Up-front Credit Default Swaps

Fixed Income Quantitative Credit Research 7

August 2003

Up-front Credit Default Swaps Dominic O’Kane and Saurav Sen When bonds are distressed, protection is often quoted as an up-front payment rather than a “running” spread paid until the earlier of default or maturity. This article examines the pricing, risk profile and performance of up-front credit default swaps and compares them to the standard running trades.

Lehman Brothers

| Quantitative Credit Research

QCR Quarterly, vol. 2003-Q3

Up-front Credit Default Swaps Dominic O’Kane +44-20-7260-2628 [email protected] Saurav Sen +44-20-7260-2940 [email protected]

When bonds are distressed, protection is often quoted as an up-front payment rather than a “running” spread paid until the earlier of default or maturity. This article examines the pricing, risk profile and performance of up-front credit default swaps and compares them to the standard running trades. We also address the issue of how an investor can decide 1 whether to trade on an up-front or a running basis.

1.

INTRODUCTION

In the standard credit default swap (CDS), the protection buyer pays for protection by making regular spread payments to the protection seller until the earlier of a credit event or maturity of the contract. We call this a running CDS as the protection payments run throughout the life of the contract. However, when the reference credit is distressed, protection-sellers quote prices on an up-front basis. This means that protection buyers make only a single up-front payment at initiation in return for protection against a credit event (typically these are bankruptcy, failure to pay and restructuring) until the contract maturity date. These contracts are also usually entered into for short maturities, i.e. up to one year. Paying for protection up-front has the effect of changing the risk profile of the default swap contract in two fundamental ways. First, it front loads the timing of cashflows to the start of the trade, which has implications in terms of the interest rate risk, funding and carry. Second, it removes the credit risk in the payment of the premium in the standard CDS, which terminates following a credit event. It should be noted that buying a bond and buying protection on the same face value is not a credit-neutral strategy when the bond is trading away from par. See O’Kane and McAdie (2001) for a discussion. While a default swap offers principal protection, the net P&L of a trade, which incorporates coupons, funding costs and the cost of protection, is strongly dependent on the timing of the credit event that triggers protection. This effect is most pronounced when the issuer is distressed, and applies to both running and up-front trades. The aim of this article is to set out in detail the mechanics and risks of the up-front CDS contract. In particular, we wish to highlight the differences between up-front and running CDS so that the investor can easily see which is more suitable for expressing a specific credit view. We begin by describing the precise mechanics of the up-front contract. Following this, we set out the model for pricing up-front protection and discuss calibration issues. We then examine the risk sensitivities of up-front CDS and compare these to running CDS. We also discuss default swap basis trades using up-front CDS and consider when up-front may be preferred to running.

2. UP-FRONT PROTECTION 2.1. Mechanics of Up-front Protection An up-front CDS is a contract that enables an investor to buy protection against the risk that an underlying reference entity suffers a credit event. Unlike a standard CDS, where it costs nothing to enter into the contract, the protection buyer in an up-front CDS makes a single initial payment to the seller, in return for protection until a specified maturity date.

1

August 2003

Reprinted from Quantitative Credit Research Quarterly, Volume 2003-Q3.

Please see important analyst(s) certifications at t he end of this report.

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Equally, an investor can use the up-front contract to sell protection in return for a single initial payment. The investor is then assuming the credit risk of the reference credit until the maturity date of the contract. If default occurs before maturity, the protection buyer typically delivers assets with a face value equal to that of the protection, to the protection seller in return for the face value amount in cash. Alternatively, the protection may be settled in cash for mat, exactly as in a standard running CDS. The value of protection delivered is equivalent to par minus recovery, where recovery is the price of the cheapest-to-deliver (CTD) asset in the basket of deliverables2. More details on the exact definitions of credit events and the mechanics of delivering protection can be found in O’Kane (2001). Figure 1 shows a schematic representation of the cashflows in running and up-front CDS contracts. Figure 1: Comparison of cashflows for running and up-front default swaps. Protection Seller Up-front Payment Running Spread

