Credit Derivatives

• A credit derivative is a financial instrument designed to transfer credit risk between parties without necessarily transferring the underlying debt assets. It functions primarily as a form of insurance against the loss arising from a credit event such as default, restructuring, or bankruptcy of a reference entity like a corporation or government.
• Valuation: The value of a credit derivative is closely linked to the creditworthiness of the reference entity and the likelihood of a credit event occurring. These derivatives take into account various risk factors, including the financial health of the entity and market conditions.
• Trading and Structure: Unlike many financial instruments, credit derivatives are traded over-the-counter (OTC), which means they are not bought and sold on formal exchanges but rather through dealer networks. This setup allows for more customized arrangements that cater to specific risk management needs of contracting parties.
• Functionality and Use: The primary function of credit derivatives is to enable financial institutions, like banks, to manage and hedge against credit risks. They can maintain relationships with valuable but risk-prone customers by transferring part of the risk via these derivatives. Institutions may choose to retain some risks on their balance sheets while using credit derivatives to insure against potential losses on others.
• Market Impact: The market for credit derivatives expanded significantly in the late 1990s and early 2000s, peaking before the financial crisis of 2007-2008. This rapid growth underscored their rising importance in global finance. However, the market contracted post-crisis, reflecting changes in regulation and market sentiment towards these instruments.
• Risks and Regulation: Credit derivatives can be complex and opaque, particularly instruments like collateralized debt obligations (CDOs) that played significant roles in the financial crisis. This complexity necessitates stringent regulation and transparency to mitigate systemic risks and ensure stability in the financial system.
• Current Relevance: Despite challenges, credit derivatives continue to be crucial in modern finance, allowing for the effective management and trading of credit risks. Their ongoing use underscores the need for robust oversight to prevent misuse and contribute to financial instability.

Credit Default Swap

• A credit default swap (CDS) is a type of financial derivative that allows the transfer of credit exposure on fixed income products to another party. It operates much like insurance against the default of a debtor.
• Structure and Payments
  • A CDS involves two parties: the buyer of protection and the seller of protection. The buyer makes periodic payments to the seller, and in return, receives the promise of a payoff if a credit event (like default or bankruptcy) occurs concerning the reference entity (the debtor whose credit risk is being hedged).
  • Payments are typically made quarterly and in arrears. The total amount paid annually to buy protection is termed as the CDS spread, expressed in basis points.
• Credit Events
  • The credit events that can trigger a payout include bankruptcy, failure to pay, and restructuring of the reference entity’s debt. The specifics of what constitutes a credit event can vary between contracts.
• Settlement Types
  • Physical Settlement: The buyer of protection delivers defaulted debt instruments to the seller in exchange for the face value of the bonds.
  • Cash Settlement: More commonly used, involves a payment from the seller to the buyer based on the difference between the par value of the bonds and their market value post-credit event, often determined through an auction process.
• Market Role and Dynamics
  • CDSs provide a way for financial institutions to manage and hedge credit risk without needing to divest from bonds or loans that they might otherwise have to write off. This instrument is crucial in the financial markets for managing the risk profiles of investment portfolios and balancing the risk-reward equation in credit-intensive environments .

Credit Default Swap – Example

• Consider an example where a telecommunications company has recently issued $150 million in corporate bonds. An investment bank, looking to protect itself against potential default on these bonds, decides to enter into a CDS contract.
• Parties Involved
  • Buyer of Protection: The investment bank, which seeks to hedge against potential default on bonds it holds.
  • Seller of Protection: An insurance company that specializes in credit derivatives, agreeing to compensate the buyer in the event of a credit event.
  • Reference Entity: The telecommunications company whose issued bonds are being hedged. Although not a party to the CDS contract, this entity is central to the CDS contract as the occurrence of a credit event concerning this entity would activate the swap.
• Terms of the CDS
  • Notional Principal: $150 million (the face value of the bonds issued by the telecommunications company)
  • CDS Spread: 100 basis points per annum
• Payments – The investment bank agrees to pay the insurance company 100 basis points (1%) of the notional principal annually. This equates to an annual payment of $1.5 million, which is typically divided into quarterly payments of $375,000.
• Credit Events Specified
  • Bankruptcy: The telecommunications company goes bankrupt.
  • Failure to Pay: The telecommunications company misses a payment on its debt obligations.
  • Restructuring: The terms of the telecommunications company’s debt are altered detrimentally to creditors.
• Settlement Type
  • Cash Settlement: In the event of a credit event, rather than physically delivering the bonds, the contract stipulates a cash settlement. The payment would be calculated based on the market price of the bonds after the credit event, as determined through an auction process.
• Example Outcome – Suppose two years into the CDS contract, the telecommunications company undergoes restructuring due to financial difficulties, triggering a credit event under the terms of the CDS. The market price of the bonds drops to 40% of their face value. Hence Current Market Value of Bonds after Credit Event = $150m x 0.4 = $60𝑚
  • Settlement Calculation: The insurance company would owe the investment bank a cash settlement to cover the difference between the face value and the reduced market value of the bonds. Loss on Bonds is given by

    Face Value – Current Market Value

    = $150 million – $60 million = $90 million

    Thus, the insurance company would pay the investment bank $90 million, providing financial compensation for the decline in the bonds’ value due to the credit event.

