CR 13 – Credit Derivatives

A credit derivative is a contract whose payoff is driven by the credit standing of one or several companies or sovereign borrowers. If a named borrower defaults, or in some contracts if a portfolio of borrowers suffers losses, money changes hands; otherwise the contract simply runs its course. These instruments first appeared as over-the-counter contracts toward the end of the 1990s and grew at an extraordinary pace. Outstanding notional principal stood at about $800 billion in 2000 and had reached $50 trillion by the time the financial crisis broke in 2007. The market then shrank considerably, and by December 2019 the total notional was about $7.5 trillion.

The economic point of these contracts is that credit risk becomes something that can be bought and sold separately from the loan or bond that created it, much as interest rate or currency risk has long been traded on its own. Before credit derivatives existed, a lender that had made a loan could do little except hold it and hope the borrower stayed solvent. Now a bank can keep some exposures, lay off others to a counterparty, and reshape the credit profile of its book without selling a single loan. Historically, banks have bought more protection than anyone else, while insurers have been the dominant sellers.

Single-name and multi-name contracts

Credit derivatives fall into two broad families. A single-name contract depends on one borrower, and the dominant example is the credit default swap (CDS), in which a buyer of protection is compensated by a seller of protection if a named company or country defaults. A multi-name contract depends on a group of borrowers. The leading multi-name structure is the collateralized debt obligation (CDO), where losses on a portfolio of debt are allocated to several classes of investors according to fixed rules. Structures of this kind built from residential mortgages played a central part in the run-up to the 2007 crisis; this lesson concentrates on structures whose underlying credit risk comes from corporations and countries.

Who ends up bearing the credit risk

Banks have grown unwilling to hold loans to maturity, because once regulatory capital is charged against a loan, its return often compares poorly with other uses of that capital. Securitization was one route for passing loans and their credit risk to investors. Credit derivatives, used heavily in the late 1990s and early 2000s, became another. The consequence is a separation of roles: the institution that checked the borrower’s credit at origination is frequently not the institution that suffers if the borrower defaults. The crisis that began in 2007 showed that this separation is not always healthy for the financial system as a whole, since the party bearing the loss may know much less about the borrower than the party that originated the loan.

How the CDS market developed

The International Swaps and Derivatives Association (ISDA) produced a standard documentation framework for over-the-counter CDS trading in 1998 and 1999, and volumes grew rapidly afterwards. A CDS resembles an insurance policy, with one important difference: insurance pays for a loss on an asset that the policyholder actually owns, whereas the buyer of a CDS does not need to own any bond issued by the reference company. So it is common for the volume of CDS written on a company to exceed the company’s outstanding debt, and cash settlement then becomes unavoidable. When Lehman Brothers defaulted in September 2008, roughly $400 billion of CDS referenced the firm, against about $155 billion of Lehman debt. The ISDA auction set the cash payout to protection buyers at 91.375% of principal.

During 2007 and 2008, regulators worried that CDS exposures could transmit systemic risk: the failure of one large financial institution could cause heavy losses for its CDS counterparties and trigger further failures. Those fears were sharpened by AIG, which had sold a large volume of protection on AAA-rated mortgage tranches, suffered enormous losses on it, and was rescued by the U.S. government. Even so, while many credit products stopped trading altogether during the turmoil, CDSs kept trading actively, although the price of protection rose sharply. Their straightforward design gave them a transparency that complex securitized products lacked.

The asymmetric information problem

Most over-the-counter derivatives depend on variables such as interest rates, exchange rates, equity indices or commodity prices, and there is little reason to think one trader knows more about these than another. A CDS spread is different, because it depends on the chance that one particular company defaults over a particular period. A bank that lends to a company, advises it and underwrites its securities is likely to know more about its credit standing than a counterparty with no relationship. Economists call this an asymmetric information problem. Banks respond that protection-buying decisions are taken by risk managers and are not based on confidential knowledge held elsewhere in the institution.

A CDS is protection against default by one named borrower, called the reference entity; the default itself is called a credit event. The buyer of protection acquires the right, when a credit event happens, to sell bonds issued by the reference entity at their face value, and the seller of protection agrees to buy them at that price. The total face value of bonds covered is the notional principal of the swap. In exchange, the protection buyer pays the seller a regular premium until the contract matures or a credit event occurs, whichever comes first. The standard arrangement is quarterly payments in arrears, although monthly, semiannual, annual and in-advance schedules are sometimes seen.