Time Running Up-front Protection Buyer 100-R on Credit Event

2.2. Up-front versus Running Trades The difference between up-front and running CDS is in the premium leg: in the former case the cost of protection is delivered as a single amount paid at initiation, whereas in the latter it takes the form of a risky coupon stream which lasts until a credit event or maturity, whichever occurs sooner. This difference in the timing of cashflows enables investors to take a view on the timing of default, as mentioned in the Introduction, and also changes the risk profile of up-front trades relative to running. As we explain below, up-front protection is priced so that, on a present value basis, an investor should be indifferent between running and up-front trades. The up-front price equals the arbitrage-free expected value of the risky coupon stream in a running CDS. However, the different timing of cashflows changes the distribution of outcomes; hence, the decision to trade on running or up-front basis also depends on the investor’s risk preferences. These concepts are illustrated with examples in the next section. There are a number of reasons why dealers prefer to quote distressed credits in up-front format:

2

See “Valuation of Restructuring Credit Event in Credit Default Swaps”, O’Kane, Pedersen and Turnbull, Lehman Brothers Quantitative Credit Research Quarterly, May 2003.

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1.

They eliminate any uncertainty about the size and timing of the payment for protection.

2.

The distribution of outcomes from an up-front contract is typically narrower for an up-front CDS than for a running CDS. This is especially true when spreads are wide. For this reason, risk aversion makes protection sellers and buyers prefer up-front to running. This will be explained in detail later.

3.

Dealers may feel more uncomfortable quoting spreads in excess of 1000bp, which is what would be required for many distressed credits. This is also the regime in which bonds begin to trade on a price basis. It is no surprise that the same should happen to CDS.

4.

Very wide spreads may also pose problems for analytics unless carefully implemented. For example, the payment of the accrued premium following a credit event has a significant effect and must be implemented correctly.

An investor’s preference for up-front or running trades, therefore, also depends on the valuation and price sensitivity of up-front trades. Before discussing these, we describe some examples to fix ideas and motivate the remainder of the article. These examples illustrate the mechanics of up-front protection and also highlight the market views and risks implicit in such trades.

3. UP-FRONT TRADES As with running CDS, up-front CDS can be used to implement basis trades between the cash and CDS market. They can also be used to express views on the likely timing of default and expected spread movements. To discuss these, it is best to use an example and for this we will use a 5-year maturity, 6% coupon bond which pays annually. Suppose initially this bond was distressed, trading at a clean price of $75 with an up-front premium quoted at $33 on a face value of $100. As we will show in the next section, the model-implied CDS spread corresponding to this up-front price, assuming a 40% recovery rate, is 1050bp. The generic basis trade will consist of an investor being long the bond and long protection on the same face value via either a running or up-front CDS. For simplicity, we will assume that the bond and running CDS cashflow dates are synchronized. We also assume that the funding rate and reinvestment rates are Libor flat at a constant 3%. We now examine the differences between running and up-front trades in terms of cashflows, carry and MTM for a range of scenarios.

3.1. Scenario I: Reference Credit Survives to Maturity Table 1 shows the protection buyer’s cashflows in the event that the bond does not default, when protection is bought on an up-front basis. Cashflows to the investor are shown with a positive sign.

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Table 1: Long Bond + Long Up-front Protection (No-default Scenario) Time (Y)

Bond ($)

Upfront CDS ($) Funding ($)

NET ($)

Reinvested carry ($)

0

-75.00

-33.00

108.00

0.00

0.00

1

6.00

0.00

-3.24

2.76

2.76

2

6.00

0.00

-3.24

2.76

5.60

3

6.00

0.00

-3.24

2.76

8.53

4

6.00

0.00

-3.24

2.76

11.55

5

106.00

0.00

-111.24

-5.24

6.65

As Table 1 shows, buying up-front protection results in a positive carry trade. In each period, the net carry is $2.76 per $100 face, which is the difference between the 6% coupon earned on the bond, and the 3% funding paid on a total initial borrowing of $108 ($75 for the bond + $33 up-front protection). The last payment is negative, as the investor has to pay back the funding principal, but the total reinvested carry over the life of the trade is still positive. Contrast this with Table 2 below, which shows the cashflows when protection is bought on a running basis. This is a negative carry trade, since the total payments in each period, consisting of $10.50 running protection and $2.25 bond funding, exceed the $6 coupon income from the bond. Even though the last cashflow is positive to the investor, the net reinvested carry is still negative. Table 2: Long Bond + Long Running Protection (No-default Scenario) Bond ($)

