credit risk methods to estimate losses using linear ... · average quarterly bp (3bp/4) 116.9...

November 2011 The RMA Journal

Credit Risk

by Peter A. Smith

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••This article outlines how linear regression analysis can be used to calculate the allowance for loan and lease losses.

Nature and Purpose of the Allowance for Loan and Lease Losses (ALLL)1 The ALLL represents one of the most significant estimates in an institution’s financial statements and regulatory re-ports. Because of its significance, each institution has a responsibility for developing, maintaining, and document-ing a comprehensive, systematic, and consistently applied process for determining the amounts of the ALLL and the provision for loan and lease losses (PLLL). To fulfill this responsibility, each institution should ensure controls are in place to consistently determine the ALLL in accordance with GAAP, the institution’s stated policies and procedures, management’s best judgment, and relevant supervisory guidance.

As of the end of each quarter, or more frequently if warranted, each institution must analyze the collectability of its loans and leases held for investment and maintain an ALLL at a level that is appropriate and determined in accordance with GAAP. An appropriate ALLL covers es-timated credit losses on individually evaluated loans that are determined to be impaired as well as estimated credit losses inherent in the remainder of the loan and lease portfolio. The ALLL does not apply, however, to loans carried at fair value, loans held for sale, off-balance-sheet credit exposures (e.g., financial instruments such as off-balance-sheet loan commitments, standby letters of credit, and guarantees), or general or unspecified business risks.

Methods to Estimate Losses Using Linear Regression Analysis


The RMA Journal November 2011 61

The ALLL consists of two components, Accounting Stan-dards Codification (ASC) 450, formerly known as Financial Accounting Statement (FAS) No. 5 (Accounting for Contin-gencies), and ASC 310, formerly known as FAS No. 114 (Accounting by Creditors for Impairment of a Loan).

The “classified” or “bad” portfolio is analyzed for impair-ment on a loan level basis in accordance with ASC 310. For loans determined to be impaired, a specific loan loss reserve is calculated. For collateral-dependent loans, the reserve is typically based on the fair market value of the collateral (as-is appraised value less costs to sell); otherwise, the reserve is based on either observable market transactions or a net-pres-ent-value discounted cash flow analysis. If it is determined that a loan is impaired but there is no dollar impairment, the loan remains in the ASC 310 portion of the ALLL. If a loan is determined not to be impaired, it is migrated back to ASC 450 and included in the appropriate pool.

The purpose of the ASC 450 calculation is to estimate the dollar amount of potential losses embedded within the “unclassified” portfolio—that is, the “pass” or “good” por-tion of the portfolio. Loans are aggregated into homogenous pools that exhibit similar risk and performance profiles. Each pool is analyzed separately. Regulators typically require banks to use their actual annualized (pool level) historical loss data covering a two- or three-year period (preferably three) and will consider weighting different periods more than others if there is logic to support it.

To round out the analysis, banks are required to include internal and external metrics. Other than in general terms, regulators do not provide specific guidance on how to derive or apply these metrics. However, regardless of which metrics are used, they must be logical and have a causal influence on future potential losses. Moreover, the end results, when taken as a whole (ASC 450 and ASC 310), must be within regulatory guidelines.

This article specifies the use of historical data as a part

Methods to Estimate Losses Using Linear Regression Analysis

of the ASC 450 calculation. It examines the analytical pro-cedures currently in use and outlines how linear regression analysis can be used. The regression model combines historical loss data with external metrics—in this case, U.S. unemployment and FDIC loan loss data—to forecast future losses. The purpose of this article is to help banks explore the concepts described here using their own data and discuss their results with both their regulators and accountants. As with any methodology, it will need to be justified, documented, and incorporated into the bank’s ALLL policies and procedures. The guidance here does not address accounting requirements for financial statement disclosure.

Several decisions are required in organizing historical loss data for analysis, including those related to data collection and analysis, pool definition, and risk-grade definition.

