“For any securitized product with loss severities below 50%, the cost of capital for the bank improves when the pace of liquidation accelerates.”

Clearly, this is something that a strategist or credit modeler would like to know as she or he considers liquidation time lines… but how can one make such a statement based on a capital rule that does not refer to time, or LGD, or a 50% threshold? Well, those are the types of conclusions that you can derive by going the extra mile on SSFA. Let’s take a look!

God used beautiful mathematics in creating the world.
– Paul Dirac

## Banks risk charges for the uninitiated

A healthy banking system is a key driver for growth and wealth in open economies. In developed countries, financial regulators watch after this system to protect its stability and help maintain confidence in its structure.
In this context, capital adequacy ratios are one of the main tools at their disposal to limit the leverage and the riskiness of banks’ balance sheets. Those ratio state the minimum amount of equity capital required for certain amount of assets, adjusted for risk (the so-called risk-weighted assets). Because of competitive pressures between countries and the high level of integration of international banking systems, most central banks around the world chose to participate in the Bank for International Settlements (BIS). In turn, this organization started to develop, among other things, regulatory capital and liquidity standards named Basel I, II and III.

In particular, following the adoption of the Basel III accord, the three U.S. federal banking agencies revised the US regulatory capital requirements for securitization holdings (ABS, CLOs, CMBS, RMBS… ). To come up with the amount of required capital, the rule sidelined rating agencies and internal credit models in favor of the SSFA, a simplified version of the supervisory formula approach used in Basel II.

It is this very formula and its implications that we will be discussing in the rest of this blog.

## The Simplified Supervisory Formula Approach

Stated simply, the purpose of the Simplified Supervisory Formula Approach (SSFA) is to calculate the risk weight of a securitized asset given the risk weight of its underlying collateral pool.

We are giving here a condensed version of this capital adequacy formula but a curious reader can find the detail of the rule at 12 CFR section 3.43 for its OCC version, at 12 CFR section 217.211 for its Federal Reserve version and at 12 CFR 324.43 for its FDIC version.

First, let’s consider the inputs of the SSFA formula: $K_G$ is the weighted-average capital requirement for the underlying collateral pool of the securitization. It is calculated as the product of 8% times the weighted-average risk weight for those assets. $w$ is the ratio of collateral that is 90-days past due or worse (including foreclosure, REO, bankruptcy or in default). Certain deferred payments on student loans and consumer loans may not be counted in this number though.  $A$ is the attachment point of the securitized asset. It is the smallest loss ratio on the collateral that would create some amount of principal losses on the securitization asset. $D$ is the detachment point of the securitized asset. It is the smallest loss ratio on the collateral that would create a 100% principal loss on the securitized asset. $p$ is a supervisory calibration parameter equal to 0.5 for standard securitizations and 1.5 for resecuritizations (securitizations where the pool contains other securitized assets).

In turn, the SSFA formula is as follows: $K_{SSFA}(A,D)=\frac{e^{\alpha u }-e^{\alpha l}}{\alpha(u-l)}$

with: $\alpha=-\frac{1}{pK_A}$, $u=D-K_A$, $l=(A-K_A)^+$, $K_A=(1-w)K_G+0.5w$

provided however that any portion of the securitization asset below the $K_A$ credit support will be accounted for as if $K_{SSFA}$ was equal to $100\%$ and that in no instance the final risk weight should be less than 20%.

## The implications for delinquencies

Clearly, the SSFA formula incorporates the level of losses only through the $K_A$ term, which in turn is defined by $K_A$ as $K_A=(1-w)K_G+0.5w$.

We can derive two key remarks from this setup:

• Delinquent assets have the same treatment as performing assets that have a 50% $K_G$ ( or 625% risk weight). Any pool asset with a risk weight higher than 625% will implicitly receive a favorable treatment in a securitization once it becomes delinquent. Typical value for loan risk weights is around 100% so the situation is probably rare.
• In a securitization with heterogeneous credits, the formula does not differentiate if the delinquencies are occurring on the better or on the worse collateral assets.
If we compare two pools with the same aggregate $K_G$, the pool with more heterogenous credit is more likely to have some level of delinquency over its life. As a result, heterogeneous pools should produce higher capital costs.
(In that respect, the formula could be easily improved if the risk weight of each collateral asset was substituted with 625% risk weight when it turned delinquent in the calculation of the $K_G$ and then discarding the $K_A$ calculation).

