lecture 141

8/2/2019 Lecture 141

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Lecture 14

Credit Derivatives

Options, Futures, Derivatives / March 24, 2008 1

8/2/2019 Lecture 141


Credit Derivatives

Credit derivatives allow companies to trade credit risks in the same way that they trade marketrisks.

Companies can enter into credit derivative contracts to protect themselves from credit defaults of their lenders/borrowers.


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Credit Default Swaps

The most popular credit derivative is a credit default swap (CDS)

• A CDS contract provides insurance against the risk of a default by a particular company.• The company is known as the reference entity and a default by the company is known as a

credit event .• The buyer of the insurance obtains the right to sell bonds issued by the company for their face

value when a credit event occurs• The sellers of the insurance agrees to buy the bonds for their face value when a credit event

occurs.• The total face value of the bonds that can be sold is known as the credit default swap’s

notional principal .


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Features of a CDS

• Buyers of a CDS makes periodic payments to the seller until the end of the life of the CDS oruntil a credit event occurs.

• The payments are typically made in arrears every quarter, every half year , or every year.• The settlement, in the event of a default, involves either physical delivery of the bonds or a

cash payment.


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Example of a CDS

Consider a typical deal. There are two parties entering into a 5-year CDS on March 1, 2004.Assume that the notional principal is $100 million and the buyer agrees to pay 90 basis points(0.9%) annually for protection against default by the reference entity.

• If the reference entity does not default, the buyer receives no payoff and pays $900,000 onMarch 1 of each years, 2005, . . . , 2009.

• If there is a credit event, a substantial payoff is likely. Suppose that the buyer noties the sellerof a credit event on June 1, 2007, (quarter of the year into 2008).

• If the contract species physical settlement, the buyer has the right to sell bonds issued by thereference entity with a face value of $100 million for $100 million.

• If the contract requires cash settlement, an independent calculation agent will poll dealers todetermine the mid-market value of the cheapest-to-deliver bond a predesignated number of days after the credit event. Suppose this bond is worth $35 per $100 of face value. the cashpayoff would be $65 million.


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The regular quarterly, semiannual, or annual payments form the buyer of protection to the seller of protection cease when there is a credit event. However, because these payments are made inarrears, a nal accrual payment by the buyer is usually required.

In the example a buyer would be required to pay to the seller the amount of the annual payment

accrued between Mar. 1 , 2007 and June 1, 2007

• The total amount paid per year, as a percent of the notional principal is known as the CDSspread .

• Many large bands are market makers in the credit default swap market.• When quoting on a new 5-year CDS on Ford, a market maker might bid 250 basis points and

offer 260 basis points.


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CDS and Bond Yields

A CDS can be used to hedge a position in corporate bonds.

• Suppose that an investor buys a 5-year corporate bond yielding 7% per year for its face value• At the same time the investor enters into a 5-year CDS to buy protection against the issuer of

the bond defaulting.• Suppose that the CDS spread is 2% per annum. The effect of the CDS is to convert the

corporate bond to an approximately risk-free bond.• If the bond issuer does not default, the investor earns 5% per year. If the bond does default the

investor earns 5% up to the time of the default.• Under the terms of the CDS, the investor is then able to exchange the bond for its face value.• This face value can be invested at the risk-free rate for the remainder of the 5 years.


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The n -year CDS spread should be approximately equal to the excess of the par yield on ann -yearcorporate bond over the par yield on an n -year risk-free bond.

If it is markedly less, an investor can earn more than the risk-free rate by buying the corporatebond and buying protection.

If it is markedly greater, an investor can earn more than the risk-free rate by selling the corporatebond and selling protection.

CDS spreads can be used to imply the risk-free rate used by market participants≈

LIBOR/swap rate minus 10 basis points.


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Cheapest-to-Deliver Bond

The recovery rate on a bond is dened as the value of the bond immediately after default as apercent of its face value.

• Therefore, the payoff from CDS isL (1 − R ) where L is the notional principal and R is therecovery rate.

• Usually a CDS species that a number of different bonds can be delivered in the event of adefault.

