Master Chapter Four

Credit Derivatives - Credit Default Swaps

As we kickstart this subject let’s begin with the most liquid product in this space, the Credit Default Swap.

What is a CDS contract?

A credit default swap as the name suggests is a derivative contract that facilitates the swapping/exchange of credit risk in an underlying reference entity between two counterparties.

- The arrangement involves the seller of the credit risk (CDS/protection buyer) to make a nominal fee for an exchange of a contingent payment triggered in the event of default on the reference entity obligations.

- One can think of the CDS payment made by the buyer as an insurance premium paid to hedge against the losses on default by the reference entity.

- In the event of a default that – could occur due to failure to pay the underlying or other reference obligations, repudiation, moratorium or conditions defined in the CDS contract – the settlement could happen either physically or in cash. In case of the less frequent physical settlement the CDS seller receives the ‘cheapest to deliver’ of the reference obligation from the buyer and in turn pays par value for it. The more popular cash settlement mechanism replicates the same economics wherein the CDS seller pays par minus recovery value to the buyer in the event of default.

- Underlying documentation – ISDA master documentation as is standard for other financial derivatives is also used for CDS contracts.

Benefits of trading Credit Default Swaps

- The significance of this derivative to the credit world is its ability to isolate and trade in and out the credit risk of an underlying investment from other market risks.

- Furthermore the sensitivity around selling/transferring issuer credit risk are much better managed as the CDS contract has no issuer involvement (as opposed to transferring a loan which gets recorded in the issuers’ books), alongside helping with better price discovery.

- Those with higher funding costs who would normally not buy a lower yielding bond (effectively selling credit protection without any compensation/negative carry) could use CDS to take exposure synthetically to higher grade credits.

In effect CDS would behave like a synthetic long bond position funded via fixed rate repo (to maturity) removing the interest rate risk inherent in a cash bond trade to isolate the credit risk.

- Conversely banks with lower funding costs can fund their own purchase of cash bonds and benefit from a spread higher than that implied by the CDS. This would be a function of the bank’s funding cost and a potential basis between the CDS and bond spread (negative basis) as covered later.

- CDS also helps hedge credit risks on bond positions holding back the compulsion of having to physically sell the bond in times of stress. Adverse taxation and accounting treatments (that get triggered at the time of sale) can be avoided in the same vein.

Valuation/Pricing of a CDS

Intuition behind pricing the credit protection (CDS) on a certain reference entity/asset is to think about the compensation on it at the time of default (Par minus Recovery Value) and the Probability of that Default. These are the two key inputs for any CDS pricing model, also used to arrive at the present value of an investment that has both risk free and (credit) risky cash flows. For bonds in general:

- Current Price of an asset = Present Value of [Cash flows x (1- Probability of Default) + Cash flows x Recovery value x Probability of default)]. Given that the risky cash flows have already been assigned a probability of default and a recovery rate, much like risk free cash flows they should also be discounted at the risk free rate.

- Hence from the above, given current market prices one can back out the probability of default for different recovery value scenarios.

- The present value of a 1 year zero coupon bond can therefore be computed as:

P V = {100 \times (1 - P_{d}) + 100 \times P_{d} \times R V} \times e^{- r t}

where P_d is the Probability of Default, RV is the recovery value, e^-rt is the risk free discount factor. When P_d = 0, it becomes a risk-free bond.

Drawing a parallel with the options world – probability of default for a CDS is akin to an asset’s implied volatility, while recovery value can be thought of as the strike. This is because a more volatile credit would intuitively have a higher probability of default and triggering the CDS in the event of default would be like exercising the option to receive par minus recovery value.

