CR 22 – Structured Credit Risk

A structured credit product is a bond or credit derivative paid from a pool of loans or other debt claims, with the pool’s cash flows and credit losses reshaped before they reach investors; the family is also called portfolio credit products. These securities financed much lending before 2007, sat at the centre of the subprime crisis, and helped build the market-based or “shadow banking” system.

A ladder of pooled-loan securities

Loan-backed securities can be ordered by how much structure each adds, which is also roughly the order in which they appeared.

Covered bonds, issued mostly by European banks in Germany and Denmark, are secured on a cover pool of mortgage loans. The pool stays on the issuing bank’s balance sheet but is ring-fenced, so if the bank fails the covered bondholders are paid from the pool before general creditors can touch it. Coupons and principal come from the bank’s general cash flows, backed by the bank’s own obligation, and because the assets never leave its balance sheet a covered bond is not a full securitization.

Mortgage pass-through securities are genuine securitizations. The mortgages are removed from the originators’ balance sheets, and a servicer collects interest and principal and passes them to bondholders. Most pass-throughs are agency MBS carrying an explicit or implicit U.S. federal guarantee, so default risk is small. The real exposure is prepayment risk: homeowners repay early at their own discretion, so the bond amortizes at an uncertain pace instead of repaying in one bullet at maturity.

Collateralized mortgage obligations (CMOs) were built to manage that prepayment risk and to carve shorter and longer bonds out of one amortizing mortgage pool. In the simplest sequential-pay design, Class A takes every principal repayment until it is retired, then Class B starts receiving principal, and so on. The early classes carry less prepayment uncertainty than a pass-through and the later classes carry more.

Structured credit products add one further step: credit losses, not just principal, are allocated in sequence. The underlying loans or bonds are genuinely credit-risky, and the tranches are designed to have different degrees of credit risk. Losses write down the junior tranches first, and a senior tranche loses nothing until everything beneath it has been written down to zero. Combining this with sequential principal payment, and routing particular slices of the pool’s cash flows to particular bonds, produces very complex deals.

Alongside securitization sits a related funding channel: vehicles that issue asset-backed commercial paper (ABCP) against receivables or against securitization bonds.

The special purpose vehicle as a robot company

A structured product behaves like a company that owns a balance sheet and does nothing else. It is set up as a special purpose entity (SPE) or special purpose vehicle (SPV), often called a trust. The loans are its assets and the tranches are its liabilities. The legal separation makes the vehicle bankruptcy remote, so investors can judge the loans themselves rather than the lender that made them or the firm that administers them.

Names follow the collateral. Deals backed by consumer or other receivables are asset-backed securities (ABS), deals backed by mortgages are mortgage-backed securities (MBS), and deals backed by leveraged loans are collateralized loan obligations (CLOs). A collateralized debt obligation (CDO) is a securitization of other securitization bonds, such as ABS, MBS or CLO tranches; collateralized bond obligations (CBOs) are the bond-backed counterpart. A third layer exists as well: a CDO-squared is backed by tranches of other CDOs.

Three ways to classify a deal

The first dimension is the underlying asset class: beneath every deal lie ordinary debts such as real estate loans, credit card balances, auto and student loans or corporate bonds, occasionally nondebt claims such as fee income, and their credit quality and prepayment behaviour drive everything above them.

The second is the type of structure: how many tranches are carved from the liability side and how large each is, and how many layers of securitization separate the investor from actual loans.

The third is how the pool changes over time, and each answer brings its own risk management challenges.

• A static pool holds a fixed set of loans that amortize, prepay or default, so the deal and its bonds wind down gradually. Auto loans and residential mortgages, which have fixed terms at origination, are typical.
• A revolving pool keeps total assets at a target level during a revolving period by adding new loans from the originator as old ones repay. Credit card debt, which borrowers draw and repay at will, suits this design. When the revolving period ends the pool is frozen and winds down as balances are repaid or charged off.
• A managed pool gives the manager discretion to sell loans and buy replacements. CLOs are the usual example. Managers are hired to find loans paying more spread than their quality warrants and to sell weakening names in the syndicated loan market; since such loans are often repaid early in a lump, management also keeps the pool fully invested.

Pool sizes differ sharply. A CLO or commercial mortgage-backed security (CMBS) pool may hold around 100 loans of several million dollars each, while a large residential mortgage-backed security (RMBS) can hold several tens of thousands of mortgages averaging $200,000.

Finally, the assets need not be cash loans at all. In a synthetic securitization the vehicle holds credit derivatives, most often CDS, written on a reference portfolio that usually consists of securitization bonds rather than loans. Deals holding actual debt instruments are called cash or cash-flow securitizations.

Tranching splits the liabilities of the vehicle into a capital structure in which each class of note has its own coupon or spread and its own place in the queue for losses. The rule is simple: senior tranches have first claim on interest and principal, and junior tranches absorb credit losses first. A deal may have a handful of tranches or many dozens, but they fall into three groups.