The annual premium, expressed as a percentage of notional principal, is the CDS spread. Large dealer banks make markets in these spreads. A dealer quoting a new 5-year contract at a bid of 250 basis points and an ask of 260 basis points stands ready to buy protection for 2.5% of principal per year and to sell it for 2.6% per year.

Maturities, standard dates and the first payment

Contracts on a large range of companies and sovereigns trade. Five-year protection is by far the most common, although 1-, 2-, 3-, 7- and 10-year maturities are also used. Contracts normally end on one of four standard dates each year (the 20th of March, June, September or December). So the true life of a new contract is close to, but not exactly, its quoted tenor. A dealer asked on November 18, 2021 for 5-year protection would probably set maturity at December 20, 2026, with a first payment on December 20, 2021 that covers only the short period from November 18. If the first standard date is less than a month away, the first payment is usually pushed to the second standard date instead.

What counts as a credit event

The definition of a credit event is a central term of the contract. It usually covers failure to pay an amount when it falls due, a restructuring of the debt, and bankruptcy. North American contracts sometimes exclude restructuring, especially where the reference entity’s debt already carries a high yield.

Buying a corporate bond and buying protection on its issuer at the same time leaves an investor holding something close to a risk-free bond. Take a 5-year corporate bond bought at face value with a yield of 7% per year, combined with a 5-year CDS on the issuer at 200 basis points, or 2%, per year. If the issuer survives, the investor collects 7% and pays away 2%, netting 5% per year. If the issuer defaults, the investor still earns that 5% up to the default date and then hands the bond over under the CDS and receives par, which can be reinvested at the risk-free rate for what remains of the 5 years. Either way the result is approximately a risk-free investment.

So, by arbitrage, the yield spread of an n-year bond over the risk-free rate should be roughly equal to the issuer’s n-year CDS spread. If the bond spread were well above the CDS spread, an investor could buy the bond, buy protection and lock in more than the risk-free rate. If the bond spread were well below the CDS spread, an investor could short the bond and sell protection, which amounts to borrowing below the risk-free rate.

The arbitrage argument suggests a basis close to zero, but observed values have drifted on both sides. The basis tended to be positive in the years before 2007 and negative through the 2007 to 2009 crisis, and since then its sign has varied from one reference entity to another.

Recovery rate and the payoff formula

The recovery rate on a bond is its value just after default, expressed as a percentage of face value. Since the protection buyer receives face value in exchange for a bond worth only the recovery fraction, the payoff is the notional principal multiplied by one minus the recovery rate.

The cheapest-to-deliver option

A CDS normally allows any of several bonds to be delivered after a credit event. They usually share the same seniority, yet they need not trade at the same percentage of face value after default. One reason is that the claim in bankruptcy is generally face value plus accrued interest, so bonds carrying a lot of accrued interest tend to fetch more. Another is that the market may expect some bondholders to come out of a reorganization better than others. The protection buyer will naturally deliver the cheapest eligible bond, which gives the buyer a cheapest-to-deliver bond option. The ISDA auction is designed to find the value of that cheapest-to-deliver bond, and it is this value that sets the payoff to the buyer of protection.

A CDS has two legs. The premium leg is the stream of spread payments the protection buyer makes while the reference entity survives, and the protection leg is the payoff the seller makes if default occurs. The fair, or mid-market, spread is the one that makes the present values of the two legs equal. Both legs depend on the probability of default in each period, so the starting point is a model of default timing.

The simplest model assumes a constant hazard rate, λ, which is the instantaneous default intensity for a firm that has survived so far. With a constant hazard rate the probability of surviving to time t is an exponential, and the unconditional probability of defaulting during a given year is the survival probability at the start of the year minus the survival probability at its end.

Consider a 5-year CDS on a reference entity with a hazard rate of 2% per year throughout. Survival to 2 years has probability e^−0.02×2 = 0.9608 and survival to 3 years has probability e^−0.02×3 = 0.9418, so default during the third year has probability 0.9608 − 0.9418 = 0.0190. To keep the arithmetic transparent the example also assumes that spread payments are made once a year at each year end, that any default happens exactly halfway through a year, that the risk-free rate is 5% with continuous compounding, and that the recovery rate is 40%. Notional principal is $1 and the unknown spread is s per year.

