Past articles: CDO Pricing and the Abacus

This article appeared in Capital page of The Edge Malaysia, Issue 808, May 31-June 6, 2010

Lessons from the Abacus
By Jasvin Josen
Fabulous Fab” Fabrice Tourre and Goldman Sachs have been hogging the news all month. Perhaps it is time to present some facts and technical features of the synthetic CDO so that we better understand these world-changing events.


Chart 1 – A Typical CDO Structure
CDOs are like bonds whose income payments and principal repayments are dependent on a pool of securities such as bonds, loans and so on. For the purpose of this article, I have skipped the basic description of the CDO and suggest interested readers refer to “Understanding credit default swaps: CDS spawns the even murkier CDO” (Issue 790, Jan 25) and “The insurance that did not protect” (Issue 788, Jan 11).

In the case of a synthetic CDO, the pool of securities is not owned by the protection buyer seeking credit protection. The transaction appears more like a speculation by the protection buyer and the protection seller, expecting a fall and increase in value of the securities respectively.

The uniqueness of the CDO is founded in its ability to allow investors with different credit risk appetites to invest in appropriate tranches accordingly. The riskiest equity tranche takes the first loss followed by the mezzanine, senior and “Super Senior Tranche” at the top [see Chart 1].

The Abacus deal

From the many, many pieces of literature attempting to study the Goldman case, I find the best explanation at the following link: http://www.interfluidity.com/v2/814.html.

The gist of the transaction: Goldman sold protection to Paulson, a hedge fund, on the 45% to 100% tranche of a synthetic CDO referencing a portfolio of RMBS (residential mortgage backed securities). In other words, Goldman was short protection and long the risk. Then Goldman bought protection from ACA Capital (then an arm of ABN Amro) on the 50% to 100% tranche of the same CDO. Here, Goldman was long protection and short the risk. On a net basis, Goldman was effectively exposed to the 45% to 50% tranche of the CDO as a protection seller.

A week after the deal was closed, Goldman wrote down most of the value of the CDO tranche in its books. Not long after that, the underlying CDO mortgage loans started defaulting and the defaults rapidly crawled up beyond the 45% tranche, triggering massive payouts payable by ACA to Goldman, and Goldman to Paulson.

The Securities and Exchange Commission charge on Goldman is on ethical grounds, where the investment bank is said to have acted on instructions of Paulson, who was eager to bet on a decline of subprime loans. Goldman allegedly allowed Paulson to play a major role in the asset-picking process, which caused the underlying securities of the CDO to comprise mainly close-to-default home loans.

The Abacus CDO only had senior tranches that begin at around the 45% mark. The equity tranche did not exist. In retrospect, with such risky loans, it is very unlikely Goldman or any other party wanted to take on the default risk there.

From a technical perspective, there are three interesting aspects to this:

• Funded and unfunded CDOs have different counterparty risk profiles

• Correlation and portfolio loss distributions are crucial in understanding the risks a CDO carries

• Pricing of a CDO is centred on modelling the dependent default structure of underlying assets in the reference portfolio.

Funded and unfunded CDOs

Chart 2 : Funded and Unfunded transactions

In most cases, CDOs are funded. A funded transaction is just another way of stating that the counterparty risk of the protection seller (PS) is taken care of. The protection buyer (PB) is exposed to a huge counterparty risk, if upon default of the CDO (tranche), the PS is unable to pay the PB. So at inception, the PB pays the notional of the CDO (if 100% funded) to the PB in the form of a Credit Linked Note, a kind of a bond with the embedded CDO. If default occurs, the PB will keep the losses on default and return the rest to the PS.

In the Goldman case, apparently the trade between Goldman and ACA was unfunded. As the defaults occurred, Goldman was put at risk in the ability of getting a payout from ACA.

It is quite common to have unfunded super senior tranches (say 85% to 100%) as the tranche is almost never expected to default. However, with hindsight, having known the quality of the underlying securities for the Abacus deal, the 50% to 100% tranche was hardly a super senior tranche despite being awarded an AAA grade by the Rating Agencies.

Summing up the above, funded CDOs should be a preference even when a senior tranche appears to be safe with an AAA rating.

Portfolio loss distributions and correlation

We know that CDO tranches carry different degrees of default risk. How the default risk varies for each tranche depends on how correlated the underlying securities are. This can be illustrated with a portfolio loss distribution, which shows the percentage loss probability of the portfolio.

