Introduction: Structured Finance Models

Models used to govern the valuation and risk management of structured finance instruments take a variety of forms. Unlike conventional equity investments, structured finance instruments are often customized to meet the unique needs of specific investors. They are tailored to mitigate various types of risks, including interest rate risk, credit risk, market risk and counterparty risks. Therefore, structured finance instruments may be derived from a compilation of loans, stocks, indices, or derivatives. Mortgage-backed securities (MBS) are the most ubiquitous example of this, but structured finance instruments also include:

  • Derivatives
  • Collateralized Mortgage Obligations (CMO)
  • Collateralized Bond Obligations (CBO)
  • Collateralized Debt Obligations (CDO)
  • Credit Default Swaps (CDS)
  • Hybrid Securities

Pricing and measuring the risk of these instruments is typically carried out using an integrated web of models. One set of models might be used to derive a price based on discounted cash flows. Once cash flows and corresponding discounting factors have been established, other models might be used to compute risk metrics (duration and convexity) and financial metrics (NII, etc.).

These models can be grouped into three major categories:

  • Curve Builder and Rate Models: Market rates are fundamental to valuing most structured finance instruments. Curve builders calibrate market curves (treasury yield curve, Libor/Swap Rate curve, or SOFR curve) using the market prices of the underlying bond, future, or swap. Interest rate models take the market curve as an input and generate simulated rate paths as the future evolution of the selected type of the market curve.

  • Projection Models: Using the market curve (or the single simulated rate path), a current coupon projection model projects forward 30-year and 15-year fixed mortgage rates. Macroeconomic models project future home values using a housing-price index (HPI). Prepayment models estimate how quickly loans are likely to pay down based on mortgage rate projections and other macroeconomic projections. And roll-rate models forecast the probability of a loan’s transitioning from one current/default state to another.

  • Cash Flow Models and Risk Metrics: Cash flow models combine the deal information of the underlying structured instrument with related rate projections to derive an interest-rate-path-dependent cash flow.

The following illustrates how the standard discounted cash flow approach works for a mortgage-related structured finance instrument:

Cash Flow Models and Risk Metrics

Most well-known analytic solutions apply this discounted cash flow approach, or some adaptation of it, in analyzing structured finance instruments.

Derivatives introduce an additional layer of complexity that often calls for approaches and models beyond the standard discounted cash flow approach. Swaption and interest rate cap and floors, for example, require a deterministic approach, such as the Black model. For bond option pricing, lattice models or tree structures are commonly used. The specifics of these models are beyond the scope of this presentation, but many of the general model validation principles applied to discounted cash flow models are equally applicable to derivative pricing models.

Validating Curve Builder and Rate Models

Curve Builders

Let’s begin with the example of a curve builder designed for calibrating the on-the-run U.S. Treasury yield curve. To do this, the model takes a list of eligible on-the-run Treasury bonds as the key model inputs, which serves as the fitting knots[1]. A proper interpolator that connects all the fitting knots is then used to smooth the curve and generate monthly or quarterly rates for all maturities up to 30 years. If abnormal increments or decrements are observed in the calibrated yield curve, adjustments are made to alleviate deviations between the fitting knots until the fitted yield curve is stable and smooth. A model validation report should include a thorough conceptual review of how the model carries out this task.

Based on the market-traded securities selected, the curve builder is able to generate an on or off-the-run Treasury yield as well as LIBOR swap curve SOFR curve, or whatever is needed. The curve builder serves as the basis for measuring nominal and option‐adjusted spreads for many types of securities and for applying spreads whenever spread is used to determine model price.

A curve builder’s inputs are therefore a set of market-traded securities. To validate the inputs, we take the market price of the fitting knots for three month-end trading dates and compare them against the market price inputs used in the curve builder. We then calibrate the par rate and spot rate based on the retrieved market price and compare it with the fitted curve generated from the curve builder.

To validate curve builder’s model structure and development, we check the internal transition between the model-provided par rate, spot rate and forward rate on three month-end trading dates. Different compounding frequencies can significantly impact these transitions. We also review the model’s assumptions, limitations and governance activities established by the model owner.

Validating model outputs usually begins by benchmarking the outputs against a similar curve provided by Bloomberg or another reputable challenger system. Next, we perform a sensitivity analysis to check the locality and stability of the forward curve by shocking the input fitting knots and analyzing its impact on the model-provided forward curve. For large shocks (i.e., 300 bp or more) we test boundary conditions, paying particular attention to the forward curve. Normally, we expect to see forwards not becoming negative, as this would breach no-arbitrage conditions.

For the scenario analysis, we test the performance of the curve builder during periods of stress and other significant events, including bond market movement dates, Federal Open Market Committee (FOMC) dates and treasury auction dates. The selected dates cover significant events for Treasury/bond markets and provide meaningful analysis for the validation.

