Look at Default Correlation
We can use equation (9.6) to take a preliminary look at how default corre-
lation a¬ects the pricing of the FTD basket. Let us examine ¬rst the case of
very high default correlation between XYZ and AZZ, a default correlation
close to 1. Suppose

ω X > ωA

i.e., XYZ is more likely to default than AZZ. It can be shown”we
will see this in Part IV”that as the default correlation approaches its
maximum at 1, the probability of both XYZ and AZZ defaulting in
one year™s time approaches the default probability of the higher-quality
entity (AZZ in this example). Intuitively, if the ¬nancially stronger entity
has defaulted, chances are that the lower-quality ¬rm has almost surely
defaulted if the default correlation involving the two companies is very high.
Thus, if ωX > ωA ,

ωX&A ≈ ωA when the default correlation ≈ 1

and, given (9.6),

Sbasket ≈ ωX

i.e., the FTD basket premium approaches the default probability of the
entity with the lowest credit quality (XYZ) as the default correlation
between the names in the basket approaches 1.
We will also seen in Part IV that, when (i) default correlation is close to
zero and (ii) the product of individual default probabilities is su¬ciently
small, ωX&A will be low. Indeed, equation (9.6) implies that the FTD
basket premium will approach the sum of the individual credit default
swap premiums of the reference entities as ωX&A approaches zero. Using
the above notation:

ωX&A ≈ 0 when conditions (i) and (ii) above are met

and, given equation (9.6),

Sbasket ≈ ωX + ωA

To sum up, and generalizing to baskets with more than two reference
entities, protection bought through a ¬rst-to-default basket will be more
expensive when the degree of default correlation among the entities in the
basket is low and cheaper when that correlation is high. In plain words,
with low default correlation, the protection seller in the above example
9.4 Variations on the Basic Structure 105

is exposed to two largely uncorrelated sources of risk and must be com-
pensated accordingly with a higher FTD basket spread. In contrast, when
default correlation is close to one, the protection seller is exposed mainly
to one type of risk, the common factor driving the fortunes of the refer-
ence entities. Intuitively, the FTD basket premium will approach the CDS
spread of the entity that is most vulnerable to that common factor, i.e.,
the one with the highest default probability. For intermediate values of
default correlation, the FTD basket premium will fall in between the high-
est (single-name) CDS premium in the portfolio and the sum of all the
CDS premiums.
One ¬nal point: Note that when the probability that both XYZ and
AZZ will default is zero or very nearly so”ωX&A ≈ 0”the cost of buy-
ing protection through the basket approaches that of buying protection
separately through two credit default swaps, one written on XYZ and the
other on AZZ, where each contract has a notional amount of $1, the same
notional amount associated with each reference entity included in the bas-
ket. Under such circumstances, the protection buyer may be better o¬ with
the two separate CDS contracts. Why? Together, the two CDS contracts
cost about the same as the basket, but they e¬ectively provide protection
against defaults by both XYZ and AZZ, whereas the basket only provides
protection against the ¬rst default. Thus, it makes more sense to buy
protection through an FTD basket when there is some default correlation
among the reference entities.



9.4 Variations on the Basic Structure
There are several variations around the basic structure discussed thus far
in this chapter. The most straightforward one is the second-to-default
swap, which, as the name suggests, is a contract where a payment by
the protection seller is triggered only by the second default in the basket.
For instance, for a basket where each reference entity is associated with a
notional amount of $1, that payment will be equal to $1 minus the recov-
ery value of the debt instruments of the second entity to default. Third-,
fourth-, and, more generally, nth-to-default swaps are de¬ned accordingly.
From the perspective of investors, these products still o¬er the opportunity
to take a leveraged exposure to credit, albeit at a lower risk (and thus with
a lower yield).4
Unlike the simple example discussed in the previous section, which
involved only two reference entities, basket swaps tend to reference ¬ve to

4
Nth-to-default baskets have similarities to tranched structures, which we discuss in
Chapter 14.
106 9. Basket Default Swaps

ten entities. Nonetheless, the main pricing results derived above regarding
default correlations can be generalized for baskets written on a larger num-
ber of entities, although the required computations become substantially
more complicated. We will address these more realistic cases in Part IV of
this book.
10
Portfolio Default Swaps




Portfolio default swaps are similar to basket swaps (Chapter 9) in that
they transfer portions of the credit risk associated with a portfolio from
a protection buyer to a protection seller. A key di¬erence is that the risk
transfer is speci¬ed in relation to the size of the default-related loss in the
reference portfolio instead of in terms of the number of individual defaults
among the reference entities. For instance, whereas protection sellers in a
¬rst-to-default basket are exposed to the ¬rst default in the reference bas-
ket, protection sellers in a “¬rst-loss” portfolio default swap are exposed to
default-related losses that amount up to a prespeci¬ed share of the reference
portfolio.
In Chapter 9 we introduced some key ideas about the role of default
correlation in the pricing of multi-name credit derivatives. We continue
to highlight this role here and introduce another key concept for valu-
ing derivatives that reference more than one entity: the loss distribution
function. Lastly, by discussing the basics of portfolio default swaps, this
chapter lays some of the groundwork for discussing synthetic collateralized
debt obligations (CDOs), which are the subject of Chapter 14.



