Transcript Document

Credit Derivatives
Advanced Methods of Risk Management
Umberto Cherubini
Learning objectives
•
In this lecture you will learn
1. The difference between product specific
and issuer specific credit derivatives
2. How to compute the asset swap spread of a
bond
3. How to boostrap the probability of default of
an issuer using credit default swaps
Credit Derivatives
• Asset Swap (ASW): swap of cash flows of a
security against floating payments plus
spread.
• Credit default swap (CDS): purchase and
sale of insurance against default of a issuer
• Total rate of return swap (TRORS): swap of
the overall return (coupons + appreciation/
depreciation) against fixed/float premium
• Credit spread options (CSO): options to buy
or sell bonds at a given spread over others.
Hedging credit exposure
•
•
•
Assume you buy a defaultable security and
buy protection in a credit derivative (CD)
Avoiding arbitrage requires
Titolo Defaultable = Risk-free – CD
Notice:
–
–
Default amounts to a position in a derivative and
makes every product a structured product.
Default risk can be synthetically built
independently of “paper” issued by an obligor.
Swap contracts
• Swaps are standard tools to transfer risk from one party
to the other. The idea is simply to exchange flows of
payments. Each flow is called “leg” of the contract.
• In the market everything can be swapped: typical
examples are
– Fixed vs floating payments plus spread (plain vanilla swap)
– Payments denominated in different currencies (currency swap)
– Floating payments in same currency but indexed to yield
curves of different currencies (quanto swap)
– Asset swap, total return swap, credit default swap…
Interest rate swaps (plain vanilla)
• In a plain vanilla IRS
– The party which is long pays a cash flow of fixed payments
equal to a percentage c of the notional value, expressed in
annual basis
– The short party pays a cash flow which is indexed to a
market rate (typically Euribor, Libor)
• Value of the fixed leg:
m
c ti  ti 1 vt , ti 
i 1
• Value of the float leg:
m
1  vt , tm    vt , ti ti  ti 1  f t , ti 1 , ti 
i 1
The parameters in a swap contract
• The value of a swap contract can be expressed
as :
– Net-present-value (NPV), or upfront; it is the
difference between the present value of cash flows
that will be received and those that will be paid.
– Running premium (swap rate): the value of a fixed
leg payment such that the present value of the fixed
be equal to the value of the float leg.
– Spread: value of a fixed percentage payment (on a
annual basis) to be added to (o subtracted from) the
floating payments to equate the present value of the
float leg to the present value of the fixed leg.
Indexed coupons
• An index coupon is determined based on an index, typically an
interest rate, observed at a date , defined reset date
• The typical case, denoted natural time lag consists of a coupon
with
– accrual period from time  to time T
– reset date  and payment date T
– reference rate for the determination of the coupon
i( ,T) (T –  ) = 1/v ( ,T) – 1
• Notice that typically the interest rate used is with simple
compounding, for the market convention that uses this
compound rule for payments with less than one year maturity.
Replicating portfolio
• What is the replicating portfolio of an indexed
coupon?
• Notice that at time  the value of the coupon,
determined at time  and paid at time T, given by
v ( ,T) i( ,T) (T –  ) = 1 – v ( ,T)
• The replicating portfolio which is natural to select is
– A long position (investiment) of one unit of cash
available at time 
– A short position (funding) for one unit of cash
available at time T
Cash flows of a indexed coupon
•
Using the argument above, a indexed coupon is actually a
derivative. A derivative is actually a porfolio of long and short
positions. In this case the long position is on a zero coupon
bond expiring on the reset date and the short position is on a
zero coupon bond expiring at time T. An indexed coupon then
hides a debt position (leverage)
C
t

