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.