Data Collection and Analysis: Quarterly (or monthly) historical data collected for individual pools over a two- to three-year period is typical. Depending on the complexity and method chosen, risk-grade tracking may be required.

Pool Definition: Homogenous pools can be defined based on call report codes—for example, 1A1 1-4 Family Construction, 4A Commercial & Industrial, etc. To fur-ther granulate the process, sub-pools can be created. For example, Owner Occupied Nonfarm Nonresidential (call code 1E1) could be further subdivided based on property type/size (office, retail, warehouse, industrial, over/under 50,000 square feet, etc.) and/or by geography (state, city, suburban, central business district, etc.).

Regardless of which metrics are used, they must be logical and have a causal influence on future potential losses.


Risk-Grade Definition: The following is an example of a typical risk-grading system (the higher the risk grade, the higher the risk). Risk grades are a function of the character of the borrowers, their capacity to do what they say they can, the amount of borrower capital, the quantity and quality of the collateral, and lender conditions. In addition, some banks supplement their risk grades with sub-risk-grade categories. • Pass—Riskgrades1to5•Watchlist—Riskgrade6• Substandard—Riskgrade7• Doubtful—Riskgrade8• Loss—Riskgrade9

There are several methodologies for modeling future losses based on historical data. Each of these calculations can be made more granular by defining and subdividing loan pools. In addition, adjustments based on trend analy-sis and weighting factors can be used to reflect where the current economy is within the business cycle. Some of the methods are as follows:•Quarterly actual losses, weighted evenly or weighted un-

evenly (see example below). • Risk-grade migration.• Risk-grade migration probability analysis and attendant

probability of migration to higher and lower grades, coupled with probability of loss (migration to default status) and expected loss upon default.

• Linear regression analysis that 1) defines the relationship of the historical record of a specific economic factor to historical FDIC loss data and/or 2) regression analysis that defines the relationship of the bank’s actual loss experience to the factor in question.The quarterly actual loss model is relatively straightfor-

ward. Table 1 shows an example.

The resultant factors are applied only to the pass portion of the portfolio plus any migration of nonimpaired loans from the classified risk grades. The logic is that at the time the loans were made they were all good loans; however, as a portfolio

they have embedded losses that are reflected in the loss history.

The quarterly factor is annualized and then multiplied times the current quarter end balance for the pool in ques-tion. The results for each pool are summed to complete the process.

The risk-grade migration method is similar, except that losses are tracked and quantified by risk grade. Table 2 shows an example:

Foreachriskgrade,an“average”BPoflosscanbecalcu-lated (as outlined above) and applied to the risk-grade strips of the pool in question. This method provides insight as to the risk associated with each risk grade. Because the classified loans are analyzed individually for impairment pursuant to ASC 350, the “classified” risk-grade factors are applied only to those loans that have been determined not to be impaired.

The risk-grade migration probability model is more com-plex and seeks to 1) determine the probability that a given risk grade will migrate to a default status and 2) quantify the loss given default. The critical steps are as follows:• Constructarisk-grademigrationchartbasedonthebe-

ginning and ending risk grade of a loan over a defined period of time (one to two years) to determine the prob-ability that any one risk grade will migrate into a default status.

• Determinetheamountoflossgivendefault.Table 3 shows an example of the results of a risk grade

(RG) migration analysis for one risk grade within a desig-nated pool.

62

Table 1Owner-Occupied Nonfarm Nonresidential (call code 1E1)

Quarter 1Q210 2Q2010 3Q2010 4Q2010 Total

Balance (millions) 800 900 1,000 1,250 3,950

Loss (millions) 10 15 8 12 45

Quarterly BP* 125.0 166.7 80.0 96.0 45

Weighting 15% 15% 30% 40% 100%

Average Total BP (3Balance/3Losses) 113.9

Average Quarterly BP (3BP/4) 116.9

Weighted Average BP (3Balance x BP x Weight/3Balance x Weight)

103.8

*BP refers to basis point; 100 BP equals 1.0%.