• Equity tranche. Usually pays no fixed coupon and takes the first loss from defaults in the pool. It is a note with a stated notional amount that receives whatever cash is left once every other obligation of the vehicle has been met. Its notional is small relative to the collateral, which is why it is described as thin.
• Mezzanine (junior) tranche. Pays a fairly generous fixed spread and takes losses next once the equity is gone; it is usually thin too.
• Senior tranche. Earns a relatively low coupon but is shielded by everything beneath it. Senior bonds normally make up the bulk of the liabilities, and this is essential to the economics: if most of the collateral cannot be funded with cheap senior debt, the securitization is generally not viable.

Attachment points and credit enhancement

The boundary between two adjacent tranches, measured as a percentage of total liabilities, is the attachment point of the upper tranche and the detachment point of the lower one. So the equity tranche has only a detachment point and the most senior tranche has only an attachment point.

Everything below a bond in the capital structure is its subordination or credit enhancement: the fraction of the pool that can be lost before that bond loses a cent. More senior bonds enjoy more of it. Credit enhancement is not fixed. Default losses eat into it over time, while accumulated excess spread, the collateral interest left after paying the liabilities and the deal’s fees and expenses, can build it up.

Hard and soft overcollateralization

A securitization is a device for long-term funding of the pool, which works only if the senior tranche is large and cheap, so deals add features that make it safe enough to sell at a low coupon. The most important is overcollateralization: issuing bonds with a smaller par amount than the collateral, which protects every bond tranche.

Deals also contain reserve accounts that must be filled to set levels before the mezzanine and equity notes receive anything. They can be filled slowly from excess spread or immediately through overcollateralization at issue, and many deals do both. Overcollateralization present at issue is hard credit enhancement; excess spread trapped over time is soft credit enhancement, because it does not exist at the start. Revolving deals usually carry an early amortization trigger that stops the pool from being replenished once a default threshold is breached.

Maturity, weighted average life and the risks around it

Collateral usually mixes maturities, amortizes, and can prepay after any initial lockout period, so the senior bonds amortize as the loans do. A senior bond may have a distant legal final maturity yet a much shorter and uncertain effective life. Analysts work instead with the weighted average life (WAL), the average number of years each dollar of par is expected to stay outstanding, which only has meaning under a stated prepayment assumption; some asset classes have conventional standard assumptions.

Combining sequential pay with credit tranching produces several senior bonds with different WALs, a feature called time tranching, as distinct from the seniority tranching that governs losses. Bondholders still face prepayment risk and extension risk, the risk that loans repay faster or slower than expected or are extended past maturity when borrowers are in distress. There is also refinancing risk: at the deal’s maturity there may not be enough principal in the pool to redeem the bonds at par unless the remaining loans can be refinanced, and that risk is greater if principal has leaked to the equity note along the way. Gradual amortization of time-tranched senior bonds is one defence.

Default for a tranche is a matter of degree

A corporate or sovereign default is binary, triggered by a missed payment, usually against a hard maturity date. A securitization bond loses only what pool losses, filtered through the waterfall, push through its credit enhancement, and the legal maturity is so distant that it seldom triggers anything. The loss given default of a tranche can therefore be anything from tiny to total.

Because of this graded behaviour, rating agencies and investors speak of material impairment rather than default. A common definition is interest left unpaid and uncured beyond a few months, or collateral performance so poor that interest or principal payments are likely to stop.

The waterfall is the rulebook for distributing the collateral’s cash, named for the way payments cascade from the top: fees and expenses usually first, then everything owed to the senior bonds, and only then anything for lower tranches. Because of the equity note, the bonds fund less than the whole pool, so every deal begins with some hard overcollateralization. Soft mechanisms then use part of the collateral interest to retire senior debt early or fill a reserve for the senior tranches, front-loading protection into the early years when collateral cash flows are strongest. Detailed overcollateralization triggers spell out when excess spread must be diverted into which reserve.

The example CLO

The rest of this lesson follows one stylised CLO. Its collateral is 100 identical leveraged loans of $1,000,000 par each, bought at par, paying a floating rate of one-month Libor plus a fixed spread of 3.5 percent. The deal pays no fees of any kind. Loans and bonds all mature in five years, every payment is annual at year-end, and the Libor swap curve is flat at 5 percent.

The interest lines follow directly from the coupons: the pool pays 0.085 × 100,000,000 = $8,500,000, the mezzanine bond is owed 0.10 × 10,000,000 = $1,000,000 and the senior bond 0.055 × 85,000,000 = $4,675,000. Together the bonds are owed B = $5,675,000 a year, leaving $2,825,000 of excess spread when nothing defaults. The mezzanine bond’s initial credit enhancement is the $5 million equity note; the senior bond’s is the $15 million of equity plus mezzanine. There is no overcollateralization at the start, and the coupons are set so that both bonds price at par.

