5.1.1. The Vašíček model

Oldrich Vašíček first considered the probability of loss on loan portfolios in 1987 (Vašíček, Reference Vašíček1987). Starting from Merton’s model of a company’s asset returns (Merton, Reference Merton1974), the question Vašíčekwas seeking to answer was relatively simple: what is the probability distribution of default for a portfolio of fixed cashflow assets?

Vašíček required several assumptions for the portfolio of assets:

• All asset returns are described by a Wiener process. In other words, all asset values are lognormal distributed, similar to Merton’s approach.

• All assets have the same probability of default p.

• All assets are of equal amounts.

• Any two of the assets are correlated with a coefficient ρ (rho).

The starting point in Vašíček’s model was Merton’s model of a company’s asset returns, defined by the formula:

(1) $$ln\;A(T) = ln\;A + \;{\rm \mu} T- {1 \over 2}{\sigma ^2}T + \sigma \sqrt TX\;$$

where T is the maturity of the asset, W(t) is standard Brownian motion, asset values (denoted A(t)) are lognormal distributed, $\mu \;$ and ${\sigma ^2}$ are the instantaneous expected rate and instantaneous variance of asset returns respectively, and X represents the return on a firm’s asset. In this setting, X follows a standard normal distribution, given by $X = {{W(T) - W\left( 0 \right)} \over {\sqrt T}}$ .

The next step in Vašíček’s model was to adapt Merton’s single-asset model to a portfolio of assets. For a firm denoted i (with i = 1, …, n), Equation (1) can be rewritten as:

(2) $$ln\;{A_i}( T) = ln\;{A_i} + \;{\mu _i}T - {1 \over 2}{\sigma _i}^2T + {\sigma _i}\sqrt T{X_i}$$

Given the assumptions above, and the equi-correlation assumption for variables X_i, it follows that the variables X_i belong to an equi-correlated standard normal distribution (equi-correlation means all assets in the portfolio are assumed to have the same correlation with one another). Any variable X_i that belongs to an equi-correlated standard normal distribution can be represented as a linear combination of jointly standard normal random variables $Z$ and Y_i such that:

(3) $${X_i} = Z\sqrt \rho + {\rm{\;}}{Y_i}\sqrt {1 - \rho }, {\rm where} \;i = 1, \ldots, n$$

Equation (3) is a direct result of statistical properties of jointly equi-correlated standard normal variables, which stipulates that any two variables X_i and ${X_j}$ are bivariate standard normal with correlation coefficient ρ if there are two independent standard normal variables Z and Y for which ${X_{\rm{i}}} = Z$ and ${X_{\rm{j}}} = {\rm{\rho }}Z + \sqrt {1 - {{\rm{\rho }}^2}} Y$ , with ρ a real number in [−1, 1]Footnote ² .

Note that it can be shown that the common correlation of n random variables has a lower bound equal to $ - {1 \over {n - 1}}$ . As n tends to infinity, ${\rm{\rho }}$ will have a lower bound of 0, also known as the zero lower bound limit of common correlation. In other words, for (very) large portfolios, firms’ assets can only be positively correlated, as is their dependence on systematic factors.

With each firms’ asset return ${X_{\rm{i}}}$ of the form ${X_i} = Z\sqrt \rho + {\rm{\;}}{Y_i}\sqrt {1 - \rho }, $ variable Z is common across the entire portfolio of assets, while Y_i is the i^th firm’s specific risk, and independent from variable Z and variables ${Y_j}$ , where j <> i.

As a parenthesis, the covariance between two firms’ asset returns X _i and X _j is determined by ${\rho _{ij}} = {{cov\left( {{X_i},{X_j}} \right)} \over {\sigma \left( {{X_i}} \right)\;\sigma \left( {{X_j}} \right)}}$ , where $\sigma \left( {{X_i}} \right)$ and $\sigma \left( {{X_j}} \right)$ are the standard deviations of each firm’s asset returns. For a fixed ρ, a higher variance of asset returns requires a higher covariance of asset returns and vice-versa. For standard normal variables, $\sigma \left( {{X_i}} \right) = \;\sigma \left( {{X_j}} \right) = 1$ , and hence ${\rho _{ij}} = \;cov\left( {{X_i},{X_j}} \right)$ .