Running CDS ($)

Funding ($)

NET ($)


0

-75.00

0.00

75.00

0.00

0.00

1

6.00

-10.50

-2.25

-6.75

-6.75

2

6.00

-10.50

-2.25

-6.75

-13.70

3

6.00

-10.50

-2.25

-6.75

-20.86

4

6.00

-10.50

-2.25

-6.75

-28.24

5

106.00

-10.50

-77.25

18.25

-10.84

Time (Y)

In this scenario, a protection buyer who expects the reference credit to survive but wishes to hedge his downside “just in case”, would prefer to pay for protection in up-front form rather than a running spread, in order to avoid locking in a high contractual spread for the life of the trade.

3.2. Scenario II: Reference Credit Defaults After One Year Table 3 shows the cashflows for a protection buyer in a basis trade if a credit event occurs one year after the trade date, and the issuer defaults with 35% recovery. The protection has been purchased in up-front format. We have assumed that the credit event happens just after the first coupon date.

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Table 3: Long Bond + Long Up-front Protection (Default Scenario) Time (Y)

Bond ($)

Upfront CDS ($) Funding ($)

NET ($)


0

-75.00

-33.00

108.00

0.00

0.00

1

6.00

0.00

-3.24

2.76

2.76

35.00

65.00

-108.00

-8.00

-5.24

DEFAULT

Following the credit event, the protection buyer receives $35 from the sale of the distressed bond and $65 from the up-front CDS, but has to repay the funding principal of $108, resulting in a net negative cashflow of $8. The reinvested carry from the trade is therefore a negative -$5.24. Compare this to a protection buyer who chooses to trade on a running basis. This is illustrated in Table 4 below in the same scenario as above. Table 4: Long Bond + Long Running Protection (Default Scenario) Bond ($)

Running CDS ($)

Funding ($)

NET ($)


0

-75.00

0.00

75.00

0.00

0.00

1

6.00

-10.50

-2.25

-6.75

-6.75

35.00

65.00

-75.00

25.00

18.25

Time (Y)

DEFAULT

The protection payout following the credit event is the same as in the up-front case, but the funding principal to be repaid is $75, which nets a positive cashflow of +$25 to the protection buyer. The net reinvested carry from the trade is therefore $18.25. The protection buyer does better in the running CDS format. As a result, protection buyers who have a view that default is almost certain should prefer to trade on a running CDS format. Protection sellers who view default as imminent should prefer to trade on an up-front basis.

3.3. Scenario III: Bond Rallies Sharply In Three Months Consider now a scenario where the bond price rallies from $75 to $92 in three months. Suppose the quoted running spread for the i ssuer now stands at 495bp, which corresponds to an up-front price of $18.50. To illustrate this, Table 5 shows the unwind value of the trade three months after the trade date, if protection is p urchased on an up-front basis. Table 5. Time (Y)

Unwind Value of Up-front Trade (Bullish Scenario) Bond ($)

Upfront CDS ($)

Funding ($)

0

-75.00

-33.00

108.00

0.25

92.00

18.50

-108.81

MTM ($)

UNWIND

17.00

-14.50

-0.81

1.69

We see that the gain in the value of the bond has been almost exactly offset by the fall in the value of the up-front CDS and the funding. Compare this with the case of running protection. The presence of a risky premium leg makes the unwind value of a running CDS more sensitive to spread movements than an August 2003

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up-front CDS. For this reason, the adverse impact of spread tightening is more pronounced as shown below in Table 6. Table 6.

Unwind Value of a Running Trade (Bullish Scenario)

Time (Y)

Bond ($)

Running spread

0

-75.00

1050 bp

0.25

92.00

495 bp

UNWIND

17.00

-$19.58

PV01

Funding ($) 75.00

3.5277

-75.56

MTM ($)

-0.56

-3.14

Here we see that the loss due to the spread tightening is greater than the increase in the bond price. It shows that a protection buyer would have been better off buying up-front protection on the trade date since this exhibits lower spread sensitivity than a running CDS.