Identical maturities are a big simplification. Term funding is one of the main reasons to securitize, yet perfect maturity matching is rare in practice, and maturity transformation remains a pervasive problem in finance.

Why the timing of defaults matters

If defaults strike at a constant rate and defaulted loans are not replaced, the number of defaults each year shrinks along with the pool, so cumulative defaults rise ever more slowly. At 5 percent a year, a pool of 100 loans would expect about 5 defaults in the first year, but by the time only 80 loans are still performing, the same rate implies about 4. Whatever the default rate, collateral cash flows (interest and amortization from survivors plus recoveries) are larger early in a deal’s life than late.

The timing of defaults strongly shapes tranche returns. Accelerating defaults raise the risk of a principal shortfall at maturity for the bonds, and decelerating defaults lower it. Other things equal, the equity tranche does relatively better when defaults arrive late, because it has already collected several years of excess spread before losses bite.

A securitization is brought to market by a cast of specialised firms, and almost every one of them faces incentives that do not line up perfectly with those of the investors who end up holding the risk.

Loan originator

The originator makes the loans that end up in the pool. It is often a bank (for bank loans or card receivables, say), though specialty finance companies and mortgage lenders also originate. An intermediary that originated most of the pool is called the sponsor or seller.

Underwriter

The underwriter, also called the arranger and frequently a large financial firm, assembles the loans, designs the structure and markets the bonds, which makes it the issuer of the securities; the legal term for this role is depositor. While loans are being accumulated the underwriter carries warehousing risk, the danger that the deal falls through and the collateral sitting on its balance sheet loses value. This mattered early in the subprime crisis, when markets realised how many “hung loans” intermediaries were holding. Underwriting in its narrow, classical sense means holding the finished bonds until investors buy them and bearing the risk that some cannot be sold at par.

Rating agencies

Agencies rate the liabilities, and for securitizations that job includes setting attachment points and the amount of subordination each rating needs. Unlike a corporate rating, where the agency comments on creditworthiness it cannot change, a structured rating shapes the deal itself. Issuers usually pay for ratings, so an agency’s wish to win business can collide with its duty to be objective, and its role in the design makes this worse. Whether the agency tells the issuer how much enhancement an investment-grade top tranche needs, or the issuer guesses beforehand, the pressure runs toward less enhancement and a larger slice of highly rated bonds. Those senior bonds carry the lowest spreads and the widest investor base, so they are central to the economics. Investors can protect themselves by demanding wider spreads or doing their own credit work.

Some bonds earn higher ratings from a third-party guarantee or wrap rather than from the pool alone. Such guarantees came mainly from monoline insurers, highly rated and well-capitalised firms that earned fees by lending their rating. Wraps were common until the subprime crisis brought heavy losses and widespread downgrades to the monolines.

Servicers and managers

The servicer collects interest and principal from borrowers, pays the bondholders and the fees of the underwriter and itself, may have to advance payments when loans are in arrears, and often decides how distressed loans are resolved. Those decisions can pit the servicer against bondholders, or one class of bondholders against another.

A distressed CMBS loan shows the tension. Foreclosing and selling the property, even at a loss, typically hands the senior bond an early return of principal and no writedown, while the junior bond is written down at once. Extending the loan spares the junior bond an immediate loss and keeps alive some chance of recovery, but exposes the senior bond to a deeper eventual loss that erodes its credit enhancement. The servicer must maximise the loan’s total present value, yet any choice will favour some classes over others.

Managers of actively managed pools face similar issues, because investors delegate credit monitoring to them and need incentives aligned. One common remedy, applied to static pools too, is to make the manager hold a first-loss piece of the deal, an idea written into the Dodd-Frank Act reforms. Conflicts are sharper where, as often with mortgages, the servicer is not the originator, and a third-party servicer is one more firm whose soundness investors must check. The trustee and custodian, economically minor, keep records, verify documents, move cash between deal accounts and pay noteholders.

The example CLO is designed around an expected default rate of 2 percent a year. Much higher defaults could cost the bonds money; much lower defaults would make the equity worth more than investors need to hold it at par. The analysis runs in two parts, because the first four years behave very differently from the final year.

The overcollateralization account and the other assumptions

Instead of paying all excess spread to the equity, the deal diverts up to $1,750,000 a year into a reserve called the overcollateralization account, which earns the money market rate of 5 percent. The name is a slight misnomer, because the account is soft rather than hard enhancement. If excess spread falls short of $1,750,000, whatever there is goes to the account; anything above $1,750,000 is paid to the equity. The account pays bond interest whenever collateral interest cannot, and whatever remains is released to the equity only at maturity. This is not a strong defence for the senior bond, but unless defaults are heavy early on, the account fills while defaults are still low.

