# How to Determine Capital Calculation

PrecisionLender’s pricing solution is a RAROC based tool, where risk adjusted net income is divided by risk adjusted capital. We use the simplified term ROE (return on equity) to indicate that ratio. The numerator is the after-tax net income generated by the loan or deposit. The denominator is risk adjusted capital that is necessary to support the opportunity being priced.

Over the years, the solution has expanded, and several new features have been incorporated that effect our capital calculation process.

We’ll explain how the capital component is determined within PrecisionLender and discuss some of the new features incorporated within the solution as related to capital calculations. We’ll provide capital calculation examples for an amortizing loan and a line of credit (LOC) – using the Multifactor approach and Probability of Default/Loss Given Default (PD-LGD) for each.

Before reading this article, we would suggest reviewing “How Does the Math Work?” That provides a broad view of the general calculations within the solution. It goes into the capital calculation, but not to the same level as within this article. This is meant to be a more detailed examination of the methods used.

## Two Different Capital Measures

The capital component within the solution can be broken out into two main areas, the economic capital and the minimum, or regulatory capital.

Economic capital (EC) is a combination of the credit capital and the unmitigated capital (operational and market risk).

The credit capital takes into account the risk of the loan, particularly as indicated through the Probability of Default (“PD” and also referred to as Loan Loss) value. This is typically used in a Basel III equation to determine a time-weighted capital percentage. This percentage can then be applied against the exposure at default (EAD) or adjusted EAD (aEAD) to determine the amount of credit capital required each month of the loan.

The unmitigated capital is typically a factor, multiplied by the average loan amount or the net interest income generated by the loan.

Regulatory capital (RC) is a set percentage that is generally applied against the EAD each month. This set percentage can vary by risk rating. In addition, depending on the exposure type  ̶  for example, owner occupied residential mortgage compared to high volatility commercial real estate  ̶  there can be an adjustment factor. The percentage used is determined by the bank.

Unfortunately, there is no set answer to what this factor should be, especially since there are at least four regulatory capital measures. For most “pass” type loans, many clients use the Tier 1 Risk measure for well capitalized banks or currently 8%. Similarly, there can be a minimum loan loss allowance factor used. This is also often scaled by the risk rating of the loans.

## Multifactor Compared to PD-LGD

For almost all PrecisionLender clients, there are two options that can be used to determine EC , either a Multifactor approach (collateral and guarantees) or PD-LGD (borrower rating and facility rating).

PD-LGD tends to be used by some of our larger clients, but the end results can be similar in both methods. Both methods require a one-year PD and duration adjusted capital percentage levels. The main difference is in how collateral and guarantees are handled within the capital and loan loss calculations.

### Multifactor

The Multifactor approach focuses on the amount, type, and recovery factor of collateral and guarantees, as an offset to the EAD. An advantage of this method is that small changes in the amount of collateral can affect the amount of capital and loan loss requirements.

In using this approach, there needs to be an assumption regarding the net recovery (after all expenses, including lost interest) on the collateral and guarantees. This net recovery should vary, based on the type of collateral  ̶  e.g., the rate for multifamily commercial real estate is generally higher than accounts receivables. The type and financial standing of the guarantor will affect the likely recovery rate, while the risk rating of the guarantor will also affect how the guarantee will reduce the exposure.

### PD-LGD

The PD-LGD method uses the stated EAD. However, typically the collateral and guarantees used to support a loan are evaluated and placed into an LGD category (facility) with an associated percentage. This percentage is then multiplied by the EAD. There is often some internal bank or vendor supplied model to determine what facility rating or category to use, based on the amount and type(s) of collateral and guarantees.

There is a formula to convert a net recovery value to LGD, depending on the loan to value (LTV) percentage. This is an extremely accurate representation at the beginning of the loan, although it can lose some validity later in the term.

If LTV > Net Recovery%, then LGD = 1 – (Net Recovery% /LTV), otherwise LGD = 0%. Thus, if the Net Recovery% = 50% and the LTV on the loan is 80% the LGD would be 1- (50% / 80%) = 1 – 62.5% = 37.5%.

Another way of looking at this: If the loan is \$1 million, the collateral value is \$1.25 million (LTV = 80%), and the net recovery rate is 50%, then LGD = (Loan Amt – Net Recovery of the collateral)/ Loan Amt or (\$1 million –(\$1.25 million *50%))/\$1 million or \$375,000/\$1 million or 37.5%.

Of course, on an amortizing loan the question might be: “Do we use the LTV at the time the loan is originated, or the average expected LTV over the life of the loan?” Further questions on the determination of LGD is beyond the scope of this article and is dependent on the internal model used by the bank.

## First Step: Calculating Loan Loss and Capital

The first step used in determining these values is to calculate the Loan Loss or PD Percentage and Credit Capital Percentage, based on the various duration at the different risk rating.