Interest Rate Models

An interest rate model is a mathematical model that is mainly used to describe the future evolution of interest rates. Its principal output is a simulated term structure, which is the fundamental component of a Monte Carlo simulation. Interest rate models typically fall into one of two broad categories:

  • Short-rate models: A short-rate model describes the future evolution of the short rate (instantaneous spot rate, usually written).
  • LIBOR Market Model (LMM): An LMM describes the future evolution of the forward rate, usually written. Unlike the instantaneous spot rate, forward rates can be observed directly from the market, as can their implied volatility.

This blog post provides additional commentary around interest rate model validations.

Conceptual soundness and model theory reviews are conducted based on the specific interest rate model’s dynamics. The model inputs, regardless of the model structure selected, include the selected underlying curve and its corresponding volatility surface as of the testing date. We normally benchmark model inputs against market data from a challenger system and discuss any observed differences.

We then examine the model’s output, which is the set of stochastic paths comprising a variety of required spot rates or forward LIBOR and swap rates, as well as the discount factors consistent with the simulated rates. To check the non-arbitrage condition in the simulated paths, we compare the mean and median path with the underlying curve and comment on the differences. We measure the randomness from the simulated paths and compare it against the interest rate model’s volatility parameter inputs.

Based on the simulated paths, an LMM also provides calibrated ATM swaption volatility. We compare the LMM’s implied ATM swaption volatility with its inputs and the market rates from the challenger system as a review of the model calibration. For the LMM, we also compare the model against history on the correlation of forward swap rates and serial correlation of a forward LIBOR rate. An LMM allows a good choice of structures that generate realistic swap rates, whose correlation is consistent with historical value.

Validating Projection Models

Projection models come in various shapes and sizes.

“Current Coupon” Models

Current coupon models generate mortgage rate projections based on a market curve or a single simulated interest rate path. These projections are a key driver to prepayment projection models and mortgage valuation models. There are a number of model structures that can explain the current coupon projection, ranging from the simple constant-spread method to the recursive forward-simulation method. Since it has been traditionally assumed that the ten-year part of the interest rate curve drives mortgage rates, a common assumption involves holding the spread between current coupon and the ten-year swap or treasury rates constant. However, this simple and intuitive approach has a basic problem: primary market mortgage rates nowadays depend on secondary-market MBS current-coupon yields. Hence, current coupon depends not just on the ten-year part of the curve, but also on other factors that affect MBS current-coupon yields. Such factors include:

  • The shape of the yield curve
  • Tenors on the yield curve
  • Volatilities

A conceptual review of current coupon models includes a discussion around the selected method and comparisons with alternative approaches. To validate model inputs, we focus on the data transition procedures between the curve builder and current coupon model or between the interest rate model and the current coupon model. To validate model outputs, we perform a benchmarking analysis against projections from a challenger approach. We also perform back-testing to measure the differences between model projections and actual data over a testing period, normally 12 months. We use mean absolute error (MAE) to measure the back-testing results. If the MAE is less than 0.5%, we conclude that the model projection falls inside the acceptable range. For the sensitivity analysis, we examine the movements of the current coupon projection under various shock scenarios (including key-rate shocks and parallel shifting) on the rate inputs.

Prepayment Models

Prepayment models are behavioral models that help investor understand and forecast loan portfolio’s likely prepayment behavior and identify the corresponding major drivers.

The prepayment model’s modeling structure is usually econometric in nature. It assumes that the same set of drivers that affected prepayment and default behavior in the past will drive them in the future under all scenarios, even though the period in the past that is most applicable may vary by scenario in the future.

Major drivers are identified and modeled separately as a function of collateral characteristics and macroeconomic variables. Each type of prepayment effect is then scaled based on the past prepayment and default experience of similar collateral. Assumed is that if the resulting model can explain and reasonably fit historical prepayments, then it may be a good model to project the future, subject to a review of the future projections after careful assessment.

Prepayment effects normally include housing turnover, refinancing and burnout[2]. Each prepayment effect is modeled separately and then combined together. A good conceptual review of prepayment modeling methodology will discuss the mathematical fundamentals of the model, including an assessment of the development procedure for each prepayment effect and comparisons with alternative statistical approaches.

Taking for example a model that projects prepayment rates on tradable Agency mortgage collateral (or whole-loan collateral comparable to Agencies) from settlement date to maturity, development data includes the loan-level or pool-level transition data originally from Fannie Mae, Freddie Mac, Ginnie Mae and third-party servicers. Data obtained from third parties is marked as raw data. We review the data processing procedures used to get from the raw data to the development data. These procedures include reviewing data characteristics, data cleaning, data preparation and data transformation processes.