  • Final Accrual Payment Calculation: Regular payments from the buyer to the seller of protection stop when a credit event occurs. Since these payments are in arrears, a final accrued payment is necessary. So, for example, if the credit event happens about one month after the last payment, the buyer owes an accrued payment for that month. Given the quarterly payment of $375,000, one month’s payment, being about one-third of the quarter, amounts to approximately $125,000. This final payment compensates the seller for the risk period up to the credit event, after which no more payments are due.

Credit Default Swaps And Bond Yields

• A CDS can be utilized to hedge risk in a corporate bond investment. An example can be used to demonstrate this.
• Scenario: An investment fund purchases a 10-year corporate bond issued by a technology company. The bond has a face value of $100 million and offers an annual yield of 6%.
• Simultaneous CDS Purchase: Alongside the bond purchase, the fund enters into a 10-year CDS contract to hedge against the risk of the issuer defaulting. The CDS spread is set at 250 basis points, or 2.5% per annum.
• Hedging Dynamics:
  • Bond Yield: The corporate bond yields 6% per year, equating to $6 million in annual interest.
  • CDS Cost: The CDS costs 2.5% of the face value per year, which is $2.5 million annually.
• Net Effect of the Hedge: Subtracting the CDS cost from the bond yield, the net yield is 3.5% per year, equivalent to $3.5 million. This adjusted return is nearly risk-free, assuming no default occurs, as it mimics the characteristics of a risk-free investment by offsetting potential losses from the bond with the CDS.
• If the Bond Issuer Defaults:
  • The fund has earned 3.5% annually up to the point of default.
  • Under the CDS terms, if the issuer defaults, the fund can exchange the bond for its full face value of $100 million. This amount can then be reinvested at a risk-free rate for the remainder of the 10-year period.
• CDS-Bond Basis: The relationship between the bond yield spread and the CDS spread is crucial. The bond yield spread is traditionally measured as the difference between the bond yield and the relevant LIBOR or swap rate. The CDS-bond basis is calculated as follows:

  CDS bond basis = CDS spread – Bond yield spread

  This basis should theoretically be close to zero, indicating no arbitrage opportunities, which means that the yield on the bond adjusted for the CDS cost should align closely with the risk-free rate. However, discrepancies can occur due to market conditions or asymmetric information.

• Arbitrage Opportunities:
  • If the CDS-bond basis is positive (bond yield spread is less than the CDS spread), it suggests that an investor can earn above the risk-free rate by buying the bond and the corresponding CDS protection.
  • Conversely, if the basis is negative, it implies that borrowing costs (via shorting the bond and selling CDS protection) are below the risk-free rate, indicating another potential arbitrage opportunity.
• Asymmetric Information and Market Dynamics: The decision to purchase protection via a CDS is typically made based on public information, despite potential private insights that financial institutions might have due to their closer relationships with the company. Thus, while the CDS market incorporates broad market perceptions of credit risk, individual institutions may possess more detailed knowledge that isn’t reflected in public CDS pricing.
• Cheapest-to-Deliver Option: In the event of a default, a CDS contract often allows for the delivery of one of several specified bonds. These bonds, while similar in seniority, might not trade at the same price post-default, enabling the protection buyer to deliver the “cheapest” bond (the one with the lowest market value relative to face value). This value is typically determined through an auction process organized by the International Swaps and Derivatives Association (ISDA), ensuring the buyer of protection receives the appropriate payoff.

Total Return Swap

• A Total Return Swap (TRS) is a financial derivative agreement in which one party (the total return payer) agrees to pay the total economic performance, including income from interest and fees, realized capital gains, and losses of a referenced asset. In return, the other party (the total return receiver) pays a set rate, which could be a fixed or floating rate, or the total return of a different asset. This arrangement allows for the transfer of total economic performance of an asset from one party to another without the need for actual ownership transfer of the asset.
• TRS can be structured around a variety of underlying assets, including loans, bonds, and equities. It is used by financial institutions to manage exposures to various kinds of assets or market risks without altering the physical assets in their portfolios. For example, if a bank holds bonds that it wants to retain in its long-term investment portfolio but seeks to eliminate the credit risk associated with the bonds, it can enter a TRS agreement where another party agrees to cover any credit losses incurred from the bonds in exchange for the income generated by them.
• The swap effectively allows parties to hedge against risks or gain exposure to asset classes without needing to hold the actual assets. This can lead to capital relief and regulatory arbitrage opportunities under certain financial regulations.