The three present values

The premium leg is the sum over years of the survival probability to each payment date, times s, discounted from that date. The third payment, for example, is made with probability 0.9418, so its expected value is 0.9418s and its present value is 0.9418s × e^−0.05×3 = 0.8106s. The protection leg pays 1 − R = 0.6 at mid-year if default occurs in that year: for the third year the expected payoff is 0.0190 × 0.6 × 1 = 0.0114, discounted from 2.5 years to 0.0114 × e^−0.05×2.5 = 0.0101. Finally, because premiums are paid in arrears, a default at mid-year triggers an accrual payment of half a year’s premium, 0.5s. In the third year its expected value is 0.0190 × 0.5s = 0.0095s, with present value 0.0095s × e^−0.05×2.5 = 0.0084s.

Mid-year discount factors apply at times 0.5, 1.5, 2.5, 3.5 and 4.5 years, when defaults are assumed to occur.

Placing every default at the midpoint between payment dates is a simplification, but it usually works well. It can be relaxed by allowing default at more points within each period, with the probabilities split accordingly. In market practice the same structure is run with quarterly payments and a full term structure of hazard rates, but the logic of matching two expected present values does not change.

Like other derivatives, a CDS is revalued every day, and its value can be positive or negative for either side. The valuation reuses the two legs just computed. Once a contract is in place its spread is fixed, so its value is the present value of the expected payoff less the present value of the fixed premiums it still requires, both measured with current default probabilities.

Risk-neutral, not real-world, default probabilities

The probabilities used to value a CDS must be risk-neutral default probabilities rather than real-world ones. Real-world probabilities are estimated from historical default experience and describe how often default actually happens. Risk-neutral probabilities are the ones implied by market prices of credit-sensitive instruments, and they are larger, because investors demand a premium for bearing credit risk. Using real-world probabilities would understate the protection leg and so understate the value of protection to its buyer; equivalently, it would produce a spread below what the market charges.

Risk-neutral default probabilities can be backed out of bond prices. An alternative, and in practice the more common route for CDS work, is to imply them from CDS quotes themselves. This mirrors the options market, where volatilities are implied from actively traded options and then used to value less liquid ones.

Implying a hazard rate from a quoted spread

Suppose the default probabilities in the valuation table are unknown, but a newly issued 5-year CDS on the reference entity is quoted at a mid-market spread of 100 basis points. Running the same calculation in reverse, a numerical search finds the hazard rate at which the premium leg and protection leg balance at s = 0.01. The answer is an implied hazard rate of 1.63% per year (the search gives 1.626%). With quotes at several maturities, the same procedure strips out a term structure of hazard rates, one segment at a time, and the process also works in reverse to produce spreads from a given hazard rate curve.

A binary credit default swap works like a regular CDS except that its payoff on default is a fixed dollar amount rather than face value less recovery. Using the same reference entity, hazard rate and interest rate as before, suppose the binary contract pays $1 on default. The survival probabilities, the premium leg and the accrual leg are unchanged, because they do not involve the recovery rate. Only the protection leg changes: each expected payoff is now simply the probability of default in the year.

Why the recovery rate matters less than it seems

Whether default probabilities are implied from bond prices or from CDS spreads, a recovery rate must be assumed. Yet the value of a regular CDS, or the spread it should carry, is not very sensitive to that assumption, provided the same recovery rate is used both to imply the default probabilities and to value the CDS. The reason is a cancellation. Implied default probabilities are roughly proportional to 1/(1 − R), because a given market spread must compensate for expected loss, and the loss per default is 1 − R. The CDS payoff is proportional to 1 − R. So in the product of the two the recovery rate approximately drops out.

A binary CDS does not enjoy this cancellation. Its implied default probabilities still scale with 1/(1 − R), but its payoff is fixed and does not depend on R at all, so a wrong recovery assumption feeds straight into its value. This difference is useful: if both a regular CDS spread and a binary CDS spread are observed for the same reference entity, the two can be solved jointly for the recovery rate and the default probability.