Chart 3: Portfolio Loss Distribution for a large portfolio at 0%, 20% and 95% correlation

When the portfolio is fully diversified and correlation is close to zero, we can expect a skewed bell curve. As correlation increases, the curve becomes monotonically decreasing. At this stage, the probability of larger losses increase and at the same time, the probability of smaller losses also increases. For a very high level of correlation, the distribution becomes U-shaped. Here, at maximum default correlation, all the probability is gathered at the two ends of the distribution. The portfolio either all survives or it all defaults. It behaves like a single asset.

The Abacus reference portfolio was made up of residential mortgages in four areas, where three were just next to each other — California, Nevada and Arizona. All the borrowers were of subprime quality, meaning that they either had poor credit quality or were refinancing their houses based on a previous almost-default case. The loans inadvertently were highly correlated with each other.

A senior tranche only looks safe assuming imperfect correlation. With correlation being almost perfect in this case, the losses very quickly crept up to the 45% level and started to crawl its way through.

What we can take from the above is that parties to a CDO must be able to know the loss distribution and the impact of correlation to appreciate the extent of default risk they are faced with.

What next

In the heat of the CDO era, investors were so eager to collect attractive coupons that they overlooked what they were actually going into. Nobody questioned the credit quality of the underlying securities, relying purely on the supercilious grading awarded by the rating agencies who also believed that defaults were remote in the good times.

How were they pricing the CDO in their books? Surely the parties could see that the tranches were not worth the value? This will be reviewed in the next article.
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This article appeared in Capital page, The Edge Malaysia, Issue 809, Jun 7-13, 2010.

Pricing of a collateralised debt obligation
By Jasvin Josen


From the previous article, we begin to appreciate the complexity of the collateralised debt obligation (CDO). However, its most complex aspect by far would be its pricing. A CDO is constructed with a pool of securities, thus, its valuation is dependent on every asset in the pool and the correlation between them. Numerous papers have been (and still are being) written about the pricing of the CDO, primarily modelling the dependent default structure.


CDO models

While the first CDOs were issued as early as 1987, they emerged in the securities market only in the late 1990s. As such the basic infrastructure for CDO pricing is at its infancy.

CDO pricing is a function of the correlation between the default risks of all assets in the reference portfolio. Theories differ on how to estimate or model these correlated defaults. Copula models attempt to get implied correlation from the market to price bespoke CDO tranches. Others prefer to dynamically model the loss of the bespoke portfolio with stochastic models using calibrated parameters.

Copulas are one of the methodologies that first caught the attention of most market practitioners to model the dependence structure in the asset portfolios. Despite its drawbacks, the main reason for its wide usage is its ease of implementation and calibration.

CDO pricing steps

The focus here however is to discuss the methodology in pricing a CDO and not its model choice and pitfalls. For ease of explanation, the 1-factor Gaussian Copula is chosen. Readers who are not interested in the nitty gritty of the pricing may skip to the paragraph titled “The Abacus Case” to catch the highlights of CDO valuation.

Pricing of a CDO is explained by the following steps:

i) Obtain the implied correlation from the market

ii) Model the dependent default times

iii) Compute the pool Expected Loss (EL)

iv) Compute the Tranche EL

v) Price the CDO tranche



(i) Obtain the implied correlation from the market

Standardised index CDOs in the market like the CDX and iTraxx have quoted bid and offer prices. The tranche prices are put in the Gaussian Copula Model and the model works in reverse to back out the implied correlations. In this model, correlation is flat, meaning that the correlation is constant in time and constant across assets.


(ii) Model the dependent default times

The whole crux in valuing a CDO is estimating when the assets in the CDO will default. For example, if the CDO is of 5-year maturity and most assets are estimated to default only between 7 to 20 years, then the tranches have low risk and would command low spreads.

The objective of the Gaussian Copula model is to arrive at dependent default times. Dependent because correlation exists between the assets in the pool, hence one asset’s default could be dependent on the other. We start with generating random numbers between 0 and 1, representing survival rate of assets. Then we impose the implied correlation obtained from the market to obtain correlated normal random numbers. The normal numbers are once again transformed into uniform numbers to represent correlated survival rates of the assets in the pool. Finally we translate these correlated survival rates into default times using the individual asset’s credit curves. The whole depiction is illustrated in Chart 1.