- Market credit spread on a bond is effectively the additional compensation a buyer demands over the risk-free rate assuming a certain default probability and a finite recovery value. Simply put – if 100 dollars are lent as 1 dollar each to 100 individuals with a 5% probability of default i.e. 5 individuals do not return the money at all (recovery value = zero), one would expect the remaining 95 to pay up ~$ 1.0526 each (5.26% additional over risk free rate) to make whole the $100 principal lent. In other words, for a 5% probability of default with no recovery value the breakeven spread to be charged on the risk free rate would be higher than the probability of default. Any non-zero Recovery Value would bring down this spread below P_d but it would still be above zero as long as the recovery (in the event of default) is less than par. In the practical world of finance this is the thinking behind credit risk related compensation sought by lenders.

- The intuition above can be explained by simply enhancing the discount rate of a risky bond by its credit spread i.e.:
  - - PV of a 1y Zero coupon Risky Bond with

$ 100 p r i n c i p a l = \frac{100}{(1 + R_{f}) \times (1 + s)}

note that the discount factor (function of the risk-free rate) here is assumed to be multiplicatively enhanced by the credit spread’s’.

- - - This should now equate to the Zero coupon PV notation above but using annual compounding to keep it uniform:

\frac{100}{(1 + R_{f}) \times (1 + s)} = \frac{100 \times (1 - P_{d}) + 100 \times P_{d} \times R V}{1 + R_{f}}

Solving this we get to

\frac{1}{(1 + s)} = (1 - P_{d}) + P_{d} \times R V

- - - For a bond with zero Recovery Value then

P_{d} = \frac{s}{1 + s} o r s = \frac{P_{d}}{1 - P_{d}}

- - - While for a bond with RV not equal to zero we can approximate Credit Spread or Credit default swap spread:

s = P_{d} \times (1 - R V)

- - - This notation even though an approximation (as we have not accounted for compounding and convexity here) cements the concept of linking the probability of default with the credit spread. Just like higher interest rates imply lower discount factors, higher credit spreads would have the same effect as they increase the discount rate/lower discount factors.

- - - Market pricing of the additional compensation (over the risk-free rate) to be charged on risky bonds/loans/reference obligation should adjust for any ‘guaranteed’ principal or interest components as it would be backed out of a joint probability of default of the guarantor and issuer/obligor.

Brady Bonds are a relatable reference that come to mind to think about how ‘below market rate’ coupons could pose as a restructuring solution for the defaulted debt during the Latin American crisis. Given the illiquidity of emerging market dollar debt back in the 80s the intention was to convert the existing (defaulted) debt on the balance sheets of the lenders (with an unavoidable haircut of course) to a more tradable instrument. The tradability/enhanced liquidity of the debt was made possible by guaranteeing and or collateralising the principal/interest component of the debt. The quality of collateral was usually as good as a Zero Coupon 30y US treasury bond that was purchased by the debt country and the country’s own foreign reserves. Interest payments too were sometimes collateralised by a much better-rated (than the issuing country) debt, which meaningfully brought down the joint default probability of the restructured debt. This in turn afforded a below market coupon to the Latin American bonds.

- Hazard Rate/Default Intensity – is the common academic reference to the Rate of probability of default (Pd) over a certain time period that’s conditional on the underlying credit/issuer having survived till the beginning of that time period. In theory, to calculate the present value of CDS payments and expected pay-offs pre and post default we use the unconditional counterpart of P_d (let’s call it UP_d,t to denote unconditional probability of default in time t) denoted as:

U P_{d, t} = P_{d, t} \times P_{s, t - 1}

Where

P_{s, t - 1}

is the probability of survival at the end of time

t - 1

(same as beginning of time t)

- Let’s illustrate the computation of the credit default swap spread. Few things to remember: a) Hazard rates apply ‘during’ a time period and determine the probability of survival at the end of that time period b) unconditional probability of default is computed as in the notation above c) defaults are assumed to occur at the end of every year, and for year 1 the unconditional probability is the same as the hazard rate.