Collateralized Debt Obligation (CDO)

• Collateralized Debt Obligations (CDOs) are structured finance products that pool various cash-flow-generating assets and repackage them into tranches with different risk and return levels. Each tranche of a CDO has a different priority in the cash flow from the pool, influencing its risk profile.
• Key Components of CDOs:
  1. Asset Pool – Includes diverse debt types like loans and bonds, which are often higher in risk and return.
  2. Tranches– Cash flows from the assets are divided into tranches that vary in risk and return. Senior tranches are prioritized for repayment and have lower yields but higher credit ratings, whereas junior tranches offer higher yields with increased risk exposure.
  3. Waterfall Mechanism – Describes how cash flows are distributed to tranches, starting with senior tranches down to equity tranches, after covering administrative costs and interest.
  4. Special Purpose Vehicle (SPV) – CDOs are issued by an SPV, which holds the asset pool and issues the securities, isolating the CDO’s financial risks from the originator.
• CDOs redistribute the risk among investors based on their risk appetite, but their complexity can lead to significant losses if the underlying assets underperform.

CDS Valuation And Credit Risk

• The Credit Default Swap (CDS) spread depends on the following factors:
  1. CDS Maturity: This refers to the duration over which credit protection is provided.
  2. Risk-Free Rate: A higher risk-free rate reduces the present value of expected future cash flows, including those from a credit event.
  3. Hazard Rate (Default Intensity): Represented as 𝜆, the hazard rate indicates the likelihood of the reference entity defaulting per year.
  4. Recovery Rate: This percentage reflects the expected recovery of the bond’s par value in the event of a default.
• The relationships between these inputs and the probabilities of default and survival are typically framed in terms of the hazard rate. As discussed, and derived in previous chapters, if hazard rate 𝜆 is constant, then the probability of survival upto time t is . Also, the probability of default in year t is determined by subtracting the probability of survival at time t from the probability of survival at time t-1, which becomes .
• CDS Valuation Example: Consider the following scenario –
  • Notional Principal: $1
  • CDS Contract Term: 5 years
  • Risk-Free Rate: 4% per annum with continuous compounding
  • Recovery Rate: 35%
  • Hazard Rate: 1.5% per annum for the entire 5-year duration of the CDS

Table CR13-1: Unconditional Default Probabilities and Survival Probabilities

Year	Probability of surviving to year end	Probability of default during year
1	e-0.015×1 = 0.9851	1 − 0.9851 = 0.0149
2	e-0.015×2 = 0.9704	0.9851 − 0.9704 = 0.0147
3	e-0.015×3 = 0.9560	0.9704 − 0.9560 = 0.0144
4	e-0.015×4 = 0.9418	0.9560 − 0.9418 = 0.0142
5	e-0.015×5 = 0.9277	0.9418 – 0.9277 = 0.0140

It will be assumed that defaults always occur halfway through the year and that payments on the CDS are made annually at the end of each year. The forthcoming Table CR13-2 displays the computation of the present value of expected payments on the CDS, based on the assumption that payments occur annually at a rate of s and the notional principal amounts to $1. For instance, the probability of making the fourth payment of s is 0.9418. Consequently, the expected payment is 0.9418s, Based on the given risk-free rate of 4% p.a. with continuous compounding , its present value is calculated as 0.9418𝑠×𝑒^(−0.04×4)=0.8025𝑠. The total present value of the expected payments is 4.2523𝑠.

Table CR13-2: Calculation of the Present Value of Expected Payments

Payment = s per annum.
Time (Year)	Probability of survival	Expected payment	PV of expected payment
1	0.9851	0.9851s	0.9465s
2	0.9704	0.9704s	0.8958s
3	0.9560	0.9560s	0.8479s
4	0.9418	0.9418s	0.8025s
5	0.9277	0.9277s	0.7596s
Total			4.2523s

• Table CR13-3 examines the accrual payment made in the event of a default. For instance, there is a 0.0144 probability that a final accrual payment will be required halfway through the third year due to a default. The accrual payment in this scenario is 0.5s. Therefore, the expected accrual payment at this time is 0.0144 x 0.5s = 0.0072s. The corresponding discount factor is calculated as
  . Thus, its present value is 0.0072s x 0.9048 = 0.0065s. The total present value of the expected accrual payments is 0.0327s.
• Hence, from the tables, CR13-2 and CR 13-3 the present value of total expected payments is 4.2523s + 0.0327s = 4.285s

Table CR13-3: Calculation of the Present Value of Accrual Payments

Time (Year)	Probability of default	Expected Accrual Payment	Discount factor	PV of expected Accrual Payment
0.5	0.0149	0.00745s	0.9802	0.0073s
1.5	0.0147	0.00735s	0.9418	0.0069s
2.5	0.0144	0.00720s	0.9048	0.0065s
3.5	0.0142	0.00710s	0.8694	0.0062s
4.5	0.0140	0.00700s	0.8353	0.0058s
Total				0.0327s

