1. Credit Default Swap Pricing A Market Approach
Moorad Choudhry

2. Topics Covered The original approach – using Asset Swaps to price CDS
Link between CDS and asset swaps
The basis
Credit risk and credit spreads
Term structure of credit spreads
Obtaining default probabilities from credit spreads
Pricing a Credit Default Swap
Default and survival probabilities
Using default probabilities to obtain CDS premiums
Bootstrapping default probabilities from CDS premiums
Marking CDS to market

3. Using Asset Swaps
Conventional IRS quote a fixed rate (the swap rate) in exchange for LIBOR flat
The difference between the swap rate and government bond yields is called the swap spread
Corporate bonds trade at a higher yield than government bonds – this difference is called the credit spread
Asset swaps are special IRS where:
The fixed rate is raised to match the higher coupon paid by a corporate bond
An up-front payment may optionally be included if the corporate bond is trading away from par
The floating rate is raised to ensure that the resulting swap has an initial NPV of zero
The LIBOR margin is then the asset swap spread
Corporate bond yield
Asset swap spread
Swap yield
Credit spread
Swap spread
Government bond yield

4. Link Between Asset Swaps and CDS
Consider an investor who:
Raises funds at LIBOR flat
Buys a corporate bond at price DP and coupon C
Enters into an asset swap to convert initial cash flow to 100 and yield of LIBOR + x bps
Buys a CDS on the same bond, with identical maturity, costing y bps p.a.
Net income is therefore x – y bps p.a.
In the event of default:
Investor exchanges bond for payment of par
Uses this cash to repay borrowing
Investor therefore earns x – y bps p.a. with no credit risk
To avoid riskless arbitrage profits, x = y
Therefore, in theory at least:
CDS Premium = Asset Swap Spread

5. Example of CDS Pricing using Asset Swaps
Data:
Reference entity: XYZ plc
Rating: Baa2
Term: 7 years
Coupon: 6.75%
IRS rate: 5.875%
Asset Swap rate: LIBOR + 93 bp
Risk-free rate: LIBOR – 18.5 bp
CDS rate: 93 less –18.5 = bp
Asset Swap
6.75%
LIBOR + 93bp
Default pmt
Reference Bond
Protection Seller
CDS Counterparty
6.75%
111.5bp
To hedge a CDS, an investor could:
Sell the CDS
Sell the reference bond
Buy a risk-free bond
Enter into an asset swap
LIBOR – 18.5bp
Risk-free Bond

6. Practical Example of Asset Swap and CDS Rates
Air Products & Chemicals 6.5% Jul 2006
Asset Swap: 41.6 bp
CDS: bp
The difference is the basis…

7. CDS spread – Asset Swap spread
The CDS Basis
The difference between CDS and Asset Swap rates is called the basis
Basis is defined as:
CDS spread – Asset Swap spread
Example quotations (May 2003):
Reference credit
Credit rating
CDS spread
Asset swap spread
Basis
Financials
Ford Motor Credit
A2 / A
59.3
51.1
+8.2
Household Finance
72.2
57.2
+15.0
JPMorgan Chase
Aa3 / AA-
89.0
66.9
+22.1
Merrill Lynch
108.1
60.4
+47.7
Industrials
AT & T Corp
Baa2 / BBB+
224.0
217.6
+6.4
FedEx Corp
499.0
481.2
+17.8
General Motors
A3 / BBB
205.1
237.7
-32.6
IBM (6-yr callable bond)
A1 / A+
27.2
8.2
+19.0
IBM (4-yr callable bond)
33.3
11.0
+22.3

8. More Examples of CDS and Asset Swap Prices
Data: November 2002

9. Why Asset Swap Rates Differ from CDS Prices - I
Technical Factors
Delivery option:
Asset swaps are for a specific bond
CDS allows delivery of any deliverable obligation of the reference entity (basis more)
Funding costs
Repo rates influence asset swap prices – higher repo rates imply higher asset swap prices
If repo rates are higher, basis will be lower
If repo rates are lower, basis will be higher
CDS spreads are always positive
… but some highly-rated bonds (e.g. US agencies) trade at sub-LIBOR in the asset swap market
Default events
CDS can be triggered by technical default events that do not affect cash bonds so much
Additional risk is reflected in higher CDS premiums (basis more)