• The bonds typically have the same seniority, but they may not sell for the same percentage of face value immediately after default. This gives the holder of a CDS a cheapest-to-deliver bondoption.

• When a default happens the buyer of protection will review alternative deliverable bonds andchoose for delivery the one that can be purchased most cheaply.


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Credit Indices

Participants in credit derivatives markets have developed indices to track credit default swapspreads. In 2004, there were agreements between different producers of indices which led to someconsolidation. Two such indices:

1. The 5- and 10-year CDX NA IG indices tracking the credit spread for 125 investment gradeNorth American companies; and

2. The 5- and 10-year iTraxx Europe indices tracking the credit spread for 125 investment gradeEuropean companies

In addition to monitoring credit spreads, indices provide a way for market participants can easilybuy or sell a portfolio of credit default.

Example : An investment bank, acting as market maker might quote the CDX NA IG 5-yearindex as bid 65 basic points and offer 66 basis points. An investor could then buy $800,000of 5-year CDS protection on each the 125 underlying companies for a total of $660,000 peryear.The investor can sell $800,000 of 5-year CDS protection on each of the 125 underlyingnames for a total of $650,000 per year.When a company defaults, the annual payments is reduced by $660,000/125=$5,280.


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Valuation of Credit Default Swaps

Mid-market CDS spreads on individual reference entities can be calculated from default probabilityestimates.

We go through a detailed example:

Suppose that the probability of a reference entity defaulting during a year conditional on noearlier default is 2%. Suppose we have the following survival probabilities as seen at time

zero for each of the 5 years:Time Default Survival

in years probability probability1 0.0200 0.98002 0.0196 0.96043 0.0192 0.9412

4 0.0188 0.92245 0.0184 0.9039


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We have:• The probability of default during the rst year is 0.02 and the probability the reference

entity will survive until the end of the rst year is 0.98.• The probability of a default during the second year is 0.02 × 0.98 = 0 .0196 and the

probability of survival until the end of the second year is0.98 × 0.98 = 0 .9604 . Theprobability of default during the third year is 0.02 × 0.9604 = 0 .0192 , etc.

We will assume that defaults always happen halfway through a year and that payments on the CDSare made once a year, at the end of each year.

We also assume that the risk-free LIBOR interest rate is 5% per annum with continuous

compounding and the recovery rate is 40%.


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• We rst calculate the present value of the payments made on the CDS assuming that paymentsare made at the rate of s per year and the notional principal is $1. For example there is a0.9412 probability that the third payment of s is made. The expected payment is therefore0.9412 s and its present value is

0.9412 se− 0.05 × 2.5

= 0 .0102

The total present value of the expected payments is 4.0704 s .


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• Next we calculate the expected present value of the payoff assuming a notional principal of $1.Assuming that defaults always happen halfway through a year. For example, there is a 0.0192probability of a payoff halfway through the third year. Given that the recovery rate is 40% theexpected payoff at this time is

0.0192 × 0.6 × 1 = 0 .0115

The present value of the expected payoff is

0.0115 e − 0.05 × 2.5 = 0 .0102

The total present value of the expected payoffs is $0.0511.


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• The nal step entails computing the accrual payment made in the event of a default. Forexample, there is a 0.0192 probability that there will be a nal accrual payment halfwaythrough the third year. The accrual payment is 0.5s . The expected accrual payment at thistime is therefore

0.0192 × 0.5s = 0 .0096 s

Its present value is0.0096 se − 0.05 × 2.5 = 0 .0085 s

The total present value of the expected accrual payments is 0.0426 s .


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• Therefore, the expected payments is

4.0704 s + 0 .0426 s = 4 .1130 s

From our second table the present value of the expected payoff is 0.0511. Equating the two, we

obtain the CDS spread for a new CDS is

4.1130 s = 0 .0511

or s = 0 .0124 . The mid-market spread should be 0.0124 times the principal or 124 basispoints per year.

In practice we are likely to nd that calculations are more extensive than those in Table21.2.


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Marking to Market a CDS

At the time it is negotiated, a CDS, like most other swaps, is worth close to zero. Later it mayhave a positive or negative value.