Quarter 1Q210 2Q2010 3Q2010 4Q2010

Pass (RG 1-6):

Balance 700 800 900 1,200

Loss 1 0 1 1

BP 14.3 0.0 11.1 8.3

Substandard (RG 7)

Balance 75 65 70 40

Loss 2 3 3 6

BP 266.7 461.5 428.6 1,500.0

Doubtful (RG 8)

Balance 25 25 30 10

Loss 7 12 4 5

BP 2,800.0 3,428.6 1,333.3 5,000.0

Total Balance 800 900 1,000 1,200

Total Losses 10 15 8 12


Ending Risk Grade/Outcome Probability Over Migration Analysis PeriodBeginning RG RG1 RG2 RG3 RG4 RG5 RG6 RG7 RG8 RG9 Default Payoff Total

RG 4 0% 0.01% 1.97% 56.73% 24.97% 3.82% 2.92% 0.22% 0.01% 0.08% 9.27% 100.00%


To be included in the analysis, a loan must be a part of the sample for the entire period unless it defaults and a loss is experienced. The “payoff” category accounts for loans that were withdrawn from the sample with no end state recorded. There are a variety of explanations: credit was no longer required, the debt was refinanced, or the bank may have declined to renew. In the above analysis, the “payoff” category is included in the total; alternatively, these loans could be excluded and the risk-grade migration probabilities adjusted accordingly.

Once all of the risk grades have been analyzed, a prob-ability- of-default schedule for each risk grade within the pool is created (Table 4).

Next, the loss given default that occurred during the analysis period is estimated for each pool by dividing the sum of losses experienced for all loans that defaulted by the sum of the maximum loan amounts that occurred during the analysis period. For example, if 25 owner-occupied loans defaulted, resulting in $2,000M of losses, and the sum of themaximumloanamountswas$32,733M,thelossgivendefault would be 6.11%.

The final step is to apply the factors to the current RG balances (dollar balance x probability of default x loss given default) within each pool and sum the results for each risk grade. For example, if the RG 4 balances within the owner-occupiedpoolwere$64,382M,thentheestimatedALLLlosswouldbe$3,147;$64,382Mx0.08%x6.11%.Again,because the classified loans are analyzed individually for impairment pursuant to ASC 350, the “classified” risk-grade factors are applied only to loans that have been determined not to be impaired.

Another approach is linear regression analysis. Once the appropriate data is selected and collected, it is relatively easy to use Microsoft Excel® or other readily available software to construct a scatter diagram, perform the regression analysis, plot the trend line, and compute the R2 and the equation for the forecast.2

Figure 1 shows a regression analysis of the percent of U.S. unemployment versus the annual percent (reported quarterly) of net charge-offs based on FDIC data for all real estate loans.

An R2 equal to 0 means there is no correlation between the two factors. An R2 equal to +1.0 means a perfect direct correlation, and an R2 equal to –1.0 means a perfect inverse correlation. An R2inthe80%+/–rangeisdesirable.

The equation establishes the relationship between x, the percent of U.S. unemployment (or some other appropriate

factor), and y, the percent of net charge-offs, and enables the calculation of net charge-offs given a forecast for un-employment.

Note in Figure 1 that if U.S. unemployment goes below 3% +/–, the forecast for net charge-offs becomes negative (that is, the “linear” line crosses the x or percent of U.S. unemployment axis). The reason is that the true relationship over a broad range of unemployment is cur-vilinear. The above analysis is valid only over a range of unem-ployment, in this case roughly between 6% and 10%, where the relationship is predominately linear. In order to define the relationship at higher or lower ranges, additional data points are required.

Regression models can be constructed for specific call code pools with similar results. These models are particu-larly useful for banks that do not already have a two- or three-year loss history.

