Credit Default Swap Pricing Model

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CREDIT DEFAULT SWAP PRICING – A PREMIERE

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Credit Default Swap Pricing – A Premiere

Prepared By

DAVID STEINBERG

Portfolio Manager | Capital Structure Arbitrage Portfolio

Carnegie Hall Tower | 152 West 57th Street | 4th Floor | New York, NY 10019

http://platinumlp.com/

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Introduction

Q: What is a Credit Default Swap? A: CDS is a contract that obligates the

protection seller to compensate the protection buyer for losses that will incur as a result of a credit Event in the reference obligation. • Similar to an insurance policy on a house; if the

house burns down, the seller of the insurance policy is obligated to compensate the buyer of the policy for the losses he incurred as a result of the fire to the underlying asset.


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Cash Flows The protection buyer pays the seller an annual premium in addition to an upfront fee. In return, the protection seller unconditionally guarantees to compensate the buyer for any losses he incurs due to a credit event in the reference obligation.


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Reference Obligation

A reference obligation is a specific debt or loan of a company that will be due in the future. This can be a corporate bond or bank loan of a company.

All CDS contracts specify a specific debt obligation of a company for which a credit event has to occur in order for the protection seller to be obligated to make good on his guarantee.


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Credit EventA credit event occurs when a company fails to make scheduled payments on the reference obligation. This includes scheduled interest and/or principle payments. When a company misses a scheduled payment of a debt obligation, the company is usually forced into bankruptcy by its lenders.


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Protection Payment

Once a credit event occurs, the protection seller is obligated to compensate the buyer for all losses that the buyer incurred on the reference obligation as a result of the credit event.

This means that since debt obligations are typically worth something even in bankruptcy, the protection seller only needs to compensate the buyer for the LOSS in value of the reference obligation.


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Two More Points


1. Once a credit event occurs, the protection buyer ceases to make the annual premium payments.

2. However, the protection buyer must make a final premium payment for any accrued time from the previous premium payment until the credit event.

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Illustration

CDS Contract Details

Investor A buys a 5 year CDS on $10MM from Bank B on Ford Motor

Credit Co.

Reference Obligation: Ford Motors Bond; F 12 2015

CUSIP: 345397VH3

Investor A pays 100bps of notional value annually or $100,000.00

Additionally, Investor A must pay an upfront fee to Bank B. The Upfront Fee and the annual

premiums make up the Premium Payments.

Contract Maintenance

So long as there is no credit event, Investor A will continue to pay the

annual premium of $100,000

The CDS expires at the stated maturity, in this case 5 years.

Investor A makes his final premium payment and the CDS

contract expires.

Credit Event

If Ford Motors Credit fails to make scheduled payments on the reference obligation, a credit

event is declared

Bank B must compensate Investor A for any losses the Investor

incurred as a result of the credit event

If, for example, the reference obligation is valued at 40 cents on

the dollar in bankruptcy, then Bank B must compensate for the

missing 60 cents

In our example of a $10MM CDS contract, Bank B would be

obligated to pay $6MM to Investor A. This is the Protection Payment.


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Cash Flow Chart


Premium Payments

Upfront Fee

Year #1:Annual Premium of

100bps of CDS contract









Protection Payment

Protection Payment =

Notional Amount of CDS Contract –

Recovery Rate of Reference Obligation

Subtracted by any accrued premium due to the seller from the

previous premium payment until the

credit event

Without a Credit Event

After a Credit Event

Credit Default Swap Pricing – A Premiere 11

Model

Pricing

CDS

To arrive at an accurate market price for a CDS contract

We need to understand…

The value of the guarantee to

compensate the protection buyer…

in the event of a default.

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Net Present Value


In an efficient market place, the amount investors will pay for a security, is only what they believe is to be it’s net present value.

Solving for NPV requires three steps Knowing all possible cash flows Determining the probability of receiving each

cash flow Discounting each cash flow to adjust for the

time value of money; at an appropriate discount rate for each cash flow

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Net Present Value


In our example, Investor A will only pay for Bank B’s guarantee the equivalent of the guarantee's NPV.