10. Why Asset Swap Rates Differ from CDS Prices - II
Technical Factors (cont)
Counterparty default risk
Protection buyer is exposed to default of CDS counterparty (basis less)
Accrued interest
Some CDS contracts also require delivery of accrued coupon (basis more)
Assets trading away from par
CDS usually require delivery of defaulted asset against par
For bonds trading at a discount – protection seller loses more when default occurs (basis more)
For bonds trading at a premium – protection seller loses less when default occurs (basis less)

11. Why Asset Swap Rates Differ from CDS Prices - III
Market Factors
Supply and Demand
Demand from protection buyers (e.g. banks, investors) will drive basis higher
Demand from protection sellers (e.g. insurers) will drive basis lower
Asset swap rates are affected by repo market rates
… especially if the bond is under Special Collateral
Liquidity
CDS liquidity can differ from bond liquidity…
Long maturities or high-yield cash bonds are often less liquid than CDS (basis less)
Difficulty in shorting the cash asset
Loans are typically illiquid…
Investors or speculators wishing to short an illiquid asset can buy protection instead
This drives CDS premiums higher and asset swap prices lower (basis more)
Link with CDOs
Synthetic CDOs typically sell single-name or portfolio CDS
Hedging drives CDS premiums down (basis lower)

12. Basis for Different Ratings
Basis tends to increase for higher ratings
Sub-LIBOR funding lowers asset swap prices (basis more)
Basis also tends to increase for low ratings
Cheapest-to-deliver creates more risk for protection sellers (basis more)
This leads to a basis “smile”
Data: November 2002

13. An intuitive market approach to CDS Pricing
We adopt the no-arbitrage approach to CDS pricing in the same way as we would to price interest-rate swaps
This states that at inception, the price of the IRS is:
PV Fixed leg = PV Floating leg
Therefore for a CDS we set
PV Premium leg = PV Contingent leg
The PV of the premium leg is straightforward, especially if there is no credit event during life of CDS.
Of course the Contingent leg is just that – contingent on credit event. Hence we need to determine the value of the premium leg at time of the credit event.
This requires us to use default probabilities.
We can use historical default rates to determine default probabilities, or back them out – implied probabilities – using CDFS spread rates.

14. Default probabilities
To price a CDS, we need the answers to two basic questions:
What is the probability of a credit event?
If a credit event occurs, how much is the protection seller likely to pay? This revolves around an assumed recovery rate.
We may also need to know:
If a credit event occurs, when does this happen?
Let us consider first the probability of default.
One way to obtain default probabilities is to observe credit spreads
in the corporate bond market…

15. Credit Risk and Credit Spreads
Riskless investments establish a benchmark riskless interest rate
Lenders and investors expect to receive a higher return from risky investments
The difference between the risky and riskless rates is the
Credit Spread
This will vary with:
Credit quality (e.g. credit rating)
Maturity
The charts overleaf provides an example of credit spreads…

16. Example of Credit Spreads – Investment Grades

17. Example of Credit Spreads – All Grades

18. Term Structure of Credit Spreads
The examples clearly show that credit spreads depend on maturity
The term structure of credit spreads usually has these features:
Lower-quality credits trade at a wider spread than higher-quality credits
Longer-dated obligations normally have higher spreads than shorter-dated ones
2yr AA: 20bp
5yr AA: 30bp
10yr AA: 37bp
… but there can be exceptions:
2yr CCC: 11%
5yr CCC: 7.75%
10yr CCC: 7%
In the case of the CCC rating:
Higher default risk right now gives rise to high spread
If company survives first few years, risk of later default is much lower, giving rise to lower spreads