Three more assumptions complete the model. A loan that defaults during a year pays no interest that year, and interest is paid only at year-end by survivors. Each defaulted loan recovers 40 percent of par, and that recovery is deposited in the account to earn 5 percent as well. Escrowing recoveries protects the senior bond, since recoveries passed down the waterfall would let the equity profit from defaults; in real deals recoveries usually pay down the senior bonds, then the mezzanine, with time tranching steering early recoveries to short bonds. Writedowns are deferred to maturity.

With defaults, interest from surviving loans and recoveries for each interim year t = 1, …, T − 1 are

Excess spread is L_t − B. When defaults are heavy it turns negative, and the bond coupon must then be paid partly or wholly from the account.

Even in this simple structure the custodian cannot just pay out cash; at the end of every interim year it runs a short sequence of tests to decide how much excess spread goes into the overcollateralization account. The tests form a two-level decision tree with two branches at each level.

When excess spread is positive and at least K, exactly K is trapped and the rest goes to the equity (nothing is left over only if excess spread equals K exactly). When it is positive but below K, the whole amount is trapped and the equity gets nothing; if it is exactly zero, the coupon is just covered and nothing enters the account. When excess spread is negative, the custodian checks whether the account balance, grown at 5 percent, plus this year’s recovery covers the coupon gap B − L_t. If so, the account pays the gap; if not, the whole balance plus recovery is paid out and the account is emptied. Writing A_t for the account balance at the end of year t, the rules are

The interim cash flows in three default scenarios

Running the rules for constant annual default rates of 2.0, 7.5 and 10.0 percent of the surviving pool, with default counts rounded to whole loans, gives the table below. Every row was recomputed from the formulas; the last column is the account balance.

Excess spread shrinks every year in every scenario as survivors dwindle. At 2 percent the cap of $1,750,000 is met in full each year and the equity collects something every year. At 7.5 and 10 percent the cap stops binding after year 1, so all remaining excess spread is trapped and the equity’s interim income stops. In the 10 percent case excess spread turns negative in year 4, and the account pays the $65,000 coupon gap: its balance of 15,119,875 grows to 15,875,868.75, loses 65,000, and gains 2,800,000 of recovery to reach $18,610,869. Because recoveries are held back and the account can always fund the coupon in these scenarios, the bonds cannot “break” before maturity. Real trust agreements usually allow an early unwind in extreme cases to protect the bonds from further loss.

In year 5 the deal stops trapping cash and everything available goes to pay the final coupons and redeem the bonds in order of priority. Four sources of funds arrive at the end of the final year: interest from surviving loans, par redemptions of those loans, recovery on loans that default in year 5, and the account balance grown by one more year of interest. Adding them gives the terminal available funds.

The bonds are owed $100,675,000 at maturity: $85,000,000 of senior par plus its $4,675,000 coupon, and $10,000,000 of mezzanine par plus its $1,000,000 coupon. The custodian tests the senior claim first, then the mezzanine claim on whatever is left.

Redemptions of surviving loans dominate the available funds. At 2 percent both bonds are paid in full and the equity receives a large final payment. At 7.5 percent, F covers the senior claim, but only 93,701,525 − 89,675,000 = $4,026,525 is left for a mezzanine claim of $11,000,000, so the mezzanine bond loses its $1,000,000 final coupon and $5,973,475 of principal. At 10 percent, F falls $3,318,588 short of the senior claim, which is less than the senior coupon of $4,675,000, so the senior bond loses part of its final coupon but none of its principal, while the mezzanine bond loses its entire $11,000,000 claim.

Measuring the equity’s credit risk through its IRR

The equity is owed nothing, so a shortfall measure does not apply. Its risk shows up instead as a lower internal rate of return, possibly below zero if not even the original investment comes back. An equity investor has a hurdle rate, the target return that compensates for risk given other uses of capital, and a return that is positive but below the hurdle is still an economic loss. Discounting the equity cash flows at a chosen hurdle rate gives a dollar price, so price and IRR can be used more or less interchangeably. The initial outlay is $5,000,000, assumed bought at par at a market-clearing expected return. If the originator also sponsors the deal and keeps the equity, that $5,000,000 is simply the part of the loans not funded by the bonds.

These scenarios are deliberately extreme, and the arithmetic of the waterfall is only half the job. The other half is judging how plausible a 10 percent annual default rate is: whether it is unlikely but possible, or close to impossible. That judgement needs a credit assessment of how the specific loans would default across economic conditions, and a decision about how adverse an environment to plan for, which depends on economic analysis and on the investor’s risk appetite.

Fixed default scenarios say nothing about default correlation, the tendency of defaults to bunch together, and without it there is no probability distribution of tranche losses. Simulating correlated default times for every loan in the pool, and pushing each simulated history through the waterfall, produces the whole distribution, which serves both valuation and risk measurement. The method is the copula simulation used for small credit portfolios, scaled up to 100 loans.