Solving for the NPV of the Protection payment requires three stepsKnowing all

possible cash flows

For our 5yr CDS for 10MM on Ford Motors, there is a possibility of receiving a protection payment in each of the 5 years. Assuming a 40% recovery rate on the reference obligation

after the credit event, the Protection Payment will be $6,000,000.00. Historically, IG companies bonds’ have a 40%

RR. Determining

the probability of receiving each cash

flow

Here we need to decide on the probability of Ford missing a debt payment in any of the five years , thereby triggering the $6MM cash flow. This in the only input that is subjective and

will vary from dealer to dealer

Discounting each cash

flow to adjust for the time

value of money

Accepted practice in the CDS market is to discount each cash flow by its corresponding Interest Rate Swap Rate. Standard practice is to use yesterday’s end of day rates. The IR Swap Curve for CDS is published daily, and can be found on the

Bloomberg at YCSW0260 Index <GO>

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Knowing all

possible cash flows

For our 5yr CDS for 10MM on Ford Motors, there is a possibility of receiving a protection payment in each of the 5 years. Assuming a

40% recovery rate on the reference obligation after the credit event, the Protection Payment will be $6,000,000.00. Historically,

IG companies bonds’ have a 40% RR. Year 1

Year 2

Year 3

Year 4

Year 5

Notional CDS

Contract Amount

$10,000,000

$10,000,000

$10,000,000

$10,000,000

$10,000,000

Standard Recovery

Rate for IG Bonds

40%

40%

40%

40%

40%

Protection Payment

=$10MM * (1- .40)

=$10MM * (1- .40)

=$10MM * (1- .40)

=$10MM * (1- .40)

=$10MM * (1- .40)

Net Possible

Cash Flow from

Protection Payment$6,000,000

$6,000,000

$6,000,000

$6,000,000

$6,000,000

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Determining the

probability of receiving each cash

flow

Here we need to decide on the probability of Ford missing a debt payment in any of the five years , thereby triggering

the $6MM cash flow. This in the only input that is subjective and will vary from dealer to dealer

The probability of a company missing a scheduled debt payment depends on many factors• The total amount of current and long liabilities that are coming

due• The cash flow from operations of the company• The amount of cash and cash equivalent securities a company

has on its balance sheet

Investors can look at the Yield to Maturity of a company’s bonds to gain insight on the market’s view of the credit worthiness of a company.

In all, this aspect of the CDS Pricing Model is the only input that is somewhat subjective and can vary from dealer to dealer. (However, the level of variance in this input cannot be that large, as that would present, what is know as a Basis Arbitrage trade.)

For the purpose of this presentation, we will use the default probabilities found on the VCDS <GO> page on the Bloomberg for the Ford Motors Bond.

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Determining the


flow



We can see the Default Probability for the next month through the year 2020.

For our 5yr CDS, we only need the probability of default for next 5 years.

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Determining the


flow



Year 1

Year 2

Year 3

Year 4

Year 5

Net Possible

Cash Flow from

Protection Payment$6,000,00

0

$6,000,000

$6,000,000

$6,000,000

$6,000,000

Default Probability

0.0529

0.1131

0.1639

0.2214

0.2733

Default Probability per Period

0.05290

0.06020

0.05080

0.05750

0.05190

Probability Adjusted

Cash Flow

$317,400

$361,200

$304,800

$345,000

$311,400

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Discounting each cash flow

to adjust for the time value

of money

Accepted practice in the CDS market is to discount each cash flow by its corresponding Interest Rate Swap Rate. Standard practice is to use yesterday’s end of day rates. The IR Swap Curve for CDS is published daily, and can be

found on the Bloomberg at YCSW0260 Index <GO>

The third step in solving for the NPV of the protection payment is to discount each of the probability adjusted cash flows to their present value.

CDS market conventions use the USD IR SWAP Curve to discount each cash flow.

Each cash flow occurs in a different period, and therefore needs to be discounted to using the period’s corresponding SWAP rate in the IR SWAP Curve. Meaning, the cash flow occurring at the end of year one needs to be discounted by the 1yr SWAP rate, the cash flow occurring at the end of year two will be discounted by the 2yr SWAP Rate, and so on.

To avoid the intraday volatility in the IR SWAP Rate market, CDS market conventions use the previous day’s end of day rate.

The IR SWAP curve rates for CDS can be found on the Bloomberg at YCSW0260 Index <GO>

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USD ISDA CDS SWAP Curve


The Description page for the USD ISDA CDS SWAP Curve

Type:YCSW0260 Index DES <GO>

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USD ISDA CDS SWAP Curve


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of money



The mathematics for discounting at different discount rates for each period varies from the typical PV formula of

Instead, the PV of a cash flow combining different discount rates is calculated by creating a discount factor. The discount factor is just another way of expressing the present value of $1.00 received at period X, using multiple discount rates.

For the first period, the discount factor resembles the typical PV formula; , when is the corresponding rate on the IR SWAP Curve.