For banks that already have a sufficient loss history, ana-lyzing their historical loss data versus a chosen economic factor using regression models is a possibility. However, this analysis may not produce satisfactory results in terms of a high R2(80%+/–).Apotentialreasonforthisisabank’s higher volatility relative to the volatility of that for all institutions taken as a whole. One solution is to perform a regression analysis that relates FDIC cumulative loss history with the cumulative loss history of the bank.

The following is an explanation of how to incorporate cumulative loss history into the ASC ALLL calculation. Here are the sequential steps, followed by an example and supporting data:1. Using Excel, perform a regression analysis of quarterly

annualized FDIC net charge-offs versus the percent of U.S. unemployment by call code pool (see Figure 1).


Probability of Default

RG1 RG2 RG3 RG4 RG5 RG6 RG7 RG8 RG9

0.01% 0.01% 0.02% 0.08% 0.27% 1.85% 3.70% 5.69% 6.06%

Regression Analysis of U.S. Unemployment versus Net Charge-offs

Figure 1

For banks that already have a sufficient loss history, analyzing their historical loss data versus a chosen economic factor using regression models is a possibility.

3.0%

2.5%

2.0%

1.5%

1.0%

0.5%

0.0%

%Net

Charg

e–Of

fs–FD

IC

%U.S. Unemployment0 2 4 6 8 10 12

y=0.0033x – 0.0109R2 = 0.8908 Series 1

Linear (Series 1)


2. Again using Excel, perform a regression analysis of cu-mulative FDIC net charge-offs and the bank’s cumulative net charge-offs for eight quarters (see Figure 2).

3. Forecast future FDIC net charge-offs using step 1 above based on a forecast of unemployment one year forward.

4. Add the results to the current cumulative FDIC net charge-offs.

5. Using the results in steps 4 and 2, calculate the bank’s future cumulative net charge-offs.

6. From the results in step 5, subtract the bank’s current cumulative net charge-offs to obtain the forecast for the current quarter.Figure 2 shows an example using all real estate loans:

1. Data and regression analysis of the FDIC’s and the bank’s cumulative net charge-off data:

2.Assuminginterestratesatora forecastof7.7%,theforecast for FDIC net charge-offs is 1.45% (0.0033 x 7.7–0.0109).

3. The forecast for FDIC cumulative net charge-offs is 17.59%(1.45%+16.14%).

4. The forecast for the bank’s cumulative net charge-offs is 21.34(1.2157x17.59–0.0398).

5. The quarterly forecast for the bank is 2.26% (21.34 –19.08).

6. The quarterly forecast is multiplied times the current risk grade 1–6 pool balance.Whiletheaboveanalysiswasappliedtoallrealestate

loans, the same methodology can be applied to any call

code pool. The above model can also be used as the basis tobuildaBPrisk-gradematrixusingtheforecastasatarget.This in turn can then be used to calculate the ALLL at the risk-grade level. Here are the mechanics of how to build the matrix in two steps:

Step 1:• Data:Moody’sandS&Phistoricaldefaultratesforcor-

porate bonds by rating category.• Analysis:•DeterminetherelationshipbetweentheBPdefaultrate

and that for the next higher category (see table below: 73/40=1.825,155/73=2.123,etc.).

•Usetheresultstobuildarisk-grademodel.

Step 2: The methodology used to build the model as shown below is as follows:• RG1isassignedaBPriskof0.• InitialguessfortheBPriskatRG2(seetablebelow:initial/finalguessis10.17).

• UsetheRGfactors(seeabovetable)tobuildtheBPriskfor the remaining RGs.

• Computetheweighted-averageresults(seetablebelow:RG3=10.17x2.5,RG4=25.425x4.0,etc.).

• Iftheresultsarehigher,lowertheinitialguess;iftheyare lower, increase the guess.

• Continueadjustingtheguessuntilweighted-averagere-sults are approximately equal to the forecast (see table below:theforecastwas2.26%or226BP.225.03BP).