10.1 How Does It Work?
Consider a hypothetical bank with a large portfolio of loans. The bank
wants to reduce its exposure to the credit risk embedded in the portfolio,
but does not want to sell or transfer the underlying loans. In addition, the
108 10. Portfolio Default Swaps




FIGURE 10.1. Diagram of a Simple Portfolio Default Swap with a First-loss Piece
of 10 Percent (Size of Portfolio = $50 million; Premium = 100
basis points, paid annually)


number of reference entities represented in the portfolio is large enough
that simple basket products, such as a ¬rst-to-default basket, would o¬er
only very limited protection. The bank could, of course, buy protection in
several baskets”e.g., ¬rst-, second-, third-to-default baskets . . .”or even
enter into several single-name credit default swaps to achieve its goals.
Alternatively, the bank could buy protection through a single portfolio
default swap.
The basic features of a simple portfolio default swap are illustrated in
Figure 10.1. Suppose the bank feels that the chances that it will experience
default-related losses in excess of, say, 10 percent over the next year are
su¬ciently small that it is willing to bear that risk. The bank can enter
into a one-year portfolio default swap with an investor who is willing to
sell protection against the ¬rst 10 percent in default-related losses in the
portfolio. The investor (the “¬rst-loss protection seller”) will be exposed to
however many individual defaults are necessary to produce a 10 percent loss
in the reference portfolio. To be more speci¬c and simplify things further,
suppose the total face value of the portfolio is $50 million and that there
are 50 reference entities represented in the portfolio, each with a face value
of $1 million and a zero recovery rate. As shown in Table 10.1, given the
assumed recovery rate and the absence of counterparty credit risk, it would
take 5 defaults in the portfolio before the bank actually becomes exposed
to credit risk.
In exchange for the protection provided, the protection buyer agrees to
make periodic premium payments to the protection seller, much like in
a single-name credit default swap. The premium payments made by the
protection buyer amount to

premium payment = (PDS premium) — (size of the ¬rst-loss piece)
10.1 How Does It Work? 109

TABLE 10.1
Loss Associated with 5 Defaults in the Bank™s Portfolio
($ Millions)


Initial value of the portfolio $50
$50 ’ 5 — $1 = $45
Value after 5 defaultsa
Percent default-related loss 10 percent
a Assuming no deterioration in the credit quality of remaining solvent
entities in the portfolio and no change in market interest rates.




where the portfolio default swap (PDS) premium is usually quoted in terms
of basis points per annum. For instance, assume that the PDS premium
in the above example is set at 100 basis points and that payments will be
made annually (in more realistic cases, the payments are more likely to be
made quarterly). Even though the protection seller is initially exposed to
defaults by any of the 50 entities in the portfolio”a total notional amount
of $50 million”the protection sold will cover a maximum of 5 defaults.
Thus, the annual premium payments received by the protection seller will
be $50,000 or 100 basis points times $5 million.
What happens when defaults take place? As with a single-name credit
default swap, and assuming that the contract calls for cash settlement
upon default events, the protection seller will pay the di¬erence between
the par and recovery values of each defaulted asset, provided, of course,
these payments do not exceed the original size of the ¬rst-loss piece.
Note that, as defaults occur, the size of the ¬rst-loss piece is reduced
accordingly. For instance, carrying on with the numerical example, if the
protection seller has already paid $1 million as a result of the loss incurred
with the ¬rst default, the future payments in the event of additional defaults
are now capped at $4 million. It is common for the premium payment made
by the protection buyer to be adjusted to re¬‚ect the new size of the ¬rst-
loss piece. Given the PDS premium of 100 basis points, the new annual
premium after the ¬rst default becomes $40,000 (= 100 b.p. — $4 mil.).
The process of paying for defaults and resetting the premium continues
until the payments made by the protection seller max out, in which case the
contract with the ¬rst-loss protection seller is terminated. At that point,
the size of the ¬rst-loss piece, as well as the premium paid by the protection
buyer, would have reached zero.
We have thus far focused on a ¬rst-loss contract, but the structure of,
say, a second-loss contract is entirely analogous. For instance, suppose the
bank had bought protection in an additional portfolio default swap, one
where the protection seller would cover all default-related losses beyond
the ¬rst $5 million. This protection seller is essentially long the second-loss
110 10. Portfolio Default Swaps

piece of the portfolio. The second-loss protection seller will start covering
default-related losses once the payments made by ¬rst-loss protection seller
are maxed out. (If the bank had chosen not to enter into this second
contract, its position would be akin that being long the second-loss piece.)



10.2 Common Uses
From the perspective of investors, portfolio default swaps allow one to take
a substantially leveraged exposure to credit risk”and thus earn a higher
premium”with only limited downside risk. In the above example, the ¬rst-
loss protection seller was exposed to defaults in the entire loan portfolio of
$50 million, but the maximum loss was capped at $5 million. At the same
time, less aggressive investors might be attracted to the relative safety of
higher-order-loss products, such as the second-loss piece, which will only
sustain losses after the protection provided by the entire ¬rst-loss piece is
exhausted.
Portfolio default swaps are attractive to protection buyers because they
allow the transfer of a substantial share of the credit risk of a portfolio
through a single transaction, as opposed to a large number of individual
transactions. In addition, similar to basket swaps, a portfolio default swap
can be a cost e¬ective way of obtaining partial protection against default-
related losses in one™s portfolio.
Lastly, we should note an important general use of portfolio default
swaps. As we shall see in Chapter 14, they are the basic building blocks for
synthetic collateralized debt obligations, which make up a rapidly growing
sector of the credit derivatives market.



10.3 Valuation Considerations
Issues of counterparty credit risk aside, portfolio default swaps cannot
increase or reduce the overall degree of credit risk in the reference port-
folio. What portfolio default swaps do is redistribute the total credit risk of
the portfolio among di¬erent investors”e.g., ¬rst-, second-, and third-loss
protection sellers.
Intuitively, it is straightforward to see that the premium on the ¬rst-loss
piece depends importantly on how much of the total credit risk of the port-
folio is borne out by the ¬rst-loss protection sellers. To illustrate this point
we start by looking at a highly arti¬cial but instructive example. Consider
the case where the hypothetical portfolio examined in Section 10.1 is made
up of ¬ve reference entities that are rated well below investment grade and
10.3 Valuation Considerations 111

45 entities that are very highly rated.1 Under these circumstances, most of
the credit risk of the portfolio resides in the ¬ve lowly rated entities. Now
let us go back to Table 10.1, which illustrates a scenario where ¬ve of the
reference entities in the portfolio default while the swap is still in force.
Given what we have just assumed about the composition of the portfolio,
the chances that such a scenario come to unfold are likely high. At the same
time, the probability of any losses beyond 10 percent of the portfolio would
be rather small. What we have here then is a situation where the ¬rst-loss
protection sellers end up absorbing most of the credit risk of the portfolio.
What do they get in return? They are compensated in the form of a high
¬rst-loss premium. Looking beyond the ¬rst-loss piece, the likelihood that
second-loss protection sellers will ever have to cover a default-related loss
in the portfolio is small in this particular example, and so would be the
second-loss premium.