Credit indices track the level of CDS spreads across a fixed set of names and, more usefully, let a trader buy or sell protection on the whole set in one transaction. Several competing index producers consolidated in 2004, and two portfolios now dominate. CDX NA IG covers 125 investment grade companies in North America, and iTraxx Europe covers 125 investment grade names in Europe. Both portfolios are rebalanced twice a year, on the 20th of March and of September, when names that have fallen below investment grade are removed and new investment grade names are brought in. Each refresh creates a new series: by September 20, 2020, when Series 34 of iTraxx Europe and Series 35 of CDX NA IG were defined, the European portfolio had been updated 33 times and the North American one 34 times.

A dealer might quote the 5-year CDX NA IG at 65 basis points bid and 66 basis points ask. This quote is called the index spread, and loosely it means that a trader can buy protection on each of the 125 companies for 66 basis points per company per year, or sell protection on each for 65 basis points.

Maturities and how long a 5-year index trade lasts

Five years is the most common index maturity, with 3-, 7- and 10-year contracts also traded. Index contracts usually mature on December 20 or June 20. Because a new series is launched every six months and trades until the next one, a contract described as 5-year actually runs for somewhere between 4¾ and 5¾ years, depending on when within the roll it is entered.

Why the index spread sits slightly below the average spread

Roughly speaking, the index spread is the average of the CDS spreads of the names in the portfolio, but in fact it is slightly lower than that average. Picture a two-name portfolio in which the first company trades at 1,000 basis points and the second at 10 basis points. The simple average is 505 basis points, yet buying protection on both would cost slightly less than 505 per name. The high-spread company is far more likely to default early, after which its premium stops, so its 1,000 basis points will on average be paid for a shorter time than the 10 basis points and deserves less weight. A further wrinkle affects CDX NA IG but not iTraxx Europe: the index definition of default includes restructuring, while the single-name CDS contracts on the constituents may not.

In practice, CDS and index contracts do not pay whatever spread happened to prevail on the trade date. For each underlying and each maturity, the market fixes a standard coupon and a standard recovery rate, and any difference between the quoted spread and the coupon is settled as an upfront price. The price is derived from the quoted spread in four steps.

First, payments are assumed to be made four times a year in arrears. Second, a hazard rate is implied from the quoted spread using calculations of the kind shown earlier for CDS valuation, with an iterative search for the hazard rate that reproduces the quote. Third, a “duration” D is computed for the spread payments: a multiplier that converts a spread into the present value of the stream of spread payments. In the 5-year valuation earlier, that multiplier was 4.1150. This use of the word duration has nothing to do with bond price sensitivity. Fourth, the price per 100 of notional is set as follows.

A protection buyer pays 100 − P per $100 of remaining notional at the start, and the seller receives it. When 100 − P is negative, which happens when the quoted spread is below the coupon, the cash goes the other way. After that, on every payment date the buyer pays the fixed coupon applied to whatever notional is still live. For a single-name CDS that live notional equals the original amount up to a default and falls to zero once default occurs. On an index it is the principal per name multiplied by the number of names that have not yet defaulted. Default payoffs are calculated in the usual way. The benefit is that every contract on a given index and maturity carries the same periodic payments whenever it was traded, so contracts trade like bonds, with price doing the work that the spread used to do.

The direction of the upfront payment makes intuitive sense. The protection buyer will pay a 40 basis point coupon for protection the market values at only 34 basis points, so the seller compensates the buyer at the start for the extra coupon the buyer is committing to pay.

Once the CDS market was liquid, dealers began trading forward contracts and options on CDS spreads. A forward credit default swap commits the parties to buy or sell a specified CDS on a specified reference entity at a future date T, and if the reference entity defaults before T the forward simply lapses. A bank could, for example, agree today that, one year from now, it will sell 5-year protection on a company at 280 basis points; should the company default within that year, the forward would disappear and the bank would owe nothing.

A credit default swap option gives its holder a right, but no obligation, to buy or sell a specified CDS on a specified reference entity at a future time T, and its premium is paid up front. The right to buy 5-year protection in 1 year at 280 basis points is a call. It is exercised if the 5-year spread in 1 year is above 280 basis points, since protection would then be bought below its market cost. The right to sell 5-year protection in 1 year at 280 basis points is a put, exercised if the spread then is below 280 basis points. Like forwards, CDS options are normally written so that they lapse if the reference entity defaults before the option matures, which separates the spread view from the default event itself.