T
C
In remembrance of swap rate
In a fixed/float swap it is market convention
that at origin
Value of fixed leg = Value of floating leg
m
swap rate ti  ti 1 vt , ti   1  vt , t m 
i 1
swap rate
1  vt , t m 
m
 t
i 1
i
 ti 1 vt , ti 
Swap rate: definitions
1. Fixed running payment, expressed in
percentage of the notional value and in
annual terms, equivalent to a flow of
indexed payments.
2. A weighted average of forward rates, with
weigths given by the discount factors.
3. The internal rate of return of a coupon bond
issued at par (par yield curve)
Bootstrapping procedure
Assume that at time t the market is structured on m periods
with maturities tk = t + k, k=1....m, and swap rates are
observed on these maturities. Since swap rates corresponds to
par yields, and that for par yields coupon rates are equal to the
yield, we recover the same bootstrapping procedure defined
for fixed rate bonds.
k 1
vt , t k  
1  swap ratet,tk  vt , ti 
i 1
1  swap ratet,tk 
Asset Swap (ASW)
• L’asset swap is a package made up by
– A bond
– A swap contract
• The two parties of the contract pay
– The cash flows of the bond plus the difference
between par and the market value of the bond, if
the different is positive positive
– Floating payments plus a spread (that may be
positive or negative) and the difference between
market value and par, it the difference is positive.
Asset Swap (ASW)
• Asset Swap on bond DP(t,T;c)
• Fixed leg value:
m
max1  DPt , T ; c ,0  c ti  ti 1 vt , ti 
i 1
• Float leg value:
m
maxDPt , T ; c   1,0  1  vt , tm   spread vt , ti ti  ti 1 
i 1
Asset Swap (ASW) Spread
• Spread is recovered equating the value
of legs
spread  c  tasso swap 
1  DPt , T ; c 
m
 t
i 1
i
 ti 1 vt , ti 
• Notice that spread is zero iff
m
DPt , T ; c    cti  ti 1 vt , ti   vt , T 
i 1
Credit Default Swap
• A credit default swap is an exchange contract in
which the protection buyer receives insurance
against loss on a set of bonds issued by a reference
entity, called a “name” in jargon, against a flow of
fixed payments, typically on a running basis. The flow
of payments stops at the maturity of the contract or
the date of default of the name, whatever comes first.
• The value of fixed payment is determined in such a
way as to equate the value of the contract to zero at
the origin of the contract.
Credit Default Swap
• The underlying asset of a CDS is not a bond, like for ASW, but a
“name”, that is the issuer. The payment can be done either way
between: cash settlement or physical delivery. The latter is the
rule rather than the exception and implies the presence of a
delivery option.
• Default isd defined among a set of credit events, specified in the
ISDA standars
–
–
–
–
–
–
Bankrupcy
Obligation acceleration
Obligation default
Failure to pay
Moratorium/repudiation
Restructuring
• The protection seller pays: (1 > t(i-1) – 1  > t(i) ) LGD
• The protection buyer pays: bp premium * 1 > t(i-1).
Payment arrangements
• The evaluation of a CDS is based on survival
probability Q(T).
• We can assume that the payment of the
premium for period t occurred in full if the
name defaults between time t – 1 an t
N 1
N 1
 vt, T  1QT   QT  1  c  vt, T  1QT 
T 0
N
T 0
or that it did not happen at all
N 1
N 1
 vt, T  1QT   QT  1  c  vt, T  1QT  1
T 0
N
T 0
Accrued premium payment at
default
• The most common payment structure is that, in
case of default at time , the protection buyer
pays accrued premium until that date and the
protection seller pays the LGD at the time of
default.
• In the case of a 1 year we have
1
1
0
0
LGD   v0, u d 1  Qu   c1  v0, u ud1  Qu 
A CDS for N years
• The N year generalization of a CDS is given by
N 1 T 1
  v0, u  T d 1  Qu  T 
T 0
cN  LGD  N 1 T T1
  v0, u  T ud1  Qu  T 
T 0 T
• The simplified versions with payments at reset dates
represent good approximations.
Default probability boostrap
• Credit default swap su Fiat 25/01/2002
Maturity
Bid
Ask
1
1.45
1.55
3
1.70
1.75
5
1.80
1.83
10
1.95
2.15
Bootstrap 1
• Assume that the premium be paid in toto at
the end of the default period
T = 1: Q(1) = 1 – c1/LGD
P0,1 c1  c2
c2 

T = 2: Q(2) = P0,2 LGD  Q11  LGD 


cN 1  cN N 1
cN 

T = N: Q(N) = P0, N LGD  P0, i Qi  1  QN  11  LGD 
i 1
Bootstrap 2
LGD
Q1 
c1  LGD

c1  c2 P 0,1Q 1
LGD
Q2  
 Q1
P0,2 c2  LGD
c2  LGD
.
N 1
c N 1  c N
LGD
QN  
P 0, i Qi   Q  N  1