19. Default Probability and Corporate Bond Spreads
Suppose that the corporate bonds of a particular issuer trade at the following yields*:
Maturity t
Risk-free rate z
Corporate bond yield z+r
Corporate spread r 1
4.00%
4.25%
0.25% 2
4.50%
0.50% 3
4.70%
0.70% 4
4.85%
0.85% 5
4.95%
0.95%
The rate of return on a risk-free asset is then:
… while the rate of return on the risky asset is:
* Yields quoted at as continuously compounded rates

20. Calculating Default Probabilities – Zero Recovery
Let’s assume that there is no recovery value following default.
If the probability of default is p, then an investor should be indifferent between an expected return of:
on the risky corporate bond, and:
Setting these two expressions equal, and solving for p gives:

21. Calculating Default Probabilities – Examples
Using this formula…
We can therefore calculate the following probabilities of default from credit spreads:
Maturity t
Risk-free rate z
Corporate bond yield z+r
Corporate spread r
Cumulative probability of default
Annual probability of default 1
4.00%
4.25%
0.25% % 2
4.50%
0.50% % % 3
4.70%
0.70% % % 4
4.85%
0.85% % % 5
4.95%
0.95% % %
e.g.
(cumulative probability of default over five-year period)
(probability of default over in year five)

22. Calculating Default Probabilities – Recovery Rate R
Now let’s assume that the recovery rate following default is R.
If the probability of default is p, then an investor should now be indifferent between an expected return of:
on the risky corporate bond, and:
Again, setting these two expressions equal, and solving for p gives:

23. Calculating Default Probabilities – With Recovery
Using this formula…
… and assuming a recovery rate of 40%, we can calculate the following default probabilities:
Maturity t
Risk-free rate z
Corporate bond yield z+r
Corporate spread r
Cumulative probability of default
Annual probability of default 1
4.00%
4.25%
0.25% % 2
4.50%
0.50% % % 3
4.70%
0.70% % % 4
4.85%
0.85% % % 5
4.95%
0.95% % %
e.g.
(cumulative probability of default over five-year period)

24. Default and Survival Probabilities
Consider what happens to a risky asset over a specific period of time
There are just two possibilities:
There is a credit event, and the asset defaults
There is no credit event, and the asset survives
Let’s call these outcomes:
D (for default) having probability q
S (for survival) having probability (1-q)
We can represent this as a binary process:
(1-q) S S q D

25. PSN = (1-q1) × (1-q2) × (1-q3) × (1-q4) × … × (1-qN)
Multiple Periods
Consider this binary process over multiple periods:
(1-q1)
(1-q2)
(1-q3)
(1-q4)
(1-qN) S q1 q2 q3 q4 qN D D D D D
The probability of survival to period N is then:
PSN = (1-q1) × (1-q2) × (1-q3) × (1-q4) × … × (1-qN)
while the probability of default in any period N is:
PSN-1 × qN = PSN-1 – PSN
Given these formulas, we can now go on to price a CDS…

26. Pricing a Credit Default Swap
Given a set of default probabilities, we can calculate the fair premium for a CDS
To do this, consider a CDS as a series of contingent cash flows…
… the cash flows depending upon whether a credit event occurs:
No Default s s s s s
Default s s s
ks
where:
s is the CDS premium
k is the day count fraction when default occurred
R is the recovery rate
(1-R)

27. Valuing the Premium Stream – No Default
The expected PV of the stream of CDS premiums over time can be calculated as:
where:
PVSND is the expected present value of the stream of CDS premiums if there is no default
s is the CDS spread
vj is the discount factor for period j
PSj is the probability of survival through period j
Tj-1,j is the length of time of period j (expressed as a fraction of a year)

28. Valuing the Premium Stream – If Default
ks
(1-R)
If a default occurs half-way through period C, and the CDS makes the default payment at the end of that period, the expected PV of the fees received are:
and the value of the default payment is:
where:
PVSD is the expected PV of the stream of CDS premiums if there is default in period C
PDj is the probability of default in period j
R is the recovery rate
… and other terms are as before

