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hanks to a lack of liquidity in the secondary markets, many companies have been left hold-ing a portfolio of structured products, includ-ing collateralized loan obligations (CLOs). Decreased liquidity, combined with recent ac-counting rule changes, has prompted a number

of companies to develop models to help them determine the fair value of these securities.

These models are highly dependent on inputs, which can be quite difficult to determine, as many are not readily observ-able in the market and are prospective in nature. Additionally, there are a number of modeling techniques available for valu-ing these securities, and choices must be made based on the strengths and weaknesses of each model, familiarity with the credit behavior of the collateral and the overall sophistication of the valuation group.

This article will demonstrate how CLO securities can be priced using both static and stochastic modeling approaches, contrast the results of these two common CLO valuation methodologies and discuss their respective limitations.

CLO Market OverviewA CLO is a special purpose vehicle (SPV) that issues debt and equity commonly collateralized by bank loans. Unlike mort-gage-backed securities (MBS), a collateral manager actively manages the collateral pool in a CLO during what is referred

Pros and Cons of Different CLO Models

Every firm that holds a portfolio of illiquid collateralized loan obligations (CLOs) must figure out the best way to price these structured products.

Both stochastic and static default models can be used to value CLOs, but what are the respective strengths and weaknesses of these approaches?

By Anthony Sepci, DuShyAnth KriShnAmurthy AnD chriStiAn eDer

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to as the revolving period, typically ranging from five to seven years, after which the principal paydown of the liability struc-ture begins.

To value CLO securities, market participants commonly use a discounted cash flow approach. Given the potential for diversity in the collateral pool, varying collateral contractual cash flow terms, complex credit and prepayment forecasting (as well as structures that revolve), this valuation approach can be complex. Further, the model also needs to account for deal-specific payment rules, excess interest tests, subordination tests and credit trigger events dynamically.

The more common method for modeling these transactions is a static approach where a single best estimate of collateral cash flows, prepayments and defaults is made. An alternative method is to model the cash flows for these securities using a stochastic default model that involves Monte Carlo simula-tions that generate multiple default scenarios by modeling the default behavior of the underlying collateral.

CLO StructuresAs mentioned earlier, a collateral manager actively manages a CLO’s collateral pool by purchasing and selling loans as per-mitted in the deal’s indenture. Collateral loans may consist of term loans, revolving credit agreements, acquisition or equip-ment lines, bridge loans, second-lien loans and covenant-lite loans. Some CLO structures allow the collateral manager to

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as the applicable reinvestment income. All fees and expenses must be accounted for because they reduce available collateral cash flows. Also, valuation dates, current security balances and collateral amortization terms need to be updated to reflect am-ortizing conditions.

Step #2: Collateral Cash Flows. The model should at-tempt to amortize each loan according to its contractual terms, but adjust the amortization for projected voluntary prepay-ments, defaults, default recoveries, interest-rate term structure and reinvestment income. Prepayments and defaults are usu-ally specified as constant prepayment rate (CPR) curves and constant default rate (CDR) curves, respectively. These curves are a vector of percentages that indicate the percentage of current collateral expected to prepay or default in that period. The percentages are specified as a yearly amount and must be converted to a periodic equivalent before being applied to the collateral balance. The recovery assumption determines the amount of principal recovered from defaulted loans and the timing of those recoveries. The model must also make as-sumptions on the type of collateral that will be purchased dur-ing the revolving period.

Step #3: Bond Cash Flows. The model needs to allo-cate accurately the projected collateral cash flows to the liabili-ties based on the specified payment rules, trigger events and fees/expenses, as detailed in the deal indenture. Structures vary from one CLO to the next, resulting in differences in tranching and payment rules.

Step #4: Bond Analytics. Once security cash flows are projected, a number of analytics can be calculated such as du-ration, weighed-average life, convexity, yield, discount margin and (ultimately) fair value. To calculate the appropriate fair value or net present value, an appropriate discount rate/rate of return must be selected. A discount rate should account for the risk-free rate, credit risk, model risk and liquidity premium. For Monte Carlo simulated values, the average of all prices is taken.