The final step in Vašíček’s model is the derivation of a firm’s probability of default, conditional on the common factor Z. This is relatively straightforward:

(4) $$P\left( {firm\;{X_i}\;defaults\;|\;Z} \right) = \;\Phi \left( {{{{x_i} - Z\sqrt \rho } \over {\sqrt {1 - \rho } }}} \right)$$

Finally, although Vašíček has not considered credit ratings in his setting other than the default state, from Equation (4) it follows that, conditional on the value of the common factor Z, firms’ loss variables are independent and equally distributed with a finite variance.

The loss of an asset portfolio (e.g. a portfolio represented in a transition matrix) can thus be represented by a single variable on the scaled distribution of variable X_i. Although overly simplistic, this setup is helpful for analysing historical data (such as historical default rates or transition matrices) and understanding the implications of asset distributions and correlations in credit risk modelling. In the following Section, we consider a method that applies a firm’s conditional probability of default to historical transition matrices.

5.1.1.1. Vašíček model calibration – Belkin

Belkin et al. (Reference Belkin, Suchower and Forest1998) introduced a statistical method to estimate the correlation parameter ${\rm{\rho }}$ and common factors $Z$ in Vašíček’s model based on historical transition matrices. The starting point in their model is the one-factor representation of annual transition matrices (denoted by variables X_i, with i representing year):

$${X_i} = Z\sqrt \rho + {\rm{\;}}{Y_i}\sqrt {1 - \rho } $$

where Z, Y_i and $\rho $ are as per Vašíček’s framework, and X_i is the standardised asset return of a portfolio in a transition matrix.

The method proposed in Belkin employs a numerical algorithm to calibrate the asset correlation parameter ρ and systematic factors $Z$ subject to meeting certain statistical properties (e.g. the unit variance of Z on the standard normal distribution), using a set of historical transition matrix data.

This approach allows a transition matrix to be represented by a single factor, representing that transition matrix’s difference from the average transition matrix. Matrices with more downgrades and defaults are captured with a high negative factor; matrices with relatively few defaults and downgrades have a positive factor.

Representing a full transition matrix (of 49 probabilities) with a single factor inevitably leads to loss of information. More information can be captured in a two-factor model, which is introduced in Section 5.1.2.

The Belkin implementation uses the standard normal distribution. If this distribution was replaced with a fatter-tailed distribution, it could be used to strengthen the calibration of extreme percentiles (e.g. the 99.5^th percentile). However, other moments of the distribution are also expected to be impacted (e.g. the mean) which would need to be carefully understood before implementation. This approach has not been explored in this paper, but it would not be expected to change the directional results seen in Section 6.

5.1.2. The two-factor model

5.1.2.1. Two-factor model description

The Vašíček model includes just a single parameter used to model transition matrices and some of the limitations of the model arise from an oversimplification of the risk. In this Section, a two-factor model is described to capture two important features of the way transition matrices change over time, particularly in stress. This model is based on a description given by Rosch and Scheule (Reference Rosch and Scheule2008).

Rosch and Scheule (Reference Rosch and Scheule2008) use two defined features of a transition matrix they describe as “Inertia” and “Bias.” In this paper we use the terms “Inertia” and “Optimism” as the term bias is widely used in statistics, potentially causing confusion with other uses such as statistical bias in parameter estimation.