Suppose, for example, the credit default swap in our example had been negotiated some time agofor a spread of 150 basis points, the present value of the payments by the buyer would be

4.1130 × 0.0150 = 0 .0617

and the present value of the payoff would be 0.01511 as above. The value of the swap to the sellerwould therefore be 0.0617 − 0.0511 = 0 .0106 times the principal. Similarly the mark-to-marketvalue of the swap to the buyer of protection would be -0.0106 times the principal.


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Estimating Default Probabilities

The default probabilities used to value a CDS should be risk-neutral default probabilities, notreal-world default probabilities. Risk-neutral default probabilities can be estimated from bondprices or asset swaps as earlier.

An alternative is to imply them from CDS quotes. The latter approach is similar to the practice inoptions markets of implying volatilities from the prices of actively traded options.


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Binary Credit Default Swaps

A binary credit default swap is structured similarly to a regular credit default swap except that thepayoff is a xed dollar amount.

Supposed that in the example we have considered the payoff is $1, instead (1 − R ) dollars, andthe swap spread is s . Tables 1, 21.2, 21.4 the same but Table 21.3 is replaced by

The CDS spread for a new binary CDS is given by

4.1130 s = 0 .0852

so that the CDS spread, s , is 0.0207 or 207 basis points.Options, Futures, Derivatives / March 24, 2008 19

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Importance of Recovery Rates

Whether we use a CDS spreads or bond prices to estimate default probabilities we need anestimate of the recovery rate. However, provided that we use the same recovery rate for (a)estimating risk-neutral default probabilities and (b) valuing a CDS, the value of the CDS is notvery sensitive to the recovery rate.

This is because the implied probabilities of default are approximately proportional to 1/ (1 − R )and the payoffs from a CDS are proportional to 1 − R .

This argument does not apply to the valuation of binary CDS. Implied probabilities of default arestill proportional to 1/ (1 − R ) . However, for a binary CDS, the payoffs from the CDS areindependent of R . If we have CDS spread for both a plain vanilla CDS and a binary CDS, we canestimate both the recovery rate and the default probability.


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CDS Forwards and Options

Once the CDS market was well established, it was natural for derivatives dealers to trade forwardsand options on credit default swap spreads.

Forward credit default swap :

• This is the obligation to buy or sell a particular credit default swap on a particular referenceentity at a particular future time T .

• If the reference entity defaults before time T , the forward contract ceases to exist. Thus a bank

count enter into a forward contract to sell 5-year protection on Ford Motor Credit for 280 basispoints starting in 1 year.• If Ford defaults during the next year, the bank’s obligation under the forward contract ceases to

exist.

Credit default swap option

• This is an option to buy or sell a particular CDS on a particular reference entity at a particularfuture time T .

• For example, an investor could negotiate the right to buy 5-year protection on Ford MotorCredit start in 1 year for 280 basis points. This is a call option.

• If the 5-year CDS spread for Ford in 1 year turns out to be more than 280 basis points, theoption will be exercised; otherwise it will not be exercised. The cost of the option would bepaid up front.


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• An investor might negotiate the right to sell 5-year protection on Ford for 280 basis pointsstarting in 1 year. This is a put option.

• If the 5-year CDS spread for Ford in 1 year turns out to be less than 280 basis points the optionwill be exercised; otherwise it will not be exercised. The cost of the option would be paid upfront.

Like CDS forwards, CDS options are usually structured so that they will cease to exist if thereferecne entity defaults before option maturity.


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Total Return Swaps

A total return swap is a type of credit derivative. It is an agreement to exchange the total returnon a bond for LIBOR plus a spread. The total return includes coupons, interest, and a gain or losson the asset over the life of the swap.