• Now let’s consider the payoff in the case of default. Table CR13-4 demonstrates the calculation of the present value of the expected payoff, assuming a notional principal of $1 and a recovery rate of 35%. As previously mentioned, it is assumed that defaults always occur halfway through the year. For instance, there is a 0.0144 probability of a payoff halfway through the third year due to a default in that year. Given the recovery rate is 35%, resulting in an LGD (Loss Given Default) of 65%, the expected payoff at this time is 0.0144×0.65×1=0.0094. The corresponding discount factor, as calculated earlier, is 0.9048. Therefore, the present value of the expected payoff is 0.0094× 0.9048= 0.0085. The total present value of the expected payoffs is $0.0426.

Table CR13-4: Calculation of the Present Value of Expected Payoff

Notional Principal = $1 and RR = 0.4
Time (Year)	Probability of default	Expected Payoff ($)	Discount factor	PV of expected payoff ($)
0.5	0.0149	0.0097	0.9802	0.0095
1.5	0.0147	0.0096	0.9418	0.0090
2.5	0.0144	0.0094	0.9048	0.0085
3.5	0.0142	0.0092	0.8694	0.0080
4.5	0.0140	0.0091	0.8353	0.0076
Total				0.0426

• To value the CDS, the main idea is to equate the present value of the expected payments to the present value of the total expected payoffs. This ensures that the cost of protection (CDS spread) reflects the risk of default. Hence

  4.285s=0.0426

  ⇒ s = 0.0426/4.285 = 0.0099

  or, 99 basis points per year

• The mid-market CDS spread for the 5-year deal considered should be 0.0099 times the principal, or 99 basis points per year. The calculations assume that defaults occur only at points midway between payment dates. This simple assumption typically yields good results but can be easily adjusted to consider more default times if necessary.
• Note – An excel has also been created and provided to demonstrate all calculations.

Marking To Market A CDS

• The process of marking-to-market a Credit Default Swap (CDS) involves revaluing the contract to reflect current market conditions. This ensures that both parties involved in the CDS have an accurate assessment of the contract’s current value.
• If the credit default swap in the example had been negotiated some time ago for a spread of 120 basis points, the present value of the payments by the buyer, given the current calculation of a mid-market spread of 0.0099 times the principal, or 99 basis points per year, would be 4.285s, which is determined by multiplying 4.285 by 0.0120, resulting in 0.0514. The present value of the payoff, as previously calculated, would be 0.0426. Therefore, the value of the swap to the seller would be 0.0514 minus 0.0426, equating to 0.0088 times the principal. Consequently, the marked-to-market value of the swap to the buyer of protection would be -0.0088 times the principal.

Estimating Default Probabilities

• The default probabilities used to value a CDS should be risk-neutral default probabilities, rather than real-world default probabilities. Risk-neutral default probabilities can be estimated from bond prices, or they can be implied from CDS quotes. This method is analogous to the practice in options markets where implied volatilities are derived from the prices of actively traded options and then used to value other options.
• In an example scenario, instead of knowing the default probabilities, the mid-market CDS spread for a newly issued CDS is provided. By reverse-engineering the calculations, potentially using Excel in conjunction with Solver, the implied hazard rate can be determined.

Binary Credit Default Swaps

• A binary credit default swap is structured similarly to a regular credit default swap, but the payoff is a fixed dollar amount. Generally, the payoff on default is the full notional amount, regardless of the recovery rate (RR), which means that RR is assumed to be zero. In the previous example, if the payoff is $1 instead of 1 minus the recovery rate (RR) and the swap spread is denoted as s, then Tables CR13-1, CR13-2, and CR13-3 remain unchanged. However, Table CR13-4 is replaced by Table CR13-5. The CDS spread for a new binary CDS is given by the equation 4.285s = 0.0655, so, the CDS spread s is 0.0153, or 153 basis points.

Table CR13-5: Calculation of the Present Value of Expected Payoff for a Binary CDS

Notional Principal = $1 and RR = 0
Time (Year)	Probability of default	Expected Payoff ($)	Discount factor	PV of expected payoff ($)
0.5	0.0149	0.0149	0.9802	0.0146
1.5	0.0147	0.0147	0.9418	0.0138
2.5	0.0144	0.0144	0.9048	0.0130
3.5	0.0142	0.0142	0.8694	0.0123
4.5	0.0140	0.0140	0.8353	0.0117
Total				0.0655

CDS And Recovery Rates

• Whether CDS spreads or bond prices are used to estimate default probabilities, an estimate of the recovery rate is necessary. However, provided that the same recovery rate is used for both
  1. estimating risk-neutral default probabilities, and
  2. valuing a CDS,

  the value of the CDS (or the estimate of the CDS spread) is not very sensitive to the recovery rate. This is because the implied probabilities of default are approximately proportional to 1/(1−𝑅𝑅), and the payoffs from a CDS are proportional to 1−𝑅𝑅.