3.4. Variation in Outcomes: Running versus Upfront Table 7 summarizes the relative performance of running and up-front trades for various scenarios including the ones described above. Two clear conclusions can be drawn from the various outcomes. First, we see that buying a bond and buying protection is not a credit-neutral strategy, and the timing of the credit event has a significant influence on the net P&L of a trade. Indeed we see that a trade can switch from being positive value-ondefault to negative value-on-default depending on the timing of a credit event. Additionally, we see that the absolute size of the gain or loss is greater for running than for up-front trades. This means that investors should choose to trade on a running or an up-front basis depending on the strength of their view and their r isk preferences. Table 7.

P&L of Trades: Upfront versus Running (Various Scenarios)

Scenario

Running ($)

Upfront ($)

Issuer survives to maturity

-10.84

6.65

Credit event in 1 year

18.25

-5.24

Credit event in 4 years

-7.42

3.55

Spreads tighten

-3.14

1.69

Spreads widen

1.84

-1.81

In each case shown the investor has hedged the bond’s face value and so has full principal protection. However, the longer the issuer survives, the better the performance of an upfront trade relative to running for a protection buyer, and vice-versa for a protection seller. If protection is triggered in one year, the running trade has a higher P&L than the up-front trade. As the timing of the credit event recedes, the performance of the up-front trade improves relative to running, as the running spread paid eventually exceeds the initial cost of up-front protection. Beyond a point, the P&L of the up-front trade exceeds that of the running trade. We now turn to the formal valuation and marking to market of up-front protection trades and discuss their sensitivity to various market factors.

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4. VALUATION OF UP-FRONT PROTECTION One starting point for valuing up-front protection is to attempt to use the cash market as a reference. For example, suppose that a 5-year bond with a 5% coupon trades at a price of $85 when risk-free rates are 3%. How much should an investor pay for up-front protection? To investigate the economics of the trade, suppose the investor was quoted $15 for upfront protection. The initial cost of the bond plus protection would be $100. T here are two possible outcomes: Credit Event: The investor receives all the coupons up to the time of the credit event and then receives par (in return for delivering the defaulted asset to the pr otection seller). No Credit Event: The investor receives all of the remaining coupons plus par.

In both cases, the investor has received 200bp over the risk-free rate for assuming no credit risk (we ignore the counterparty risk of the protection seller). The only difference is that there is uncertainty regarding the timing of the principal payment since this is at the maturity date of the contract or the time of the credit event. This trade presents an arbitrage as it implies that the investor should pay more than $15 for up-front protection. To determine how much more, we need a valuation model which can reconcile both the pricing of bonds and up-front CDS.

4.1. Valuation Model The value of the up-front CDS is the expected present value of the contingent payment of (100%–R) made on the face value of the protection following a credit event. This is simply the value of the protection leg in the standard CDS, where we define R as the expected price of the Cheapest To Deliver (CTD) obligation following a credit event. Let the current date be time t and consider the problem of pricing an up-front default swap maturing at time T. The value of up-front protection is then given by the discounted expectation of (100%-R) at the time of the credit event. T his can be written as: T   U (t , T ) = E (1 − R) ∫ Z (t , s )Q (t , s )λ ( s) ds  t   Q

(1)

where 





λ (s)

is the hazard rate, the instantaneous probability of default in the period [s,s+ds] conditional on surviving to time s. This is usually assumed to be deterministic and independent of interest rates. See O’Kane and Turnbull (2003) for a discussion. Q(t,s) is the arbitrage-free survival probability of the reference entity from valuation time t to time s. Z(t,s) is the Libor discount factor from valuation date t to time s.

The next step is to calibrate this model in order to value the up-front protection. There are essentially three ways to do this, which we describe next.

4.2. Calibration of Survival Probabilities Calibration within a risk-neutral framework involves determining the term structure of survival probabilities which refits the market prices of traded assets. There are a number of ways to do this.