P 0,2 c2  LGD i 0
c N  LGD
/0
12 1/2
/0 01
21 1/2 1
/0 01
01 1/2 1
/0 01
10 2/2 1
/0 01
21 2/2 1
/0 01
02 2/2 1
/0 01
11 3/2 1
/0 01
22 3/2 1
/0 01
31 3/2 1
/0 01
11 3/2 1
/0 01
20 4/2 1
/0 01
29 4/2 1
/0 01
10 4/2 1
/0 01
19 5/2 1
/0 01
30 5/2 1
/0 01
08 5/2 1
/0 01
17 6/2 1
/0 01
28 6/2 1
/0 01
07 6/2 1
/0 01
18 7/2 1
/0 01
27 7/2 1
/0 01
05 7/2 1
/0 01
16 8/2 1
/0 01
25 8/2 1
/0 01
05 8/2 1
/0 01
14 9/2 1
/0 01
23 9/2 1
/0 01
04 9/2 1
/1 01
13 0/2 1
/1 01
24 0/2 1
/1 01
02 0/2 1
/1 01
1/ 1
20
11
03
Italy
0,4
0,35
0,05
0
0,7
0,6
0,3
0,5
0,25
0,4
0,2
10 Year
5 Year
0,3
0,15
0,1
0,2
0,1
0
03
/0
12 1/2
/0 01
1
21 1/2
/0 01
1
01 1/2
/0 01
1
10 2/2
/0 01
1
21 2/2
/0 01
1
02 2/2
/0 01
1
11 3/2
/0 01
1
22 3/2
/0 01
1
31 3/2
/0 01
11 3/2 1
/0 01
20 4/2 1
/0 01
1
29 4/2
/0 01
1
10 4/2
/0 01
1
19 5/2
/0 01
1
30 5/2
/0 01
1
08 5/2
/0 01
1
17 6/2
/0 01
1
28 6/2
/0 01
1
07 6/2
/0 01
1
18 7/2
/0 01
1
27 7/2
/0 01
1
05 7/2
0
/0 1
1
16 8/2
/0 01
1
25 8/2
/0 01
1
05 8/2
/0 01
1
14 9/2
/0 01
1
23 9/2
/0 01
1
04 9/2
/1 01
1
13 0/2
/1 01
1
24 0/2
/1 01
1
02 0/2
/1 01
1/ 1
20
11
Spain
0,35
0,3
0,05
0
0,6
0,5
0,25
0,4
0,2
0,3
0,15
0,1
0,2
0,1
0
5 Year
10 Year
01/11/2011
24/10/2011
14/10/2011
06/10/2011
28/09/2011
20/09/2011
12/09/2011
02/09/2011
25/08/2011
17/08/2011
09/08/2011
01/08/2011
22/07/2011
14/07/2011
06/07/2011
28/06/2011
20/06/2011
10/06/2011
02/06/2011
25/05/2011
17/05/2011
09/05/2011
29/04/2011
21/04/2011
13/04/2011
05/04/2011
28/03/2011
18/03/2011
10/03/2011
02/03/2011
22/02/2011
14/02/2011
04/02/2011
27/01/2011
19/01/2011
11/01/2011
03/01/2011
Portugal
0,7
0,9
0,6
0,8
0,7
0,5
0,6
0,4
0,5
5 Year
10 Year
0,3
0,4
0,3
0,2
0,2
0,1
0,1
0
0
/0
12 1/2
/0 01
21 1/2 1
/0 01
01 1/2 1
/0 01
10 2/2 1
/0 01
21 2/2 1
/0 01
02 2/2 1
/0 01
11 3/2 1
/0 01
22 3/2 1
/0 01
31 3/2 1
/0 01
11 3/2 1
/0 01
20 4/2 1
/0 01
29 4/2 1
/0 01
10 4/2 1
/0 01
19 5/2 1
/0 01
30 5/2 1
/0 01
08 5/2 1
/0 01
17 6/2 1
/0 01
28 6/2 1
/0 01
07 6/2 1
/0 01
18 7/2 1
/0 01
27 7/2 1
/0 01
05 7/2 1
/0 01
16 8/2 1
/0 01
25 8/2 1
/0 01
05 8/2 1
/0 01
14 9/2 1
/0 01
23 9/2 1
/0 01
04 9/2 1
/1 01
13 0/2 1
/1 01
24 0/2 1
/1 01
02 0/2 1
/1 01
1/ 1
20
11
03
Ireland
0,7
0,4
0,3
0
0,9
0,6
0,8
0,7
0,5
0,6
0,5
5 Year
10 Year
0,4
0,3
0,2
0,2
0,1
0,1
0
/0
1
11 /2
/0 01
1 1
19 /2
/0 01
1 1
27 /2
/0 01
1 1
04 /2
/0 01
2 1
14 /2
/0 01
2 1
22 /2
/0 01
2 1
02 /2
/0 01
3 1
10 /2
/0 01
3 1
18 /2
/0 01
3 1
28 /2
/0 01
3 1
05 /2
/0 01
4 1
13 /2
/0 01
4 1
21 /2
/0 01
4 1
29 /2
/0 01
4 1
09 /2
/0 01
5 1
17 /2
/0 01
5 1
25 /2
/0 01
5 1
02 /2
/0 01
6 1
10 /2
/0 01
6 1
20 /2
/0 01
6 1
28 /2
/0 01
6 1
06 /2
/0 01
7 1
14 /2
/0 01
7 1
22 /2
/0 01
7 1
01 /2
/0 01
8 1
09 /2
/0 01
8 1
17 /2
/0 01
8 1
25 /2
/0 01
8 1
02 /2
/0 01
9 1
12 /2
/0 01
9/ 1
20
11
03
Greece
1
0
1
0,9
0,9
0,8
0,8
0,7
0,7
0,6
0,6
0,5
0,5
0,4
0,4
0,3
0,3
0,2
0,2
0,1
0,1
0
5 Year
10 Year
17
01
1
/0
1/
20
31
11
/0
1/
2
14 01
1
/0
2/
2
28 01
1
/0
2/
20
14
11
/0
3/
2
28 01
1
/0
3/
20
11
11
/0
4/
2
25 01
1
/0
4/
2
09 01
1
/0
5/
20
23
11
/0
5/
2
06 01
1
/0
6/
20
20
11
/0
6/
2
04 01
1
/0
7/
2
18 01
1
/0
7/
20
01
11
/0
8/
2
15 01
1
/0
8/
20
29
11
/0
8/
2
12 01
1
/0
9/
20
26
11
/0
9/
2
10 01
1
/1
0/
2
24 01
1
/1
0/
20
07
11
/1
1/
20
11
/0
1/
2
03
5 Year CDS
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
ITALY
IRELAND
PORTUGAL
SPAIN
SPAIN