• This argument does not apply to the valuation of binary CDS. While implied probabilities of default remain approximately proportional to 1/(1−𝑅𝑅), the payoffs from a binary CDS are independent of 𝑅𝑅. If a CDS spread is available for both a plain vanilla CDS and a binary CDS, both the recovery rate and the default probability can be estimated.

Credit Indices

• CDS indices work a lot like stock indices, but instead of giving exposure to the performance of multiple stocks, they give exposure to the credit risk of multiple companies at once. In these indices, each company’s risk is equally weighted, and the total amount of protection bought (notional principal) is spread evenly across all the companies in the index.
• Credit indices have become vital tools in credit markets, providing benchmarks for credit default swap (CDS) spreads. Two significant standard portfolios are:
  1. CDX NA IG: A portfolio of 125 investment-grade companies in North America.
  2. iTraxx Europe: A portfolio of 125 investment-grade names in Europe.

  These indices are updated semiannually on March 20 and September 20, ensuring that only current investment-grade companies are included. Companies downgraded below investment grade are replaced by new investment-grade companies.

• Example Scenario – Suppose a trader wishes to buy protection on the CDX NA IG index, with the following parameters –
  1. • Quoted Index Spread: Bid – 70 basis points or 0.0070, and Ask – 71 basis points, or 0.0071 per dollar of notional principal
     • Protection Amount: $1,000,000 per company
     • Total number of companies: 125

  The calculations can be carried out as follows –

  • Annual cost of protection = 0.0071 × $1,000,000 × 125 = $887,500
  • Annual income from selling protection = 0.0070 × $1,000,000 × 125 = $875,000
  • Default Event: When one company defaults, the protection buyer receives the usual CDS payoff based on (1 – RR) on notional of $1,000,000. The annual payment is reduced by $887,500/125 = $7,100. Alternatively, the reduction can be calculated as 0.0071 × $1,000,000 = $7,100. So, the annual payment is then adjusted to $880,400, reflecting a reduction of $7,100 to account for the default, and the contract is continued with the remaining 124 companies.
• Maturities: The most common maturity for an index CDS is 5 years. Contracts also trade with maturities of 3, 7, and 10 years. The maturity dates are typically December 20 and June 20.

Advantages Of Credit Indices

• Diversification: Credit indices pool multiple entities, reducing idiosyncratic risk. Example: The CDX NA IG index includes 125 companies, spreading the risk across various sectors.
• Liquidity: Indices are typically more liquid than single-name CDS, facilitating easier trading. Example: A trader can quickly buy or sell protection on the entire index, benefiting from the high trading volumes.
• Standardization: Indices provide a standardized framework, simplifying the trading and hedging processes. Example: The fixed spread and regular updates ensure consistency and predictability.
• Cost Efficiency: Trading in indices can be more cost-effective than managing multiple single-name CDS contracts. Example: Buying protection on the entire CDX NA IG index costs $887,500 annually, which is more efficient than negotiating separate contracts for each company.
• Market Sentiment: Indices reflect overall market sentiment regarding credit risk, offering a snapshot of market health. Example: A widening spread on the iTraxx Europe index indicates increased perceived risk among European investment-grade companies.
• In practical terms, if an investor holds a diversified portfolio of North American investment-grade bonds, they can hedge against credit risk by buying protection on the CDX NA IG index. This approach provides a broad hedge, covering all companies in the index, and leverages the liquidity and cost efficiency of trading a single instrument.
• In practical terms, if an investor holds a diversified portfolio of North American investment-grade bonds, they can hedge against credit risk by buying protection on the CDX NA IG index. This approach provides a broad hedge, covering all companies in the index, and leverages the liquidity and cost efficiency of trading a single instrument.