Chart 1: Copulas to generate correlated default times

iii) Compute the Pool EL

In Chart 2 (see page 41), say we have a two-asset portfolio example. Assume that the default time of the assets in Column A are already modelled. Now the EL of each asset is calculated as [Notional* (1-recovery rate)]. The EL of both assets is combined to obtain the pool EL in Column C. From the simulated default times, many simulations of pool EL is obtained. If we were to plot all the simulated pool EL in a graph, we would get a typical portfolio loss distribution discussed in my previous article.

Chart 2: Computation of Pool and Tranche EL




iv) Compute the Tranche EL

Let us look at a CDO with a total notional of 100m and its 10%-25% tranche (see Chart 3). As defaults occur, there can be one of three consequences for the tranche:

a) No losses reaches the 10%-25% tranche

b) Losses reaches the 10%-25% tranche

c) Losses exceed the 10%-25% tranche

Chart 3

Back to the table in Chart 2, say we have a 10%-25% tranche. For simulation (1) both default times occurred after the 5-year maturity of the CDO, so default is zero for both the pool and the tranche. In simulation 2, both asset 1 and 2 happen to default before five years. The pool loss is $62 million; this means that the equity and 10-25% tranches are fully wiped out. In simulation 3, only Asset 1 defaults before the CDO maturity which causes the 10-25% tranche EL to be affected partially to the tune of $4 million ($14 million - $10 million equity tranche loss)

We can summarise the tranche EL into one algorithm to capture the EL of a tranche in all three situations:

Tranche EL (i) = max [(min(A (1-rec)), det) – att), 0]

Where

i = the tranche

A = Pool EL

Rec = recovery rate

Det = notional at detachment point (in this case $25m)

Att = notional at attachment point (in this case $10m)

v) Price the CDO Tranche

A CDO tranche is priced by simply computing the present value (PV) of its two legs, the premium leg (paid by the protection buyer) and the contingent leg (paid by the protection seller in the event of a default). For each simulated tranche loss in Chart 2 (column D), the PV of the fee and contingent leg will be calculated using the formulae given:
 Where

t = point in time

H = Notional at the detachment point = $25m

L = The Notional at the attach ment point =$10m

s = fixed coupon of the tranche

df = Fee Leg: discount rate for each payment period/Contingent Leg: discount rate(s) at the estimated default dates

Intuitively the PV of the premium leg is just the premium rate multiplied by the notional of the tranche for every payment period. For each payment period, the notional is evaluated to see if it has decayed when defaults occurred. As for the contingent leg, the PV is simply the EL of the tranche that the PS has to make good to the PB upon default. The calculation takes place in a continuous time period (and not discrete) thus the use of integration in the formulae.

The Abacus case

If we go back to the Abacus’s case, Goldman wrote down its unhedged 45%-50% portion about a week after the transaction was completed. Going back to the above valuation principles, it must be the case where the steep individual default curves of the subprime loans coupled with high correlation led to very near default times in the model. Events have shown that correlation tends to get stronger as asset quality worsens.

This triggered massive pool EL which ate through the 45%-50% tranche very quickly. As GS was the protection seller, the value of the CDO tranche in its books would be + PV (premium leg) – PV (contingent leg). The high negative PV of the contingent leg pushed down the tranche’s mark-to-market value.

It is a wonder how the 50%-100% tranche of the same CDOs was valued in ACA Capital’s books, considering that it willingly entered into such a deal.

Besides admitting to the cumbersome process involved in pricing a CDO, the above points out that the main driver of CDO valuation is how dependent the defaults are in the reference portfolios. This in turn depends on the steepness of default curves (credit quality) and correlation among the assets.

Conclusion

The Abacus event was tragic but at least it has taught us some interesting technical aspects. We acknowledge how funded transactions are crucial in minimising counterparty risk. One can also appreciate the significance of loss distributions in figuring out how risky a CDO is. We also have come to terms with pricing of a CDO to recognise how the main drivers, defaults and correlation act together to impact its value.

After the financial crisis, many practitioners realised how far off their pricing models were, when values of CDOs plunged in the market. One of the issues was that the correlation skew, the difference between the modelled and market correlation (just like the volatility smile in the Black Scholes world) was not captured properly.

More focus is being given to other copulas that capture better tail risk (or remote events) and dynamic models with dynamic correlation surface to produce better models. After an eventful decade of a rise and a fall, hopefully the CDO will now mature into an instrument with robust pricing and risk management.

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