Table 1 – Present Value of Expected payments for CDS = s per annum

Time period	Hazard rate	Survival probability	Discount Factor	PV of Expected payment
00		100%
1	2.0%	98.0%	95.2%	1.0s
2	2.8%	95.3%	90.7%	0.9s
3	3.5%	92.0%	86.4%	0.8s
4	2.5%	89.7%	82.3%	0.8s
5	1.0%	88.8%	78.4%	0.7s
Total				4.1s

Source: Pandemonium.

Table 2 – Present Value of Expected pay-off in the event of default

Time period	Probability of default	Expected pay-off	Discount Factor	PV of expected pay-off
1	2.0%	0.4	95.2%	0.0
2	2.7%	0.4	90.7%	0.0
3	3.3%	0.4	86.4%	0.0
4	2.3%	0.4	82.3%	0.0
5	0.9%	0.4	78.4%	0.0
Total				0.1

Source: Pandemonium.

Contractually then, payments made in case of no default should equal (or be an insurance for) the pay-off that’s received at the time of default. Hence:
4.12s = 0.0589, i.e. s = 142.96 basis points

Real world and Risk neutral probability of default

Real world and Risk neutral probability of default – is an important differentiation to consider when thinking of what default probability estimate to use. P_d that’s backed out from the market traded credit spreads (that have a recovery value assumption) is the risk neutral probability of default. But these can be different from the historical pattern of default rates or Real world P_d for the same obligor as markets and macro environment changes. The term ‘Risk-Neutrality’ can also be understood to suggest the default metric that would imply a credit spread to neutralise the current credit risk exposure. Hence traders use the risk-neutral P_d to value their credit portfolios/pnl for a fair reflection of default adjusted cash flows. But estimating let’s say the Credit VAR (Value at Risk – to be covered in future write-ups on Risk) which entails figuring out the potential future losses based on a historical pattern of default, Real world P_d comes in handy.

You can think of the risk neutral P_d metric as what’s implied by the live market credit spread curve, quite analogous to the implied volatility backed out of the market traded price of an option. Or extending the analogy further, real world P_d can be thought of as realised vol while risk neutral P_d is akin to the implied vol.

As for the differences between Real World P_d and Risk-Neutral P_d, the following reasons can be attributed: a) illiquidity of corporate bonds which warrants a higher credit spread b) recency bias i.e. traders might be more inclined to price in a credit blow-up depending on how recently it has happened vs history c) default correlation risks or contagion risks where credit risk from a similar industry/sector are prone to defaults as they react similarly to changing market variables.

CDS is a swap of two floating rate bonds

CDS cash flows work like fixed coupon interest rate swaps i.e. the payments made by the CDS buyer to the seller in practice are valued at a fixed coupon of 1% for investment-grade debt and 5% for high-yield debt. You can further break down the cash flows (assuming Investment Grade debt) as an exchange of two floating rate bonds. One, a Credit linked Floating Rate Note (CLN), (bearing a coupon of risk free floating rate + credit spread of 1%) sold by the protection buyer at a premium or discount to par depending on the reference obligation’s yield and the other sold by the protection seller (to the buyer) bearing just a risk free floating rate (zero credit spread) coupon. In other words, an exchange of a risky floating rate bond with a risk-free floating rate bond.

- - - At the start of the swap, the protection seller would pay the buyer the present value of the purchased CLN obtained by discounting it by its YTM (reference obligation’s yield). The Protection seller in turn would sell a risk-free floating rate bond which would be purchased by the buyer at par (cash flows discounted at the risk-free floating rate always). Net, the counterparties exchange the premium/discount of the CLN upfront (at the start of the swap) as par values on the respective bonds cancel out (discount paid to the protection seller if CDS spread >1% or premium received from the seller if CDS spread < 1%).

- - - During the tenor of the swap the protection buyer pays a coupon of floating rate + 1% while the Protection seller pays Floating rate. Effectively the protection buyer is paying only the net cash flow of 1% annualised.