10.3.1 A First Look at the Loss Distribution Function
In the above example, we considered a portfolio with very particular
characteristics”a few very risky assets combined with lots of very highly
rated assets. We used that example to discuss a situation with one speci¬c
feature: very high odds of losses of up to 10 percent of the initial par value
of the portfolio and very low odds of losses beyond 10 percent. In e¬ect,
what we did was consider a particular loss distribution function for the
portfolio. The loss distribution function is a key concept when it comes
to valuing a portfolio default swap. Intuitively, it tells us the probabilities
associated with di¬erent percentage losses in the portfolio.
Figure 10.2 shows loss distributions for two hypothetical portfolios.
To highlight the importance of the loss distribution in the pricing of port-
folio default swaps, both distributions correspond to portfolios with the
same expected loss of 5 percent. Assuming a ¬rst-loss piece of 10 percent,
the loss distribution of portfolio A (shown as the solid line) is such that it is
nearly certain that virtually all the losses will be borne out by the ¬rst-loss
investors: The ¬gure shows a probability of nearly 100 percent of losses less
than or equal to the ¬rst-loss piece and tiny probabilities of larger losses.
In contrast, the loss distribution of portfolio B (the dashed line) shows a
much higher probability of total losses in the portfolio exceeding 10 percent,
suggesting the total credit risk associated with portfolio B is more widely
spread between ¬rst- and second-loss protection sellers.2 In terms of the
pricing implications for portfolio default swaps written on these portfolios,

1
Assume further that the default correlation between the lowly and highly rated
assets is zero.
2
Given the loss distributions depicted in Figure 10.2, the probability of losses greater
than 10 percent in each portfolio, which corresponds to the sum of the probabilities
112 10. Portfolio Default Swaps




FIGURE 10.2. Loss Distributions for Two Portfolios, Each With an Expected
Loss of 5 Percent




one would expect that while ¬rst-loss protection sellers would receive most
of the credit risk premium associated with portfolio A, they would have to
share more of that premium with second-loss protection sellers in portfolio
B, despite the fact that the two portfolios have identical expected losses.
What are the main factors that determine whether the loss distribution
of actual portfolios will look more like that of portfolio A or that of port-
folio B? We have thus far relied on a case that highlights how the credit
quality of the individual reference entities in the portfolio can a¬ect the
loss distribution. In more realistic examples, however, it will not generally
be the case that the reference entities in the portfolio can be so neatly
assigned to nearly opposite ends of the credit quality spectrum. Indeed, in
many instances, the portfolio may well be composed of debt instruments
issued by entities with similar credit quality. What would determine the
shape of the loss distribution in such instances? To put it di¬erently, how
much of the total credit risk in the portfolio is being shifted, say, to the
¬rst-loss protection sellers? To answer these questions, we need to revisit a
theme ¬rst introduced in Chapter 9: default correlation.


associated with losses larger than 10 percent, is virtually zero for portfolio A and about
14 percent for portfolio B. We will examine loss distributions more closely in Part IV.
10.3 Valuation Considerations 113

10.3.2 Loss Distribution and Default Correlation
Let us again go back to the example discussed in Section 10.1”a portfolio
composed of 50 reference names, each with a $1 million notional amount
and a zero recovery rate”but this time we will assume that all of the enti-
ties represented in the portfolio have the same credit quality. In particular,
suppose that each of the reference entities has a risk-neutral default proba-
bility of 6 percent. We also continue to consider two portfolio default swaps
written on the portfolio, one with a ¬rst-loss piece of 10 percent and the
other with a second-loss piece that encompasses the remaining losses in the
portfolio.
We will consider ¬rst the case of very high default correlation among
the 50 names in the portfolio. In this case, as we argued in Chapter 9, the
reference entities in the portfolio tend to default or survive together. As
a result, the portfolio behaves more like one single asset. Let us assume
the polar case of perfect default correlation. What is the expected loss
in the portfolio? Note that there are only two possible outcomes in this
case, either all entities represented in the portfolio default together or they
all survive. Thus, with 6 percent probability all reference entities default,
resulting in a total loss of $50 million, and with 94 percent probability
there are no losses. The expected loss in the portfolio is

.06 — $50 million + .94 — $0 = $3 million

which results in an expected loss of 6 percent for the entire portfolio.
What is the expected loss in the ¬rst-loss piece? With 6 percent proba-
bility the entire ¬rst-loss piece is wiped out; with 94 percent probability it
remains intact. The expected loss in this piece is

.06 — $5 million + .94 — $0 = $0.3 million

which amounts to an expected loss of 6 percent of the ¬rst-loss piece.
How about the expected loss in the second-loss piece? If all ¬rms in
the portfolio default, second-loss investors absorb the residual loss of $45
million, which corresponds to an expected loss of $2.7 million:

.06 — $45 million + .94 — $0 = $2.7 million

or 6 percent of the second-loss piece. Thus, with perfect default correlation,
the ¬rst- and second-loss investors in this example should, in principle, earn
the same protection premium, although, naturally, the premium payments
will be scaled to the sizes of the ¬rst- and second-loss pieces.
To illustrate the e¬ect of default correlations on expected returns on the
¬rst- and second-loss pieces we now examine another limiting case, one
114 10. Portfolio Default Swaps