The Use Of Fixed Coupons

• The use of fixed coupons in CDS and CDS index transactions introduces a standardized approach that simplifies trading and valuation. The detailed procedure for calculating the price from the quoted spread involves several steps to determine the accurate cost and payments for the protection buyer and seller.
  • Assumption of Payment Schedule: Payments are assumed to be made four times per year, in arrears. This regularity simplifies the calculation of cash flows and the present value of payments.
  • Implying a Hazard Rate: The hazard rate is implied from the quoted spread. An iterative search determines the hazard rate that aligns with the quoted spread, representing the annualized probability of default.
  • Calculating Duration: Duration 𝐷 is calculated for the CDS payments. This duration is used to determine the present value of the spread payments. In the previous example, it was 4.285
  • Calculating Price: The price 𝑃 is given by the formula P = 100-[100×D×(s – c)], where 𝑠 is the spread and 𝑐 is the coupon, both expressed in decimal form. This formula adjusts the price based on the difference between the spread and the coupon, reflecting the net cost of the protection.
  • Calculating Up-front Premium: The up-front premium in percentage terms is D×(s – c) and in dollar terms it is D×(s – c) per $100 of the notional.
• The use of fixed coupons in CDS and CDS index transactions introduces a standardized approach that simplifies trading and valuation. The detailed procedure for calculating the price from the quoted spread involves several steps to determine the accurate cost and payments for the protection buyer and seller.
  • Assumption of Payment Schedule: Payments are assumed to be made four times per year, in arrears. This regularity simplifies the calculation of cash flows and the present value of payments.
  • Implying a Hazard Rate: The hazard rate is implied from the quoted spread. An iterative search determines the hazard rate that aligns with the quoted spread, representing the annualized probability of default.
  • Calculating Duration: Duration 𝐷 is calculated for the CDS payments. This duration is used to determine the present value of the spread payments. In the previous example, it was 4.285
  • Calculating Price: The price 𝑃 is given by the formula P = 100-[100×D×(s – c)], where 𝑠 is the spread and 𝑐 is the coupon, both expressed in decimal form. This formula adjusts the price based on the difference between the spread and the coupon, reflecting the net cost of the protection.
  • Calculating Up-front Premium: The up-front premium in percentage terms is D×(s – c) and in dollar terms it is D×(s – c) per $100 of the notional.
• When protection is bought by a trader, 100 – P per $100 of the total remaining notional is paid by the trader, and this amount is received by the seller of protection. (If 100 – P is negative, the buyer of protection receives money, and the seller of protection pays money.) The buyer of protection then pays the coupon times the remaining notional on each payment date.
• For a CDS, the remaining notional is the original notional until a default occurs, after which it becomes zero.
• For a CDS index, the remaining notional is the number of names in the index that have not yet defaulted multiplied by the principal per name. The payoff in the event of a default is calculated in the usual manner. This arrangement facilitates trading because the instruments trade like bonds. The regular quarterly payments made by the buyer of protection are independent of the spread at the time the contract is entered into.

Example –

The 5-year iTraxx Europe CDS has a fixed coupon rate of 90 bps. If the CDS spread is 60 bps, and the duration of the 5-year CDS is 4.2, calculate the up-front premium for a contract with $150,000 notional per company.

Price = 100-[100×4.2×(0.0060 – 0.0090)]=100+1.26=101.26, or

Up – front premium = 1.26%

Because the price is greater than 100, the protection seller pays the protection buyer. This is logical, given that the protection buyer is agreeing to a higher coupon of 90 bps compared to the justified spread of 60 bps.

The amount the protection seller pays to the buyer is calculated as follows:

Up – front payment by seller = 125×150,000×0.0126 = $236,250

The Use Of Fixed Indices

• When protection is bought by a trader, 100 – P per $100 of the total remaining notional is paid by the trader, and this amount is received by the seller of protection. (If 100 – P is negative, the buyer of protection receives money, and the seller of protection pays money.) The buyer of protection then pays the coupon times the remaining notional on each payment date. For a CDS, the remaining notional is the original notional until a default occurs, after which it becomes zero.
  For a CDS index, the remaining notional is the number of names in the index that have not yet defaulted multiplied by the principal per name. The payoff in the event of a default is calculated in the usual manner. This arrangement facilitates trading because the instruments trade like bonds. The regular quarterly payments made by the buyer of protection are independent of the spread at the time the contract is entered into.
• Example –
  The 5-year iTraxx Europe CDS has a fixed coupon rate of 90 bps. If the CDS spread is 60 bps, and the duration of the 5-year CDS is 4.2, calculate the up-front premium for a contract with $150,000 notional per company.

  

  or

  Up – front premium = 1.26%

  Because the price is greater than 100, the protection seller pays the protection buyer. This is logical, given that the protection buyer is agreeing to a higher coupon of 90 bps compared to the justified spread of 60 bps.
  The amount the protection seller pays to the buyer is calculated as follows:

  

CDS Forwards And CDS Options

• CDS Forwards: A forward credit default swap (CDS) is a contract to buy or sell a specific CDS on a particular reference entity at a predetermined future date (time T). The key aspects of a CDS forward include:
  • The obligation to enter into a CDS transaction at a specified future date.
  • If the reference entity defaults before the future date (time T), the forward contract ceases to exist.

  Example: A bank could enter into a forward contract to sell 5-year protection on a company for 190 basis points, starting in 1 year. If the company defaults before the 1-year mark, the forward contract would no longer be valid.