- - - In case of a default of the underlying reference entity/obligation (as defined by the CDS contract) the protection buyer would only need to pay back the recovery value of the CLN in return for par value that the seller would pay back on the risk-free floating bond. Net, amounting to the protection buyer receiving par minus recovery value in case of a default.

- - - For instance, the price of 5 year CDS trading at 50bps would be equivalent to a 5y CLN bearing a coupon of SOFR + 1% sold by the protection buyer (with the CLN’s credit linked to the underlying reference entity) at a YTM equal to SOFR + 50bps to the seller. For the sake of simplicity (ignoring discounting effects) the protection seller would pay 2.5% upfront against receiving 1% per annum from the protection buyer.

- - - Valuing the CDS prices and cash flows in the above manner tries to mimic exactly how floating rate bonds of the reference issuer/ obligor would trade based on the current market.

Extending the risky and risk-less floating rate bonds analogy you can now tie up the economics of an asset swap trade and that of a CDS. Going long a credit by selling its CDS in principle should yield similar to (it isn’t exactly the same as discussed in cash-bond basis below) putting on an asset-swap or a total return swap trade. The TRS or an asset swap buyer is going long a floating rate risky bond and being short a floating rate risk-free bond (akin to paying funding cost) net yielding the credit-linked return on the underlying bond.

- - - Credit default swap spread oftentimes is used interchangeably with the credit spread on a corporate bond. In theory bonds can be delivered on CDS contracts in case of a default which should imply that credit spreads and CDS spreads should not diverge as that could create arbitrage opportunities. For instance, if Credit spreads widen a lot but CDS doesn’t, arbitrageurs could buy the bond and buy CDS protection to earn the basis till maturity. In case of default, they can sell the bond at recovery value and get compensated for their loss by the CDS, earning the risk-free locked-in basis of the trade.

- - - CDS vs Cash Bond basis can persist – Though simply described above, this arbitrage (CDS spreads tighter than bond credit spreads is called negative basis) is not easy to lock in for reasons like a) it’s expensive to source matching tenors on the bond and the CDS b) balance sheet constraints i.e. shortage of capital/higher funding costs eats into the arbitrage opportunity c) different investor playfields or mandates/market accessibility. Similarly, if CDS trades wider than bond credit spread (called positive basis), the inability to short the bond/borrow it on repo in a timely manner would make it challenging to lock in this basis and even if the repo is secured the risk of the bond going ‘special’ needs to be monitored.

Also, terms around reference obligations in a CDS contract are crucial in determining its level as a wider spectrum of cheapest to deliver instruments many of which could be illiquid would trade at a deeper concession to the liquid ones nudging the CDS spread higher.

- - - Credit spread risk of a portfolio is typically denoted as CS01 i.e. the change in the present value of the portfolio due to a basis point change in the credit spread. One could think of this sensitivity as the change in the price of a floating rate bond due to a basis point change in credit spread. If you recall, we assumed above that the credit spread of a risky bond has a multiplicative effect over the risk-free discount rate for computing PV of its cash flows. For instance if R_f is the risk free interest rate and s is the credit spread for the corporate bond, the discount rate to arrive at the bond’s present value is (1+R_f) x (1+s). In practice, since market professionals denote the difference between the fixed rate risky yield and fixed rate risk free yield as the credit spread, they effectively assume an additive impact of s on R_f i.e. the discount rate denoted as (1+R_f+s). Mathematically then the dollar value of change in yield (DV01) would be less than the dollar value of change in credit spread when the assumption is for a multiplicative spread over risk free rate to discount cashflows, while it would be the same if the spread is additive.

