Alphidem co-founder Ken Sutter outlines how the quantitative modelling of capital relief trades can help investors assess the risks of their investments

In light of the current regulatory and interest rate environment, it could be assumed that capital relief trades enjoy great popularity, even outside of the dedicated investor community. However, despite growth and positive market trends, the asset class still leads a niche existence and sticks to its esoteric label. This is even more astonishing considering that many alternative credit managers struggle to address increased volume requirements for their target sector, while the desired exposure sits on bank balance sheets waiting to be tapped.

This circumstance can partly be explained by the structural complexity and the actuarial nature of the CRT asset class. Therefore, the subsequent sections of this article shall briefly discuss the quantitative modelling of a generic CRT, thereby alluding to some of the basic elements relevant for due diligence.

It should be noted that this essay does not aim for an academic discussion on individual modelling approaches and their peculiarities or limitations. Rather, it is about showing what insights such modelling yields in practice and to what extent it can help investors to better assess the risks.

Setting the scene
We start by defining a generic transaction structure, as illustrated in Figure 1. While detachment point D of the tranche sold is linked to the estimated unexpected loss of the reference portfolio, the specific shape of the portfolio loss distribution remains vague. Apart from that, the even greater uncertainty for investors arises from the securitisation structure and its terms regarding amortisation, replenishment, call options and so forth.

Furthermore, there is also a time component involved. So, the key question is, to what extent all these elements affect the noteholder’s performance and risks, and how can they be represented? For the sake of simplicity, we use the note’s estimated IRR and loss distribution and its associated metrics as a measure to assess the above.

Our generic portfolio consists of 1000 corporate loans, with each reference entity having the same weight of 0.1%. We sample the probability of default (PD) and loss given default (LGD) of each loan from two normal distributions, namely:

What is still missing is an assumption regarding the correlation structure of the portfolio. Correlation is notoriously hard to estimate, yet the Bank for International Settlements (BIS) proposed some references as part of its advanced internal ratings-based approach (A-IRB).

Under this approach, the mutual correlation ρ between two loans depends on the type of exposure (corporate, mortgages, retail etc) and the PD of the loans. For two corporate loans with same PD, it is given by:

We will use the above approach to derive our portfolio’s correlation structure.

What still needs to be defined are some terms of the securitisation; most notably: the attachment (A) and detachment (D) point of the tranche sold, the coupon paid (CP), the amortisation type (sequential, pro-rata or hybrid), replenishment conditions and call options. While many of the above terms are subject to negotiations, there are those that are largely specified by the regulator.

Accordingly, we will first define detachment point D=8%, which we approximate based on BIS’ SEC-IRBA securitisation framework. Furthermore, we opt for a static portfolio without replenishment and sequential tranche amortisation. That is not to say that other structures cannot be modelled; rather, it is about keeping this generic example as elementary as possible.

With this in mind, we also choose our maturity structure to be nine years non-call three (9YNC3), knowing that there is a high chance the transaction will be called soon after the non-call period. Furthermore, we assume a bullet repayment of the reference loans.

What remains to be specified is the coupon paid and attachment point of the tranche; probably the most discussed terms in practice. This is also the reason why we include these two parameters in our sensitivity analysis later, although we would otherwise just focus on parameters that exhibit uncertainty by nature (e.g. PD and LGD). By default, we set A=0% and CP=10% p.a.

The following section will briefly illustrate the modelling results based on the above assumptions before we conduct some sensitivity analysis. Generally, it should be noted that we do not cover any issuer side related topics, such as the efficiency of the risk transfer. Furthermore, we assume that all the above data will be made available to investors as part of their due diligence process and subjected to careful examination.

Modelling results
As previously mentioned, we will consider the estimated probabilistic IRR and loss distribution of the notes to assess the transaction outlined. Figure 3 depicts the two corresponding distributions, together with their summary statistics.

A transaction tenor t=3 years has been chosen, assuming the issuer calls the structure soon after the non-call period. Due diligence wise, this is a rather conservative assumption.

We know the risk profile would likely improve for longer durations, as the amortising loans raise the detachment point of the tranche, as depicted in Figure 4. This duration thematic will be discussed more extensively in the specific sensitivity analysis section.

The metrics marked with a percentage prefix (e.g. 95% IRR) in Figures 3 and 5 represent the limit of the respective confidence interval and should therefore contribute to a better assessment of the downside risk. Figure 5 depicts the estimated cumulative tranche loss over time.

The slopes of the two lines are equal to the annual losses already reported in Figure 3, which is straightforward to verify. The linear behaviour itself is what we would expect for the portfolio, given its granularity and correlation structure.

It should be mentioned that the shape of the IRR and loss distribution shown in Figure 3 can be different in the general case. It is predominantly driven by the underlying portfolio’s granularity and correlation structure, as discussed in more detail in the respective sensitivity analysis section. Consequently, the shape of the distributions should also be considered when comparing different scenarios or transactions with one another.

In the case of the subsequent sensitivity analysis, comparisons are permissible because the range of the parameters that are examined is relatively narrow. However, the subject is pointed out where necessary.

