Loans with Partial Guarantees as Securitizations

As noted in American Bankers' Association's (ABA) comment letter to regulators, several financial institutions received guidance during regulatory examinations directing them to treat as synthetic securitizations all commercial loans for which there are guarantees covering losses up to a limit (limited guarantees). This is a relatively vague area in regulatory definitions where one can argue either way whether partially guaranteed loans (PGL) should be treated in the securitizations capital framework. ABA's comment letter argued against this treatment from the securitization definition perspective. In this post, I will look at this matter from the capital calculation perspective, discussing some counter-intuitive capital results associated with applying securitization framework to PGLs.

Consider a US advanced approach bank with a 50mm commercial loan with a guarantee up to 5% as an example, other parameters include BB- S&P equivalent rating, one-year probability of default (PD) of 1.7% (Moody's Corporate Default and Recovery Rates, 1920-2010), 40% loss given default (LGD), zero delinquencies, and maturity of 2 years. When applying the securitization framework, the bank is economically exposed to the 5-100% senior tranche of this loan, therefore is required to hold capital for the 5-100% tranche with attachment point at 5% and detachment point at 100%.

With these inputs, the SFA dollar capital charge calculated on the 5-100% tranche is 4.9mm with capital per unit of tranche being 10.3%, greater than the dollar capital requirement of 4mm with K(IRB) of 8% calculated on the entire 50mm principal of the loan unsecuritised in the wholesale framework. That means, under the SFA framework, consider the guarantee on this loan would incur a higher capital charge than if the guarantee is ignored (which is supposed to be more conservative). This apparently is counter-intuitive. Additionally, the SSFA capital calculated on this loan is 7.4%, lower than the SFA charge, which again contradicts the notion that SFA is generally more capital favorable than SSFA.

These counter-intuitive results are caused by the underlying mathematical construction of the SFA/SSFA formulas. In one of the previous posts Comparing SSFA and SFA, I illustrated the instances when SSFA can be more capital favorable than SFA. Here I will explore comparing SFA capital on the senior tranche versus IRB capital on the underlying exposures unsecuritised by re-visiting the SFA formula. Specifically, if the tranche attaches below K(IRB) and detaches above K(IRB) (which is the case of our example), SFA can be written as:

With respect to our example, K(IRB) = 8%, L = 5% (attach) , L+T = 100% (detach), SFA charge of 10.3% is composed of the three components (K(SFA)=3.2%+6.8%+0.3%=10.3%):
1). The first component represents the dollar-for-dollar capital charge for the tranche portion that lies below K(IRB), equal to (8% - 5%)/95%= 3.2%.
2). The second component represents the capital attracted by the tranche portion above K(IRB) based on the ULP K[.] capital function, calculated as (K[100%] - K[8.07%])/95% = (8% - 1.5%)/95%= 6.8%.
3). Lastly, the supervisory marginal capital floor is computed to be 0.3%.

Then K(SFA) of 10.3% is multiplied by 47.5mm to obtain the $ capital charge of 4.9mm.

The first two components can be illustrated in the chart below:

Similarly, the SSFA capital for the tranche portion below and above K(A) are also illustrated above (K(SSFA)=3.2%+4.2%=7.4%).

It is now clear to see how the SFA capital for the tranche ends up greater than IRB capital for the unsecuritised underlying. It is also worth noting that both the 2nd and 3rd components above have inverse relationships with the number (N) of underlying exposures, that is, the smaller the N, the bigger the values. Subtracting K(IRB) from the equation above, it shows that when the following equation is positive, SFA charge would exceed underlying IRB capital.

Again, when this is the case, the securitization capital would be capped at IRB capital charge according to the maximum capital charge provision in the Basel rulesets, a mechanism to resolve the disparity in capital charge for the same pool of exposures before and after securitization.