with zero default correlation. The computation of expected losses for the
¬rst- and second-loss investors in this case is a bit more involved than that
for the case of perfect default correlation.
Whereas in the perfect correlation case there were only two possible
default outcomes, either all ¬rms default together or all survive, we now
have 51 possible default outcomes (no defaults, one default, two defaults,
. . ., 50 defaults), each with a di¬erent probability attached to it. This may
sound complicated, but, with zero default correlation it is not hard to show
that the expected loss of the ¬rst-loss investors will amount to approxi-
mately $2.88 million, or about 57.5 percent of the ¬rst-loss piece, and that
of the second-loss investors will be about $0.12 million or approximately
0.27 percent of the second-loss piece.3
Naturally, with zero default correlation, ¬rst-loss investors in this exam-
ple will only risk losing over half of the ¬rst-loss piece if the premium
that they receive from the protection buyer is su¬ciently high. As for
the second-loss investors, because their expected loss is very small, the
second-loss premium will be very low.
Table 10.2 summarizes the main results obtained thus far regarding the
two alternative default correlation scenarios just discussed. The table shows
that the expected loss of the second-loss investors goes from 0.27 percent
in the case of zero correlation to 6 percent in the case of perfect corre-
lation. As more of the credit risk in the portfolio is transferred to the
second-loss investors in the perfect-correlation case, the premium that they
receive should also increase. The reverse happens, of course, to the ¬rst-loss
investors. Their expected loss falls from 57.61 percent in the case of zero
default correlation to 6 percent in the case of maximum default correlation,

3
One way to arrive at these results for the expected losses of the two classes of
investors under the assumption of uncorrelated defaults is to make use of binomial
distribution. In particular, let q(i) denote the probability that there will be i defaults in
the portfolio within the next 12 months. Under the binomial distribution, the current
example is such that
50!
.06i .9450’i
q(i) =
i!(50 ’ i)!
and we can, for instance, write the expected loss of ¬rst loss-investors as
50
q(i)Min[L(i), 5]
i=0

where L(i) is the total loss, in $ millions, in the portfolio when there are i defaults,
and the notation Min[L(i), 5] indicates that the ¬rst-loss investors™ loss is capped at
$5 million in this case. The expected losses of the overall portfolio and of second-loss
investors can be computed in an analogous way.
We will get back to the binomial distribution and the modeling of expected portfolio
losses in Part IV of this book. Some background on the binomial distribution is provided
in Appendix B.
10.3 Valuation Considerations 115

TABLE 10.2
Loss Distribution and Default Correlation

zero perfect
correlation correlation

Expected loss in portfolioa 6 percent 6 percent
($3 million) ($3 million)
Expected loss of ¬rst-loss investors 57.61 percent 6 percent
($2.88 million) ($0.3 million)
Expected loss of second-loss investors 0.27 percent 6 percent
($0.12 million) ($2.7 million)
a Size
of portfolio: $50 million. Sizes of ¬rst- and second-loss pieces are $5 million and
$45 million, respectively.




and the premium that they receive is reduced accordingly to re¬‚ect that
the total risk of the portfolio is now more evenly distributed between the
¬rst- and second-loss pieces.4
For the more realistic cases of intermediate degrees of default correlation,
the expected losses of the ¬rst- and second-loss investors fall somewhere
in between the two polar cases shown in Table 10.2. In general, as we
shall see in greater detail in Part IV, as the extent of default correlation
in the portfolio increases, more of the total credit risk embedded in the
portfolio is shared with the second-loss piece, which, as in the example just
examined, will then earn a higher protection premium than in instances of
lower default correlation. At the same time, as less of the total risk in the
portfolio is borne out by the ¬rst-loss protection sellers alone, the premium
that they earn decreases as default correlation rises.
In terms of the loss distribution, Table 10.2 suggests that, other things
being equal, portfolios with higher default correlations have higher prob-
abilities of larger losses than portfolios with lower default correlations.
Intuitively, as default correlations increase, so does the likelihood that
the reference entities represented in the portfolio will default together,
leading to a greater chance of larger default-related losses in the overall
portfolio.

4
With nonzero recovery rates, the ¬rst- and second-loss investors will not generally
earn the same premiums in the case of perfect default correlation. For instance, with a
50 percent recovery rate, the second-loss investors™ expected loss would be

.06 — $(25 ’ 5) million + .94 — $0 = $1.2 million

which amounts to about 2.7 percent of the second-loss piece.
116 10. Portfolio Default Swaps

10.4 Variations on the Basic Structure
Before we move on, we should note that the terms of portfolio default swap
agreements are less standardized than, say, those of single-name credit
default swaps. Thus, although the example discussed above was meant to
be illustrative of the basic portfolio default swap structure, one is bound
to ¬nd actual contracts that will di¬er in one or more dimensions from the
example provided.
Variations around the example examined in this chapter include physical
settlement upon default, immediate vs. deferred default settlement, the
timing and manner of resetting the premium, and the de¬nition of credit
events. Also, in many situations where the protection buyer is a bank, the
bank retains a small ¬rst-loss piece and enters into various portfolio default
swaps (e.g., second-, third-, and fourth-loss products) with investors, either
directly or through an intermediary bank.
One last point: In the examples explored in this chapter, the reference
portfolio was often described as a collection of loans or bonds held by the
institution seeking to buy protection in the portfolio default swap. We
could just as well have described the reference portfolio as a collection of
individual credit default swaps in which the institution sold protection.
In this case, the institution uses one or more portfolio default swaps to
transfer at least some of the credit risk acquired via the single-name credit
default swaps.
11
Principal-Protected Structures




Principal-protected structures are coupon-paying ¬nancial products that
guarantee the return of one™s initial investment at the maturity of the struc-
ture, regardless of the performance of the underlying (reference) assets.
The coupon payments themselves are stopped in the event of default by
the reference entity. Principal-protected structures can be thought of as a
form of a funded credit derivative.