This first step actually involves several parts:

1. A credit migration is provided by the bank in the form of a risk rating migration matrix.

2. This is used to determine a first year PD or Loan Loss Percentage by risk rating (Exhibit 1) 3. The first-year PD or loan loss % can then be duration-adjusted for periods longer than one year. In the Multifactor method, we tend to use a cumulative provision for credit loss (PCL) calculation for loan loss% on loans with longer terms. This is based on a bank’s risk rating migration matrix. Exhibit 2 shows an example of the cumulative PCL. This can be adjusted into an annual PCL, Exhibit 3. In the PD-LGD method, often the first year PD is used for each year regardless of the term of the loan, this is also an option in the Multifactor method. 4. Once the first year PD is determined, many banks, as well as PrecisionLender, uses either the Basel III capital formula (see- International Convergence of Capital Measurement and Capital Standards- A Revised Framework) or a variation. This formula is shown in Exhibit 4. In this same exhibit we show the variation of the formula typically used within the solution when using the PCL method. 5. When the Multifactor method is being used, there is an adjustment on guarantees for the risk rating of the opportunity and borrower (if different), this uses as a base formula similar to the Basel calculations shown in Exhibit 4.

6. The capital % and guarantee factor % can be calculated for each year, for each risk rating. Typically, at PrecisionLender we determine all these values, but only directly use those associated with the 12-month, 60-month and 120-month points. PrecisionLender will interpolate the values in between these levels. Exhibit 5 shows what these values might be for a risk rated 4 (average pass) loan and borrower. 7. Besides credit capital, as discussed above, a second component of capital is related to unmitigated risk, such as operational and market risk. This can be measured as a percentage of the loan size and/or the net interest income.

## Calculating Loan Loss and Capital: Some Examples

In all these examples we will use the values shown in Exhibit 5. However, Exhibit 6 shows the interpolation of the data on a monthly basis, over an initial 36-month period. Let assume the initial loan size is \$1 million, with a rate of 5% (on an actual/365 basis). There is collateral of \$1.25 million (80% LTV), with a net recovery percentage value of 50%. There is a personal guarantee for the full loan amount with a net recovery percentage of 5%. There is a minimum capital requirement of 6% and loan loss of 0.15%. Unmitigated capital is assumed to be 1% of the average loan balance. Finally, it should be noted that the tables that follow are in spread sheet format - this is for demonstration purposes only. Actual calculations within PrecisionLender are through the use of dynamic programming languages.The following examples examines using both Multifactor and PD-LGD. While it might be helpful to review all the examples, if your institution just uses one of these methods the description on the other can be skipped. ## Amortizing Loan – Multifactor

In the first example, assume an amortizing loan with a term of 36 months and an amortization period of 120 months. PrecisionLender does monthly calculation and creates a balance sheet for each month of the loan. Thus, in this example there would be 36-monthly balance sheets. These would be combined to determine an average capital balance and loan loss.

For the Multifactor approach, the EAD is adjusted by the net recovery value of collateral and guarantees. In the case of an unsecured loan with no guarantees, this adjusted EAD (aEAD) is the average expected loan level for each month. Thus, at 36 months remaining - or at the start of the 3-year loan - the aEAD is \$1 million, while at one month remaining it would be \$757,882, or the final principal payment, which is the same as the EAD.

If there is just collateral but no guarantees the first month aEAD is \$1 million less the net recovery value of collateral (50% * \$1.25 million) or \$375,000. The net recovery value would stay the same throughout the loan term, so with one month left the aEAD (assuming no guarantees) would be: \$757,882 less \$625,000 = \$132,882.

Including a \$1 million loan guarantee effects the aEAD for capital calculations by reducing it using the formula (1 – Guarantee Factor%) * Guarantee Amount * Net Recovery%. Note the Guarantee Factor % generally declines as the loan term goes down. Exhibit 7 and 8 show the calculation of the Capital and Loan Loss for the first several months of the loan and the last three months. (The complete printout is in the appendix.

Column I in Exhibit 8 shows the calculation of credit capital and column J the unmitigated capital. The unmitigated capital is simply the EAD (column A) times the unmitigated capital percentage (1% in this example). The combination of both of these columns  ̶  or Column K  ̶  is the Economic Capital. The average over the 36 months is \$42,457. Column L shows the monthly calculation of the minimum or regulatory capital, the average over the term is \$52,908. PrecisionLender allows the bank to choose which type of capital to use in the calculation of the ROE (RAROC)  ̶  either the Economic Capital, the Minimum (Regulatory) Capital, or the monthly greater of the two.  The latter  is the general approach that most clients use. In Exhibit 8, in column M, this averages to \$55,174.

As is shown in the appendix, for the first 10 months the Economic Capital is larger, thereafter the Minimum is greater. Whether the economic or minimum is greater and when the switch occurs, depends on several factors. In some cases, the Minimum Capital is always greater.  In others it does not have any influence until late in the loan life.

The same can be true with the minimum loan loss. The loan loss calculation is not affected by the guarantee factor shown in exhibit 5. However, the facility risk rating can be different than the guarantor’s risk rating. In that case there would be different loan loss percentages used. In column N of Exhibit 8 the loan loss percentage for the facility is shown, while column O shows the guarantor’s loan loss percentage. In this example they are the same.