Generally, the SFA capital charge for the securitised tranche can be higher than the underlying exposures unsecuritised if either or combination of the following holds:
a). the loan has a high PD or LGD resulting in a high unsecuritised capital charge relative to the guarantee/subordination level - in this case the K(IRB) of 8% exceeds the 5% guarantee. If this loan were rated BBB with a PD of 0.17%, the resultant SFA charge (2.4%) would be lower than the unsecuritised IRB capital charge of 2.62%, and SSFA at 7.4%.
b). the number of underlying exposures/loans is small, because SFA by design penalizes the concentration/lack of granularity of the underlying pool. Particularly, the partially guaranteed loan example has only one single exposure (N=1). If we were looking at a 5-100% tranche securitised out of 5 underlying loans with the same BB- rating/PD, the SFA charge would be 7.2%, lower than both IRB charge (8%) and SSFA charge (7.4%).
c). the loan has a very low PD or LGD resulting in a SFA capital charge below the supervisory floor of 1.6%. If the loan in our example was rated single A with a PD of 0.06%, same LGD, the SFA capital would be 1.6%, higher than the IRB capital of 1.43% due to the supervisory capital floor.

In summary, due to the mathematical construct of SFA/SSFA formulas, for US banks, particularly advanced approach banks, treating partially guaranteed loans as securitizations, could result in counter-intuitive capital charges. The actual result depends on the individual loan's specific credit profile. When calculated SFA capital exceeds the unsecuritised IRB capital, the capital charge is effectively the unsecuritised IRB capital ignoring the guarantee as a result of the maximum capital provision. If SFA does not provide capital benefit after the hassle of meeting all the data and operational requirements, banks should probably take unsecuritised IRB capital charge on the entire loan balance without counting the guarantee, which is theoretically more conservative than considering the guarantee as in securitizations.

Capital Implications of SSFA

In the earlier posts, I explored the underlying math of SSFA (here) and derivation of SSFA from SFA (here). Although there are different capital implications between the two (here), given the similar construct of the formula, SFA and SSFA share some similar capital implications. This post will use SSFA (since it is simpler) to explain some capital implications of the securitization supervisory formula. Before considering the maximum capital requirement provision of the rules, we generally expect:

• The total capital requirement for individual securitization tranches is higher than the capital requirement for the underlying exposure had the exposure not been securitized
• The finer the tranching (i.e. more tranches), the higher the capital charge for the tranches combined (i.e. if one held a mezzanine 8%-12% and a senior 20%-100% from the same securitization structure, the capital charge for the two tranches will be higher than the bond 8%-100% of the same underlying had it been securitized that way)

Let's start with a really simple 2-tranche structure, to simplify, assume the junior tranche detaches at KA, and the senior tranche is the entire piece above KA:

 

Capital charge for the junior tranche = 100% * KA = KA

Capital charge for the senior tranche = KSSFA(KA, 1)*(1-KA) 

(see notations here. Also the 1.6% capital floor is ignored for simplification)

Capital charge for the underlying pool  = KA



Apparently, the capital charge for the two tranches is greater than that of the underlying pool had it not been securitized. This is by design, as a result of the stochastic loss prioritization framework that supervisory formula is built upon.

Next, let's further tranche the piece above KA as follows:

Following the same notation as here, without the capital floor, the capital charge for a tranche with attachment point A and detachment point D can be written as:
KSSFA(A, D)*(D-A) = f(D) - f(A)

Taking into account of the floor at 1.6%, we will have the following:
capital charge for Mezzanine = Max[f(A), 1.6%] = f(A) + Max[0, 1.6%-f(A)]
capital charge for Senior = Max[f(D) - f(A), 1.6%] = f(D) - f(A) + Max[0, 1.6%-(f(D) - f(A)]
capital charge for Super Senior =  Max[f(1) - f(D), 1.6%] = f(1) - f(D) + Max[0, 1.6%-(f(1) - f(D)]
Recall f(1) = KSSFA(KA, 1)*(1-KA), which is the capital charge without the capital floor for the entire senior piece (KA, 1) before tranching into 3 tranches above.



If f(1) <= 1.6%, for any of the tranche within (KA, 1) the capital charge would be <=1.6%, then
Total capital = 1.6% + 1.6% + 1.6% = 3.2% >= Max[f(1), 1.6%]= 1.6%



If f(1) > 1.6%, add the capital charges for the above three tranches, we have:
Total capital = f(1) +  Max[0, 1.6%-f(A)] + Max[0, 1.6%-(f(D) - f(A)] + Max[0, 1.6%-(f(1) - f(D)] >= f(1)

Therefore, the combined capital charge for Mezzanine, Senior and Super Senior tranches above is higher than the capital charge for the combined exposure had the exposure not been further tranched, even though the credit exposure is the same.

