Credit derivatives and structured products: CDS, MBS, ABS, CDOs, tranching and 2008

Level 1 Plain words

For every reader: the idea in plain words, no formulas.

Insurance on a borrower

A credit default swap (CDS) works like an insurance policy on a company's or a country's debt. The buyer of protection pays a yearly fee, quoted in basis points of the amount covered. If the borrower defaults, the seller of protection pays the buyer the loss: the face value of the debt minus what the debt is still worth after default. If no default happens before the contract ends, the seller keeps the fees and pays nothing.

Unlike home insurance, you do not need to own the debt to buy protection. A bank that has lent to a company can buy protection to hedge. An investor who thinks the company is in trouble can buy protection to profit if it fails. An insurer or fund that thinks the risk is overpriced can sell protection and collect the fee. Because the CDS price moves every day with the market's view of the borrower, it works as a live credit rating.

Pooling loans into bonds

A bank that makes thousands of mortgages can keep them, or it can sell them to a trust that issues bonds backed by the mortgage payments. That process is securitisation. Buyers of the bonds receive the homeowners' monthly payments, passed through after fees. The bank gets its money back to lend again.

The bonds that result are mortgage-backed securities (MBS). Pools of car loans, credit-card balances and student loans become asset-backed securities (ABS). In the United States, most MBS carry a guarantee from Ginnie Mae, Fannie Mae or Freddie Mac, so their buyers bear the timing risk of the payments and not the default risk. MBS without that guarantee, called private-label, carry both.

A mortgage holder in the United States can repay early, usually without penalty, by selling the house or refinancing. That right to prepay is what makes MBS unusual, as Level 2 shows.

Slicing the pool

A pool can also pay its bondholders in order of rank. The most senior bonds get paid first and take losses last. The most junior take the first losses and earn the highest yield for it. Each layer is a tranche, from the French word for slice. A collateralised debt obligation (CDO) is a pool of bonds or loans sliced this way. Some CDOs in the 2000s were pools of tranches from other pools, which made the risk much harder to see.

Why it mattered in 2008

In the mid-2000s lenders made many mortgages to borrowers with weak credit, often with little or no down payment. Those loans went into pools, the pools went into CDOs, and the senior tranches received AAA ratings. Banks, insurers and funds around the world bought them. When US house prices fell nationwide, losses went through the layers much further than the ratings had allowed for, and the firms holding the senior tranches, or insuring them through CDS, found they held the risk they thought they had passed on.

Level 2 How it works

For readers who want the mechanism, with the diagrams.

Beginner path stops here

The two legs of a CDS

A CDS has a premium leg and a protection leg. Protection buyers pay the premium leg: a quarterly fee on the notional until the contract ends or the borrower defaults, whichever comes first. The seller pays the protection leg: nothing unless a credit event happens, then the notional times one minus the recovery rate.

Industry rules, set by the International Swaps and Derivatives Association, define the credit events: failure to pay, bankruptcy and, for some contracts, restructuring of the debt. Since the 2009 changes to the standard contract (the "Big Bang" protocol in North America), an auction run after each credit event fixes one recovery price for all contracts on that borrower, and contracts trade with standard coupons of 100 or 500 basis points, with an upfront payment making up the difference from the market spread.

Index CDS, such as the CDX indices for North American companies and iTraxx for European ones, bundle protection on a set list of names, often 125, into one contract. They trade in large size and are the main way investors hedge broad credit risk.

The CDS–bond basis

A bond holder who also buys CDS protection on the same issuer has, in theory, a riskless position: if the issuer defaults, the CDS pays the loss on the bond. So the CDS spread should be close to the bond's spread over a risk-free rate. Traders call the gap between the two the CDS-bond basis. When the CDS costs less than the bond's spread, the basis is negative, and a trader can buy the bond, buy protection and earn the difference.

In late 2008 the basis on many investment-grade bonds went deeply negative. On paper the trade was cheap, but buying the bond needed funding, and funding was what nobody could get. Dealers who already held basis trades had to sell bonds to raise cash, which widened the gap further. It was another case of the lesson from Chapter 15: a hedge that is right on paper can still fail for want of cash.