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Survival Probabilities from CDS Spreads This most straightforward approach, which should enforce consistent pricing within the CDS market, is to bootstrap a term structure of survival probabilities using the term structure of spreads in the CDS market as described in O’Kane and Turnbull (2003). This assumes the existence of quoted CDS spreads which may not be the case when the credit is trading distressed. Survival Probabilities from Bond Prices An alternative is to use bond prices. However, we need a valuation formula for bonds which allows us to do so, and this is shown in t he appendix. In this case, it is also possible that a bootstrapping approach may not be appropriate, especially when bonds exist with similar maturities but different coupons and prices. Calibration in these circumstances may be best achieved using some best-fit approach. Care should be taken to remove default swap basis effects. Survival Probabilities from Up-front Prices The observed price of up-front protection can also be used to extract survival probabilities. In this case, we use Equation (1) to invert prices and extract survival probabilities. This can allow us to price up-front CDS to other maturities.

4.3. Example Assuming a recovery rate of 40%, a flat 3% risk-free rate, a flat 3 hazard rate, a 5-year bond trading with a 5% annual coupon and a price of $85 implies a term structure of survival probabilities. Substituting these results into the up-front pricing formula gives us a value for the up-front protection of $22. Observe that this is greater than par minus the full price of the bond, i.e. $15 (= $100–15). This is exactly what we expected from our observation in the example at the start of this section, where we noted that an up-front value of $15 presents an arbitrage. The value of the up-front implicitly should be greater to take into account the coupon payments on the bond which would be paid. As a result, the investor who buys the bond and buys up-front must pay $85+$22=$107. The $7 represents the expected present value of the coupon payments on the bond. Just to be clear, the valuation equation (1) of the up-front does not explicitly know about the coupons on the bond. However, it implicitly knows about the bond since the survival probabilities have been calibrated to the bond price.

5. SENSITIVITY ANALYSIS OF UP-FRONT PROTECTION The inputs into the valuation of up-front protection are the interest rate term structure; the CDS spread curve and a recovery rate assumption. The question to be answered next is: what is the sensitivity of the up-front protection value to t hese inputs?

5.1. Sensitivity to Interest Rates The value of up-front protection is the value of (100%-R) paid following a credit event. The value of this decreases with increasing interest rates as shown in Figure 2.

3

This is the simplest assumption we can make, but it may not be realistic as a distressed credit typically has a downward sloping hazard rate term structure.

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Figure 2.


Dependence of Up-front Price on Interest Rates

21.00

20.00

e 19.00 c i r P t 18.00 n o r f p U 17.00

16.00

15.00

% 0 0 . 0

% 0 0 . 1

% 0 0 . 2

% 0 0 . 3

% 0 0 . 4

% 0 0 . 5

% 0 0 . 6

% 0 0 . 7

% 0 0 . 8

% 0 0 . 9

% 0 0 . 0 1

Interest Rate

5.2. Sensitivity to CDS Spreads Figure 3 shows the dependence of the up-front price on the CDS spread level. As expected, the up-front price increases with increasing CDS spread, since the protection seller has to be compensated for the greater likelihood of default. As spreads increase and default becomes inevitable, the cost of up-front protection tends to (100%-R). This is clear in Figure 4, where we have shown recovery rates of 30% and 50%. Figure 3.

Up-front Price vs. CDS Spread for 5-year maturity

80.00

R = 30% 70.00

60.00

e c 50.00 i r P t 40.00 n o r f p 30.00 U

R = 50%

20.00 10.00 0.00

0

0 0 0 1

0 0 0 2

0 0 0 3

0 0 0 4

0 0 0 5

0 0 0 6

0 0 0 7

0 0 0 8

0 0 0 9

0 0 0 0 1

Spread

5.3. Sensitivity to Recovery Rates Figure 4 shows the up-front price as a function of recovery rate, where we have assumed a flat term structure of CDS spreads at 500bp and one at 1500bp. As the recovery rate increases, two things happen. First, the default probability increases since we have fixed the CDS spread (at low spreads the probability of default is roughly equal to S/(1-R)).

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However, the protection amount paid on default (1-R) decreases. These effects tend to cancel out for low spreads, though for high spreads (>1000bp) the effect can be material. Figure 4.