11.1 How Does It Work?
It may be useful to start by reminding ourselves about the mechanics of
traditional (nonprincipal-protected) debt instruments. For instance, assum-
ing a ¬xed-rate bond is valued at par, an investor in that security hands
over the face value of the bond to the issuer and, in exchange, the issuer
promises to return the full par amount of the note to the investor at the
maturity date of the note and to make intervening coupon payments until
that date. Should the issuer run into ¬nancial di¬culties and default on its
debt obligations, however, the investor loses both its initial investment (or
part of it in the case of a nonzero recovery rate) and any remaining future
coupon payments that would otherwise be made by the note.
We shall focus on single-name principal-protected notes (PPNs), the
simplest form of a principal-protected structure. Such notes have some
similarities with traditional ¬xed-rate bonds, but there are a few key dif-
ferences. As with the bond in the previous paragraph, PPNs are funded
118 11. Principal-Protected Structures

instruments, generally sold at par, that promise regular coupon payments
at prespeci¬ed dates and the return of principal at the maturity date of
the note. Unlike traditional debt instruments, however, PPNs are generally
issued by highly rated third parties, rather than by the reference entities
themselves. In that regard, a PPN is very much like a credit-linked note, in
that its cash ¬‚ows are contingent on default events by a reference entity.1
Where a PPN di¬ers both from traditional bonds and simple credit-linked
notes is in the event of default by the reference entity. Upon default, typical
bonds and CLNs terminate with investors getting only their corresponding
recovery values. With PPNs, the stream of future coupon payments ter-
minates, but the PPN does not. In particular, the repayment of the par
value of the PPN at its maturity date is una¬ected by the reference entity™s
default.
Table 11.1 uses a simple numerical example to illustrate the main features
of PPNs, and how they compare to par bonds (and, implicitly, to typical
CLNs). The table shows the cash ¬‚ows of two instruments: a four-year ¬xed-
rate bond issued by XYZ Corp. and a four-year PPN that references XYZ
Corp., issued by a highly rated ¬nancial institution.2 Both instruments are
assumed to pay coupons annually and are initially valued at par ($100).
The bond pays a coupon of 9 percent, and the PPN™s coupon is 6.8 percent.3
If XYZ does not default during the four-year period covered by the notes,
the holders of the bond and the PPN receive the cash ¬‚ows shown in the
upper half of the table, the only di¬erence between them being the size of
their respective coupons.
The lower panel of Table 11.1 shows the cash ¬‚ows of the bond and the
PPN in the event of default by XYZ Corp. at year 2, immediately after
the coupon payments are made. We assume a recovery rate of 50 percent.
The bond terminates with the investor™s position valued at $59, the coupon
payment of $9 just received plus the recovery value of $50. The PPN lives
on, but makes no further coupon payments; it terminates only at its original
maturity date, when it pays out its par value of $100.
We mentioned above that principal-protected structures are generally
issued by highly rated entities. In addition, it is not uncommon for the
principal guarantee to be collateralized. The main idea, of course, is to
minimize the PPN investor™s exposure to any credit risk associated with
the PPN issuer.

1
Credit-linked notes were introduced in Chapter 1 and are discussed further in
Chapter 12.
2
Many of the assumptions made in this example can be easily relaxed without loss
of generality. For instance, as we shall see later in this chapter, to value a PPN we need
not have a ¬xed-rate note with the same maturity as the PPN.
3
Why does the PPN have a lower coupon than the bond? The investor essentially
has to forego some yield in order to obtain the principal-protection feature.
11.3 Valuation Considerations 119

TABLE 11.1
Cash Flows of a PPN and a Par Bonda
(Assuming that the PPN references the bond issuer)


Years Fixed-rate Principal-
from now bond protected note

(1) (2) (3)

A. Assuming no default by the reference entity

’100 ’100
0
1 9 6.8
2 9 6.8
3 9 6.8
4 109 106.8

B. Assuming default by the reference entity at year 2b

’100 ’100
0
1 9 6.8
2 59 6.8
3 0 0
4 0 100
a From the investor™s perspective. Par value = $100; recovery rate = 50 percent.
b Assuming default occurs immediately after the coupons are paid at year 2.



11.2 Common Uses
PPNs appeal to investors seeking some exposure to credit risk, but who
want to protect their initial investment. As such, PPNs can be used to make
sub-investment-grade debt instruments appealing to conservative investors.
As the example in Table 11.1 illustrated, investors may have to give up a
substantial portion of the credit spread associated with the reference entity
in order to obtain the principal-protection feature. As a result, a PPN that
references a highly rated entity would have very limited interest to some
investors as its yield would be very low. On the other hand, conservative
investors might welcome the additional safety that a PPN would provide
even to investment-grade instruments.



11.3 Valuation Considerations
For valuation purposes, it is helpful to decompose a PPN into two
components, the protected principal and the (unprotected) stream of
120 11. Principal-Protected Structures

coupon payments.

PPN = protected principal + stream of coupon payments

For simplicity, we start by assuming that counterparty credit risk is com-
pletely dealt with via full collateralization and other credit enhancement
mechanisms so that the PPN buyer has no credit risk exposure to the
PPN issuer. This implies that we can think of the protected principal as
being akin to a riskless zero-coupon bond that has the same par value and
maturity date as the PPN.
As for the stream of coupon payments, it can be characterized as a risky
annuity that makes payments that are equal to the PPN™s coupon and on
the same dates as the PPN. We have thus decomposed the PPN into two
simpler assets

PPN = Riskless zero-coupon bond + Risky annuity

where the payments made by the annuity are contingent on the reference
entity remaining solvent. Indeed, using now familiar terminology, the zero-
coupon bond and the annuity constitute the replicating portfolio for this
PPN, and, as a result, valuing a PPN is the same as determining the market
prices of the bond and the annuity.
Valuing the zero-coupon bond is straightforward, especially if one is will-
ing to assume that it involves no credit risk. In practice, one might want to
discount the repayment of principal based on the discount curve of the PPN
issuer. For instance, if that issuer is a large highly rated bank, one might
want to derive a zero-coupon curve from short-term LIBOR and interest
rate swap rates, which embed the average credit quality of the large banks
that are most active in the interbank loan and interest rate swap markets.
If the PPN matures N years from today, the value of the corresponding
zero-coupon bond can be written as:

V ZCB (0, N ) = D— (0, N )F (11.1)

where, using the same notation introduced in Chapter 5, D— (0, N ) denotes
a discount factor derived from the LIBOR/swap curve, and F is the face
value of the PPN.
The future payments of the risky annuity, which are contingent on the
¬nancial health of the reference entity, should be more heavily discounted
than the principal payment if the reference entity has a credit quality lower
than that of participants in the LIBOR market. Let D(0, j), j = 1 to
N , correspond to the discount factors derived from the reference entity™s
11.3 Valuation Considerations 121

yield curve. The value of the risky annuity can be written as

N
A
V (0, N ) = D(0, j)CP P N (11.2)
j=1


where CP P N is the coupon payment made by the PPN.
Given the above equations, we can write an expression for the market
price of a PPN:

N

PPN
V (0, N ) = D (0, N )F + D(0, j)CP P N (11.3)
j=1


PPNs are typically issued at their par value so pricing a brand new PPN
amounts to ¬nding the value of CP P N that makes V P P N (0, N ) in (11.3)
equal to F .
Equation (11.3) tells us that the basic ingredients for pricing a new PPN
are the zero-coupon bond prices that correspond to the credit quality of
the PPN issuer and the reference entity. These are typically not directly
observable in the marketplace, but they can be inferred from related market
quotes such as the LIBOR/swap curve and CDS spreads.4 As an exam-
ple, suppose those quotes give you the zero-coupon bond prices shown in
Table 11.2. If you use these numbers in equation (11.3), you will ¬nd that

TABLE 11.2
Zero-Coupon Bond Prices Used in PPN Valuationa

Maturity “Riskless” Reference
(years) bonds entity™s bonds

(1) (2) (3)

1 0.943396226 0.895426927
2 0.88999644 0.801789381
3 0.839619283 0.717943802
4 0.792093663 0.642866212
a Based on an assumption of ¬‚at riskless and risky yield curves.
The riskless rate is set at 6 percent, and the reference entity™s
risk spread and recovery rate are assumed to be 300 bps and 50
percent, respectively. Face value of the bonds = $1.



4
In Appendix A, we show how to derive zero-coupon bond prices from observed prices
of coupon-paying bonds.
122 11. Principal-Protected Structures

the PPN coupon that is consistent with the PPN being valued at par is
6.80 percent, the same coupon shown in Table 11.1. Indeed, the numbers in
that table correspond to the zero-coupon bond prices shown in Table 11.2,
and we can now verify that the PPN described in that example was indeed
valued at par.



11.4 Variations on the Basic Structure
We have thus far limited ourselves to the simplest type of principal-
protected structure. Other more complex structures do exist and are not
uncommon, such as principal-protected structures that reference more than
one entity or that pay ¬‚oating coupons. One straightforward extension of
the setup analyzed in this chapter are PPNs that o¬er only partial principal
protection and thus provide a higher yield to investors. For instance, the
PPN may guarantee only 50 percent of the investor™s original principal in
the event of default by the reference entity. The main points highlighted in
the preceding sections are generally applicable to these and other variations
of the basic principal-protected structure.
12
Credit-Linked Notes




Credit-linked notes are essentially securities structured to mimic closely, in
funded form, the cash ¬‚ows of a credit derivative. Credit-linked notes have
a dual nature. On the one hand, they are analogous to traditional coupon-
paying notes and bonds in that they are securities that can be bought
and sold in the open market and that promise the return of principal at
maturity. On the other hand, they can be thought of as a derivative on
a derivative, as a credit-linked note™s cash ¬‚ow is tied to an underlying
derivative contract.
Credit-linked notes play an important role in the credit derivatives
market as they have helped expand the range of market participants. In par-
ticular, some participants are attracted to the funded nature of a CLN,
either because of their greater familiarity with coupon-bearing notes or
because they are prevented from investing in unfunded derivatives contracts
by regulatory or internal restrictions.



12.1 How Does It Work?
To illustrate the basic workings of a CLN, we will go back to one of the
simplest credit derivatives, the credit default swap (Chapter 6). Consider an
asset manager who is seeking exposure to a given reference entity but who
wants that exposure to be in funded form. A credit derivatives dealer may
buy protection against default by that reference entity in a vanilla credit
default swap and essentially securitize that contract, passing the resulting
124 12. Credit-Linked Notes




FIGURE 12.1. Diagram of a Simple Credit-Linked Note



cash ¬‚ows to the asset manager, who ultimately buys the newly created
securities. Alternatively, the dealer may sell protection against default by
the reference entity to a special purpose vehicle, which then issues the notes
to the investor.1
Figure 12.1 uses the example just discussed to illustrate the mechanics of
a dealer-issued CLN. Imagine that the asset manager (the investor) wants
to invest $100 million in the reference entity™s debt. The dealer sells to
another party $100 million worth of protection in a CDS contract that
references that entity. At the same time, the dealer issues $100 million
of notes to the asset manager, where the notes will pay a predetermined
spread over LIBOR for as long as the reference entity does not trigger the
underlying credit default swap.
The spread over LIBOR paid by the note may di¬er from the premium
paid under the CDS owing to administrative costs incurred by the dealer
and counterparty credit risk considerations. Here it is important to note
that, even if the whole CDS premiums were passed along to the investors,
they would fall short of what the investor would consider an “adequate”
coupon for the CLN. (Recall that the CLN is a funded instrument, but
the CDS is not”see Chapter 6.) To make up for this shortfall, the dealer
may invest the proceeds of the note sale in high-grade securities and use
the income generated by these securities, along with the CDS premiums,
to fund the coupons owed to the investor. These high-grade securities may
also be used as collateral against the dealer™s obligations under the PPN.
At the maturity date of the CLN, assuming the underlying CDS was not
triggered, the dealer pays out the last coupon and returns the investor™s

1
Special purpose vehicles are discussed in the next chapter. They are essentially
entities with high credit ratings created speci¬cally to issue CLNs and other securitized
products.
12.2 Common Uses 125

initial investment ($100 million in this case) and both the CDS and CLN are
terminated. Indeed, from the investor™s perspective, the whole arrangement
is very much akin to investing in a debt instrument issued by the reference
entity, although, as we shall discuss later in this chapter, CLNs involve
certain risks that are not present in traditional notes.
In the event of default by the reference entity, the investor bears the
full brunt of the loss. Suppose that the recovery rate associated with the
reference entity is 30 percent, and that a credit event does take place.
The dealer pays its CDS counterparty the di¬erence between the notional
amount of the contract and the recovery value, $70 million, and the CLN
is terminated with the asset manager receiving only $30 million of the $100
million that it had originally invested. Here again the cash ¬‚ows to the
investor mimic those of a traditional note issued by the reference entity.