The risk-grade distribution (percentage of pool) can be established in a number of ways:• Existing (that is, the distribution of the current

portfolio).

64

2009-10 Cumulative Net Charge-offs

Figure 2

Quarter FDIC Cumulative NC Bank Cumulative NC

2010:Q4 16.14% 19.08%

2010:Q3 14.17% 12.93%

2010:Q2 12.34% 8.87%

2010:Q1 10.40% 7.61%

2009:Q4 8.35% 5.26%

2009:Q3 5.73% 1.21%

2009:Q2 3.50% 0.37%

2009:Q1 1.44% 0.44%

Risk Grade Bond Rating BP Average Default Bond FactorBond Factor

Range RG Factor Used

RG 1 Aaa 40

RG 2 AA 73 1.825

RG 3 A 155 2.123 1.9–2.6 2.5

RG 4 Baa 555 3.581 3.5–3.8 4.0

RG 5 Ba 1,902 3.427 2.9–4.6 4.0

RG 6 B 3,933 2.068 1.3–2.6 2.0

Existing Portfolio:

Risk Grade % of Pool RG Factor RG BP Factor Weighted Average = 3RG times %

RG1 0.20% 0 0.00

RG2 0.40% 2.50 10.17 0.04

RG3 22.90% 4.00 25.425 5.82

RG4 45.90% 4.00 101.7 46.68

RG5 18.80% 2.00 406.8 76.48

RG6 11.80% 813.6 96.00

100.00% 225.03

25%

20%

15%

10%

5%

0%

-5%

Bank

FDIC0% 5% 10% 15% 20%

y=1.2157x – 0.0398R2 = 0.9069 Series 1

Linear (Series 1)


• Historical.• Establishedbymanagementasdesirable.

If the existing distribution is used, the ALLL forecast willbeequaltotheforecast—inthiscase,225BPtimesthe RG 1–6 balance. The individual RG notwithstanding, risk-grade factors can be used to analyze and measure the impact of changes in the risk profile as loans move from one risk-grade bucket to another. In some cases, the exist-ingdistributionmaybeskewedinsuchawaythattheBPfactors for individual risk grades in a specific pool relative to other pools are not consistent, even though the weighted average for the pool is okay. In those cases, consideration should be given to adjusting the distribution factors.

Historical distributions should eliminate any anomalies due to skewing. However, to the extent that the historical distribution differs from the existing distribution, the overall calculated ALLL will be different. In the following example, if the factors are based on historical balances and then applied to the existing balances, the factor for the pool increases the weighted-averagefactorfrom225BPto289BP.Ofcourse,within the entire portfolio, there will be instances where the reverse is true so that there will be some offsetting effects.

If management establishes the risk-grade factors annu-ally based on the above analysis and holds them constant for four quarters, the impact of risk-grade migration will automatically be captured, provided the factors are applied at the risk-grade level. In this case, a quarterly regression analysis can be used to supplement the analysis by calcu-lating the incremental change as a result of improving or declining unemployment and incorporating the results as an external factor/adjustment.

In the previous model, the bank will have to decide which rate of unemployment to use. In any event, you will probably want to use that which aligns most closely with your lending area. Next, the question is whether to use the current rate or a forecast for the future. To the extent a future rate is used, it becomes embedded in the forecast as an exter-nal factor.The “risk grade 7”

factor for nonimpaired loans could be devel-oped by expanding the default matrix above to the next level below a “B” rating. Alternatively, it could be established as a multiple of the fore-cast. A multiple of 4.0 +/– is suggested; this is consistent with regulatory guidelines that RG 6 loans have reserves of 200BPto700BPandRG7loans700BPto2,500BP,orratiosof3.6to5.0.Intheaboveexample,theRG7factorwould be .1,028BP.

Even if a bank chooses to stay with a more traditional historical loss calculation, regression analysis can be used, as mentioned above, to supplement the calculation by calculating the incremental change due to changes in un-employment.