Inertia is defined as the sum of the leading diagonal of the transition matrix. For a transition matrix in our setting with probabilities ${p_{ij}}$ , where i denotes row and j denotes column:

$$Inertia = \mathop \sum \limits_{i = 1}^7 {p_{ii}}$$

Optimism is defined as the ratio between the upgrade probabilities and downgrade probabilities summed over all seven rows and weighted by the default probabilities in each row:

$$Optimism = \sum\limits_{i = 2}^6 {\left( {{{\sum\nolimits_{i \lt j} {{p_{ij}}} } \over {\sum\nolimits_{i \gt j} {{p_{ij}}} }}*{p_{iD}}} \right)} /\sum\limits_{i = 2}^6 {\left( {{p_{iD}}} \right)} $$

$p_{iD}$ is the default probability for each row and this value is summed across. $\mathop \sum \nolimits_{i \lt j} {p_{ij}}$ is the sum of all upgrades in each row. $\mathop \sum \nolimits_{i \gt j} {p_{ij}}$ is the sum of all downgrades in each row. Appendix B gives a sample calculation of Inertia and Optimism for a given matrix.

Any historical matrix can be characterised by these two factors and the base transition matrix (a long-term average transition matrix used as a mean in the best estimate) can be adjusted so that its Inertia and Optimism correspond to those in an historical transition matrix. This allows the generation of an historical time series for Inertia and Optimism based on our historical data set; then to fit probability distributions to the values of these parameters that can then be combined with a copula.

It would also be possible to extend the model by weighting the values of Inertia and Optimism by the actual assets held in the portfolio (but this is not explored in this paper). In this paper, Optimism has only been calculated based on AA-B ratings; but in practice, this could be changed to be closer to the actual assets held.

This means that with probability distributions for Inertia, Optimism and a copula all calibrated from historical data, a full probability distribution of transition matrices can be produced.

The base transition matrix can be adjusted so that its Inertia and Optimism are consistent with the parameters of a historical matrix or parameters outputted from a probability distribution by using the following steps (BaseInertia and StressInertia and BaseOptimism and StressOptimism are used for the base Inertia and Optimism and the Inertia and Optimism from the matrix the base matrix is adjusted to have the same values as):

1. 1. Multiply each of the diagonal values by StressInertia/BaseInertia.

2. 2. Adjust upgrades and downgrades so the rows sum to 1, by dividing them by a single value.

3. 3. Adjust upgrades and downgrades so that their ratio is now in line with StressOptimism.

The adjustments required for steps 2 and 3 above are now defined. To do this the matrices required to calculate the adjustments are first defined.

• Elements of the base transition matrix are defined as $p_{ij}^{\left( 1 \right)}$ for the i^th row and j^th column from this matrix.

• Elements of the matrix after step 1 (having the diagonals adjusted by StressInertia/BaseInertia) are defined by $p_{ij}^{\left( 2 \right)}$ .

• Elements of the matrix after step 2 (upgrades and downgrades adjusted to sum to 1) are defined by $p_{ij}^{\left( 3 \right)}$ .

• Elements of the matrix after step 3 (upgrades and downgrades adjusted so optimism is equal to StressOptimism) are defined by $p_{ij}^{\left( 4 \right)}$ .

Following step 1, upgrades and downgrades in step 2 are given by:

$$p_{ij}^{\left( 3 \right)} = p_{ij}^{\left( 1 \right)}*{{1 - p_{ii}^{\left( 2 \right)}} \over {\sum\nolimits_{k = 1\left( {k \ne i} \right)}^7 {p_{ik}^{\left( 1 \right)}} }}$$

This gives a matrix which has the same Inertia as the StressInertia, but the Optimism is still not the same as the StressOptimism.