• CDS Options: A credit default swap (CDS) option provides the right, but not the obligation, to buy or sell a CDS on a particular reference entity at a future date (time T). CDS options can be categorized as either call options or put options.
  • CDS Call Option: It gives the right to buy protection at a specified future date and spread. For example, a trader could negotiate the right to buy 5-year protection on a company starting in 1 year for 125 basis points. If the 5-year CDS spread for the company in 1 year is more than 125 basis points, the option will be exercised; otherwise, it will not. The cost of the option is paid upfront.
  • CDS Put Option: It gives the right to sell protection at a specified future date and spread. For example, an investor could negotiate the right to sell 5-year protection on a company starting in 1 year for 160 basis points. If the 5-year CDS spread for the company in 1 year is less than 160 basis points, the option will be exercised; otherwise, it will not. The cost of the option is paid upfront.

  Both CDS forwards and CDS options are typically structured so that they cease to exist if the reference entity defaults before the option or forward contract’s maturity date.

Synthetic CDOs

• Asset-backed securities (ABSs) with underlying assets as bonds are known as collateralized debt obligations (CDOs). These CDOs use a “waterfall” structure for interest and principal payments, ensuring that senior tranches are more likely to receive payments than junior ones.
• Synthetic CDOs are created using credit default swaps (CDSs) rather than bonds. Here’s how they work:
  • Portfolio and Maturity:
    1. The originator selects a portfolio of companies and a maturity (e.g., 5 years).
    2. CDS protection is sold on each company, matching the maturity of the structure.
    3. The synthetic CDO principal is the sum of the notional principals of the CDSs.
  • Cash Inflows and Outflows:
    1. Cash inflows come from CDS spreads.
    2. Cash outflows occur when companies default.
  • Tranche Structure:
    1. Equity Tranche: Responsible for the first 5% of payouts. Earns 1,000 basis points per year.
    2. Mezzanine Tranche: Responsible for payouts between 5% and 20%. Earns 100 basis points per year.
    3. Senior Tranche: Responsible for payouts above 20%. Earns 10 basis points per year.
• Example: For a $100 million synthetic CDO:
  1. Equity tranche: $5 million, earning 1,000 basis points.
  2. Mezzanine tranche: $15 million, earning 100 basis points.
  3. Senior tranche: $80 million, earning 10 basis points.

  If $2 million in defaults occur in the first year, the equity tranche covers this, reducing its principal to $3 million. If another $4 million in defaults occur later, the equity tranche is exhausted, and the mezzanine tranche covers the next $1 million, reducing its principal to $14 million.

• Investment and Collateral: Cash CDOs require initial investments by tranche holders. In contrast, synthetic CDO holders do not need an initial investment but must post the tranche principal as collateral. When a tranche has to make a CDS payoff, the amount is deducted from the collateral, which usually earns interest.

Valuing Synthetic CDOs Using Spread Payments

• The process of valuing a synthetic Collateralized Debt Obligation (CDO) using the spread payments approach involves calculating the present value of expected payments and payoffs for the different tranches.

1. Identify Payment Dates and Expected Tranche Principal: Define the payment dates of the CDO tranche as . Let be the expected tranche principal at time , and let be the present value of $1 received at time .
2. Calculate Present Value of Expected Spread Payments: If the spread s on the tranche is paid on the remaining tranche principal, then the present value of the expected regular spread payments on the CDO is given by sA, where

   

3. Calculate Present Value of Expected Payoffs:
   The expected loss between times . Assume that the loss occurs at the midpoint of the time interval (i.e., at time ) The present value of the expected payoffs on the CDO tranche is

   \(C = \sum_{j=1}^{m} (E_{j-1} – E_j) v(\tau_{j-0.5})\)

4. Calculate Accrual Payment:
   
   The accrual payment due on the losses is given by 𝑠𝐵, where

   

5. Determine Value of the Tranche and Breakeven Spread: The value of the tranche to the protection buyer is 𝐶− 𝑠𝐴−𝑠𝐵. The breakeven spread on the tranche occurs when the present value of the payments equals the present value of the payoffs or 𝐶=𝑠𝐴+𝑠𝐵. The breakeven spread is therefore

   

Valuing Synthetic CDOs Using Gaussian Copula

• The Gaussian copula model is a popular method for pricing CDS tranches in synthetic CDOs, as it accounts for the correlation of default times among different entities in the collateral pool. Here is an overview of the process:

1. Define the Collateral Pool: A synthetic CDO involves a pool of reference entities, typically companies, whose default probabilities are to be modeled.
2. Homogeneous Probability Distribution: The Gaussian copula model assumes a homogeneous probability distribution for the time to default across all reference entities. This means that the time to default probability distribution is the same for all companies in the CDO collateral pool.
3. Correlation Modeling: The model incorporates the correlation between default times of different entities. This correlation, known as the copula correlation, is assumed to be constant for any pair of companies in the CDO.
4. Unconditional Probability of Default: Calculate the unconditional probability of default for each reference entity over time. This is the basic probability that an entity will default by a given time, not considering any correlations with other entities.
5. Conditional Probability of Default: Using the Gaussian copula, convert the unconditional default probabilities into conditional probabilities of default, given a common factor (such as the overall economic environment).
6. Default Distribution: Use a binomial distribution to determine the probability of a specific number of defaults within the collateral pool by a certain time, given the conditional default probabilities.
7. Expected Loss Calculation: Calculate the expected losses for each tranche of the CDO. Each tranche absorbs a different level of loss based on its seniority and the number of defaults occurring in the collateral pool.
8. Integrate Over Correlation: Integrate the conditional expected losses over the entire distribution of the common factor to obtain the unconditional expected losses for each tranche. This step accounts for the variability in default correlations.
9. Determine the Breakeven Spread: Calculate the breakeven spread for each tranche. This is the spread that ensures the present value of the expected premium payments equals the present value of the expected losses.

By following these steps, the Gaussian copula model provides a structured way to value synthetic CDOs, taking into account the correlated risk of defaults among the entities in the reference portfolio. This method allows for a detailed assessment of the risk and return for each tranche of the CDO.

Implied Correlation Measures

• Implied correlations are essential in the pricing and risk management of synthetic CDOs, particularly when market participants derive them from market quotes for tranches. There are two main measures of implied correlation: compound (tranche) correlation and base correlation.
  1. Compound (Tranche) Correlation: is the value of the correlation parameter that, when used in the Gaussian copula model, leads to the model-generated spread matching the market spread for a specific tranche. It is determined through an iterative search process. This measure ensures that the model’s output aligns with observed market prices for individual tranches.
  2. Base Correlation: – Base correlation is derived for a tranche defined from zero up to a specific attachment point. It represents the value of the correlation parameter that makes the Gaussian copula model consistent with market pricing for the tranche from 0 to the specified point. The process involves the following steps:
     1. Calculate the compound correlation for each tranche.
     2. Use the compound correlation to determine the present value of the expected loss on each tranche as a percent of the initial tranche principal.
     3. Compute the present value of the expected loss for the tranche spanning from 0 to a given attachment point.
     4. Divide the computed value by the attachment point to get the C-value for the tranche.
     5. Perform an iterative search to find the correlation parameter that matches this C-value.
• Both compound and base correlations are crucial in the valuation and hedging of synthetic CDOs, providing insights into how market participants perceive default correlations among reference entities. These measures help in understanding and managing the risks associated with different tranches of a CDO.

Alternative Approaches To Estimate Default Correlation

• Estimating default correlation is crucial in pricing and managing credit derivatives, particularly for synthetic CDOs. While the one-factor Gaussian copula model is widely used, there are several alternative approaches that address its limitations. These include heterogeneous models, other copulas, random recovery and factor loadings, the implied copula model, and dynamic models.
  1. Heterogeneous Model
     • Description: Unlike the standard homogeneous Gaussian copula model, the heterogeneous model assumes that the time-to-default probability distributions and the copula correlations are not the same for all companies.
     • Implementation: Each company has a different probability of defaulting at any given time, making the binomial calculation more complex. Numerical procedures are required to handle the increased complexity.
  2. Other Copulas
     • Description: Several alternative one-factor copula models have been proposed, including the Student t copula, the Clayton copula, Archimedean copula, and the Marshall-Olkin copula. These models offer different ways to capture the dependence structure between default times.
     • Implementation: These copulas can be created by assuming non-normal distributions for the common factor and individual default times. For instance, the double t copula assumes that both the common factor and individual default times follow a Student t distribution.
  3. Random Recovery and Factor Loadings
     • Description: This approach, suggested by Andersen and Sidenius, assumes that the copula correlation is a function of the common factor and that the recovery rate is negatively related to the default rate.
     • Implementation: The correlation increases as the common factor decreases, meaning that in scenarios with high default rates, the default correlation is also high. This model fits market quotes better by reflecting the empirical relationship between default rates and correlations.
  4. Implied Copula Model
     • Description: Hull and White propose a method to imply a copula from market quotes. This involves assuming an average hazard rate for all companies in a portfolio over the life of a CDO.
     • Implementation: The probability distribution of the average hazard rate is implied from the pricing of tranches, similar to how implied volatility is derived from option prices. This model captures market-implied correlations directly from observed prices.
  5. Dynamic Models
     • Description: Unlike static models, dynamic models attempt to capture the evolution of portfolio losses over time. There are three main types:
       1. Structural Models: These model the stochastic processes for the asset prices of many companies simultaneously. Defaults occur when asset prices hit a barrier. The asset prices are correlated, and implementation requires Monte Carlo simulation.
       2. Reduced Form Models: These model the hazard rates of companies, assuming jumps in hazard rates to capture realistic correlation levels.
       3. Top-Down Models: These models directly focus on the total loss of a portfolio without considering individual company defaults. They provide a macro-level view of portfolio risk.