This certainly creates disincentives for securitizations given the disparity in capital treatment for the same pool of exposures before and after securitizations. The maximum capital requirement in the Basel rulesets was introduced as a result of such inconsistency, where if the bank holds each tranch out of a securitization transaction, the total capital cannot be higher than the capital required for the underlying pool had the pool not been securitized.Since the application of this provision is rather limited, the lack of consistency in supervisory formulas still remain.

Comparing SSFA and SFA

In an earlier post, I discussed the derivation of SSFA from SFA. The conventional notion is SSFA is more capital punitive than SFA - is it always the case? Not necessarily. The detailed analysis is as follows:

As a recap, we can compare the inputs to both formulas side by side:

SFA Inputs	SSFA Inputs
K(IRB): the IRB capital charge had the underlying exposures not been securitized       -          a function of conditional PD and stressed LGD       -          within the range of (0, 100%) in general       -          apply either SSFA or SFA for securitizations' contribution to Kg in a resec	K(A): baseline capital requirement for the underlying exposures       -          a function of general risk-based capital Kg and delinquencies (W) of the underlying pool       -          bounded between (Kg, 50%) by definition       -          apply either SSFA or SFA for securitizations' contribution to Kg in a resec
Credit Enhancement (L)	Attachment (A): same as L in SFA
Tranche Thickness (T)	Detachment (D): D=L+T in SFA
Pool's effective number of exposures (N)	N is not an input to SSFA.  SSFA ignores the marginal capital term in SFA, effectively assuming N= ∞.
Supervisory parameter ω =20:  this parameter is set the same for securitization and re-securitization	Parameter p: p=1/ ω in SFA p is set to 1 for securitization; 1.5 for re-securitization
Calculated parameter d:  d falls between (0,1) in SFA	d:  set to 1 in SSFA
Floor: Final Rule: 0.56%; NPR: 1.6%	Floor: 1.6% in both Final Rule and NPR

Question: SFA apparently requires intensive data and modeling efforts, does it provide favorable capital treatment as a result of the granular modeling exercise?

The general notion is, SSFA is more capital punitive compared to SFA. This is generally the case, based on the inputs to the formulas, especially for bonds backed by good collateral and for resececuritizations.  

•  By design, K(A) is effectively floored at general-risk based capital requirement K(G) vs K(IRB) is advanced risk-based effectively floored at 0

         (eq.1)

• For tranches above the capital requirement of the underlying pool, SFA capital is generally smaller than SSFA as it decays faster to the floor due to the more negative decay parameter discussed in last post 
• SSFA is more punitive for re-securitizations through parameters  p, i.e. p for re-sec is three times of securitization under SSFA while SFA does not impose additional penalty for re-secs 

 The above can be illustrated through the following examples for tranches that attach at zero (or zero credit enhancement):

That being said, there are instances that SSFA can be favorable than SFA, e.g. senior bonds backed by poor collateral.

• SSFA can be lower (<100%) than SFA (100%) for the bond that detaches above K(A) but below K(IRB) due to an implicit cap (50% in eq. 1) for K(A). This could be the case for bonds backed by poor quality collateral. See the example:

  

• SSFA can be lower than SFA for bonds backed by less granular pool as SFA has a marginal capital term that decreases with N while SSFA ignores this term. If we change the effective number of exposures in the above example to 5, the SFA capital would be higher than SSFA:
  

Based on the analysis above, it looks like SFA is indeed more risk-sensitive than its simplified version SSFA. In particular, bonds backed by high quality collateral would benefit in capital savings from SFA modeling, while poor quality collateral would suffer a higher capital charge under SFA  So, this may provide incentives for banks to invest in data and modeling infrastructure to enable SFA modeling. However, the fact that banks are still subject to capital floor under standardized approaches (i.e. SSFA) due to Collins Amendment could be discouraging. Also, the SFA floor (1.6%) in the proposed rules is about three times higher than the European rules (0.56%) - this certainly will create competitive issues between US banks and European banks; US regulators are still yet to fix this issue.