Zero Recovery CDS vs Standard CDS

Standard market convention is for the protection seller to pay par minus recovery value to the buyer in the event of default. Because the exact recovery rates are unknown – market-makers have their own assumptions – till the event of default a standard contract is also known as Floating recovery CDS. As a variation to the floating recovery contract, we also observe trades wherein the protection seller wishes to have a fixed recovery value built into the price (typically RV = 0) and the dealer/buyer then builds that assumption to arrive at the appropriate CDS level. For instance if a standard recovery for a sovereign bond S_SR is implied at 40%, then a zero recovery CDS on the same sovereign risk (i.e. same probability of default) would trade at

S_{Z R} = \frac{S_{S R}}{1 - 0.4} = 1.67 \times S_{S R}

S_{Z R}

where

S_{Z R}

is Zero recovery CDS price. The dealer who buys a zero recovery CDS and hedges it with a floating recovery CDS for the same notional has a net long recovery value position. CLNs sold into private banking clients are normally priced as zero recovery CDS for yield enhancement on the note.

Recovery swaps

The recovery Rate on default also has a market in the form of recovery swaps that typically start trading close to the event of default. These are obviously less liquid but at least offer a mechanism to calibrate the credit portfolio pricing/valuation models.

FX risk in a CDS on cross correlation

- - - Another interesting nuance of the CDS market comes from the different currency notional of the reference entity obligation and that of the CDS contract that offers protection on it. Prima facie if we were to think of buying protection on a local currency INR obligation (INR notional exposure) and use a USD denominated CDS to buy protection on it we would be exposed to currency fluctuations wherein the amount of protection may not equal the underlying exposure at default. Please note however there is no change to the assumptions around probability of default and recovery value even while switching into a different currency for the CDS.

- - - Going a step further to assess the degree of correlation between the currency of the reference entity/issuer and its obligation please consider – Premium for a USD denominated CDS to hedge against default risk of an INR denominated SBI bond/loan should account for the movement in USDINR FX at the time of default. In other words, a USD 100 mio equivalent INR obligation for SBI would be worth less in dollar terms at the time of default assuming a sharp INR FX depreciation along with SBI’s default. A 20% expected INR depreciation for instance at the time of default would only need USD 80 mio worth of protection which would be 20% cheaper to buy versus the USD 100 mio notional on the reference obligation at the beginning. Hence, the CDS price on INR notional of SBI’s obligations would be 20% cheaper than the corresponding USD CDS.

- - - FX correlation risks on the same underlying credit are also quite common. If a dealer is long a EUR denominated CDS of a European Quasi-Sovereign but short its USD denominated CDS, the risk exposure, while mitigated on the underlying credit, still appears as correlation of the underlying with the respective currencies. If the correlation of the Quasi-Sovereign’s credit quality is high/positive with EUR FX movements, it would be natural for the market to price the EUR denominated CDS cheaper than the one in USD as in the event of default the recovery value in EUR would be much lesser than that in USD. CDSs denominated in a highly positively correlated currency denote wrong way risks as recovery value is vulnerable to FX gap risk in the event of default.

- - - Credit risk in a CDS – a protection buyer in a CDS contract has taken the view to hedge the credit risk of the reference entity/obligation but in doing so ends up taking credit risk on the seller to the extent of settling the par minus recovery value contingent on the default. On the other hand the seller also runs credit risk on the buyer to the extent of the periodic coupons, but by way of magnitude this is a much smaller risk than what the buyer runs on the seller for the full contingent settlement amount. Going a step further, credit risk for the buyer can get accentuated in case of a correlation between the credit risk of the seller and the reference entity/obligation. Buying USD denominated credit protection on the Republic of Korea from a Korean security company would now bring in correlation or wrong-way risk owing to the credit correlation between the seller and reference entity (credit gap risk). This should be priced by the buyer of the CDS akin to how a CVA charge is incorporated in bilateral non-CSA swaps.

Critique on CDS - Cheapest to Deliver and other reference obligations vs the Recovery Rate

CDS pricing theory and the recovery mechanisms seem standard when we read about it but in practice the reference obligations that come under the purview of a CDS contract convolute the valuation and hedge effectiveness of the instrument. I’ll list down some commonly discussed idiosyncrasies of a CDS contract that sometimes prove to be hurdles for the fixed income/credit world in using them as risk mitigating innovations.