12.2 Common Uses
We have mentioned already the most obvious applications for credit-linked
notes. They allow the cash ¬‚ows of derivatives instruments to be “repack-
aged” into securities that can be bought and sold in the market place. This
is especially useful for certain classes of institutional investors, such as some
mutual funds, that are precluded from taking sizable positions in unfunded
derivatives contracts. These investors would otherwise be shut out of the
credit derivatives market, and thus credit-linked notes play an important
role in diversifying the market™s investor base.
Investors who do not have master credit derivatives agreements with
dealers are attracted to credit-linked notes because they generally require
less documentation and lower setup costs than outright credit derivatives
contracts. In addition, credit-linked notes can be tailored to meet spe-
ci¬c needs of investors. For instance, they can be used to securitize the
risk exposures in portfolio default swaps, thereby broadening the pool of
potential ¬rst-loss investors. Lastly, credit-linked notes can be rated at the
request of individual institutional investors.
Credit-linked notes can help increase the liquidity of certain otherwise
illiquid assets or even create a market for assets that would otherwise
not exist in tradable form. For instance, CLNs that reference a pool of
bank loans can be traded in the open market without restrictions, whereas
actual sales of the loans might be subject to restrictions and noti¬cation
or approval by the borrowers.
Bankers/dealers too ¬nd value in the issuance of credit-linked notes, over
and above the revenue that they generate. For instance, CLNs provide
dealers with an additional vehicle to hedge their exposures in other credit
derivatives positions.
126 12. Credit-Linked Notes

12.3 Valuation Considerations
As general rule, the single most important risk exposure in a CLN is, nat-
urally, the credit risk associated with the reference entity. CLN spreads are
often wider than the spreads associated with the corresponding reference
entities, however, although the issuer may reduce the spread paid under
the CLN to cover its administrative costs.
The higher CLN spread re¬‚ects the investor™s exposure to the counter-
party credit risk associated with the CLN issuer; the investor would not
be exposed to such a risk if buying a note issued directly by the reference
entity. Counterparty credit risk is more important for CLNs issued out of
a bank or dealer, rather than from a highly rated special purpose vehicle,
which tend to make more widespread use of collateral arrangements, as we
shall see in the next chapter.



12.4 Variations on the Basic Structure
There are at least as many types, if not more, of CLNs as there are
credit derivatives. For instance, while the example discussed in this chap-
ter focused on a CDS-based credit-linked note, earlier CLNs were actually
set up to securitize asset swaps (Chapter 5). In addition, the principal-
protected notes discussed in Chapter 11 are, in essence, a member of the
CLN family.
Credit-linked notes are common features of complex structured products
such as synthetic CDOs. Indeed, the cash ¬‚ows of synthetic CDOs are com-
monly channeled to investors in the form of coupon and principal payments
made by speci¬c CLNs.
13
Repackaging Vehicles




Repackaging vehicles are special-purpose trusts or companies typically
associated with banks and derivatives dealers. In the credit derivatives
market, they are often counterparties in contracts that are subsequently
securitized and sold o¬ to investors. In this chapter, we will temporarily
deviate from the stated goal of this book, which is to discuss speci¬c types
of credit derivatives, and take a closer look at this important aspect of many
credit derivatives contracts. Repackaging vehicles are commonly backed by
high-grade collateral and thus tend to be highly rated themselves in order
to minimize investors™ concerns about counterparty credit risk. The main
buyers of structured products issued by repackaging vehicles are insurance
companies, asset managers, banks, and other institutional investors.
Repackaging vehicles are important issuers of credit-linked notes and play
a central role in the synthetic CDO structure (Chapter 14). In this chapter
we go over the basics of repackaging vehicles, also called special-purpose
vehicles (SPVs), focusing on their applications to credit-linked notes, which
were discussed in Chapter 12.1


13.1 How Does It Work?
Repackaging vehicles are trusts or companies sponsored by individual insti-
tutions, but their legal structure is such that they are “bankruptcy-remote”
to the sponsoring entity, meaning that a default by the sponsoring entity
does not result in a default by the repackaging vehicle. As a result, investors

1
Das (2000)[18] discusses repackaging vehicles in greater detail.
128 13. Repackaging Vehicles




FIGURE 13.1. Diagram of a Simple SPV Structure



who buy, say, credit-linked notes issued by a repackaging vehicle are not
directly subject to the credit risk associated with the sponsoring entity.
In the credit derivatives context, the purpose of the vehicle is to “repack-
age” the cash ¬‚ows and risk characteristics of derivatives contracts into
coupon and principal payments made by notes that can be bought and
sold in the marketplace.2 Figure 13.1 illustrates the basic workings of a
repackaging vehicle. Consider a hypothetical situation where an investment
bank (the sponsoring entity) has identi¬ed strong investor demand for, say,
¬‚oating-rate notes issued by a given reference entity (XYZ Corp.). But sup-
pose that entity has issued mostly ¬xed-rate liabilities. If the investors are
unwilling, or unable, to enter into an asset swap with the bank, the bank
can essentially use a combination of credit default swap and SPV technol-
ogy to ¬nancial-engineer a note that will mimic the cash ¬‚ows and risk
characteristics sought by the investors.
Here is how it could work. The bank enters into a credit default swap
with a repackaging vehicle especially created for the purposes of this
transaction”hence the name special purpose vehicle. The bank buys pro-
tection against default by XYZ in a contract with a notional amount of,
say, $100 million, which would correspond to the exposure desired by the
investors.3 At the same time, the SPV issues notes to the investors in the
amount of $100 million and uses the proceeds of the note sales to buy
highly rated securities. Those securities will serve as collateral for the par
value of the notes, which, absent a default by XYZ Corp., will be paid