Default ModelingDefault assumptions in a CLO valuation can be modeled as a constant default rate (CDR) or modeled stochastically us-ing advanced mathematical tools. In the case of static default (CDR) modeling, a certain percentage of the outstanding col-lateral balance is assumed to default in each period. The de-fault assumption is specified as a vector of yearly default rates called a CDR vector, which can be easily converted into a de-sired periodic rate.

However, defaulting a portion of a loan is not consistent with real world default behavior, where loans either default en-tirely or not at all (a binary event). Stochastic default modeling tries to incorporate this reality by modeling the approximate time that a loan defaults.

Simulation is the process by which one can generate possible outcomes of a random event. For example, if one wants to simulate throwing a die, one would first need the probabilities of obtaining the various possible outcomes one through six. One could then simulate the outcomes using a random number generator that produces one of the six numbers for each run, such that the probabilities are maintained (i.e., after a number of runs, on average one expects to get each number one-sixth of the time). Similarly, given a probability of default curve that specifies probabilities of a loan defaulting over time, one could simulate the time to default using a random number genera-tor, such that the probability distribution is maintained.

In order to simulate the time to default for each loan, the following is needed: (1) a default curve for each loan; (2) a cor-relation structure; and (3) a process for generating correlated time to default for each loan. (Most of the stochastic default modeling techniques discussed in this article can be found in a key paper authored by David X. Li.1)

Since the credit crisis began to unfold, there have been a number of criticisms hurled at this stochastic model. Some have even blamed the model as the cause of the credit cri-sis.2 However, though this method has serious limitations, it has, until recently, been the most widely used model for pric-ing3, and it is therefore worth examining the construction and shortcomings of this model.

Probability of Default Curves (Credit Curves)The probability of a default curve is one of the key inputs into the default model. It specifies the cumulative probability of default over time for a loan. It is general industry practice to vary the probability of a default curve based on the underlying loan credit quality. The expectation would be that higher qual-ity loans would have a lower probability of default over time when compared to loans of lesser credit quality.

Default curves can be derived using a number of methods. For example, Moody’s has a table of historic probability of default curves based on credit ratings. Alternatively, default curves can be generated from market information such as bond spreads or credit-default swap (CDS) spreads.

One can also adjust historic default probability curves using market composites, such as the North American Loan Credit

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invest in other CLO securities, resulting in the notorious “CLO Squared” securities.

CLO securities have varying capital structures, but gener-ally feature debt that is tranched using seniority related to re-ceiving interest and principal payments and the distribution of losses. A typical CLO structure may have super senior, senior, mezzanine and subordinate bonds, as shown in Figure 1 (right). Each CLO features payment rules that specify how collateral interest and principal cash flows are distributed to the securities.

Additionally, CLOs have a revolving period during which principal cash flows from the collateral are reinvested rather than paid out to the securities. During this reinvestment pe-riod, the collateral manager purchases new bank loans or other eligible investments. The CLO structure in Figure 1 contains a pro rata split for the super senior bonds. All collat-eral principal is first paid to the super senior bonds, followed by the senior, mezzanine and subordinate bonds, forming the appropriately termed “waterfall.” The credit ratings and du-rations of these securities commonly parallel this waterfall, respectively running from high to low ratings and from short to long durations.

Collateral losses of interest or principal are typically first absorbed by excess spread, then by the lowest rated bond and then by more senior bonds, in reverse order of payment seniority. Collateral losses can potentially reach the senior bonds, depending on the loss levels incurred by the collateral pool.

In addition, CLO structures typically have interest and principal credit enhancement coverage tests to further pro-tect senior bonds from losses. When any of these tests fall below predetermined levels, collateral interest and principal cash flows are commonly redirected to the most senior secu-rities outstanding in an attempt to bring levels into compli-ance.

The CLO structure in Figure 1 exhibits how a coverage test breach can impact security cash flows. In the event of a coverage test failure, coupon payments to the super senior and senior security holders are generally protected. However, mezzanine and subordinate bond holders may not receive coupon payments during any payment periods where the coverage tests fail. Less senior security interest cash flows are diverted to pay off the principal of the most senior security outstanding, resulting in an early amortization.