For the second period, instead of , which uses one rate for all periods, we need an equation that will capture the different rates for each corresponding period on the IR SWAP Curve. The exact equivalent to the typical PV formula, but that also takes into account the different rates, is

cr

1

00.1$cr

21 rate

FVPV

)1)(1(

00.1$

212

ppp crcr

df

periodrate

fvpv

1

22




of money



Discount factor for period two;

Where; = discount factor for period 2 = corresponding rate from period 1 = corresponding rate from period 2

For period three the discount factor simply adds the corresponding rate to the denominator;

For each of the remaining periods we construct a discount factor by simply adding the new period’s corresponding rate to the denominator.

Discount factor for period four;

Discount factor for period five;

2pdf

)( 1pcr

)( 2pcr

)1)(1)(1(

00.1$

3213

pppp crcrcr

df

)1)(1)(1)(1(

00.1$

43214

ppppp crcrcrcr

df

)1)(1)(1)(1)(1(

00.1$

543215

pppppp crcrcrcrcr

df

)1)(1(

00.1$

212

ppp crcr

df

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Determining the


flow



Year 1

Year 2

Year 3

Year 4

Year 5


Cash Flow

$317,400

$361,200

$304,800

$345,000

$311,400

Corresponding Rate from USD ISDA IR SWAP Curve

.8938 %

1.250 %

1.7951 %

2.2875 %

2.7095 %

Discount Factor

0.99114

0.97934

0.96207

0.94055

0.91574

pv of Protection Payment

$314,588

$353,738

$293,239

$324,491

$285,162

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Year 1

Year 2

Year 3

Year 4

Year 5


Cash Flow

$317,400

$361,200

$304,800

$345,000

$311,400

Corresponding Rate from USD ISDA IR SWAP Curve

.8938 %

1.250 %

1.7951 %

2.2875 %

2.7095 %

Discount Factor

0.99114

0.97934

0.96207

0.94055

0.91574

pv of Protection Payment

$314,588

$353,738

$293,239

$324,491

$285,162

Summing these discounted cash flows will give us the NPV = $1,571,218.52

We now know the value, or NPV, of the seller’s guarantee to reimburse the buyer for any loss resulting from a credit event.

This amount is what the buyer will be willing to pay for the guarantee, and is the market value for this CDS contract.

NVP of Protection Leg

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We now know the amount the protection buyer will have to pay to buy this CDS from the seller.

As we mentioned in the outset, the buyer does not pay for the CDS in one lump sum, but rather the payments consist of annual premiums of 100bps of the notional CDS amount, and one upfront payment. In other words, some of the money owed to the seller is paid annually for the duration of the contract, and some is paid upfront.

Due to standard CDS contract conventions, we always know that the annual payments will be 100bps of the CDS notional amount. The only remaining question is the amount of the upfront payment.

To solve for the upfront amount due to the seller, we will discount the annual premium payments , as well as any accrued premiums that may be due if a credit event occurs in between premium payments. The NPV of the possible premium payments plus the upfront premium must equal the market value, or the NPV of the protection payment.

Valuing the Premium Payments

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Solving for the premium upfront amount

In our example, we know the market value of the protection payment is $1,571,218.52. This is the amount the buyer will need to pay to receive the protection guarantee, and the NPV of all the buyer’s payments will need to equal $1,571,218.52.

By CDS market conventions the buyer will make annual premium payments of 100bps of 10MM notional or $100,000 annually for 5 years or until a credit event occurs. Additionally, the buyer will have to pay any accrued premium if a credit event occurs in between scheduled premium payments.

To arrive at the NPV of these cash flows we will again have to take into account the probability of the buyer having to make these payments, i.e. the probability of survival, and then discount each cash flow by its corresponding rate on the ISDA USD IR Swap.

The difference between the NPV of the buyer’s annual premium payments and NPV of the seller’s protection payment will be the amount the buyer will need to make in the upfront payment.

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Probability Adjusted Annual Premium

Just as we adjusted the cash flows of the protection payment by the probability of receiving them, we will do the same for the premium payments. The buyer will make these annual payments for as long as the company survives, so the probability of these cash flows is simply yprobabilitdefault 1

Year 1

Year 2

Year 3

Year 4

Year 5

Annual Premium Payment

$100,000

$100,000

$100,000

$100,000

$100,000

Default Probabilit

y

0.0529

0.1131

0.1639

0.2214

0.2733

Survival Probabilit

y (1-default prob)

0.94710

0.88690

0.83610

0.77860

0.72670

Probability

Adjusted Cash Flow

$94,710

$88,690

$83,610

$77,860

$72,670

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Probability Adjusted Accrued Premium

Besides the scheduled annual premium, we must also account for the possibility that the buyer will have to pay accrued premium if a credit event occurs in between annual payments.