To carry out step 3 no change is needed for the AAA or CCC/C categories, which have no upgrades nor downgrades respectively. For the other ratings, a single factor is found to add to the sum of the upgrades and subtract from the sum of the downgrades to give new upgrades and downgrades, which have the same ratio as the value of StressOptimism. This factor is:

$$Factor = {{\left( {StressOptimism*\sum\nolimits_{j \,= \,i + 1}^7 {p_{ij}^{\left( 3 \right)}} - \sum\nolimits_{j \,=\, 1}^{i - 1} {p_{ij}^{\left( 3 \right)}} } \right)} \over {\left( {1 + StressOptimism} \right)}}$$

The final matrix can now be found by:

$$if\;j \gt i\;\left( {i.e.\;downgrades} \right)\,p_{ij}^{\left( 4 \right)} = {{p_{ij}^{\left( 3 \right)}} \over {\sum\nolimits_{j = i + 1}^7 {p_{ij}^{\left( 3 \right)}} }}*\left( {\sum\limits_{j = i + 1}^7 {p_{ij}^{\left( 3 \right)}} - Factor} \right)$$

$$if\;j \lt i\;\left( {i.e.\;upgrades} \right)\,p_{ij}^{\left( 4 \right)} = {{p_{ij}^{\left( 3 \right)}} \over {\sum\nolimits_{j = 1}^{i - 1} {p_{ij}^{\left( 3 \right)}} }}*\left( {\sum\limits_{j = 1}^{i - 1} {p_{ij}^{\left( 3 \right)}} - Factor} \right)$$

5.1.2.2. Two-factor model calibration

This Section describes an approach to calibrating the two-factor model.

Using historical data for historical transition matrices from 1981 to 2019, as well as the 1932 matrix and a matrix from 1931 to 1935, a time series of Inertia and Optimism can be constructed. It is then possible to fit probability distributions from this data and simulate these probability distributions, combined with the approach of adjusting the base transition matrix for given Inertia and Optimism to give a full probability distribution of transition matrices.

The data is shown in Figure 5 with the time series compared to the values for 1932 and 1931–1935.

Figure 5. (a) Inertia and (b) Optimism, historical values compared to 1932 and 1931–1935.

The first four moments for this data are shown in Table 5 (including the 1932 and 1931–1935 matrices).

Table 5. First four moments of inertia and optimism.

The correlations between the two data series are given in Table 6:

Table 6. Correlation between inertia and optimism.

The main comments on these data series are:

• Inertia is negatively skewed and has a fat-tailed distribution.

• Optimism is slightly positively skewed and with a slightly fatter tail than the normal distribution.

• The two data series are correlated based on the Pearson, Spearman or Kendall Tau measure of correlation. In the most extreme tail event, both Inertia and Optimism were at their lowest values. This means that this year had the most amount of assets changing rating as well as the greatest number of downgrades relative to upgrades in that year.

To simulate from these data sets probability distributions have been fitted to the two data series using the Pearson family of probability distributions. The Pearson Type 1 distribution produces a satisfactory fit to the two data sets as shown in Figure 6, comparing the raw data (“data”) against 10,000 simulated values from the fitted distributions (“fitted distribution”).

Figure 6. Historical plots of (a) Inertia and (b) Optimism compared to fitted distributions.

The plot above shows the actual historical data values for Inertia and Optimism from each of the historical transition matrices compared to the distributions fitted to this data. Note that the darker pink indicates where both the data (blue) and fitted distribution (pink) overlap.

As well as probability distributions for Inertia and Optimism, a copula is also needed to capture how the two probability distributions move with respect to one another. For this purpose, a Gaussian Copula has been selected with a correlation of 0.5. This correlation is slightly higher than found in the empirical data. The model could be extended to use a more complex copula such as the t-copula instead of the Gaussian copula. An alternative to a one or two-factor model is to use non-parametric models and, rather than fit a model to the data, use the data itself directly to generate a distribution for the risk.

5.2.1. The K-means model

Under the K-means model the key steps are:

1. 1. Apply the K-means algorithm to the data to identify a set of groups within the data set and decide how many groups are required for the analysis.

2. 2. Assign each of the groups to real-line percentiles manually, e.g. assign a group containing the 1932 matrix at the 0.5^th percentile, assign the average matrix of the 1931–1935 matrix at 0.025^th percentile, put an identity matrix at the 100^th percentile, put a square of the 1932 matrix at the 0^th percentile, etc.