Basket credit default swaps

A basket credit default swap references several entities at once, and the variants differ in which defaults trigger a payment. An add-up basket CDS pays whenever any of the reference entities defaults. A first-to-default CDS pays only on the first default in the basket; a second-to-default CDS pays only on the second; and in general a kth-to-default CDS pays only when the kth default occurs. Each payoff is computed exactly as for a single-name CDS. Once the triggering default has been settled, the swap ends and neither party makes further payments.

The value of basket protection depends heavily on how defaults cluster, a point taken up in the section on default correlation. Intuitively, a first-to-default contract on a basket of unrelated names is close to a bundle of individual protections, because the first default is quite likely, while a high-order contract needs many defaults to happen together and is cheap unless defaults tend to arrive in groups.

A total return swap swaps everything earned on a bond, or on a portfolio of assets, against a floating rate with a spread added. The total return means every economic benefit of holding the asset: coupons, interest, and the gain or loss in its value over the life of the swap. One party, the total return payer, hands over that total return; the other, the total return receiver, pays the floating leg.

Adding the notional principal to both sides at maturity gives a neat way to see the swap. The payer delivers the cash flows of a $100 million investment in the corporate bond; the receiver delivers what a $100 million note at floating plus 25 basis points would pay. If the payer actually owns the bond, the swap transfers the bond’s credit risk to the receiver. If the payer does not own the bond, the swap leaves it effectively short the bond.

Total return swaps as a financing tool

Financing is the most common motive. Suppose the receiver wants to invest $100 million in the reference bond but needs funding. It approaches a financial institution, which becomes the payer, enters the swap and buys the $100 million of bonds itself. The receiver now has exactly the economic position it would have had by borrowing at the floating rate plus 25 basis points and buying the bond. The payer, meanwhile, keeps legal ownership of the bond throughout. Compared with lending the receiver $100 million against the bond as collateral, it carries less credit risk, because if the receiver defaults there is no legal process needed to take possession of the collateral; the payer already owns it. In this sense total return swaps resemble repos, which are also designed to keep credit risk low when securities are financed.

The 25 basis point spread is the payer’s compensation for the risk that the receiver defaults. A loss arises for the payer only when a receiver default coincides with a drop in the bond’s price, so the spread reflects how creditworthy the receiver is, how creditworthy the issuer is, and how closely the two are linked. Several variations exist. The change in the bond’s value is sometimes settled physically, with the payer exchanging the bond for the notional principal at maturity, and sometimes settled periodically rather than in a single payment at the end.

An asset-backed security whose collateral is a pool of bonds is called a collateralized debt obligation. Interest and principal from the bonds pass through a waterfall into tranches, and while the detailed rules are intricate, their purpose is simple: a more senior tranche is more likely than a junior one to receive its promised interest and principal. A CDO built this way from bonds that are actually bought is a cash CDO.

The synthetic CDO grew from the observation that owning a corporate bond carries much the same credit risk as selling CDS protection on its issuer. The originator picks a portfolio of companies and a maturity, say 5 years, and sells protection on each company with CDSs of that maturity. The synthetic CDO principal is the sum of the CDS notionals. The originator receives the CDS spreads and pays out when companies default, and these inflows and outflows are allocated to tranches by rules much simpler than a cash CDO’s waterfall.

Cash CDO investors must pay for their tranches, because the money buys the bonds. Synthetic CDO investors need not invest anything, since they only agree to the rules for cash flows; in practice they almost always post their initial tranche principal as collateral, which earns interest, and payouts are drawn from it.

Standard portfolios and single-tranche trading

A further step removed the underlying CDSs altogether. Tranches can be traded on an imaginary reference portfolio: the protection buyer pays the tranche spread, and the protection seller pays whatever losses on the reference portfolio fall into that tranche. Because one tranche can trade without any trading in the others, this is called single-tranche trading. The CDX NA IG and iTraxx Europe portfolios define standard tranches that trade actively. For iTraxx Europe they cover losses of 0 to 3%, 3 to 6%, 6 to 9%, 9 to 12%, 12 to 22% and 22 to 100%; for CDX NA IG, 0 to 3%, 3 to 7%, 7 to 10%, 10 to 15%, 15 to 30% and 30 to 100%.