The four steps

1. Estimate parameters: a default probability or default distribution for each loan, and correlations that tie the individual default distributions together.
2. Simulate default times: use the copula to draw a correlated default time for every loan in every simulation thread, then record whether and in which year each loan defaults within the deal’s life.
3. Generate cash flows: convert each thread’s defaults into collateral cash flows year by year, exactly as in the scenario analysis, except that the default counts now come from the simulation.
4. Allocate and summarise: run every thread through the waterfall to get each bond’s loss and the equity’s cash flows, giving full distributions of bond losses and equity returns from which statistics such as Credit VaR follow.

Parameters

Default probabilities may be physical or, where comparable spreads exist, risk-neutral. A pool of N loans needs up to N probabilities, plus a term structure if hazard rates vary over time. Here every loan has the same one-year physical default probability π, as if taken from a rating, and it is converted into a constant hazard rate.

Copula correlations are harder, because they have no intuitive meaning and little data to estimate them. The only restriction imposed is that they are non-negative, and the example uses one constant pairwise correlation ρ for every pair of loans. To study both inputs, π runs from 0.0075 to 0.0975 in steps of 0.0075 (13 values) and ρ takes the values 0, 0.3, 0.6 and 0.9, which makes 52 parameter pairs.

From correlated normals to default times

For each correlation, draw I = 1,000 rows from a 100-dimensional standard normal distribution whose covariance matrix has ones on the diagonal and ρ everywhere else. Each row is one simulation thread. In the implementation behind the example, with ρ = 0.30, the top-left corner of this matrix reads −1.2625, −0.3968, −0.4285, −1.0258 in the first row and −0.3778, −0.1544, −1.5535, −0.4684 in the second; any other random number generator would give different values. Each normal draw is mapped to a default time:

The first four default times in the first four threads for this parameter pair are 4.7951, 18.6433, 17.8702, 7.2705; 19.1194, 25.3727, 2.7262, 16.9309; 39.3599, 24.6542, 17.6515, 15.5915; and 12.3274, 14.5882, 18.7161, 24.9461, so two of these 16 fall within five years. One matrix of normals per correlation is enough; changing π only rescales the default times. The 52 resulting matrices are each 1,000 by 100.

Counting defaults year by year

For each thread, count how many of the 100 default times fall in each interval (t − 1, t] for t = 1 to 5, giving a five-element vector of annual defaults and its cumulative sum. In the first thread for π = 0.0225 and ρ = 0.30, the 100 simulated times produce annual counts of (11, 5, 7, 7, 7) and cumulative counts of (11, 16, 23, 30, 37), one of the grimmer threads. These vectors play the role that the assumed default counts played in the scenario tables.

A practical advantage is that changing the recovery rate, the collateral coupon or the liability structure needs no new simulations, only reprocessing. Fresh draws are needed only to add threads for accuracy, to change the number of loans, or to try a correlation outside the set already simulated.

Averaging over the 1,000 threads for each parameter pair gives mean results. For low default probabilities, the mean equity IRR is above 30 percent a year; for high default probabilities with low correlation, the equity is wiped out in a large share of threads.

Risk statistics need dollar values, so the equity is valued by discounting its simulated cash flows at an assumed hurdle rate of 25 percent, a choice similar in spirit to setting market-clearing bond coupons. At this rate the equity is worth roughly its $5,000,000 par when π = 2.25 percent and ρ = 0.30. Each thread’s value is

Bond writedowns are shown as a percentage of each bond’s par so the mezzanine and senior bonds can be compared. Because the writedown includes unpaid final coupons, the mezzanine figure can exceed 100 percent: its maximum claim is $11,000,000, or 110 percent of par.

Means do not add up across tranches even though losses add up thread by thread. At π = 0.0075 and ρ = 0.30 the senior bond shows a mean writedown of 0.05 percent although the mezzanine mean is only 1.11 percent. The senior loss comes from a handful of threads in which defaults cluster so heavily that the mezzanine is wiped out completely; in only 19 threads does the mezzanine lose anything, so its average stays low.

Four patterns in the means

Default probability hurts everyone. For any fixed correlation, a higher π raises both bond writedowns and lowers the equity’s IRR and value. This is the unsurprising result.

Correlation splits the capital structure. At low π a rise in correlation changes little, but at high π it raises the equity’s value and IRR materially while raising the senior bond’s losses. The equity benefits from correlation and the senior bond is hurt by it.

The mezzanine bond changes character. At low π, more correlation increases its losses, as for a senior bond, because reaching its 5 percent attachment point is unlikely without clustering. At high π, more correlation reduces its losses, as for the equity, because a breach is already likely and correlation adds threads with few defaults.

Convexity. At low correlation, the equity value is strongly positively convex in π: it loses value quickly as defaults first rise, then less and less once most of the investment is gone, since there is only so much left to lose. The senior bond’s losses are negatively convex at low correlation, accelerating as defaults rise. The mezzanine is negatively convex at low π and positively convex at high π. At high correlation every tranche responds more nearly linearly.