Up-front Price vs. Recovery Rate

50.00 45.00 40.00

S = 1500 bp

35.00

e c 30.00 i r P t 25.00 n o r f p 20.00 U 15.00

S = 500 bp

10.00 5.00 0.00

% % % % % % % % % % % % % 5 0 5 0 5 0 5 0 5 0 5 0 5 1 2 2 3 3 4 4 5 5 6 6 7 7

Recovery Rate

5.4. Sensitivity to Maturity Figure 5 shows the variation in up-front price with protection maturity. The value of protection clearly increases monotonically with maturity since the protection buyer is protected for a longer period. Assuming a flat spread curve, the value of protection is asymptotically equal to U

∞

=

(1 − R )

λ λ + r

,

where the hazard rate λ is approximately linked to the spread and recovery rate assumption via the Credit Triangle relationship S= λ (1-R). If the spread curve is flat, the credit triangle is a surprisingly good approximation. For example, the up-front price of protection to a maturity of 20 years, assuming a recovery rate of 40% and flat spreads of 1250bp is $52.18 per $100 face. The credit triangle approximation estimates this at $52.44. The error becomes smaller as maturity increases.

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Figure 5.


Up-front Price vs. Maturity

60.00

S = 1500 bp 50.00

40.00

e c i r P t 30.00 n o r f p U

S = 50 0 bp

20.00

10.00

0.00

0

2

4

6

0 8 1

2 1

4 1

6 1

8 1

Years to Maturity

What we have shown so far is the sensitivity of the up-front price to a number of important parameters. For those trading up-front CDS, we need to determine the sensitivity of the mark-to-market to market factors. This is addressed in the next section.

6.

VALUING AN UP-FRONT POSITION

Valuing an up-front protection trade is different to marking to market a running CDS. The difference is due to the fact that the upfront trade is funded while the running is unfunded. For a running CDS trade, ignoring the coupons paid or received, the P&L of the trade is the MTM of the contract described elsewhere (O’Kane and Turnbull 2003). However, the value of an upfront trade is the value of what is paid or received if the contract is unwound minus the value paid for the upfront. This may incorporate funding costs for the initial payment of upfront protection.

6.1. Computing the Up-Front Position Value Consider an investor who has sold protection on an up-front basis at time 0 for T years at a price U(0,T). Let t be the current valuation date. If U(t,T) is the remaining value of protection between t and T, then the value of the position is given by: UF M Short ( t, T )

=

U( t , T ) U (0, T) −

= −

UF M Long

This is because U(t,T) is the cost of entering into an offsetting position on the reference credit at time t. However this ignores the cost of funding or investing the upfront payment.

6.2. Funding Issues Unlike a running CDS, which costs nothing to set up, an upfront CDS involves an initial payment to the protection seller. The cost of funding (in case of a long protection position) or the value of reinvestment (in case of short protection) must therefore be taken into account, when evaluating the P&L of a trade.

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Consider the position of an investor selling protection on an up-front basis. At time 0, the investor receives U(0,T). Assuming that this is reinvested at Libor flat, the investor’s wealth at time t is given by U(0, T) B(0, t ) ⋅

where B(0,t) is the value of a dollar continuously reinvested at Libor between 0 and t. If, at time t, the investor enters into an offsetting contract, buying up-front protection for the remaining time from t to T at the price U(t,T), then the net gain or loss fro m this trade is PLUF S ( t , T)

=

U(0, T)B(0, t ) U( t , T) −

For a long protection position, assuming that the funding of the initial payment of U(0,T) is also done at Libor flat, we have PLUF L ( t , T)

=

U( t , T) U (0, T) B(0, t ) −

⋅

= −

M SUF ( t , T )

Clearly, different borrowing and lending rates can break this symmetry.

