Division of Applied Mathematics Seminar Report

School of Education, Culture & Communication

Mälardalen University

Box 883, 721 23 Västerås, Sweden Date: 15th January 2016

CREDIT DEFAULT SWAPS

MMA707- ANALYTICAL FINANCE II

LECTURER: JAN RÖMAN

AUTHOR:

ACHEAMPONG ARCHIBOLDNANA

ABSTRACT

This report aims at highlighting on the concept of Credit Default Swaps(CDS). It evaluates the pricing of CDS. It also describes the cash flows and settlement types of CDS. An example is illustrated using The Lehmann Brothers.

Table of Contents

1.0 Introduction ...... 3

2.0Credit Default Swaps..……………………………………..…………… ……..4

3.0Cash Flows of Credit Default Swaps………….……..……………….………..4

3.1No Credit Event Occurs...... 4

3.2Credit Event Occurs………………………….………………………………….4

4.0Settlement of Credit Default Swaps ...... 5

4.1Physical Settlement...... 5

4.2 Cash Settlement ...... 5

5.0Pricing of Credit Default Swaps...... 5

5.1 Probability Model……………...... 5

5.2No Arbitrage Model...... 7

6.0 The Lehmann Brothers Auction...... 7

7.0 Conclusion ...... 8

8.0 References ...... 8

1.0 INTRODUCTION

A swap is an agreement between two parties, say A and B. In this agreement, these two parties decide to make periodic payments to each other based on two different indices. In a plain vanilla interest rate swap for instance, counterparty A makes periodic fixed rate payments on the specified notional to counterparty B and in return, counterparty B makes floating rate payments to counterparty B on the same specified notional amount. Thus, the plain vanilla interest rate swap specifies a notional amount, the payment frequency, the floating rate, fixed rate, maturity and tenor of the agreement. The diagram below simply illustrates the concept of a swap between two parties, A and B.

Credit Default Swap (CDS) is among a range of financial instruments termed as Credit Derivatives. These Credit Derivatives were originally designed by JP Morgan. Other Credit Derivatives include Credit Default Indices, Total Return Swaps, Credit Default Swaptions, etc.

2.0 CREDIT DEFAULT SWAPS(CDS)

Credit Default Swap (CDS) is a financial swap where the buyer of the CDS makes periodic payments, called spread or fee, to the seller of the CDS. The seller on the other hand compensates the buyer in the case of a credit event such as defaults. As an example, Company A buys a corporate bond from another financial entity. To protect itself against default on the part of the financial entity, Company A buys a CDS from Company B, making periodic payments to Company B. In the event that the financial entity issuing the bond defaults on its payments to Company A, Company B will compensate Company A by paying the amount agreed upon in the contract.

The payment of spread continues until there is a default or the CDS contract expires.Individuals who do not own a bond can also purchase a CDS. These are termed “naked” CDS.

3.0 CASHFLOWS OF CREDIT DEFAULT SWAPS

3.1 No Credit Event Occurs

When no credit event occurs, the only cash flow is the periodic premium payments made by the buyer of the CDs to the seller. This premium is specified at the beginning of the transaction.

3.2 Credit Event Occurs

The CDS buyer makes the premium payments until there is a credit event. The seller then makes payment of the agreed amount to the seller.

The diagram below illustrates the cash flow of a CDS:

4.0 SETTLEMENT OF CREDIT DEFAULT SWAPS

4.1 Physical Settlement

In this form of settlement, the CDS seller pays the CDS buyer par value and takes delivery of the debt obligation of the specific entity involved.

4.2 Cash Settlement

Here, the CDS seller pays the CDS buyer the difference between the par value and market value of the debt obligation of the reference entity.

5.0 PRICING OF CREDIT DEFAULT SWAPS

There exists two basic theories for the pricing of CDS. They are the Probability model and the No Arbitrage model.

5.1 Probability Model

This model takes four inputs in pricing a CDS. They include:

Issue premium
Recovery rate, being the percentage of notional repaid in the event of default
Credit curve for the reference entity
XIBOR interest rate curve

The price of the CDS in the event of no default is the sum of the discounted premium payments. This suggests that the pricing of the CDS need to take into consideration the possibility of a default occurring.

Consider a one-year CDS with effective date to, where premium payments are made quarterly at times t1, t2, t3andt4. With the nominal of the CDS being N and the issue premium c, the size of the quarterly premium payments is Nc/4. Assuming defaults can only occur on one of the payment dates, there are five outcomes of the CDS contract:

there is no default, hence the four premium payments are made and the contract survives until maturity.
a default occurs on one of the payment dates, namely, first, second, third or fourth date.

Pricing the CDS involves assigning probabilities to the five possible outcomes and calculating the present value of each outcome. The present value of the CDS is therefore the present value of the payoffs multiplied by their respective probabilities of occurrence.

This is represented in the diagram below. At each payment date, the contract has a default event with payment of N(1-R), where R is the recovery rate, or the contract survives without default and a premium payment is made. At either side of the diagram below are cash flows up to that point.

Let piand(1-pi) be the probability of surviving over the interval ti-1totiwithout a default payment and the probability of a default occurring respectively. Given discount factor of D1toD4, the present value is calculated as follows:

The probability of no default occurring from timestto t +Δtis given by

,where s(t)is the credit spread zero curve at time t.

The total present value of the credit default swap is given by

5.2 No Arbitrage Model

Duffie and Hull-White proposed this model. It includes an assumption that there is no risk free arbitrage. Duffie and Hull-White use the LIBOR and US Treasuries as the risk free rates respectively. The Duffie approach is however often used by the market to determine theoretical prices.

In the Duffie approach, the price of a CDS can be determined by calculating the asset swap spread of a bond. If for instance a bond has a spread of 100 and the swap spread of 70 basis points, a CDS contract in this instance would trade at 30. The difference between the theoretical model and actual price of a CDS is termed the basis.

In terms of cash flow profile, a CDS is comparable with a par floating rate note funded at LIBOR or an asset swapped fixed-rate bond financed in the repo market.

The price of a CDS would include several factors, among which are:

probability of default of the reference entity and protection seller
correlation between the reference entity and the protection seller
joint probability of default of the reference entity and protection seller
swap maturity
expected recovery value of the reference asset

6.0 THE LEHMANN BROTHERS AUCTION

The procedures and systems developed to settle credit derivatives were subjected to true testing during the Lehmann Brothers failure in September 2008. The auction set a price of 8.625 cents on the dollar for Lehmann Brothers debt. This implied that sellers of protection on Lehmann Brothers CDS would have to pay 91.375 cents on the dollar to buyers of the protection in order to settle and terminate the contracts through the Lehmann Protocol auction process.

At the time of bankruptcy, Lehmann had approximately $155 billion of outstanding debt but about $400 billion notional value of CDS contracts. Since there was not enough Lehmann Brothers debt to fulfill these contracts, not all the contracts were physically settled. There was therefore the need for cash settlements.

7.0 CONCLUSION

Credit Default Swaps, as have been elaborated, can yield great returns to investors and protect them from potential loses. It can also result in great loses to financial entities.

8.0 REFERENCES

Röman, Jan, Lecture Notes For Analytical Finance II (2015)