3. 3. Interpolate any percentiles needed in between using a matrix interpolation approach.

5.2.1.1. K-means to transition data

K-means clustering is an unsupervised clustering algorithm that is used to group different data points based on similar features or characteristics. K-means clustering is widely used when un-labelled data (i.e. data without defined categories or groups) needs to be organised into a number of groups, represented by the variable K (Trevino, Reference Trevino2016). The algorithm works iteratively to assign each data point to one of K groups based on the features that are provided. The results of the K-means clustering algorithm are:

1. 1. The means of the K clusters can be used to label new data.

2. 2. Labels for the training data (each data point is assigned to a single cluster).

3. 3. Rather than defining groups before looking at the data, clustering allows you to find and analyse groups that have formed organically. The “Choosing K” Section below describes how the number of groups can be determined.

Based on the transition risk data, we apply the K-means algorithm to group each of the transition year data points into groups with a similar profile of transitions from one rating to another. We present the K-means visualisations in Figure 7. In K-means clustering, as we increase the number of clusters, generally the sum of squares within and between groups reduces. The key idea is to optimise the number of groups (i.e. value of K) such that reduction in the sum of squares within and between the groups stops reducing substantially for higher numbers of groups.

Figure 7. K-Means clustering. Examples where K = 5, 6, 7 and 8.

Further details of the K-means clustering algorithm are given in Appendix A.

As shown in Figure 7:

• 1931 and 1931–1935 average matrices are in separate groups as we increase the groups from 7 onwards.

• Below K = 7 groups, the average 1931–1935 matrix does not come out as a separate group in its own right.

As shown in Figure 8, the total within clusters sum of squares reduces significantly as we increase the number of groups from K = 2 to K = 6. The reduction in total within clusters sum of squares does not reduce materially after K = 8. We have applied equal weights to each of the transition matrices.

Figure 8. K-means clustering examples with different K values – sum of squares within clusters.

5.2.1.2. Manually assign the matrix group to percentiles

Once K-means has been run, we get each of the transition matrices assigned to a group, as shown in Figure 7. For our example, we have selected K = 8. The next step is to manually assign each of the groups to a percentile value on the empirical distribution. This is set via an expert judgement process as follows:

• Square of 1932 matrix is assigned to the 0^th percentile.

• 1932 matrix is assigned to the 0.5^th percentile.

• 1932–1935 average matrix is assigned to the 1.25^th percentile.

• A Group containing the 2002 matrix is assigned to the 8.75^th percentile.

• A Group containing the 2009 matrix is assigned to the 27.5^th percentile.

• A Group containing the 1981 matrix is assigned to the 48.75^th percentile.

• A Group containing the 2016 matrix is assigned to the 66.25^th percentile.

• A Group containing the 2017 matrix is assigned to the 81.25^th percentile.

• A Group containing the 2019 matrix is assigned to the 93.75^th percentile.

• An identity matrix is assigned to the 100^th percentile.

All other percentiles are derived using a matrix interpolation approach as follows, where ${P_1}$ is the known matrix at percentile ${p_1}$ , ${P_2}$ is the known matrix at percentile ${p_2}$ , and Q is the interpolated matrix at percentile q.

$$Interp = \left( {{{{p_2} - q} \over {p2 - {p_1}}}} \right)$$

$${E_1} = Eigen\left( {{P_1}} \right)*Diag{\left( {EigenVal\left( {{P_1}} \right)} \right)^{Interp}}*Eigen{\left( {{P_1}} \right)^{ - 1}}$$

$${E_2} = Eigen\left( {{P_2}} \right)*Diag{\left( {EigenVal\left( {{P_2}} \right)} \right)^{\left( {1 - Interp} \right)}}*Eigen{\left( {{P_2}} \right)^{ - 1}}$$

$$Q = {E_1}*{E_2}$$

5.2.2. Bootstrapping

The bootstrapping model refers to the relatively simplistic approach of sampling from the original data set with replacement. In this case, there are n transition matrices from which 100,000 random samples are taken with replacement to give a distribution of transitions and defaults.