Figure 1: Cash Flow Structure of a Typical CLO

Valuation TechniquesValuation of CLOs usually involves projecting cash flows for the underlying collateral using assumptions derived from his-torical collateral performance, market research, management’s judgment and deal-specific inputs such as fees and triggers. The payment waterfall structure in the deal indenture deter-mines how the projected collateral cash flows are allocated to the various liabilities making up the capital structure and the fee/expense requirements.

The resulting projection of cash flows for each security is then discounted using a risk adjusted discount rate to de-termine fair value. This valuation process is summarized in Figure 2 (below).

Figure 2: Steps Involved in CLO Pricing

Below, we provide a more thorough explanation for each of these important steps:

Step #1: Inputs & Assumptions. Initially the model needs to be set up to account for all deal-specific inputs. The interest rate term structure needs to be estimated to project interest yield for the floating rate loans and securities, as well

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simulations is taken.

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Default Swap (LCDX) index. The traded price of the index can be used to derive an implied probability of default. A scal-ing factor can be applied to the historic curve to match the im-plied probability of default implied by the LCDX index while maintaining the original historic curve shape.

Although a probability of a default curve can be a tool to analyze the likelihood of a loan defaulting over time, it cannot capture the correlation of defaults between loans. To do so, one can start by constructing a correlation structure (or ma-trix) that captures the correlation between each loan in the pool.

Correlation between any two loans can be estimated based on historical data, such as industry-level asset correlations. However, there is also a commonly held assumption that all assets have the same correlation, which can be thought of as the average correlation expected in the portfolio.

Once the correlation matrix has been determined, a copula function can be used to incorporate the correlation between the loans. The copula function is a multivariate distribution that makes it convenient to link marginal distributions and in-troduce dependencies between them. Details on copula func-tions can be found in the Li paper. The most commonly used copula function is the normal copula, which takes on the form of an N-dimensional standard normal4 distribution function with a correlation matrix .

Generating Time to DefaultsThe process of generating the time to default is shown in Figure 3 (right). One can start by generating a set of random numbers from an N-dimensional standard normal distribution with a correlation matrix , knowing that the generated ran-dom numbers are correlated. One can then perform an inverse mapping of the generated numbers back to the corresponding cumulative default probability. The determination of the time to default for each loan is done by mapping the cumulative default probability determined in the previous step to the cor-responding time from the loan’s probability of default curve.

Next, one can model the loan’s cash flows under the assump-tion that the loan continues to perform until the time to default is reached, at which point the entire loan goes into default. Any recovery estimate would be projected as a percentage of the aggregate defaulted amount and realized at a subsequent date, otherwise known as the recovery lag.

The remainder of the valuation process is similar to the stat-ic approach. This process results in a one-price outcome from a number of possible outcomes. In order to value the security,

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one would have to repeat the process a number of times to de-rive a distribution of possible prices for the security. The steps in the Monte Carlo simulation are summarized below:1. Generate random numbers from a correlated standard normal variable Z~MVN(0,) using a random number generator.2. Map Z’s to a cumulative probability Yi = -1 (Zi) for each of the N assets.3. Map the probability to an appropriate time-until-default as ti = F-1 (Yi) for each of the N assets.4. Project cash flows for loans and securities using assumptions and in accordance with waterfall rules. 5. Obtain a price for the security based on projected cash flows and an appropriate discount rate. 6. Repeat above steps multiple times to generate a distribution of prices. 7. Calculate the average of prices obtained from the simulations.

Figure 3: Process of Generating Time to Defaults for Collateral Loans

Hypothetical ExampleA hypothetical loan portfolio is used to demonstrate the dif-ferences between static and Monte Carlo valuations. The hy-pothetical portfolio consists of 10 B2-rated loans with varying maturity dates, equal balances, equal coupon spreads, an as-sumed recovery rate of 50 percent and an assumed prepay-ment rate of 10 percent CPR.

The CLO structure consists of five CLO bonds with a par-ity ratio of 110 percent. Figure 1 (pg. 44) summarizes the CLO structure used in this example. We further assumed that the cor-relation between any two loans is 0.75, and we used Moody’s historic default probability curve for B2-rated loans.