Mortgage pass-throughs and the prepayment option

Each month an MBS holder receives three things from the pool: interest on the outstanding balance, the scheduled principal on the level-payment mortgages, and any prepayments. Prepayments come from people who move, refinance, pay down extra, or default (in an agency pool, the guarantor buys out a defaulted loan at par, which looks like a prepayment to the holder).

Analysts measure prepayment speed as the conditional prepayment rate (CPR), the share of the remaining balance repaid in a year beyond the scheduled principal. Refinancing makes the CPR depend on rates. When mortgage rates fall well below the rate on the loans in the pool, homeowners refinance in large numbers and the CPR jumps. When rates rise above it, almost nobody refinances and the CPR falls to the slow pace of house sales.

So every homeowner holds an option to repay at par, a call on their own mortgage, and every MBS holder has sold that option. The holder receives a higher yield than a Treasury for having sold it.

Negative convexity

A Treasury bond's price rises faster as yields fall, the convexity of Chapter 8. An MBS behaves the other way near the rate on its loans. When rates fall, prepayments speed up, the holder gets cash back at par just when reinvesting pays less, and the price rises only a little. That is contraction risk. When rates rise, prepayments slow, the MBS holder is stuck with a below-market rate for longer, and the price falls faster than a Treasury's. That is extension risk. A price curve that bends the wrong way has negative convexity.

Holders who hedge MBS with Treasuries or swaps must buy duration when rates rise (their MBS are extending) and sell it when rates fall. With enough MBS hedgers in the market, that buying and selling can amplify moves in Treasury yields, as happened in 1994 and 2003.

Asset-backed securities and credit enhancement

ABS pools car loans, credit-card receivables, equipment leases or student loans. Their structures add protection for senior buyers, called credit enhancement:

1. Subordination. Junior tranches absorb losses before senior ones.
2. Overcollateralisation. The pool holds more loans than the bonds issued against it.
3. Excess spread. The loans pay more interest than the bonds, and the surplus covers losses first.
4. Reserve accounts. Cash set aside at issue pays for early losses.

CDOs: the loss waterfall

Each CDO holds a pool of bonds or loans and issues tranches with attachment and detachment points. A tranche that attaches at 3 % and detaches at 7 % takes no loss until pool losses reach 3 % of the pool, then takes every dollar of loss until they reach 7 %, when it is wiped out. Losses flow down from the top of the capital stack to the bottom like water filling a glass from the base: the equity tranche fills first, then the mezzanine, then the senior.

Cash CDOs buy the bonds. Synthetic CDOs sell CDS protection on a list of names instead and pass the premiums and the losses to their tranches. Some CDOs, called CDO-squared, held tranches of other CDOs. In the mid-2000s many CDOs held the BBB-rated tranches of subprime mortgage pools, and Level 4 shows what that did.

What correlation does

The expected loss on the whole pool depends only on how likely each loan is to default and how much the lender loses when it does. How the tranches share that loss depends on whether defaults happen one at a time or together. That is default correlation.

If defaults are independent, a large pool loses close to its expected loss every time. The equity tranche, sized to cover that, nearly always takes the hit, and the senior tranche nearly always stays safe. If defaults cluster, most years see few defaults and the equity tranche survives, while a few years see many and the losses reach the senior tranche. Raising correlation moves risk from the bottom of the structure to the top. Senior tranches are short correlation, equity tranches are long it.

The risk drivers of credit derivatives and structured products

Level 3 The math

For Series 65 and CFA readers: formulas and worked examples.

Series 65 path stops here

Hazard rates and survival

Assume a borrower defaults at a constant rate $\lambda$ per year, the hazard rate, so that in any short interval $dt$ the chance of default, given survival so far, is $\lambda \, dt$. The probability of surviving to time $t$ is then:

In words, the chance the borrower is still paying at time $t$ is the exponential of minus the hazard rate times $t$.

The credit triangle

If the protection seller expects to lose $1 - R$ per dollar of notional when default comes, and default comes at rate $\lambda$, a fair yearly fee is the expected loss rate:

In words, the CDS spread is about the hazard rate times the loss given default, so the hazard rate is about the spread divided by one minus the recovery rate.

Pricing both legs

With a flat risk-free rate $r$ and premiums paid continuously, the premium leg is worth the spread times the notional times a risky annuity, the present value of one dollar a year paid while the borrower survives:

In words, the risky annuity discounts each year's payment twice, once for time at the risk-free rate and once for the chance the borrower has already defaulted.