6.3. Comparison with Running CDS It is important to compare the value of an up-front CDS contract to the value of the equivalent running CDS contract. The present value of a running long protection position initially traded at time 0 at a contractual spread of S(0,T) with maturity T and which has been offset at valuation time t with a position traded at a spread of S(t,T) is given by the value of the protection leg minus the expected present value of the premium leg of the CDS. M RUN ( t , T) L

=

=

U ( t, T) S(0, T ) RPV 01( t , T ) −

⋅

[S( t, T) S(0, T )] RPV 01( t , T ) −

⋅

where RPV01(t,T), is the present value at time t of a 1bp premium stream which terminates at the earlier of maturity time T or default. See the references for a discussion of the MTM of running default swaps. The difference is clear. In both cases the protection buyer is long the protection which is worth U(t,T). This will change in value as spreads, interest rates and recovery rate assumptions change. The difference is in the premium leg. In an up-front contract, all the market information from the trade date to maturity is incorporated into a single initial payment U(0,T). This has no sensitivity to any subsequent market inputs and no exposure to future interest rates, credit spreads or recovery rates. In contrast, the spread leg of the standard CDS is sensitive to all of these market variables since it is the discounted expectation of risky spread payments. In other words, in an upfront trade the investor has already paid (or received payment) for the remaining protection, worth U(t,T), whereas in a running trade part of the payment, which is contractually fixed at time 0, this payment is made between t and min[T, τ] where τ is the time of the credit event. For an investor using up-front CDS to buy or sell credit risk, it is essential to understand how sensitive the valuation of the position is to changes in market variables. We now examine the sensitivity of running and upfront trades to spreads and interest rates.

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6.4. Sensitivity to Spreads Both the MTM of an up-front and running long protection CDS increase as spreads widen, since the implied default probability increases and so the value of the protection increases. However, a running CDS is more sensitive to spread changes than an up-front CDS, since both the premium and the protection legs are sensitive to spread changes. The MTM of a running CDS is dependent on two opposing factors. Recall, the MTM of a long protection position is given by M RUN ( t , T) L

=

[S( t , T) S(0, T)] RPV 01( t , T ) −

⋅

As low market spreads, the value of the MTM is negative. As the value of S(t,T) increases, the risky PV01 decreases. As the spread increases beyond S(t,T)=S(0,T), the MTM becomes positive. This is shown in Figure 8. Figure 8.

Relative Spread Sensitivity of the MTM of Up-front and Running CDS.

80.00

Running 60.00 ) n o i t c e t o r P g n o L (

40.00 20.00

Upfront

0.00

M-20.00 T M

-40.00 -60.00 0

0 0 0 1

0 0 0 2

0 0 0 3

0 0 0 4

0 0 0 5

0 0 0 6

0 0 0 7

0 0 0 8

0 0 0 9

0 0 0 0 1

Spread

We see the same sort of behavior for the up-front protection. However, the slope of the running CDS curve is different to that of the up-front CDS because the running CDS has a risky PV01 effect – at low spreads the risky PV01 is high, while for high spreads the risky PV01 is low. This makes the running protection more negative at lower spreads and more positive at high spreads. At very high spreads the upfront CDS value tends asymptotically to (1-R)-U(0,T) while that of the running CDS tends to (1-R) since the value of the premium leg tends to zero.

6.5. Interest Rate Sensitivity Figure 9 compares the interest rate sensitivity of MTM for up-front and running CDS as spreads change. The IR01 is defined as the change in the value for a long protection position for a 1bp parallel change in the Libor curve. There is a fundamental difference between running and up-front which is that in a running CDS both legs of the contract trade have an interest rate sensitivity.

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Figure 9.


Interest Rate Sensitivity of MTM for Running and Up-front Trades 0.02

0.01 ) n o i t 0.01 c e t o r P 0.00 g n o L ( -0.01 1 0 R I -0.01

Running

Upfront

-0.02

0

0 0 0 1

0 0 0 2

0 0 0 3

0 0 0 4

0 0 0 5

0 0 0 6

0 0 0 7

0 0 0 8

0 0 0 9

0 0 0 0 1

Spread

At low spreads the MTM of the running CDS is negative so that an increase in interest rates increases the value of the contract and the IR01 is positive. The interest rate sensitivity of the MTM of the up-front CDS is only to the contingent incoming payment of (100%-R) and so is negative. However, at very high spreads the sensitivity to interest rates of both contract types tends to zero as they both tend to contracts paying a certain 1R immediately.