2
This is similar to what was achieved in the case of a dealer-issued credit-linked note
example examined in Chapter 12, but there are some important di¬erences as we shall
discuss in the next section.
3
An alternative arrangement would be for the bank to sell an asset swap to the SPV
and have the SPV securitize the asset swap, selling the notes to the investors.
13.2 Why Use Repackaging Vehicles? 129

to investors at the maturity of the notes. In many instances, the collat-
eral bought by SPVs is actually selected by the investors. Along with the
protection premium that the SPV receives from the sponsoring bank, the
coupons paid by the collateral serve as a funding source for the coupons
promised by the notes and for covering the SPV™s administrative costs.4
From the investors™ perspective, the notes bought from the SPV are very
close to traditional notes sold in the capital market. For as long as the
reference entity remains solvent, the investor will collect the notes™ coupon
payments from the SPV until the maturity date. In the event of default by
the reference entity, the SPV liquidates the collateral to fund the protection
payment owed to the dealer, and the notes are terminated with the residual
proceeds of the collateral liquidation being transferred to the investors.


13.2 Why Use Repackaging Vehicles?
In Chapter 12 we discussed the basic structure of a simple credit-linked
note issued directly by a derivatives dealer and noted that dealer banks
are common issuers of such notes. Indeed, a quick look at Figures 12.1 and
13.1 will reveal remarkable similarities between the bank- and SPV-issued
structures, and all of the common uses for credit-linked notes, outlined
in Section 12.2, also apply to typical SPV-based structures in the credit
derivatives market. The question then becomes: Why the need for repack-
aging vehicles in the credit derivatives market when the dealers themselves
can, and do, issue credit-linked notes directly?
One rationale for the widespread use of SPVs relates to the issue of
counterparty credit risk. With a bank-issued CLN, the investor is exposed
to the credit risk associated both with the reference entity and the bank. In
particular, if the bank defaults on its obligations, the investor may end up
losing part or all of the principal and future coupon payments associated
with the note even if the reference entity has not defaulted. In contrast,
because the SPV is bankruptcy-remote to the sponsoring bank and is fully
backed by high-grade collateral, the investor has a potentially much smaller
exposure to counterparty credit risk.
Of course there are ways to mitigate the investor™s exposure to counter-
party credit risk in bank-issued structured notes. For instance, the bank
may pledge the proceeds of the note sales as collateral. Still, counterparty
credit risk considerations aside, there are other powerful factors behind the
popularity of SPV-based structures. In particular, SPVs allow for greater
¬‚exibility to suit the particular needs of individual institutional investors

4
Note that the sponsoring bank need not hold any debt instruments of the reference
entity. E¬ectively, in the example in Figure 13.1, the bank has a short position on the
credit quality of the reference entity.
130 13. Repackaging Vehicles

when it comes to tax and regulatory issues. Indeed, SPVs are typically
setup in jurisdictions that o¬er favorable tax and regulatory treatment
such as, in the US, the states of Delaware and New York, and, elsewhere,
in jurisdictions that include the Cayman Islands, Jersey, and Luxembourg.
Most of the examples provided above feature a bank as the SPV sponsor
and an investor who buys SPV-issued notes. But one should be aware that
the investor may just as well be, and in many cases is, another bank. For
instance, AZZ Bank may be unwilling to do a total return swap directly
with XYZ Bank, but it may feel comfortable doing that same swap with a
AAA-rated SPV sponsored by AZZ Bank.



13.3 Valuation Considerations
Not surprisingly, the credit quality of the reference entity and, in the case of
multiple reference entities, the default correlation of the entities loom large
in the valuation of SPV-based credit derivatives. In addition, the credit
quality of the SPV™s collateral and of the credit derivative counterparty
(the sponsoring bank) are also factored into the coupons paid by the SPV
as funds received both from the sponsoring bank and the collateral pool
may be used to fund the SPV™s obligations.
SPV-based structures are often rated by the major credit-rating agencies.
The factors mentioned above are key determinants of the credit rating of a
given structure. In addition, the legal structure of the SPV, such as the way
that multiple-issuance SPVs segregate collateral pools (discussed below), is
also a factor in the risk pro¬le and consequent credit rating of repackaging
vehicles.



13.4 Variations on the Basic Structure
In the simple example illustrated in Figure 13.1, the repackaging vehicle
was created speci¬cally for the purposes of issuing the credit-linked notes
sought by the investors. While this arrangement has the advantage that
the investors are the sole claimants to the SPV™s collateral (provided, of
course, that the reference entity does not default) a drawback is that such
“single-purpose SPVs” may be more costly than SPVs that issue more than
one type of security, so-called multiple-issuance structures.
Multiple-issuance SPVs can be more cost e¬ective than single-purpose
SPVs in that their associated administrative and setup costs can be spread
out among a larger number of issues. The legal structure of such vehi-
cles is such that, even though the multiple-issuance SPV is a single entity,
each note series issued by the entity has its own separate collateral pool.
13.4 Variations on the Basic Structure 131

As a result, investors only have recourse to the speci¬c pool backing
the notes that they hold. Assuming these non-recourse stipulations (often
termed ¬rewalls) are fully e¬ective, defaults in one collateral pool do not
a¬ect the recourse rights of investors in notes backed by other pools, and
investors in defaulted pools have no claim to assets held as collateral for
other notes.
An alternative form of multiple-issuance repackaging vehicles are the
so-called “umbrella” programs, whereby a separate legal entity is created
for each issue, but each individual entity is based on the same master legal
framework. Umbrella programs tend to provide a more e¬ective segregation
of collateral pools than other multiple-issuance vehicles while being more
cost e¬ective than single-purpose SPVs.
A particular type of repackaging vehicle that grew in popularity in the
late 1990s and early 2000s, when insurers and reinsurers became increas-
ingly active in the credit derivatives market, is the so-called “transformer.”
Transformers are essentially captive insurance companies established by

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