In words, the premium leg equals spread times notional times the risky annuity, and the protection leg is the loss given default times the notional times the hazard rate times the same annuity.

Setting the two equal gives back the credit triangle, $s = \lambda (1 - R)$.

Marking a CDS to market

A day later the borrower's outlook darkens and five-year protection costs 300 basis points. The buyer at 100 now holds a contract a new buyer would pay 300 for. Its value is the spread gap paid over the risky annuity at the new hazard:

In words, a CDS gains for the protection buyer the rise in the spread times the notional times the new risky annuity.

Prepayment speeds

Mortgages pay monthly, so analysts turn the yearly CPR into a single monthly mortality (SMM):

In words, the monthly prepayment rate is one minus the twelfth root of one minus the yearly rate.

A CPR of 6 % a year is an SMM of 0.514 % a month. The industry's PSA benchmark ramps the CPR up over a loan's first 30 months to 6 % and holds it there; a pool "at 200 PSA" prepays twice as fast.

MBS duration and convexity

For bonds whose cash flows change with rates, analysts use effective duration, measured by repricing at yields a little higher and lower with the prepayments re-estimated each time:

In words, effective duration is the price change between a lower and a higher yield, per unit of yield change, as a share of today's price; effective convexity is how far the average of the two moved prices sits from today's price.

We price a stylised 30-year pass-through paying 6 % at different market mortgage rates. Its prepayment speed follows an S-shaped refinancing curve: about 6 % a year with no incentive, rising toward 46 % as rates fall two points or more below the pool's rate. The last column freezes the prepayment speed at the 6 % rate level, removing the option.

Read down the duration column. At 8 % the pool behaves like a long bond. At 6 %, its own rate, the duration is moderate and the convexity is strongly negative. Below 5 % the duration collapses toward one year, because the market expects most of the pool to be repaid soon, and the price stops rising much above par, which traders call price compression.

Tranche loss arithmetic

A pool of 100 loans of $1 million each loses 60 % of a loan's value when it defaults. Tranches are equity 0–3 %, mezzanine 3–7 % and senior 7–100 %. For a pool loss $L$, a tranche with attachment $a$ and detachment $d$ loses this share of its size:

In words, a tranche loses nothing until pool losses reach its attachment point, then loses every further dollar until its detachment point, measured as a share of its thickness.

A thin tranche goes from untouched to wiped out over a narrow range of pool losses. The mezzanine tranche here spans four points of pool loss: at 5 % of loans defaulting it is whole, at about 12 % it is gone.

Correlation in the one-factor model

The standard model, from Oldrich Vasicek's work on loan portfolios and David Li's copula approach of 2000, links every borrower to one common factor $M$, the state of the economy, plus its own noise. Each borrower defaults with probability $p$ on its own. Given the economy's state $M$, the share of a large pool that defaults is:

In words, in a bad state of the economy (low $M$) the default rate rises above its average, and the correlation $\rho$ sets how far it swings with the economy.

With $\rho = 0$ the default rate equals $p$ in every state. With $\rho$ near one, the pool either mostly survives or mostly defaults. Integrating each tranche's loss over $M$ gives its expected loss. For a 5 % default probability and 60 % loss given default, so a 3 % expected pool loss:

The pool's expected loss is 3 % in every row. Only its distribution changes. At zero correlation the equity tranche loses everything and the senior nothing. As correlation rises the equity tranche's expected loss falls by more than half, and the senior tranche goes from safe to exposed, with the chance of any senior loss rising from zero to about one in eight. A rating agency that used a low correlation would have rated the senior tranche far safer than one using a high correlation, from the same loans.

Level 4 History, analysed

For readers who want past episodes run through the Level 3 math.

Falling house prices and the first losses

US house prices rose fast from the late 1990s to 2006 and then fell nationwide, the first such fall since the Great Depression, by more than a quarter on the S&P CoreLogic Case-Shiller national index by 2012. A nationwide fall mattered because the models behind subprime pools assumed that regional house markets would not all fall at once, a low default correlation.

Run the arithmetic for one borrower. A buyer pays $300,000 with 5 % down, so the loan is $285,000. A 20 % fall in prices takes the house to $240,000. The buyer's equity is now −$45,000, and the loan is 119 % of the house's value. A borrower in that position who loses a job, or whose adjustable rate resets higher, has little reason to keep paying. Defaults rise together across the country, which is the high-correlation row of the Level 3 table.