Sensitivity analysis
Because in practice our transaction is based on different assumptions and we cannot be entirely certain of PDs, LGDs, correlations and so forth, we aim to better understand the impact of such uncertainties on our risk profile. For this reason, we will briefly summarise and comment on the results of various sensitivity tests in the following sections.

PD and LGD
To check our transaction’s sensitivity regarding PD and LGD estimates, the mean of our portfolio’s PD and LGD distribution shall be varied, as illustrated in the right column of Figure 6. Since we are applying five different mean values for PD and three for LGD, we have 15 different combinations to model. It turns out that we can reduce PD and LGD to the expected loss (EL), given by: EL=PD*LGD.

This simplifies the illustration of the results, which are presented on the left side of Figure 6. We take the expected IRR and 95% IRR as an example to show the impact of different PD and LGD distributions, expressed in terms of their expected loss.

We notice that the 95% IRR is quite sensitive to changes of PD and LGD. A credit investor focused on downside risk protection would therefore put special focus on the PD and LGD estimates provided.

Correlation
Since in practice the correlation structure is particularly difficult to estimate, we pay special attention to the sensitivity of our portfolio towards it. To properly understand our results, we first examine the influence of different values of mutual correlation on the shape of a generic portfolio loss function, as depicted in Figure 7. Although the loss distribution functions shown are subject to simplified assumptions, the qualitative conclusions we draw are generally valid, including that the variance of the portfolio loss distribution increases with higher correlation.

In the two extreme cases when ρ≈0% and ρ≈100%, we therefore find the probability density to be concentrated around PD or the two poles. This behaviour explains our portfolio’s modelling results shown in Figure 8.

We first note that the 95% IRR decreases for higher values of mutual correlation, which is caused by the loss distribution’s variance increase. At one point, however, the 95% IRR improves again and climbs to even higher values than before.

We can explain this trend reversal by studying the extreme case when ρ≈100%; in other words, investors lose either all their investment or nothing on a single transaction. Statistically speaking, this means that the expected loss of the portfolio is distributed over a few individual cases which suffer a total portfolio loss.

This has a statistically positive effect on our first loss investor’s 95% IRR, as their loss is limited by the detachment point of the tranche - meaning the residual loss occurs higher in the credit hierarchy; something that is very rarely the case for lower values of mutual correlation. On the other hand, the resulting binary risk profile and thus increased total loss probability are, of course, undesirable from a risk management perspective.

Finally, it should be noted that there is an opposite interaction between correlation and granularity. An uncorrelated portfolio with low granularity also tends to have a binary risk profile. Conversely, portfolios with low granularity are also less sensitive to changes in correlation.

Transaction tenor
As we do not know the specific tenor of our transaction at inception, it makes sense to check our risk profile’s sensitivity against this parameter as well. We already mentioned that the risk profile is expected to improve for longer tenors, as the sequential amortisation raises the detachment point of our tranche over time (see Figure 4).

Figure 9 quantifies this assumption, expressed in terms of 95% IRR and 95% tranche loss. To get a better intuition about the strength of the deleveraging effect over time, one may consider Figure 10. Even though we expect the issuer to call the transaction shortly after the non-call period, it is also helpful to understand the potential influence of the non-call period and our portfolio’s loan tenors on our risk profile.

Attachment point and coupon
Even if there is no uncertainty about the attachment (A) point and coupon (CP), we still want to briefly explain the influence of these two variables on our risk profile. On the one hand, due to their substantial influence. On the other hand, because tough negotiations often precede their determination. A sensitivity analysis can therefore help to set the right priorities.

At this point, it should be noted again that this analysis only relates to the investor’s risk profile; the issuer side is not examined e.g. in relation to the capital relief efficiency. Even if the interests of the two parties largely contradict each other in this case, the fact that - depending on the investor’s risk preferences and the regulatory handling on the part of the issuer - some optimisation potential may exist should not be ruled out.

As we want to examine variations in two dimensions, we use contour plots to visualise the change of our risk metrics. Figures 11 and 12 show the results for the expected IRR and 95% IRR.

We are certainly not surprised that extreme values can be found at both poles; yet more interesting is the fact that the course of change varies for different metrics. The above analysis can be helpful for the due diligence and negotiation process, and it may even enable investors to set specific negotiation targets to compensate for other uncertainties resulting from the sensitivity analysis.

Conclusion
The aim of this short essay was to show how the quantitative modelling of capital relief transactions can help investors to assess the risks of their investments. We also explained how to deal with uncertainties regarding various risk parameters and how the downside risk can be estimated.

Even if this article only provides a rough overview and results may vary for different transactions, it is hoped that credit investors outside the core community will find pleasure in the statistical nature of the asset class and thus enrich the entire market. The prospect of gaining access to the multitude of assets sitting on bank balance sheets across the globe should be an incentive.

About the author:

Ken Sutter is co-founder of Alphidem AG, a Swiss-based company specialised in financial engineering. Alphidem develops modelling software products and advises various investors on capital relief trades. Contact details: ks@alphidem.com.