Histograms of all 1,000 simulated outcomes reveal far more than the means. The key is how correlation shapes defaults: averaged across threads, the default count is about what the default probability implies, but with high correlation the defaults come in clusters, so some threads see unusually many defaults and others unusually few. The higher the correlation, the more of these extreme threads there are.

Shapes of the tranche distributions

Equity. With low correlation, the simulated equity values form a bell curve centred higher for lower default probabilities; for high default probabilities the bell is squashed against zero, because the tranche is wiped out before it collects much. With high correlation, the distribution becomes U-shaped. Both a near-total wipeout and a long stream of cash become more likely, so the equity behaves like a lottery ticket. When defaults are expected to be low, correlation raises the chance of a disastrous state and cuts the value; when defaults are expected to be high, it raises the chance of a benign state and lifts the value. Under low correlation some defaults happen in nearly every thread, and since the equity bears the first loss, at least some equity loss is almost certain. Correlation is therefore more decisive for the equity’s value and risk than the default probability.

Bonds. Bond losses are not bell-shaped even at low correlation. Subordination piles the outcomes up at or near zero, with a long tail of loss threads; for the senior bond, nearly every thread shows a zero or small loss unless both π and ρ are high. As π and ρ rise the loss distribution tends toward a U shape, much more strongly for the mezzanine. Because the mezzanine is thin, at low correlation a modest rise in π moves the bulk of its distribution from no loss to total loss, as a comparison of π = 0.0375 with π = 0.0975 at ρ = 0 shows.

Computing Credit VaR from sorted simulations

Sort each tranche’s 1,000 simulated outcomes. The 1 percent tail is the 10th worst result and the 5 percent tail the 50th worst. Credit VaR is the distance from the mean to that quantile.

For the equity, VaR falls as π and ρ rise because so much loss is already expected that the mean sits close to the floor; in many cells the 1 percent quantile is zero and the VaR simply equals the mean value. The mezzanine’s risk resembles the equity’s at high π and ρ and the senior bond’s at low values; because it is thin, its 95 and 99 percent VaRs are very close, meaning that if it loses at all it tends to lose a lot. The senior bond is nearly riskless when defaults are uncorrelated, even at high π, but at high correlation its 99 percent VaR is on the order of half its par. At ρ = 0.90 that VaR barely moves with π, because clusters of defaults in a few threads guarantee at least 1 percent of extreme outcomes regardless.

One cell is odd: the senior VaR at (π, ρ) = (0.0075, 0.30) is negative. Almost every thread, including the 10th worst, pays the senior bond in full, yet a few threads show losses, so the average loss is larger than the loss at the 1 percent quantile and the computed VaR comes out below zero, an artifact of simulation. Measuring at 99.5 or 99.9 percent would remove it, but higher confidence levels need many more simulations to keep simulation error under control.

Higher default probabilities lower every equity value and raise every bond loss, but how strongly depends heavily on the correlation and on where π already stands. The default01 measures that responsiveness: the change in a tranche’s mean value or mean loss for a 1 basis point rise in the default probability. It plays the same role for default risk that DV01 plays for rates and spread01 for spreads, and it is computed numerically in the same way.

Shift π up and down by 10 basis points and rerun the valuation from the default-time step onward. The correlated normals z should be reused rather than redrawn, so that a change of random seed does not add simulation noise, but the default count vectors, the cash flows, the valuation and the losses must all be recomputed twice. The central difference is then scaled to one basis point:

Every default01 is positive, since values fall and losses rise monotonically with π, and each would shrink toward zero at very high default rates because once losses are enormous a few more defaults change little (the range shown is not high enough to see this for the senior bond). The default01 varies most when correlation is low. With ρ = 0, each tranche’s sensitivity peaks where pool losses approach its attachment point: at π of zero for the equity, around π = 0.0675 for the mezzanine with its peak of 0.2593, and still rising at 0.0975 for the senior bond. This is a hazard when positions are taken in a low-correlation environment or when correlation is underestimated, and underestimated default correlation was an important ingredient of the subprime crisis.

Some sensitivity curves are jagged, which carries two lessons. First, simulation is an expensive way to value credit portfolios: default counts are integers, and the loss distribution is so skewed and fat-tailed that noise worth one or two defaults moves the average noticeably, especially for the senior bond and at high correlation where defaults arrive in clusters. Only many more simulations, at a heavy cost in computing time and storage, would smooth the curves. Second, the same fat tails make credit portfolios hard to diversify: even with 100 loans, defaults remain lumpy events and idiosyncratic risk does not wash out.

Means, distributions, Credit VaR and default01 together support a few general conclusions about tranche risk, all reached by modelling the structure and waterfall while treating the credit analysis of the loans broadly.