The ABX indices, launched in 2006, let investors buy CDS protection on baskets of subprime mortgage bonds by rating. Through 2007 the BBB and then the AAA indices fell far below par. Gary Gorton has argued in "The Panic of 2007" (2008) that the ABX gave the market a price for subprime risk for the first time, and that the price revealed losses that holders of the bonds could no longer ignore.

The mezzanine CDO cliff

Many CDOs of 2005–07 held the BBB tranches of subprime pools, tranches that attached around 5 % of pool losses and detached around 8 %. A CDO built from them issued its own tranches, with the senior ones rated AAA. We take a stylised CDO whose senior tranche attaches at 25 % of its own collateral and assume every BBB tranche it holds comes from a similar pool.

Between pool losses of 6 % and 8 %, a range of two percentage points, the "AAA" tranche goes from a small loss to total loss. Tranching a tranche thins the slice twice, and stacking the same kind of collateral removes the diversification the senior rating assumed. The Financial Crisis Inquiry Commission's report of January 2011 concluded that the rating agencies were "key enablers" of the crisis, and recorded mass downgrades of the AAA mortgage securities rated in 2006.

Lehman and AIG

AIG's unit had sold protection on the safest-looking slices of mortgage CDOs, earning small fees on large notionals and posting little capital, because the parent's AAA rating stood behind it. Its contracts required collateral when the protected tranches lost value or when AIG's own rating fell. Both happened in 2008. A stylised version of the collateral math: on $10 billion of protection, a fall in the protected tranches' value from 100 to 85 requires $1.5bn of collateral, with no default in the underlying loans. Repeat across a book many times that size, add the downgrade triggers, and the cash runs out within days. The CDS mark-to-market formula of Level 3 describes the drain on any protection seller.

Lehman's own failure triggered a CDS credit event. The auction in October 2008 set recovery on Lehman bonds at under ten cents on the dollar, so sellers of protection on Lehman paid more than 90 % of notional. Payments settled in an orderly way, because dealers netted their positions; the larger damage came from not knowing who owed what to whom until it settled.

Two weeks before Lehman, on 7 September 2008, the US government had placed Fannie Mae and Freddie Mac into conservatorship, which kept their guarantee on agency MBS good. Agency MBS holders lost little to default; private-label MBS and CDO holders lost a great deal.

Greece, 2012: a sovereign credit event

CDS on governments work the same way as on companies. In March 2012 Greece used collective action clauses to force holders of its bonds into an exchange for new bonds worth far less. ISDA's determinations committee ruled that this was a credit event, and the auction that followed set the recovery value of Greek bonds at about 21.5 cents on the dollar. Sellers of protection paid buyers about 78.5 % of notional. The net amount of protection outstanding on Greece was small next to its debt, because most holders were European banks and official lenders who had not bought protection, so the payments did not spread distress. The decision mattered more as a test: had the committee ruled the exchange voluntary, protection on government debt would have been worth much less everywhere.

What changed after the crisis

The Dodd–Frank Act, signed on 21 July 2010, required standard swaps, including the main index CDS, to clear through central counterparties and to report trades, and required most securitisers to keep at least 5 % of the credit risk of what they sold, so that the originator has something to lose if the loans go bad. The 2009 standard contract changes, with fixed coupons and recovery auctions, made single-name CDS easier to net and to settle. The Federal Reserve bought agency MBS in large amounts from early 2009, which pulled mortgage rates down and set off a refinancing wave, the contraction side of the Level 3 table.

MBS extension in 2022

The other side of the table appeared in 2022. The average 30-year US mortgage rate in Freddie Mac's weekly survey roughly doubled during the year, from about 3 % to about 7 %. Nearly every outstanding mortgage now carried a rate below the market, refinancing stopped, and prepayment speeds fell to the pace of house sales. In the stylised table, moving from a market rate of 4 % to 7 % for a pool whose loans pay well below the new rate stretches duration several times over. Holders of MBS, including banks that had bought them with deposits in 2020 and 2021, found their bonds had extended just as their value fell. That combination was part of what sank Silicon Valley Bank in March 2023, as Chapter 8 discusses for duration in general.