### Securitization carpaccio

If we focus on the more interesting portion of the capital stack where $A$ is above the $K_A$ threshold and that the 100% cap risk weight does not apply, we notice that the SSFA formula is an integral: $K_{SSFA}(A,D)=\frac{1}{\alpha(u-l)}\int_{u}^{l} e^{\alpha x }dx$
applying a simple translation in the integral, we obtain the following: $K_{SSFA}(A,D)=\frac{1}{D-A}\int_{A}^{D} e^{\alpha (y-K_A) }dy$

Clearly this formula is a an averaging of the term in the integral between $A$ and $D$. Said differently, the formula is built as the average risk charge of infinitely thin slices where the capital of each thin slice is equal to $K_{SSFA}(x,x)=e^{\frac{1}{p}}.e^{\alpha x}$.

Assuming we only consider such thin slices, we can also find the attachment point $K_F$, where the risk weigh starts to floor apply: $K_F = K_A (1 - p.ln( 20\% \times 8\%) )$

For $K_A=8\%$, $w=0\%$ and $p=0.5$, $K_F \approx 24.5\%$.

We can distinguish three regimes for risk charges depending on the attachment point $x$ of the thin slice. We show this in the chart below: If we focus on the middle section where $K_{SSFA}(x,x)$ varies, we can easily demonstrate that the capital is always decreasing when $x$ increases and that the point of steepest decrease is when $x=K_A$ and the value for that steepest decrease is equal to $\alpha$.

### Balancing delinquencies and liquidations

Let’s look at what happens when some amount $\beta$ of the pool liquidates with a loss ratio $L$.

When this happens:

• the pool size changes from $1$ to $1-\beta$
• delinquencies change from $w_0$ to $w(\beta)=\frac{w_0-\beta}{1-\beta}$
• the attachment point changes from $x_0$ to $x(\beta)=\frac{x_0 - L\beta}{1-\beta}$

The variations of $K_{SSFA}(x,x)$ are then: $\frac{\partial K_{SSFA}(x,x)}{\partial \beta} = -K_{SSFA}(x,x).\frac{\partial \frac{x(\beta)}{K_A(\beta)}}{\partial \beta}$

which has the same sign as $S(\beta)=x(\beta)\frac{\partial K_A(\beta)}{\partial \beta} - \frac{\partial x(\beta)}{\partial \beta} K_A(\beta)$

in particular for $\beta=0$, $S(0) = x_0 \frac{ (w_0 - 1) (0.5-K_G) }{(1-\beta)^2} - \frac{ (x_0 - L) }{(1-\beta)^2} K_A(0)$

After simplifications, we find this has the same sign as $2L-\frac{x_0}{K_A}$.

Now remember that we assumed we were in the middle section of the chart where $K_{SSFA}$ varies and $K_A \ge$x_0\$. This means that if $L \le 50\%$, then the sign of $\frac{\partial K_{SSFA}(x,x)}{\partial \beta}$ is negative and $K_{SSFA}(x,x)$ decreases.

Said differently, if $L \le 50\%$, the bank has a better capital treatment if the collateral is liquidated instead of remaining delinquent.

This is precisely the opening statement of this blog:

“For any securitized product with loss severities below 50%, the cost of capital for the bank improves when the pace of liquidation accelerates.”

Note that the 50% severity is not a necessary condition for this improvement in capital charges, but it does ensure this behavior in the formula.

In fact this improvement remains in many other regimes when credit support is thick enough, even if loss ratios exceed the 50% threshold. In particular, when the credit support exceeds $2K_A$, risk charges always improve with liquidations, at any level of loss ratios.

For instance, in the situation when $K_G=8\%, w_0=10\%, p=0.5$, we can workout the boundary where the combination of credit support and loss ratio improves or worsen capital charges by using the equation above: 