29. Determining the CDS Premium
If a CDS is fairly priced, then the expected value of the premium stream must equal the expected value of the default payment.
As default can occur in any period j, we can therefore write:
Expected PV of stream of premium payments if no default occurs
Expected PV of accrued premium payment in period when default occurs
Expected PV of default payment in period when default occurs
Rearranging this expression gives the fair premium s for the CDS:

30. Determining the CDS Premium – Example
To illustrate the pricing of a CDS, let’s use the table of forward rates and survival probabilities in columns 2 and 4 to derive the set of CDS premiums shown in the last column:
Probability-weighted PVs
Period
Forward rates
Discount factors vj
Survival prob-ability PSj
Default prob-ability PDj
Receipt of 1 p.a. if no default
Receipt of 1 p.a. if default
Default payment if default
CDS Spread s
0.5
2.33%
0.9885
0.9305
0.0695
0.4599
0.0172
0.0481
10.08%
1.0
2.52%
0.9762
0.8656
0.0649
0.8824
0.0330
0.0924
10.10%
1.5
2.87%
0.9624
0.8151
0.0505
1.2746
0.0452
0.1265
9.58%
2.0
3.22%
0.9471
0.7730
0.0421
1.6407
0.0551
0.1544
9.10%
2.5
3.52%
0.9307
0.7432
0.0298
1.9865
0.0621
0.1738
8.48%
3.0
3.82%
0.9133
0.7145
0.0287
2.3128
0.0686
0.1921
8.07%
3.5
4.02%
0.8953
0.6990
0.0155
2.6257
0.0721
0.2018
7.48%
4.0
4.22%
0.8768
0.6837
0.0153
2.9254
0.0754
0.2112
7.04%
4.5
4.37%
0.8581
0.6806
0.0031
3.2174
0.0761
0.2131
6.47%
5.0
4.52%
0.8391
0.6776
0.0030
3.5017
0.0767
0.2149
6.00%

31. CDS Premium – 1yr CDS
On the previous page, the 1yr CDS premium is 10.10%
To see how this works out, we can calculate:
Expected PV of premium on 6mth and 12mth dates:
Survival probability × Discount factor × Premium × Day count fraction
6mth: × × × 0.5 =
12mth: × × × 0.5 =
Expected PV of accrued premium if default occurs half-way through a period
Default probability × Discount factor × Premium × Day count fraction
6mth: × × × 0.25 =
12mth: × × × 0.25 =
Total expected value of premium income =
Expected PV of default payment if payment is made at end of period
Default probability × Discount factor × (1 – Recovery rate)
6mth: × × (1 – 30%) =
12mth: × × (1 – 30%) =
Total expected value of default payment =

32. Linking Default Probabilities and CDS Premiums
The previous formulas solve for the CDS premium given the default probabilities
But which comes first?
Default probabilities, or
CDS premiums?
We have so far started with default probabilities, and then determined CDS premiums
This assumes that we can obtain reliable default probabilities…
… from bond spreads
… or from cumulative default probability tables published by the credit rating agencies
In practice, it is the CDS market that is the most liquid!
So, in terms of this “chicken and egg” problem:
CDS premiums come first
From these you can infer default probabilities

33. Obtaining Default Probabilities from CDS Premiums
The formula given previously solves for the N-year CDS premium sN in terms of:
Default probabilities PD ■ Discount factors v
Survival probabilities PS ■ The day-count fraction T
This formula can be re-arranged to solve for the default probability PD given the CDS premium s:
To allow for accrual premium:
Add PDj/2 to the first summation so that it is: sum [v × T × (PS + PD/2)]
Divide the first term inside the brackets in the denominator by 2, i.e. s × v × T / 2
Let’s examine this formula more closely…

34. Analysing the Formula for Default Probability
The formula for determining s from the default and survival probabilities PD and PS was quite straightforward…
PD and PS (the data) is on the right
s (what we are trying to solve for) is on the left
The new formula appears to be a little mixed-up!
s (the data) is on the right
PD and PS (what we are trying to solve for) is BOTH on the left and the right
In fact, this is an example of a bootstrapping formula
To solve for PDN, you need sN, as well as PDN-1 and PSN-1
To solve for PDN-1, you need sN-1, as well as PDN-2 and PSN-2
… and so on
Let’s look at a worked example