For static modeling, we examined the impact of different CDR assumptions on the CLO securities. We assumed a constant CDR rate for the life of the transaction and examined the effect of increasing the CDR on security losses and resulting prices.

Figure 4 (below) illustrates the impact of different CDRs on CLO bond losses. It is noticeable that as the CDR increases, losses are absorbed by the lowest bond first before working their way up the capital structure. Note that the bonds do not experience losses immediately as a result of overcollateraliza-tion (i.e., a 110 percent parity ratio) and excess spread. The subordinate bond experiences its first losses when the CDR exceeds 8 percent annually.

Figure 4: Impact of CDR on Bond Losses

Figure 5 (right) shows how losses impact the pricing of the various bonds. Prices were calculated by matching a discount margin to each bond’s accruing spread. As CDRs increase, triggers are breached leading to cash flows being redirected from subordinate bonds to senior bonds. This figure also high-

lights that even at a 100 percent CDR, the super senior and senior bond will retain some value, which is due to cash flows from recoveries.

Figure 5: Impact of CDR on Bond Prices

Figure 6 (below) highlights the impact of triggers within the CLO structure. For this, we tested the prices of one su-per senior bond by using a discount margin that was higher, lower, and equal to the accruing coupon margin. We noted that depending on whether the bond is valued at a premium or discount, prices decrease or increase, respectively.

The reason for this is that as coverage triggers breach, super senior bonds receive principal back faster, as additional cash is redirected from the subordinate bonds. Different coupon ac-cruing and cash flow discounting rates with faster principal payments will cause prices to vary.

Figure 6: Impact of Triggers on Super Senior Bond Prices

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Probability of Default CurveGenerate N random numbers for the N assets from an N-dimensional standard Normal Random distribution with correlation structure .

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For Monte Carlo simulation, we applied Monte Carlo anal-ysis to our hypothetical deal, using 100 default curve iterations. The distribution of losses impacting each bond is shown in Figure 7 (below). We can see that the super senior bond rarely faces severe losses in the majority of the 100 scenarios. The expected loss is the average of the 100 scenarios.

As with the static CDR approach, bonds experience losses according to seniority. The key difference is that stochastic modeling creates various loss scenarios while the static default modeling creates only one outcome.

Figure 8 (right) shows the pricing distribution for the sto-chastic valuation approach. As with the static CDR pricing example earlier, discount margins were matched with accruing spreads. The final bond price is the average of the 100 pric-ing outcomes. Again, it is noticeable that triggers protect the bonds higher up in the capital structure and that recoveries also cause super senior bonds to retain value even in severe loss scenarios.

A benefit of using a stochastic pricing approach is that it presents a picture of possible prices at different loss severity outcomes. A static approach, where we are using the best es-timate CDR curve to forecasts future collateral losses, would provide us with only one price point.

Figure 7: Gaussian Copula Model — Bond Loss Distribution

Figure 8: Gaussian Copula Model — Bond Price Distribution

Model LimitationsThe copula-driven stochastic default model described in this article makes certain simplifying assumptions regarding the behavior of the underlying loans. It assumes that any two loans in the portfolio have a fixed correlation coefficient be-tween them. However, default correlations — which in essence measure how two companies are likely to fail at the same time — are time varying in nature. Additionally, the assumption of a Gaussian Copula structure assigns low probabilities to ex-treme outcomes.

Over the past few years, market participants have moved to-ward a single-factor stochastic default model that assigns a sin-gle correlation number between all assets in the collateral pool, partly because this model made it easy to back out an implied correlation from market prices found in, for example, CDS trades (which further translated into a simplified explanation of risk and pricing). This fueled the acceptance of single-factor model as a market standard, as it allowed more people to trade these securities and it made structuring new CLO transactions — including complex instruments like CLO/CDO squares — convenient. However, as can be implied from the epic failures of the credit crisis, people did not fully consider or understand the limitations of this model completely.