We assume that if a credit event occurs, it will happen exactly half way through the payment period, therefore the buyer will have to pay for half the period or in our case $50,000.

Of course, accrued payments are only made if the company defaults, so the probability of receiving any accrued payments is the same as the default probability for the period.

Year 1

Year 2

Year 3

Year 4

Year 5

Accrued Premium

$50,000

$50,000

$50,000

$50,000

$50,000

Default Probabilit

y per Period

0.05290

0.06020

0.05080

0.05750

0.05190

Probability

Adjusted Accrued

Cash Flow$2,645.00

$3,010.00

$2,540.00

$2,875.00

$2,595.00

NET Probabilit

y Adjusted

Cash Flow$97,355.0

0

$91,700.00

$86,150.00

$80,735.00

$75,265.00

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Discounting all Premium Payments

Next we need to discount each probability adjusted cash flows by the corresponding rate of the ISDA USD IR Swap.

As we did with the protection payment, we will formulate a discount factor for each period.

Year 1

Year 2

Year 3

Year 4

Year 5

NET Probabilit

y Adjusted

Cash Flow$97,355

$91,700

$86,150

$80,735

$75,265

df

0.99114

0.97934

0.96207

0.94055

0.91574

pv of Annual

Premium Payments$96,49

3 $89,80

5 $82,88

2 $75,93

6 $68,92

3

NPV = $414,039.44

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Solving for the premium upfront amount

Year 1

Year 2

Year 3

Year 4

Year 5

NET Probabilit

y Adjusted

Cash Flow$97,355

$91,700

$86,150

$80,735

$75,265

df

0.99114

0.97934

0.96207

0.94055

0.91574

pv of Annual

Premium Payments$96,49

3 $89,80

5 $82,88

2 $75,93

6 $68,92

3

NPV of Annual Premium

Payments = $414,039.44

NPV of Protection Payment

= $1,571,218.52

Remaining premium due to

seller upfront=

$1,157,179.07

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The Equation

NPV of Annual Premium

Payments = $414,039.44

NPV of Protection Payment

= $1,571,218.52

Remaining premium due to

seller upfront=

$1,157,179.07 In other words the upfront premium payment is simply;

premiumupfrontNPVNPV paymentspremiumaccruedscheduledpaymentprotection

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The CDS Equation

The mathematically stated equation for CDS is;

)()()1()2/)(()()()( 11

11

iii

n

iiiii

n

ii tDFttDRPtDFttDPtDFtS

Where;= Survival probability for period t= Discount factor for period t= Premium payment= Default probability for period t, but not before period t= Recovery rate on reference obligation

)( itS)( itDF

P

)( 1 ii ttD

R

NPV scheduled premium payments

NPV accrued premium payments

NPV of all premium payments

NPV of protection payment

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The Spreadsheet

CDS Notional 10,000,000 Recovery Rate 40.00%Annual Premium Rate 1.000%Present Value of Protection Payment

P Protection

Payment Default

Probability Default Probability

for Period Probability Adjusted CF cr df pv

1 6,000,000.00$ 5.290% 0.05290 317,400.00 0.8938% 0.99114 314,588.21$ 2 6,000,000.00 11.310% 0.06020 361,200.00 1.2050% 0.97934 353,737.66 3 6,000,000.00 16.390% 0.05080 304,800.00 1.7951% 0.96207 293,238.94 4 6,000,000.00 22.140% 0.05750 345,000.00 2.2875% 0.94055 324,491.41 5 6,000,000.00 27.330% 0.05190 311,400.00 2.7095% 0.91574 285,162.30

NPV 1,571,218.52$

Present Value of Premium Payment

p Premium Payment

Survival Probability

Probability Adjusted CF Accrued

Probability Adjusted Accrued CF df pv

1 100,000.00$ 94.71% 94,710.00$ 50,000.00 2,645.00$ 0.99114 96,492.55$ 2 100,000.00 88.69% 88,690.00 50,000.00 3,010.00 0.97934 89,805.49 3 100,000.00 83.61% 83,610.00 50,000.00 2,540.00 0.96207 82,882.33 4 100,000.00 77.86% 77,860.00 50,000.00 2,875.00 0.94055 75,935.69 5 100,000.00 72.67% 72,670.00 50,000.00 2,595.00 0.91574 68,923.38

NPV 414,039.44$

NPV of CDS Contract 1,157,179.07

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Credit Default Swap Pricing Model

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