This is a very simple model and the main benefit of being true to the underlying data without many expert judgements and assumptions. A significant downside of this model is it cannot produce scenarios worse than the worst event seen in history; this means it is unlikely to be useful for Economic Capital models where the extreme percentiles are a crucial feature of the model. Nevertheless, this model is included for comparison purposes as it is very close in nature to the underlying data.

6.1.1. The need to reduce dimensions

Transition matrix modelling is a high-dimensional exercise. It is almost impossible to visualise the 56-dimensional distribution of a random 7 × 8 matrix. To make any progress comparing models we need to reduce the number of dimensions while endeavouring to mitigate the loss of information.

Dimension reduction is even more necessary because commonly used transition models employ a low number of risk drivers, in order to limit calibration effort, particularly the need to develop correlation assumptions with other risks within an internal model. Vašíček’s model, for example, has a single risk driver when a portfolio is large. If we have 7 origin grades and 8 destination grades (including default, but excluding NR), we could say that the set of feasible matrices under Vašíček’s model is a one-dimensional manifold (i.e. a curve) in 56-dimensional space. This is a dramatic dimension reduction relative to the historical data.

We can simplify matters to some extent by modelling the transition matrices row-by-row, considering different origin grades separately. This is possible because the investment mandates for many corporate bond portfolios dictate a narrow range of investment grades most of the time, with some flexibility to allow the fund time to liquidate holdings that are re-graded out with the fund mandate. In that case, we are dealing with a few 8-dimensional random variables; still challenging but not as intimidating as 56 dimensions.

Popular transition models generally calibrate exactly to a mean transition matrix so that the means of two alternative consistently calibrated models typically coincide. It is the variances and covariances that distinguish models.

6.1.2. Principal components analysis

Principal components analysis is a well-known dimension-reduction technique based on the singular value decomposition of a variance-covariance matrix. A common criticism of PCA, valid in the case of transition modelling, is that it implicitly weights all variances equally, implying that transitions (such as defaults) with low frequency but high commercial impact have little effect on PCA results. We propose a weighted PCA approach which puts greater weight on the less frequent transitions.

Standard PCA can also be distorted by granulation, that is lumpiness in historical transition rates caused by the finiteness of the number of bonds in a portfolio. We now describe granulation in more detail and show how a weighted PCA approach, applied one origin grade at a time, can reveal the extent of granulation.

6.2.1. Systematic and granulated models

Some theoretical models start with transition probabilities for an infinitely large portfolio (the systematic model), and then use a granulation procedure (such as a multinomial distribution) for bond counts so that, for example, each destination contains an integer number of bonds. Other models such as that of Vašíček are specified at the individual bond level and then the systematic model emerges in the limit of diverse bond portfolios.

It is possible that two transition models might have the same systematic model, differing only in the extent of granulation. It could also be that discrepancies between a proposed model and a series of historical matrices are so large that granulation cannot be the sole explanation. It is important to develop tests to establish when model differences could be due to granulation.

6.2.2. Granulation frustrates statistical transformations

When a theoretical model puts matrices on a low-dimensional manifold, granulation can cause noise in both historical matrices and simulated future matrices, which are scattered about that systematic manifold. Granulation complicates naïve attempts to transform historical transition data. For example, under Vašíček’s model, the proportion of defaults (or transitions to an X-or-worse set of grades) is given by an expression involving a cumulative normal distribution whose argument is linear in the risk factor. We might attempt to apply the inverse normal distribution functions to historical default rates and then reconstruct the risk factor by linear regression. However, when the expected number of bond defaults is low, the observed default rate in a given year can be exactly zero, so that the inverse normal transformation cannot be applied.