Example :

• Consider a 5-year agreement with notional principal of $100 million to exchange the total

return on a corporate bond for LIBOR plus 25 basis pints.• On coupon payment dates the payer pays the coupons earned on an investment of $100 million

in the bond.• The receiver pays interest at a rate of LIBOR plus 25 basis points on a principal of $100 million.• At the end of the life of the swap there is a payment reecting the change in the value of the

bond.– If the bond increases in value by 10% over the life of the swap, the payer is required to to

pay $10 million (= 10% of $100 million) at the end of 5 years.– If the bond decreases in value by 15%, the receiver is required to pay $15 million at the end

of the 5 years.– If there is a default on the bond, the swap is usually terminated and the receiver makes a

nal payment equal to the excess of $100 million over the market value of the bond.


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If we add the notional principal to both sides at the end of the life of the swap, we can characterizethe total return swap as follows:

• The payer pays the cash ows on an investment of $100 million in the corporate bond.• The receiver pays the cash ows on a $100 million bond paying LIBOR plus 25 basis points.• If the payer owns the corporate bond, the total return swap allows it to pass the credit risk on

the bond to the receiver.• If the payer does not own the corporate bond, the total return swap allows it to take a short

position in the bond.

Total return swaps are used as nancing tools:

• Consider a case in which the receiver wants nancing to invest $100 million in the referencebond. A payer, likely a large bank and agrees to a swap

• The payer then invests $100 million in the bond. This leaves the receiver in the same positionas it would have been if it had borrowed money at LIBOR plus 25 basis points to buy the bond.

• The payer retains ownership of the bond for the life of the swap and faces less credit risk thanit would have done if it used as collateral for the loan.

• If the receiver defaults the payer does not have the legal problem of trying to realize on thecollateral.


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Notes:

• The spread over LIBOR is compensation for bearing the risk that the receiver will default. Thepayer will lose money if the receiver defaults at a time when the reference bond’s price hasdeclined.

• The spread depends on the credit quality of the receiver, the credit quality of the bond issuer,and the default correlation between the two.


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Collateralized Debt Obligations

A Collateralized Debt Obligations (CDO) is a way of creating securities with widely differentrisk characteristics from a portfolio of debt instruments.

Ingredients of the current market crisis.

Consider the following example: Four types of tranches (securities) are created from a portfolio of bonds (or mortgages).

• The rst tranche has 5% of the total bond principal and absorbs all credit losses from theportfolio during the life of the CDO until they reached 5% of the total bond principal.

• The second tranche has 10% of the total bond principal and absorbs all losses during the life of the CDO in excess of 5% of the principal up to a maximum of 15% of the principal.

• The third tranche has 10% of the total bond principal and absorbs all losses during the life of

the CDO in excess of 15% of the principal up to a maximum of 25% of the principal.• The fourth tranche has 75% of the principal absorbs all losses in excess of 25% of the principal.


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The yields are the rates of interest paid to tranche holders.


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These rates are paid on the balance of the principal remaining in the tranche after losses have beenpaid.

• For the rst tranche, initially a return of 35% is paid on the whole amount invested by tranche1 holders.

• After losses equal to 1% of the total bond principal have been experienced, tranche 1 holdershave lost 20% of their investment and the return is paid on only 80% of the original amountinvested.

• Tranche 1 is referred to as the equity tranche .• A default loss of 2.5% on the bond portfolio translates into a loss of 50% of the tranche’s

principal.• Tranche 4 by contrast is usually given an Aaa rating. Defaults on the bond portfolio must

exceed 25% before the holders of this tranche are responsible for any credit losses.• The creator of the CDO normally retains the equity tranche and sells the remaining tranches in

the market.• A CDO provides a way of creating high quality debt from average quality or low quality debt.


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Synthetic CDOs

The CDO above is a cash CDO . An alternative structure is the synthetic CDO where the creatorof the CDO sells a portfolio of credit default swaps to third parties. It then passes on the defaultrisk to the synthetic CDO’s tranche holders.


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Single Tranche Trading

The portfolios of 125 companies are used to generate the CDX and iTraxx indices. The marketuses the portfolios to dene standard CDO tranches.

The trading of these standard trances is known as single tranche trading .

• A single tranche trade is an agreement where one side agrees to sell protection against thelosses on a tranche and the other side agrees to buy the protection.

• The tranche is not part of a synthetic CO buy cash ows are calculated in the same way as if itwere part of a synthetic CDO.