35. Bootstrapping Default Probabilities – Step 1
First calculate PD1:
Suppose:
s1 = 10.46% ■ R = 30%
v1 = ■ T0,1 = 0.5
Of course, the one-period survival probability PS1 is:

36. Bootstrapping Default Probabilities – Step 2
Now we have PD1, we can calculate PD2:
Suppose:
s2 = 10.48% ■ R = 30%
v2 = ■ T1,2 = 0.5
And from the previous calculation:
PD1 = ■ PS1 =

37. Bootstrapping Default Probabilities – Step 3
We therefore continue the process
As we calculate each default probability, we can use this to calculate the next default probability…
… until the table of default and survival probabilities is complete
Period
Forward rates
Discount factors vj
CDS Spread s
Default probability PDj
Survival probability PSj
0.5
2.33%
0.9885
10.46%
0.0695
0.9305
1.0
2.52%
0.9762
10.48%
0.0649
0.8656
1.5
2.87%
0.9624
9.92%
0.0505
0.8151
2.0
3.22%
0.9471
9.41%
0.0421
0.7730
2.5
3.52%
0.9307
8.75%
0.0298
0.7432
3.0
3.82%
0.9133
8.31%
0.0287
0.7145
3.5
4.02%
0.8953
7.69%
0.0155
0.6990
4.0
4.22%
0.8768
7.22%
0.0153
0.6837
4.5
4.37%
0.8581
6.62%
0.0031
0.6806
5.0
4.52%
0.8391
6.14%
0.0030
0.6776

38. Term Structure of Default Probabilities
From the example on the previous page, we can construct a term structure of default probabilities…

39. Approximating Default and Survival Probabilities
There are some approximations to the formulas given earlier:
One-Period Default Probability
This compares to the exact answer of 7.0%
Term Survival Probability
This compares to the exact answer of 67.8%

40. Marking CDS to Market
When the CDS curve changes, existing CDS contracts will exhibit a mark-to-market profit or loss
To determine this:
Use the current CDS quotes to construct a new set of default and survival probabilities
Use the current set of interest rates to construct a new set of discount factors
Value the stream of swap payments and default payments with these new curves
The difference between the value of these two legs is the mark-to-market profit
The example on the following page illustrates this…

41. Mark to Market Example
In the example below, the entire credit curve has risen by 1%
The table recalculates the default and survival probabilities, and then the mark-to-market profits…
Period (yrs) 
Discount factors
Survival probability
Default probability
Old CDS Spread
New CDS Spread
PV of spread receipt if no default
PV of contingent payment
PV for buyer on $1m
0.5
0.9885
0.9244
0.0756
10.46%
11.46%
0.0478
0.0523
4,569 1
0.9762
0.8542
0.0701
10.48%
11.48%
0.0915
0.1003
8,738
1.5
0.9624
0.7991
0.0551
9.92%
10.92%
0.1248
0.1374
12,583 2
0.9471
0.7528
0.0462
9.41%
10.41%
0.1519
0.1681
16,148
2.5
0.9307
0.7191
0.0337
8.75%
9.75%
0.1705
0.1900
19,495 3
0.9133
0.6869
0.0322
8.31%
9.31%
0.1880
0.2106
22,631
3.5
0.8953
0.6678
0.0191
7.69%
8.69%
0.1970
0.2226
25,621 4
0.8768
0.6492
0.0186
7.22%
8.22%
0.2056
0.2340
28,467
4.5
0.8581
0.6425
0.0067
6.62%
7.62%
0.2068
0.2380
31,224 5
0.8391
0.6360
0.0065
6.14%
7.14%
0.2080
0.2418
33,892
The $33,892 profit on the 5yr swap arises from paying $10,000 less per annum over a five year period.