Systematic risk. A tranche can carry heavy systematic risk even when the collateral is well diversified. High default correlation is one way of saying there is a small but real chance of a bad state in which unusually many loans default together. The senior bond is the clearest case: its expected loss may be lower than the pool’s, yet with high correlation its loss tail and Credit VaR are large, so it is heavily exposed to the economy as a whole. How exposed depends on the collateral’s credit quality and on the credit enhancement.

Tranche thinness. The gap between a tranche’s 99 and 95 percent Credit VaRs reveals how thin it effectively is. For the mezzanine that gap is small almost everywhere, and for the senior bond it narrows as π rises under high correlation. A thin tranche, conditional on losing anything, is likely to lose a large share of its par.

Granularity. A pool of a few large loans carries more risk of extreme outcomes than a pool of the same par split into many small loans with identical default probabilities, recoveries and correlations. So a more granular pool can support a somewhat larger senior tranche for the same Credit VaR. Many CMBS deals are not granular, holding relatively few mortgages on large properties, as are fusion deals that combine many small loans with a few large ones. A pool that lacks granularity, has high correlation, or both, is said to carry high concentration risk.

Tranche prices contain the market’s view of portfolio characteristics, default correlation included. Running the valuation model backwards from observed prices, instead of forwards from an assumed correlation, yields an implied credit correlation (or implied default correlation), a risk-neutral parameter available whenever portfolio credit products trade.

Credit index default swaps

Credit index default swaps, or CDS indexes, reference a portfolio of single-name CDS rather than one company. Two families dominate: CDX (CDX.NA) on North American companies and iTraxx on European and Asian companies, both administered by Markit, alongside customised indexes built for particular clients. New series are launched every six months and numbered; CDX.NA.IG.10, for example, began in March 2008. Each series offers standard maturities of 1, 3, 5, 7 and 10 years on fixed calendar dates, and credit-quality versions such as the investment-grade CDX.NA.IG, the high-yield CDX.NA.HY and narrower subsets.

Each IG series holds equal notional CDS on 125 investment-grade names, so $125,000,000 of the index contains $1,000,000 of protection on each. Membership changes between series as firms gain or lose investment-grade status or merge and spin off. Unlike single-name CDS, the index spread premium is fixed when the series starts, so later trades exchange a present value: if spreads have widened since launch, the protection buyer pays the seller an upfront amount. When a constituent defaults it leaves the index, the contract notional and premium shrink by 0.8 percent (one of 125 equal weights), and the protection seller pays 0.8 percent of notional minus the recovery value, which is usually set by auction with cash settlement.

Standard tranches

Writing a synthetic CDO on the 125 CDX constituents creates something economically close to a cash CDO holding equal par amounts of the 125 firms’ bonds. The resulting liabilities, called standard tranches, follow a fixed capital structure and trade fairly liquidly, in contrast to bespoke tranches issued to order for a client’s chosen names, maturity and slice of the capital structure. iTraxx tranches are similar with somewhat different boundaries, and arbitrage ties every tranche’s value to the constituent CDS.

Backing out the implied correlation

Standard tranches are valued with the same copula machinery, with one change. Instead of physical default probabilities, the 125 constituents’ observable CDS spreads, often along the whole term structure, give risk-neutral default probabilities, and recovery swaps may give a risk-neutral recovery rate. The one remaining unknown is the constant pairwise copula correlation. It cannot be observed directly, but because the tranches trade daily it can be solved for: for the equity tranche, take its market price (or compute it from points upfront plus running spread), derive each constituent’s risk-neutral default probability, simulate the equity cash flows, and find the unique correlation at which their present value, discounted at risk-free rates, equals the market price.

A correlation solved this way is called a base correlation, because it is tied to a particular tranche’s attachment point. Different tranches generally imply different correlations, a pattern known as correlation skew.

The example CLO illustrates the idea. Suppose the equity trades at $5 million and CDS spreads imply a risk-neutral default probability of roughly 2 percent, the 0.0225 row of the mean equity value table. A correlation of 0.3 reproduces the price (the table gives 4.98). If the price rose to $5.6 million with no change in default probability, the implied correlation would have risen to about 0.6 (5.61 in the table), signalling that the market now sees more systematic risk in the loans.

Implied correlation is as much a market-risk idea as a credit-risk one. Each standard tranche has a risk-neutral spread01 to each of the 125 constituents rather than a default01, measuring mark-to-market risk, and each depends on the implied correlation, which itself moves along with index and constituent spreads. In the cash CLO, changes in default rates and correlation alter expected cash flows and losses, that is, fundamental value; in standard tranches, moves in risk-neutral default probabilities and implied correlation show up as mark-to-market gains and losses.

The word correlation covers several different quantities in credit risk, and mixing them up is a common source of error.