• The tranche is referred to as unfunded because it has not been created by selling credit defaultswaps or buying bonds.

• In the case of the CDX NA IG index, the equity tranche covers losses between 0% and 3% of the principal. The second tranche, which is referred to as the mezzanine tranche covers lossesbetween 3% and 7%. The remaining tranches cover losses from 7% to 10%, 10% to 15%, 15%to 30%.

• In the case of iTraxx Europe index, the equity tranche covers losses between 0% and 3%, themezzanine tranche covers losses between 3% and 6%. The remaining tranches cover lossesfrom 6% to 9%, 9% to 12%, 12% to 22%.


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Example : Consider the mid-market quotes for 5-year CDX and iTraxx tranches on August 4, 2004.

On that date the CDX index level was 63.25 basis points and the iTraxx index was 42 basis points.

• The mid-market price of mezzanine protection for the CDX IG NA was 347 basis points peryear while that for iTraxx Europe was 168 basis points per year.

• Equity tranche is quoted differently from the other tranches.• The market quote of 41.75% for CDX means that the protection seller receives an initial

payment of 41.75% of the principal plus a spread of 500 basis pointsper year.• Similarly the market quote of 27.6% for iTraxx mean that the protection seller receives an

initial payment of 27.6% of the principal plus a spread of 500 basis points per year.


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Valuation of Basket CDS and CDO

The spread for the tranche of a CDO is critically dependent on the default correlation.

• If the correlation is low the junior equity tranche is very risky and the senior tranches are verysafe.

• As the default correlation increases the junior tranches become less risky and the seniortranches become more risky.

• In the limit where default correlation is perfect the tranches are equally risky.

One can use the Gaussian Copula Model of Time to Default to value a CDO.


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Convertible Bonds

A convertible bond is a bond issued by a company in which the holder has the option toexchange the bonds for the company’s stock at certain times in the future.

• The conversion factor is the number of shares of stock obtained for one bond.• The bonds are almost always callable (i.e. the issuer has the right to buy them back at certain

times at a perdetermined price.• The holder always has the right to convert the bond once it has been called.• The call feature is usually a way of forcing conversion earlier than the holder would otherwise

choose. Sometimes the holder’s call option is conditional on the price of the company’s stockbeing above a certain level.

• Credit risk plays an important role in the valuation of convertibles. If we ignore credit risk, wewill get poor prices because the coupons and principal payments on the bond will be overvalued.


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Pricing a convertible:

• Assume that the issuer’s stock follows a Wiener process except that there is a probability λ ∆ tthat there will be a default in each short period of time ∆ t .

• In the event of a default the stock price drops to zero and there is a recovery on the bond. Thevariable λ is the risk-neutral default intensity.

• We can represent the stock price process by varying the usual binomial tree so that at each node1. A probability pu of a percentage up movement of size u over the next time period of length

∆ t .2. A probability pd of a percentage down movement of size d over the next time period of

length ∆ t .3. A probability 1 − e − λ ∆ t ≈ λ ∆ t that there will be a default with the stock price moving to

zero over the next time period of length ∆ t .Parameter values are chosen, as before, to match the rst two moments of the stock pricedistribution, yield:

pu =a − de − λ ∆ t

u − d pd =

ue − λ ∆ t − a

u − du = eq ( σ 2 − λ ) ∆ t d =

1

u

and a = e ( r − q)∆ t , r is the risk-free rate and q is the dividend yield on the stock.


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• The life of the tree is set to be equal to the life of the convertible bond.• The value of the convertible bond at the nal nodes of the tree is calculated based on any

conversion options that the holder has at that time. We then work back through the tree.• At nodes where the terms of the instrument allow conversion, we test whether conversion is

optimal. We also test whether the position of the issuer can be improved by calling the bonds.• If so we assume that the bonds are called and retest whether conversion is optimal. This is

equivalent to setting the value at the node equal to

max [min( Q 1, Q 2) , Q 3]

where Q 1 is the value given by the rollback (assuming that the bond is neither converted nor

called at the node). Q 2 is the call price. Q 3 is the value if conversion takes place.

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lecture 141

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