Source quotes: Creditex Group. The final column is derived.

The index row is the cost of protection on all names. For every tranche except the equity tranche, the quote is the annual spread paid on a principal that shrinks as the tranche takes losses. The 0 to 3% tranche is quoted differently: the buyer pays an upfront amount, shown as a percentage of initial tranche principal, plus 500 basis points a year on outstanding principal. The table records how violently the crisis repriced credit. The index rose from 23 to 165 basis points in two years, partly because markets judged investment grade default probabilities to be higher and partly because protection sellers, many short of liquidity, became more risk averse and demanded larger risk premiums.

The cost of protection on a kth-to-default CDS or on a CDO tranche depends critically on default correlation, because it is the clustering of defaults, not just their average frequency, that decides how many losses reach a given layer. Take a basket of 100 reference entities, each with a risk-neutral probability of 2% of defaulting within 5 years, used to define 5-year kth-to-default contracts.

So as correlation rises from zero, the chance of at least one default falls and the chance of ten or more defaults rises. First-to-default protection loses value; tenth-to-default protection gains it.

CDO tranches respond the same way. With low correlation, losses are steady and moderate: the equity tranche is very likely to be hit and is very risky, while senior tranches are very safe. Higher correlation makes outcomes more all-or-nothing, so the probability of few defaults rises (good for equity) and the probability of many defaults rises (bad for seniors). The equity tranche becomes less risky and senior tranches more risky. With perfect correlation and zero recovery, every tranche is equally risky, because the whole portfolio either survives or is wiped out.

A tranche is valued with the same logic as a CDS, but the quantity that drives both legs is the expected tranche principal. Let the payment dates be τ₁, τ₂, …, τ_m, with τ₀ = 0. Let E_j be the expected tranche principal at τ_j, and let v(τ) be the discount factor for a dollar paid at τ. The protection buyer pays a spread s per year on the remaining principal, so the premium leg is s times a factor A. The expected loss between τ_j−1 and τ_j is E_j−1 − E_j, and it is assumed to happen at the midpoint. Discounting those losses gives the protection leg C, and an accrual payment on each loss, worth s times a factor B, completes the picture.

The tranche is worth C − sA − sB to the protection buyer. Setting this to zero gives the breakeven spread.

If the tranche instead trades with a fixed running spread s* and an upfront payment, as the equity tranche does with s* = 500 basis points, then the upfront amount as a fraction of initial tranche principal is C − s*(A + B). All three factors depend on the expected principal path. Given that path for every payment date and a zero-coupon curve, the spread follows at once. The modelling problem is therefore to produce the expected tranche principal, which requires a model of correlated default times.

The standard market model for synthetic CDOs is the one-factor Gaussian copula of time to default. Every company has the same unconditional probability Q(t) of defaulting by time t, and default times are linked through one common factor F with a standard normal distribution. Given F, defaults are independent, with a conditional probability that depends on the copula correlation ρ, which is the same for every pair of companies.

Q(t) normally comes from a constant hazard rate λ chosen so that the CDS valuation method reproduces the index spread, giving Q(t) = 1 − e^−λt. Because defaults are independent once F is known, the number of defaults by time t follows a binomial distribution.

The tranche covers portfolio losses between the attachment point α_L and the detachment point α_H. Each default costs (1 − R)/n of the portfolio, so the numbers of defaults at which the tranche starts and stops absorbing losses are n_L = α_Ln/(1 − R) and n_H = α_Hn/(1 − R). Write m(x) for the smallest integer greater than x and take initial tranche principal as 1. Principal stays at 1 while k is below m(n_L), is zero once k reaches m(n_H), and in between equals [α_H − k(1 − R)/n] / (α_H − α_L). Weighting by binomial probabilities gives the conditional expected principal at each payment date.

Substituting E_j(F) for E_j gives conditional factors A(F), B(F) and C(F). The unconditional factors are found by integrating over the normal distribution of F with Gaussian quadrature, using M weights w_k and points F_k; accuracy improves with M.

A kth-to-default CDS is valued in the same model by conditioning on F. Given F, the probability that the kth default falls between τ_j−1 and τ_j is the probability of at least k defaults by τ_j less the probability of at least k defaults by τ_j−1, each found from the binomial formula.