7. CONCLUSIONS Up-front CDS trades are common for short-dated and distressed bonds. For protection sellers, they are attractive as a means to lock in the PV of protection as a sure payment rather than a risky cashflow stream. For protection buyers, they offer the opportunity for better carry trades and a way to avoid being locked into paying high spreads. If the issuer survives, an up-front trade will outperform a running trade. The relative performance of running and up-front trades depends on whether a credit event occurs and when. Which is chosen should reflect the investor’s view. We summarise the main conclusions below. •

•

•

•

Buying a bond and buying protection is not a credit-neutral trade. The net P&L depends on the timing of the credit event, and an investor’s P&L can be significantly different depending on whether protection is bought on an up- front or running basis. A protection buyer who expects the reference credit to survive, but wishes to hedge downside risk “just in case”, should prefer to pay for protection in up-front form rather than as a running spread. This is in order to avoid locking in a high contractual spread for the life of the trade. Protection buyers who have a view that default is almost certain should prefer to trade on a running CDS format since the spread will only be paid until the credit event. For the same reason, protection sellers who view default as imminent should prefer to trade on an up-front basis. A protection buyer who is taking a strong view on a spread movement should prefer a running CDS as this exhibits a higher spread sensitivity than a n upfront CDS. Equally the downside can be greater.

Up-front trades can therefore be used to express a view on the timing of a credit event, and also take exposure to a different risk profile than r unning trades. August 2003

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REFERENCES Berd, Mashal and Wang (2003), Estimation of Implied Default and Survival Probabilities from Credit Bond Prices, Lehman Brothers QCRQ, August 2003. Jarrow and Turnbull (1995), Pricing Derivatives on Financial Securities Subject to Credit Risk, Journal of Finance, Vol 50 (1995), 53-85. O’Kane (2001) Credit Derivatives Explained , Lehman Brothers, March 2001. O’Kane and Schloegl (2001), Modelling Credit: Theory and Practice, February 2001. O’Kane and McAdie (2001), Trading the Basis, Risk Magazine, October 2001.

8. APPENDIX Given that a bond and a credit default swap are linked to the same reference entity, cross default provisions mean that they should default together and so have the same term structure of survival probabilities Q. Suppose the full price at time t of a bond issued by the same issuer as the reference credit, maturing in T years and paying an annual coupon rate of C, is given by B(t,T). If coupons are paid semi-annually, the model-based valuation formula for a bond is given by n

B( t, T) =

M

C ∑ Z( t, t j )Q(t, t j ) +Z(t, T)Q( t, T) + R ∑ Z(t, t m )(Q( t, t m 1 ) − Q( t, t m )) 2 j 1 m 1 −

=

=

The first term is the sum of the risky discounted coupons. The second term is the PV of the principal repaid at maturity, weighted by the probability that the issuer survives to maturity. The third term is the PV of the price of the bond after a credit event, R, which is realised if the credit event occurs before maturity. We have assumed the issuer defaults at M discrete times and that coupons default with no recovery. The survival probabilities can now be calibrated to a term structure of bond prices using some best-fit technique. See Berd, Mashal and Wang (2003) for details of a fitting approach.

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Lehman Brothers Fixed Income Research analysts produce proprietary research in conjunction with firm trading desks that trade as principal in the instruments mentioned herein, and hence their research is not independent of the proprietary interests of the firm. The firm’s interests may conflict with the interests of an investor in those instruments. Lehman Brothers Fixed Income Research analysts receive compensation based in part on the firm’s trading and capital markets revenues. Lehman Brothers and any affiliate may have a position in the instruments or the company discussed in this report. The views expressed in this report accurately reflect the personal views of Dominic O’Kane and Saurav Sen, the primary analyst(s) responsible for this report, about the subject securities or issuers referred to herein, and no part of such analyst(s)’ compensation was, is or will be directly or indirectly related to the specific recommendations or views expressed herein. The research analysts responsible for preparing this report receive compensation based upon various factors, including, among other things, the quality of their work, firm revenues, including trading and capital markets revenues, competitive factors and client feedback. To the extent that any of the views expressed in this research report are based on the firm’s quantitative research model, Lehman Brothers hereby certify (1) that the views expressed in this research report accurately reflect the firm’s quantitative research model and (2) that no part of the firm’s compensation was, is or will be directly or indirectly related to the specific recommendations or views expressed in this report.

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Up-front Credit Default Swaps

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