Thus, the loan loss is the product of aEAD before guarantees (column C of Exhibit 7) less the net recovery on the guarantee (Column D) and the loan loss percentage for the facility (column N) plus the net recovery value of the collateral times the loan loss % for the facility times the loan loss percentage of the guarantor.  Except in very high-risk loans, the latter part of the equation is close to zero. As noted, the minimum loan loss (if any) is compared monthly to this calculation and the greater of the two levels is used. In this example the average loans balance is \$881,792 so the capital to asset ratio would be 6.26% and the loan loss to capital is 0.17% per annum. ## Line of Credit – Multifactor

With a line of credit, an average balance is determined and typically the balance is assumed to stay constant throughout the term of the line. (Note PrecisionLender does have the ability to vary the balance over the term. This is particularly relevant for construction and development loans).

The calculation of the EAD is a little more complex than for an amortizing type loan. The full commitment is not used, nor is the project average balance. EAD for lines is dependent on the usage percent (U%) and the expected usage of the committed but unfunded balance at time of default or Usage Given Default (UGD). Thus EAD = Commitment * U% + [Commitment * (1-U%) *UGD]. (See Column A of Exhibit 9.)

The unmitigated capital uses EAD (not the full commitment) and the Unmitigated percentage. (See column K in Exhibit 10.) The Minimum Capital, which is based on regulatory guidelines, uses credit conversion (CC) rules. These rules state that the CC is 20% if the original term of the line is one year or less and 50% if greater than one year.

Thus, Minimum Capital = {Commitment * U% + [Commitment * (1-U%) *(20% or 50%)]} * Capital Percentage. (See Column L in Exhibit 10.) All other calculations are similar to those shown in Exhibit 7 and 8. The assumptions are also the same, except we assume a usage percentage of 60% and a UGD of 80%.  ## Amortizing Loan: PD-LGD

Capital can also be calculated using and Probability of Default - Loss Given Default (PD-LGD) method. Typical the loan loss is based on the one-year PD and this percentage value is constant regardless of the term of the loan. The main difference between this method and Multifactor, as mentioned, is the use of an LGD constant instead of stated values of collateral and guarantees. While the LGD percentage is based on collateral and guarantees, there is typically a separate model run to determine the level. It is not the intention of this article to explain the calculation of LGD, which are often bank dependent, or why one bucket is used compared to another for a given amount of collateral. In this and the next example we will assume an LGD value of 35%. The loan is considered a 3-year loan with a 20-year amortization. All the base assumptions are used as detailed above.

The calculations using LGD are less complex than in the Multifactor method. As shown in Exhibit 11 (Note the full exhibit is in the appendix) an EAD is determined, which is the monthly balance of the loan (column B).

The Credit Capital is simply the monthly EAD times the Credit Capital% (Column C) times the LGD (column H) as shown in Exhibit 12. The Unmitigated calculation as discussed is the Unmitigated Percentage (column D) times the EAD (Column B). The calculation of loan loss is EAD times LGD times the PD percentage (column L).

Using a 35% LGD results in similar capital to that in the Multifactor method, however, that may not always be true and, as mentioned, the internal formula to determine LGD can play a big role in the capital and loan loss calculation. PrecisionLender allows the bank to choose which type of capital to use in the calculation of the ROE (RAROC) either the Economic Capital, the Minimum (Regulatory) Capital, or the monthly greater of the two. The latter is the general approach that most clients use.  ## Line of Credit – PD-LGD

The final example is a line of credit using the PD-LGD method. This is shown in Exhibit 13 and 14. The assumptions used are similar to those used for the amortizing PD-LGD method. As discussed in the Multifactor example: The calculation of the EAD is a little more complex than for an amortizing type loan. The full commitment is not used nor is the project average balance. EAD for lines is dependent on the usage percent (U%) and the expected usage of the committed but unfunded balance at time of default or Usage Given Default (UGD).

Thus EAD = Commitment * U% + [Commitment * (1-U%) *UGD]. (See Column B of Exhibit 13.)
The unmitigated capital, however, is the product of the funded amount outstanding and the unmitigated percentage. (See column H in Exhibit 14). The minimum capital, which is based on regulatory guidelines, uses credit conversion (CC) rules. These rules state that the CC is 20% if the original term of the line is one year or less and 50% if greater than one year.

Thus, Minimum Capital = {Commitment * U% + [Commitment * (1-U%) *(20% or 50%)]} * Capital Percentage. (See Column J in Exhibit 14.)

All other calculations are similar to those shown in Exhibit 11 and 12. The assumptions are also the same, except we assume a usage percentage of 60% and a UGD of 80%.  ## Other Examples and Conclusions

We have tried to show the most likely loan types and their capital calculations. We did not show a construction loan. That would be like a line of credit, except the balance would be increasing over time and thus a larger EAD (assuming UGD is not set to 100%).

We also only included one piece of collateral and a single guarantee. The results would be similar for multiple collateral (assuming the use of a weighted average recovery percentage). They would also generally be similar for multiple guarantors, but we suggest you review the Engine Standards for the exact formula used.

Additional information, particularly on interest only loans and deposits, can be found on the PrecisionLender support site. For additional questions please contact one of the Precision Lender’s Managing Directors or your Client Success Manager.