6.2.3. Mathematical definition of granulation

For n ≥ 2, let S _n denote the n-simplex, that is the set of ordered (n+1)-tuples (x ₀, x ₁, x ₂ … x _n ) whose components are non-negative and sum to one.

We define a granulation to be a set of probability laws {ℙ _x : x ∈ S _n } taking values in S _n such that if a vector Y satisfies Y ∼ ℙ _x then:

$$\mathbb{E}\left( {{Y_i}} \right) = {x_i}$$

$${\rm{Cov}}\left( {{Y_i},{Y_j}} \right) = h{x_i}\left( {{\delta _{ij}} - {x_j}} \right) = \;\left\{ {\matrix{ {h{x_i}\left( {1 - {x_i}} \right)\quad i = j} \cr { - h{x_i}{x_j}\qquad i \ne j} \cr } } \right.$$

The parameter h, which must lie between 0 and 1 is the Herfindahl index (Herfindahl Reference Herfindahl1950) of the granulation.

6.2.4. Granulation examples

One familiar example of a granulation is a multinomial distribution with n bonds and probabilities x, in which case the Herfindahl index is n ⁻¹. In the extreme case where n = 1, this is a categorical distribution where all probability lies on the vertices of the simplex. In the other extreme, as n becomes large, the law ${\mathbb{P}_x}$ is a point mass at x.

Other plausible mechanisms for individual matrix transitions also conform to the mathematical definition of a granulation. For example, if bonds have different face values, we might measure transition rates weighted by bond face values. Provided the bonds are independent, this is still a granulation with the standard definition of the Herfindahl index. In a more advanced setting, we might allocate bonds randomly to clusters, with all bonds in each cluster transitioning in the same way, but different clusters transitioning independently. This too satisfies the covariance structure of a granulation. Transition models based on Dirichlet (multivariate-beta) distributions are granulations, with h ⁻¹ equal to one plus the sum of the alpha parameters. Finally, we can compound two granulations to make the third granulation, in which case the respective Herfindahl indices satisfy:

$$1 - {h_3} = \left( {1 - {h_1}} \right)\left( {1 - {h_2}} \right)$$

6.2.5. Granulation effect on means and variances of transition rates

We now investigate the effect of granulation on means and variance matrices of simplex-valued random vectors. Suppose that X is a S _n -valued random vector, representing the systematic component of a transition model and that Y is another random vector with $Y|X\sim {\mathbb{P}_X}$ for some granulation.

Let us denote the (vector) mean of X by

$$\pi = \mathbb{E}\left( X\right)$$

and the variance (-covariance) matrix of X by

$${{\bf{V}}^{sys}} = {\rm{Var}}\left( X\right)$$

Then it is easy to show that the mean of Y is the same as that of X

$$\mathbb{E}\left( Y\right) = \pi $$

And the variance(-covariance) matrix of Y is:

$${\bf{V}}_{ij}^{gran} = \left( {1 - h} \right){\bf{V}}_{ij}^{sys} + h{\pi _i}\left( {{\delta _{ij}} - {\pi _j}} \right)$$

6.3.1. Weighting proposal

We are now able to propose a weighted principal components approach for models of simplex-valued transition matrices.

Suppose then that we have a model with values in S_n. Its mean vector π is, of course, still in S_n. Let us denote the variance matrix by V. Our proposed weighted PCA method is based on a singular value decomposition of the matrix.

$$diag{\left( \pi \right)^{ - {1 \over 2}}}{\bf{V}}diag{\left( \pi \right)^{ - {1 \over 2}}}$$

As this is a real positive-semidefinite symmetric matrix, the eigenvalues are real and non-negative. The simplex constraint in fact implies no eigenvalue can exceed 1 (which is the limit of a categorical distribution). We can without loss of generality take the eigenvectors to be orthonormal. We fix the signs of eigenvectors such that the component corresponding to the default grade is non-positive so that a positive quantity of each eigenvector reduces default rates (and a negative quantity increases defaults). This is consistent with our definitions of Optimism and Inertia in Section 5.1.2.1.