Default correlation is the concept that bears most directly on portfolio credit risk: the correlation coefficient between the random variables describing whether each of two firms defaults over a given horizon. Asset return correlation is, strictly, the “wrong” quantity, since it is not about default, yet it is the right input in a Merton-type model in which default happens when asset value falls far enough; in a factor model it comes from the firms’ loadings on the common factor. Asset values cannot be observed directly, so equity return correlation is commonly used as a proxy.

Copula correlations fill the off-diagonal entries of the correlation matrix inside a copula model. They have no direct economic meaning and depend on which family of distributions the copula uses, although the correlation of a Gaussian copula equals the correlation in a Gaussian single-factor model. The Gaussian copula is close to an industry convention, and since standard CDS index equity tranches depend as heavily on correlation as on spreads, their values can be quoted in terms of it. Spread correlation links changes, usually in basis points, in two firms’ spreads on comparable debt and is a mark-to-market concept. Implied credit correlation is not a separate theoretical quantity but the copula correlation inferred from market prices rather than estimated; like spread correlation, it is a market concept.

Securitization happens because it serves both the originators who sell loans into the trust and the investors who buy the bonds and equity. Those motives also explain the regulatory questions it raises and its part in the subprime crisis.

Issuer motivations

The central motive is maturity matching: term funding for loans. Securitizing only makes sense if it is cheaper than the next-best alternative, which is either keeping the loans and funding them with unsecured borrowing, usually costlier than secured funding through a securitization, or selling them in the secondary market. Deals aimed mainly at capturing the gap between loan interest and liability coupons are called arbitrage CDOs, and deals done largely for balance sheet relief are balance sheet CDOs, although in practice the two are hard to tell apart.

Diversified pools and an originator’s reputation for careful underwriting lower the spreads on the liabilities, and originators whose earlier deals disappointed may pay persistently wider spreads. These same factors let the issuer reduce the credit enhancement on the cheapest senior bonds, borrowing more through the deal at a lower weighted-average cost. Residual idiosyncratic risk limits the funding advantage in some sectors: credit card and auto pools can be made highly granular, commercial mortgage pools are hard to diversify, and both commercial and residential mortgages carry systematic risk that many participants and regulators badly underestimated before the subprime crisis. If the issuer keeps the servicing rights and enjoys economies of scale in servicing, securitization also raises servicing profits, so the deal stays attractive even at higher liability rates.

Outright sale is easier for big corporate and commercial property loans, which have secondary markets with buyers prepared to monitor, than for most consumer loans. The obstacle is asymmetric information combined with the need for monitoring: loan quality is hard to judge and track, the originator usually knows the borrower best, and its very decision to sell can signal that a loan is worse than disclosed, the “lemons” problem. Retaining the equity or other subordinated pieces, or a slice of the loans, gives the originator or another capable monitor a first-loss stake that rewards careful selection and monitoring and protects senior investors too. Risk retention of this kind was written into the Dodd-Frank reforms and is expected to stop most issuers from selling every tranche. Legal representations about underwriting standards give buyers recourse if they prove false, and many such claims followed the crisis. The need for these mitigants shows how hard genuine credit risk transfer is, however robust the legal sale, and the same ambiguity in credit derivatives and off-balance sheet vehicles makes it harder for markets to see issuers’ true asset volume and leverage.

Investor motivations

Securitization lets capital-market investors hold diversified pools of mortgages, credit card and auto loans that would otherwise belong to banks alone. A pass-through offers only that diversification; credit tranching adds risk sharing, and it is investors, not issuers, who drive tranching beyond the issuer’s retained interest. Thin subordinate tranches attract investors seeking high risk and return, some tranches carry embedded leverage, and thick senior tranches attract investors who want low risk in most states and will accept severe losses in very bad states for extra yield. Higher tranches gain lower default risk at the price of greater exposure to systematic risk.

These benefits hold only for investors who do the due diligence to understand the return distribution. Institutions such as pension funds deploy large sums against return targets and want high-quality bonds yielding even slightly more than government or top-rated corporate debt, a behaviour known as reaching for yield. Much of securitization rested on that demand for senior bonds, and with regulatory safe harbours and weak governance it encouraged inadequate scrutiny of systematic risk.

Mezzanine tranches fit awkwardly. Depending on default probability, correlation and thickness, one may behave like a senior bond, with a low chance of loss but high systematic risk and a large loss if impaired, most likely in a bad economy, or like equity, with a high chance of impairment but a decent chance of only a small loss, and it may switch between the two. With no natural investor base, many were packaged into CDOs whose senior bonds were sold to yield-seeking investors buying on yield and rating alone.

Fees motivate intermediaries as well. Originating and perhaps servicing loans can pay more than holding them, and securitization moves the loans off the balance sheet. This shades into regulatory arbitrage, where an intermediary keeps much of a pool’s risk and return while sharply cutting its regulatory capital. Off-balance sheet vehicles, and through them money market mutual funds, were also major buyers of securitization bonds.