The purpose of the above discussion was simply to outline some of the methodologies a bank can choose to incorporate into its ALLL methodologies.

As mentioned earlier, in addition to using historical data, the following internal/external factors need to be incorpo-rated into the ASC 450 analysis as outlined below.• Natureandvolumeofloanportfolio:•Quarterlychangesinpool-levelweighted-averagerisk

grades.•Quarterly changes in pool-level weighted-average

delinquencies.• Concentrationsofcreditrisk:•Peer-groupanalysisbasedontheUniformBankPer-formanceReport(UBPR).

• Changesineconomicandbusinessconditions.• Competition,legal,andregulatoryrequirements.• Changesinlendingpoliciesandprocedures.• Qualityofloanreviewsystem.• Managementexpertiseandturnover.

As a final step, the ALLL results should be back-tested against regulatory guidelines for reasonableness and to ensure they are directionally consistent with portfolio per-formance. Below is a summary of regulatory guidelines. Note that the ranges are very broad and the “right” answer will depend in part on current economic conditions and

Historical Portfolio:


RG1 2.00% 0.00 0.00

RG2 5.00% 13.08 0.65

RG3 30.00% 2.50 32.70 9.81

RG4 40.00% 4.00 130.80 52.32

RG5 15.00% 4.00 523.20 78.48

RG6 8.00% 2.00 1,046.40 83.71

100.00% 224.98

Existing Portfolio:


RG1 0.20% 0 0.00

RG2 0.40% 2.50 13.08 0.05

RG3 22.90% 4.00 32.7 7.49

RG4 45.90% 4.00 130.8 60.04

RG5 18.80% 2.00 523.2 98.36

RG6 11.80% 1,046.4 123.48

100.00% 289.41

The individual RG notwithstanding, risk-grade factors can be used to analyze and measure the impact of changes in the risk profile as loans move from one risk-grade bucket to another.


forecasts. An ALLL calculation outside the defined range will be difficult to justify to the regulators.

In conclusion, linear regression analysis can refine and improve the historical component of the FAS 450 ALLL calculation as well as provide an analytical framework for improving the understanding of portfolio risks. v

••Peter A. Smith is a credit portfolio analyst at Essex Bank, Glen Allen, Virginia. He can be reached at [email protected].

Notes1.ThisexcerptistakenfromtheInteragencyPolicyStatementfor

theAllowanceforLoanandLeaseLosses(OCC2006-47,December2006). It covers key concepts and requirements included under ALLL supervisoryguidanceandGAAP.

2. R2, also referred to as R-squared, is a statistical measure of how well a regression line—in this case, a straight line—approximates real data points.

ReferencesShahram Elghanayan, “Taking Account of the Economic Cycle in ALLL,” The RMA Journal, February 2006, pp. 32–36.

FDICQuarterlyBankingProfile/LoanPerformance:http://www2.fdic.gov/qbp/index.asp

Moody’s Credit Risk Calculator: http://v2.moodys.com/cust/content/Content.ashx?source=StaticContent/Free%20Pages/Products%20and%20Services/Downloadable%20Files/CRC_Brochure.pdf

MonevatorBondDefaultProbabilitybyRating:http://monevator.com/2010/04/09/bond-default-rating-probability/

FidelityIncomeProductsCreditandDefaultRisks:http://personal.fidelity.com/products/fixedincome/risks.shtml

CRM Credit Risk Migration Analysis: http://www.creditriskmgt.com/credit-risk-migration-analysis.html

An Internal Ratings Migration Study © 2004 by RMA: http://www.defaultrisk.com/pp_other_84.htm

66

Table BP Range

Risk Grade Low High High/Low Ratio

Pass 75 125 1.67

Special Mention 200 700 3.50

Substandard 1,000 2,500 2.50

Doubtful 3,000 7,000 2.33

Loss 10,000 10,000

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