As the components of a simplex add to 1, it follows that $\bf V1 = 0$ where 1 is a vector of 1s. This implies that the weighted PCA produces a trivial eigenvector e with eigenvector zero and

$${e^{triv}} = {\pi ^{{1 \over 2}}} = diag{\left( \pi \right)^{{1 \over 2}}}\bf 1\;$$

6.3.2. Weighted PCA and granulation

Suppose now that we have a non-trivial eigenvector e of the weighted systematic matrix with eigenvalue ${\lambda ^{sys}}$ so that

$$diag{\left( \pi \right)^{ - {1 \over 2}}}{{\bf{V}}^{sys}}diag{\left( \pi \right)^{ - {1 \over 2}}}e = {\lambda ^{sys}}e$$

It is easy to show that e is also an eigenvector of any corresponding granulated model, with transformed eigenvalue shrunk towards 1.

$$\lambda^{gran} = \left( {1 - h} \right){\lambda ^{sys}} + h$$

Thus, if one model is a granulation of another, the weighted eigenvectors are the same and the eigenvalues are related by a shrinkage transformation towards 1. This elegant result is the primary motivation for our proposed weighting.

PCA usually focuses on the most significant components, that is those with the largest associated eigenvalues. In the context of a transition matrix, the smallest (non-zero) eigenvalue of a granulated model has a role as an upper bound for the Herfindahl index of any granulation. Where the systematic model inhabits a low-dimensional manifold, the smallest non-zero eigenvalue is typically close to zero, which implies that the smallest non-zero eigenvalue of a granulated model is a tight upper bound for the Herfindahl index.

Knowing the Herfindal index allows us to strip out granulation effects to reconstruct the variance matrix of an underlying systematic model.

6.3.3. PCA and independence

PCA decomposes a random vector into components whose coefficients are uncorrelated.

It is well known that lack of correlation does not imply independence. Nevertheless, in some contexts, the component loadings emerging from PCA might be analysed separately and then recombined as if they were independent. In this way, the PCA is sometimes used as a step in a model construction procedure.

Model construction via PCA does not work well for simplex-valued transition models. As the simplex is bounded, typically the component loadings have distributions on closed intervals. Recombining component models as if they were independent then produces distributions in a hyper-cuboid. The hyper-cuboid cannot represent a simplex; either some points of the hyper-cuboid poke beyond the simplex, leading to infeasible negative transition rates, or some feasible parts of the simplex poke beyond the hyper-cuboid rendering those events inaccessible to the model.

For these reasons, we do not propose PCA as a way of constructing transition models. Rather, we advocate the use of PCA to analyse historical transition matrices and to compare models that have been constructed by other means.

6.6.1. Bootstrapping

Bootstrapping involves sampling many data points from the historical data with replacement. This model is the simplest method to apply without any inclusion of expert judgement. The expert judgement is now exclusively in the selection of the data set.

6.6.2. K-means model

The K-means model involves the most amount of expert judgement of the four models compared. This is mainly due to the choice of percentiles for each of the k-means groups being selected by expert judgement. The model could easily be updated to a different choice of percentiles for each k-means group which would give different results and could be used for a different purpose. There are other judgements required including how many groups to use.

6.6.3. Two-factor model

The main expert judgements are around the distribution used to model the two factors and the copula used to model their interaction.

6.6.4. Vašíček model

The assumptions behind the Vašíček model include several expert judgements around the way the risk behaves. To pass the 1932 backtest an expert judgement overlay is required to strengthen the calibration.

6.6.5. Summary of model comparisons

The comparisons between the models are summarised in Table 12:

Table 11. The two-factor 99.5^th percentile transition matrix.

Table 12. Summary of model comparisons.

The models explored all have strengths and weaknesses and all could be used for a variety of purposes. The Vašíček model is used extensively in the insurance industry, but it does not replicate historical movements in the data as well as the K-means model or the two-factor model.