As a consequence of an extended period of benign credit performance and vast amounts of global liquidity, CLO credit loss models were calibrated to an overwhelming amount of positive economic data, resulting in low credit loss expecta-tions and lower probabilities of extreme events. Subsequently, capital markets rewarded CLO collateral pool diversification

accomplished through geography and industries (and even through varying asset-backed issuers), resulting in thinly capi-talized CLO structures and overvalued securities.

The stochastic and static default models both fail to address complex contractual loan features, such as loan conversion op-tions, refinance options, facility extension options, potential interest rate increases in the event of delinquencies and other drivers of collateral behavior.5 These features have always required a more traditional fundamental approach to credit analysis or more robust default modeling coefficients devel-oped through rigorous regression analysis utilizing full cycle data sets.

The stochastic model has also proven ineffective in hedg-ing, because it has been unable to capture large movements in price consistently. The rating downgrade of General Motors’ debt in May 2005 is one example on the ineffectiveness of delta hedging using this model.6

Closing ThoughtsIn spite of the limitations associated with the models described in this article, they are still commonly used for CLO valua-tions. Using a static CDR curve for CLO valuations is more commonly used by market participants for fair value account-ing purposes, due to its more simplistic approach and easier interpretation of results. Research is usually available to cali-brate market inputs such as losses and prepayment rates.

The stochastic modeling approach requires additional effort in setting up the model and determining the appropriate prob-ability of default curves, correlation matrices and discounting spreads. Stochastic models are more effective when one can regularly calibrate calculated prices to market prices.

Static modeling does result in reasonable valuations when assumptions are constructed appropriately based on contem-porary market consensus on credit, careful monitoring of both the collateral pool and CLO manager’s performances, corroboration (using quality industry research) and price com-parisons (using relevant trade proxies from both the secondary and primary markets).

One advantage of stochastic modeling is the consideration for distribution of outcomes. Stochastic modeling makes it pos-sible to analyze the distribution of possible prices, as apposed to a static valuation that produces a single outcome. Unlike simpler instruments such as interest-rate swaps, CLOs have complicated structures that include dynamic rules for allocat-ing cash and losses to the bonds, which can have a significant impact on both duration and pricing.

For example, a slight increase in losses could trip a trigger, thereby cutting off cash flows to a particular bond or reducing the life of another bond. A static valuation with a single out-come would not capture these nuances, but a stochastic model would be able to capture a distribution of possible loss out-comes and the implication that loan loss scenarios would have on the respective bonds in the capital structure.

There is a popular saying in statistics, “Essentially, all mod-els are wrong, but some are useful.”7 This should be kept in mind when interpreting results from any model. However, when faced with the task of choosing a model, one has to weigh complexity and cost, and make a determination on the most appropriate model for the application at hand while be-ing aware of the limitations a model presents to the user. That said, investors in structured collateralized products like CLOs can and should supplement credit loss modeling with sound, fundamental credit analysis and careful input from a collateral manager.

FOOTNOTES

1. D. X. Li. “On Default Correlation: A Copula Function Ap-proach,” The RiskMetrics Group, Working Paper (April 2000). Li’s method has become well known in the CLO industry. 2. F. Salmon. “Recipe for Disaster: The Formula That Killed Wall Street,” Wired Magazine (February 23, 2009).3. D. Duffie. “Innovations in Credit Risk Transfer: Implications for Financial Stability,” Stanford University paper (July 2007).4. A standard normal distribution is a mean zero and variance of one.5. “A Guide to the Loan Market,” Standard & Poors (October 2007).6. D. Duffie. “Innovations in Credit Risk Transfer: Implications for Financial Stability,” Stanford University paper (July 2007).7. George E.P. Box, a famous British statistician, is thought to be the original source of this saying.

Anthony Sepci is a partner in KPMG LLP’s financial risk management practice, based in Los Angeles. With more than 15 years of experience, he co-leads a team of more than 30 structured finance professionals. He can be reached at asepci@kpmg.com.

Dushyanth Krishnamurthy is a manager in KPMG LLP’s financial risk management practice. He can be reached at dkrishnamurthy@kpmg.com.

Christian Eder is a senior associate in KPMG LLP’s financial risk management prac-tice. He can be reached at ceder@kpmg.com.

The opinions expressed in this document do not necessarily represent the views of KPMG LLP.

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