- - - The concept of ‘cheapest to deliver’ (CTD) reference obligation comes into play when a CDS contract is triggered in the event of default. ‘Cheapest’ by definition is the lowest recovery value obligation at the time of default that the protection buyer can and would deliver (in case of a physical settlement) or that would be used to value the CDS (in case of a cash settlement).

- - - A CDS contract with a wide variety of reference obligations might result in over-hedging the concerned credit exposure for which the CDS protection was bought in the first place. The inherent CTD option value exercised in the event of default could lead to a net cash flow (Par minus RV of the CTD) that’s larger than the assumed net value if the protected credit exposure was delivered.

- - - A low coupon bond in a high interest rate environment is likely to become CTD for reference CDS contracts except in the event of default.

Recently talked about stalemate in the US debt ceiling debate pushed US CDS levels higher – 1y CDS shot up to ~178bps levels (on May 1, 2023) vs 72bps couple of months back. The low coupon 2050 Treasury bonds (ISINs US912810SN90 and US912810SP49) after a 500bps fed hike turned out to be the cheapest to deliver instruments. The worst-case scenario here would have been a technical default, but the lower implied recovery value of the CTD nudged the CDS levels significantly higher, even though the market implied a lower probability of default this time vs previous such episodes. One needs to be mindful of the recovery risk on the CTD here. A potential risk-off at the time of default would mean large flight to safety flows to Treasuries which might significantly erode the expected gains on the CTD.

CTD option value is the premium that CDS spreads have over the credit spreads (positive basis) of the individual bonds. Another way to think about it is that CDS spreads show divergence with observed credit spreads of underlying obligations as the CTD option value increases.

The value of the CTD option would differ depending on:

- - - The diversity and breadth of reference obligations of the underlying entity, option value is higher with higher diversity.

- - - Types of liability structures/complexity of it for the entity on which protection is being bought, for instance banks on the one end of the spectrum have complicated liability structures (more on this in the risk section) that have different levels of seniority (even inclusion) in the event of default, while corporates on the other end of the spectrum have much simpler/limited market traded liabilities.

- - - The nature of the reference entity and the treatment of their debt in the event of default. For instance, the event of default (EoD) for a sovereign typically amounts to exchange offers/Debt restructuring and differing treatments for reference obligations and the value of a CTD option is higher here than for a corporate EoD that’s oftentimes met with liquidation (all reference obligations trade at the same level).

- - - Higher the probability of default higher the value of the CTD option as the probability of exercising it also goes up.

Typical CDS transaction’s terms and conditions, including its maturity date (the Scheduled Termination Date) and the Credit Events covered in the contract, are defined in the trade “Confirmation” exchanged between the counterparties. Standard Confirmations reference the ISDA Credit Derivatives Definitions and the various supplements issued from time to time including the 2009 ISDA Credit Derivatives Determinations Committees, Auction Settlement and Restructuring Supplement. These provide the basic documentation for CDS contracts and the standard set of definitions, provisions that govern the majority of CDS transactions. For those interested in further details I’d request you to read these documents to get a better understanding of the timeline and sequence of events post the event of default in determining the settlement (Recovery) value of CDS contracts.

Master Chapter Four

Credit Derivatives - Credit Default Swaps

What is a CDS contract?

Benefits of trading Credit Default Swaps

Valuation/Pricing of a CDS

Real world and Risk neutral probability of default

CDS is a swap of two floating rate bonds

Zero Recovery CDS vs Standard CDS

Recovery swaps

FX risk in a CDS on cross correlation

Critique on CDS - Cheapest to Deliver and other reference obligations vs the Recovery Rate

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Related Resources

1: CDS Valuation and Pricing

2: CDS Hazard Rates and Default Probabilities

3: CDS Swap of Two Floating Bonds

4: CDS Bond Basis

5: Risks on a CDS

6: Critique on CDS

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Varda Pandey