Placing those defaults at the midpoint of each period, conditional payments and payoffs are valued exactly as for a single-name CDS, then integrated over F.

Market practice fixes recovery at 40% in the standard model, so the copula correlation ρ is the one unobservable input, much as volatility is in Black-Scholes-Merton. Traders therefore imply a correlation from tranche quotes. Let successive tranches run {α₀, α₁}, {α₁, α₂} and so on, with α₀ = 0; for iTraxx Europe the points are 0, 0.03, 0.06, 0.09, 0.12, 0.22 and 1.00.

Compound correlation, also called tranche correlation, is the ρ at which the model spread for a tranche {α_q−1, α_q} equals its market spread, found by iterative search. Base correlation for detachment point α_q is the ρ that prices the {0, α_q} tranche consistently with the market. It takes four steps. First, find each tranche’s compound correlation. Second, with that correlation, compute each tranche’s C, its discounted expected loss per unit of initial tranche principal; call it C_q. Third, compute the expected loss of the {0, α_q} tranche as a fraction of total portfolio principal.

Fourth, divide by α_q to get the C-value of the {0, α_q} tranche and search for the ρ that reproduces it.

These come from the January 2007 quotes, with rates flat at 3% and 40% recovery; the 23 basis point index spread implies a hazard rate of 0.382%.

The compound correlations form a correlation smile, while base correlations show a correlation skew. If the one-factor Gaussian copula described prices correctly, every tranche would imply the same correlation, so the pattern shows the market is not consistent with it.

Standard tranches of a standard portfolio need no model, since their spreads are quoted. Models are needed for nonstandard tranches, such as a 4 to 8% tranche of iTraxx Europe. One method interpolates base correlations to estimate base correlations for 0 to 4% and 0 to 8%, uses them to compute expected losses on those two tranches as a share of portfolio principal, and takes the difference as the expected loss on the 4 to 8% tranche, from which a compound correlation and breakeven spread follow.

A better method skips base correlation. Compute the expected loss of the 0 to X% tranche, as a share of portfolio principal, at each standard detachment point and interpolate that curve directly to 4% and 8%. For January 31, 2007 the curve climbs steeply to roughly 0.9% at a 3% detachment point, passes 1% by about 6%, and then flattens, ending a little above 1.1% at 100%. For no arbitrage, expected loss must increase with X at a decreasing rate. Interpolated base correlations often break this, because base correlation is a nonlinear function of expected loss. Interpolating expected losses directly can be done so that the condition holds.

The smile and skew show that the standard model does not fit market prices, and several alternatives are used.

Heterogeneous model

The standard model is homogeneous, giving every company the same default-time distribution and every pair the same copula correlation. Relaxing this is more realistic but harder, because with different default probabilities P(k, t | F) can no longer use the binomial formula and needs a numerical procedure such as those of Andersen, Sidenius and Basu (2003) and Hull and White (2004).

Other copulas and more factors

The Gaussian copula is one choice of dependence structure among many. Other one-factor copulas include the Student t, Clayton, Archimedean and Marshall-Olkin copulas. New ones arise by giving F and the idiosyncratic terms nonnormal distributions with mean 0 and standard deviation 1. Hull and White found a good fit with Student t distributions with four degrees of freedom for both, the double t copula. Another route adds factors, but the model then runs much more slowly because it integrates over several normal distributions.

Random recovery and random factor loadings

Andersen and Sidenius let the copula correlation depend on F and make recovery negatively related to the default rate. Correlation is higher when F is low, so in bad states, when defaults are frequent, correlation is also high, which empirical evidence supports. The model fits market quotes much better.

Implied copula model

Hull and White imply a copula from market quotes. In the simplest version one average hazard rate applies to all companies over the CDO’s life, and its probability distribution is implied from tranche prices, much as a stock price distribution is implied from option prices.

Dynamic models

All the models above are static: they describe the average default environment over one horizon, so 5-, 7- and 10-year CDOs each need a separate model. Dynamic models describe how portfolio losses evolve over time. Structural models simulate many companies’ asset values together, with correlated processes and default at a barrier, but need Monte Carlo simulation and are hard to calibrate. Reduced form models model each hazard rate and need jumps in hazard rates to produce realistic correlation. Top down models model total portfolio loss directly, ignoring individual companies.