Advanced Estimation and Simulation Methods for Retail Credit Portfolios: Frequentist vs. Bayesian Techniques

We consider two common examples of retail portfolios, an auto loan book, and a credit card portfolio. Performance data is collected at a vintage/cohort level with quarterly frequency (i.e., quarterly cohorts of loans/accounts observed on a quarterly basis). The target variable to model is the vintage-level default rate, defined as the ratio of the number of accounts that have defaulted to outstanding accounts. Our methodology is in line with Licari and Suarez-Lledo (2013).¹ Our target variable, (logit of) default rate, gets decomposed into three dimensions:

Parameters in (ii) and (iii) are assumed to be stochastic in the frequentist and Bayesian settings, while parameters (i) are assumed to be deterministic in both and fixed to the values found after performing the frequentist regression.

Both frequentist and Bayesian methods produce similar average values for the key parameters. For auto loans, the results from both approaches are very similar. For credit cards, the Bayesian estimation has a significantly lower standard deviation in the parameter estimation than the frequentist standard errors, resulting in parameters that are less volatile and more precise and thus presenting a lower model risk when used in portfolio loss estimations.

Bayesian methods have the added advantage of producing posterior distributions for all parameters. The figures below illustrate the statistical shape of the posterior distribution for macroeconomic drivers and how these compare with frequentist parameters and 95% confidence intervals.

The estimated models for the risk parameters are then used for the estimation of the portfolio loss distribution through a dynamic Monte Carlo simulation. Three distinct steps are considered:

1. Macroeconomic scenarios are built using a Dynamic Stochastic General Equilibrium Model to produce forward-looking “states of the world” with quarterly updates (Licari and Ordonez-Sanz (2015)²).
2. The Bayesian and frequentist models are then used to estimate risk parameters in each of these macroeconomic scenarios (e.g., conditional default rates).
3. The portfolio is simulated dynamically (multiperiod default simulation) to estimate the cumulative loss distribution over nine future quarters.

For this last step, two different comparisons between the frequentist and Bayesian approaches were performed with and without model risk. The table below highlights the set-up of these two exercises, as well as key similarities and differences.

Figure 2 illustrates the set of macroeconomic scenarios used in the first step of the process for two different factors: unemployment rate and home price dynamics.

The estimated forward-looking default rates estimated in the second step of the process are shown below for both the frequentist and Bayesian approaches for the set of macroeconomic scenarios.

The first simulation exercise (macroeconomic risk only) shows consistency across both estimation methods. The simulated default rate distributions are fairly similar for both portfolios. They generate a tailed, asymmetric density with higher values at the block of stressed scenarios. The Bayesian method for credit cards seems to produce slightly less volatile default rate projections, but the overall shape of both densities is quite similar. The key difference appears when we move from exercise 1 to 2, adding model risk dimensions such as parameter volatility and error/residual properties.

Figure 4.2 drives home the key message of this section. Once we add the uncertainty coming from model risk to the simulation mechanism, frequentist and Bayesian methods produce very different outcomes. The higher precision of the Bayesian estimators flow into more concentrated, less volatile simulated default rates while still presenting the “fat-tails” that would be expected from the impact of very severe macroeconomic scenarios.

The statistical properties of conditional default rates (conditional on a given macroeconomic scenario) influence the shape of the portfolio loss distribution. To quantify this effect, in step 3, dynamic Monte Carlo simulations are performed on both sets of conditional default rate distributions estimated in exercises 1 and 2. The charts below highlight the significant effect that these estimation results can have on the shape of portfolio losses. The severity of CCAR Adverse, Severely Adverse, ECCA S3 and S4 scenario as well as the VaR loss at 99.9% confidence levels are also shown for comparison purposes. In summary, Bayesian methods prove to be more stable, particularly after including model risk in the loss simulations.

Table 1. Macroeconomic parameters – Bayesian vs. frequentist estimations

Table 1A. Auto loan portfolio

Table 1B. Credit card portfolio

Source: Moody's Analytics

Table 2. Alternative simulation exercises – macro only vs. fully-fledged

Source: Moody's Analytics

Figure 1. Auto loan portfolio – posterior Bayesian distributions for macroeconomic parameters

Density functions (top) and box-plots (bottom). Red dots for frequentist betas, red vertical lines for frequentist 95% confidence intervals

Source: Moody's Analytics

Figure 2. Macroeconomic simulations – nine out-of-sample quarters

Source: Moody's Analytics

Figure 3. Simulation exercise 1 – macroeconomic risk only – frequentist vs. Bayesian

Figure 3.1. Auto loan portfolio – across blocks of scenarios, five quarters out-of-sample (+Q5)

Figure 3.2. Credit card portfolio – across blocks of scenarios, five quarters out-of-sample (+Q5)

Source: Moody's Analytics

Figure 4. Simulation exercise 2 – macroeconomic and model risks – frequentist vs. Bayesian

Figure 4.1. Auto loan portfolio – selected vintages, five quarters out-of-sample (+Q5)

Figure 4.2. Credit card portfolio – selected vintages, five quarters out-of-sample (+Q5)

Source: Moody's Analytics

Figure 5. Cumulative portfolio default losses – simulation exercise 1 – macroeconomic risk only – frequentist vs. Bayesian

Figure 5.1. Auto loan portfolio – after nine quarters (+Q9)

Figure 5.2. Credit card portfolio – after nine quarters (+Q9)

Source: Moody's Analytics

Figure 6. Cumulative portfolio default losses – simulation exercise 2 – macroeconomic and model risks – frequentist vs. Bayesian

Figure 6.1. Auto loan portfolio – after nine quarters (+Q9)

Figure 6.2. Credit card portfolio – after nine quarters (+Q9)

Source: Moody's Analytics