After the development data preparation, variable selection and loan segmentation become key to explaining each prepayment effect. Model developers seek to select a set of collateral attributes with clear and constant evidence of impact to the given prepayment effect. We validate the loan segmentation process by checking whether the historical prepayment rate from different loan segments demonstrates level differences based on the set of collateral attributes selected.

A prepayment model’s implementation process is normally a black box. This increases the importance of the model output review, which includes performance testing, stress testing, sensitivity analysis, benchmarking and back-testing. An appropriate set of validation tests will capture:

  • Sensitivity to collateral and borrower characteristics (loan-to-value, loan size, etc.)
  • Sensitivity to significant assumptions
  • Benchmarking of prepayment projections
  • Performance during various historical events
  • Back-testing
  • Scenario stability
  • Model projections compared with projections from dealers
  • Performance by different types of mortgages, including CMOs and TBAs

A prepayment model sensitivity analysis might take a TBA security and gradually change the value of input variables, one at a time, to isolate the impact of each variable. This procedure provides an empirical understanding of how the model performs with respect to parameter changes. If the prepayment model has customized tuning functionality, we can apply the sensitivity analysis independently to each prepayment effect by setting the other tuning parameters at zero.

For the benchmarking analysis, we compare the model’s cohort-level, short-term conditional prepayment rate (CPR) projection against other dealer publications, including Barclays and J.P. Morgan (as applicable and available). We also compare the monthly CPR projections against those of the challenger model, such as Bloomberg Agency Model (BAM), for the full stack Agency TBA and discuss the difference. Discrepancies identified during the course of a benchmarking analysis may trigger further investigation into the model’s development. However, it doesn’t necessarily mean that the underlying model is in error since the challenger model itself is simply an alternative projection. Differences might be caused by any number of factors, including different development data or modeling methodologies.

Prepayment model back-testing involves selecting a set of market-traded MBS and a set of hypothetical loan cohorts and comparing the actual monthly CPR against the projected CPR over a prescribed time window (normally one year). Thresholds should be established prior to testing and differences that exceed these thresholds should be investigated and discussed in the model validation report.

Validating Cash Flow Models and Risk Metrics

A cash flow model combines the simulated paths from interest rate, prepayment, default, and delinquency models to compute projected cash flows associated with monthly principal and interest payments.

Cash flow model inputs include the underlying instrument’s characteristics (e.g., outstanding balance, coupon rate, maturity date, day count convention, etc.) and the projected vectors associated the CPR, default rate, delinquency, and severity (if applicable). A conceptual review of a cash flow model involves a verification of the data loading procedure to ensure that the instrument’s characteristics are captured correctly within the model. It should also review the underlying mathematical formulas to verify the projected vectors are correctly applied.

Model outputs can be validated via sensitivity analysis. This often involves shocking each input variable, one at a time, and examining its resulting impacts on the monthly remaining balance. Benchmarking can be accomplished by developing a challenger model and compare the resulting cash flows.

Combining the outputs of all the sub-models, a price of the underlying structured finance instrument can be generated (and tested) along with its related risk metrics (duration, convexity, option adjusted spread, etc.).

Using MBS as an example, an option adjusted spread (OAS) analysis is commonly used. Theoretically, OAS is calibrated by matching the model price with the market price. The OAS can be viewed as a constant spread that is applied to the discounting curve when computing the model price. Because it deals with the differences between model price and market price, OAS is particularly useful in MBS valuation. It is particularly helpful in measuring prepayment risk and market risk. A comprehensive analysis reviews the following:

  • Impact of interest rate shocks on a TBA stack in terms of price, OAS, effective duration, and effective convexity.
  • Impact of projected prepayment rate shock on a TBA stack in terms of price, OAS, effective duration, and effective convexity.
  • Impact of projected prepayment rate shock on the option cost (measured as basis point, zero-volatility spread minus OAS).

Beyond OAS, the validation should include independent benchmarking of the model price. Given a sample portfolio that contains the deal information for a list of structured finance instruments, validators derive a model price using the same market rate as the subject model as a basis for comparison. Analyzing the shock profiles enables validators to conclude whether the given discounting cash flow method is generating satisfactory model performance.

Conclusion

Structured finance model validations are complex because they invariably involve testing a complicated array of models, sub-models, and related models. The list of potential sub-models (across all three categories discussed above) significantly exceeds the examples cited.

Validators must design validation tasks specific to each model type in order to adequately assess the risks posed by potential shortcomings associated with model inputs, structure, theory, development, outputs and governance practices.

When it comes to models governing structured finance instruments, validators must identify any model risk not only at the independent sub-model level but at the broader system level, for which the final outputs include model price and risk metrics. This requires a disciplined and integrated approach.

 

 

[1] Knots represent a set of predefined points on the curve

[2] Burnout effect describes highly seasoned mortgage pools in which loans likely to repay have already done so, resulting in relatively slow prepayment speeds despite falling interest rates.