Credit Risk The Loan Equivalency Factor

for Revolving Lines of

••This article proposes a framework for estimating credit conversion factors (CCFs) in measures of exposure at default. Based on the Basel II Advanced IRB approach, the proposed framework can be a reference for both credit risk practitioners and regulators.

b y Ke l i n Pa n Ba s e l II a l l o w s banks to use their own measures of credit slightly greater than 1. The empirical CCF expressed in risk to calculate appropriate levels of economic capital. equation 2 is an obligation level and can be considered Prior to implementing this Advanced IRB (internal ratings a random variable with some statistical characteristics. A based) approach, however, the bank must model three ma- statistic of CCF at the portfolio level, as expressed in equa- jor credit risk parameters: tion 1, can be used to capture such characteristics. 1. Probability of default (PD), which is the probability that The Advanced IRB approach emphasizes that banks a borrower will default in the next year. should use estimates of off-balance-sheet items and reflect 2. Exposure at default (EAD), which is the bank’s expo- the probability of additional drawings by borrowers up to sure to borrowers upon their default. and after a default event is triggered. In this case, EAD is 3. Loss given default (LGD), which is the percentage of defined as EAD that the bank ends up losing. As outlined by the Basel Committee on Banking Su- 3 EAD = NB + Ave(LEQ) x (NE – NB), pervision in June 2006, Basel II requires banks to con- sider on- or off-balance-sheet exposures in their own EAD where average LEQ is the mean of the loan equivalency fac- estimates. For on-balance-sheet exposures, the following tor in a special bucket based on rating grade and/or other formula can be adopted to estimate EAD: factors that will be determined by statistical characteris- tics or business practice. In equation 3, net exposure (NE) 1 EAD = Ave(CCF) x NB, minus net book balance (NB) is equal to the amount off balance sheet. where NB is the current net book balance for each ob- For commercial loans, when granting a revolving line ligation, and CCF is the credit conversion factor specified of credit, a bank usually provides a credit limit. A bor- for on-balance-sheet exposures. The following empirical rower obtains an immediate loan amount and the future formula can be used to estimate CCF: availability of the total loan amount. Accordingly, the corresponding exposure for the bank has two parts: the 2 CCF = (NB + Accrued Interest + Accrued Fee)/NB. outstanding balance (NB) and the current commitment (NE). The outstanding balance refers to the amount Usually, accrued interest and accrued fee are equal to drawn by the obligor, while the current commitment or greater than zero. Therefore, CCF is always equal to or includes drawn and undrawn portions that the bank has

50 June 2009 The RMA Journal The Loan Basel II requires banks to consider on- or off-balance-sheet exposures Equivalency Factor in their own EAD estimates.

for Revolving Lines of Credit in Commercial Lending

promised to lend to the borrower at his or her request. The second model can accurately describe the statistical The probability of drawing the undrawn portion in the characteristics of LEQ within the bucket and minimize the next 12 months is defined as the loan equivalency factor volatility of LEQ. (LEQ) off balance sheet. Since a probability always has values from 0 to 1, LEQ is constrained into a range of 0 Empirical Expression of LEQ to 1. The LEQ is defined as a ratio of the change in the amount A few published papers have discussed LEQ in com- drawn at default to the current undrawn commitment, mercial lending. Asarnow and Marker (1995) showed that which can be expressed as LEQ decreases with increasing facility rating. The authors 4 studied 50 facilities for Citibank covering the 1988-93 pe- LEQ=(NBtd – NBt)/(NEt – NBt), riod. They looked at LEQ for facilities with rating grades

of BB/B or worse, then extrapolated the results to facilities where td and t are time at default and one year prior, re- with better ratings. They found an inverse relationship spectively. NB and NE represent net book balance and net between LEQ and facility ratings—a conclusion based on exposure, respectively. The numerator of equation 4 rep- inference from a limited sample rather than actual obser- resents the change in the drawn amount at default in the vations for better rated facilities. next 12 months from time t. The denominator represents Araten and Jacobs (2001) estimated LEQ for revolving the current undrawn amount due to the current commit- credits and advised lines from JPMorgan Chase. The au- ment minus the drawn amount (net book balance), which thors used 408 facilities covering the period 1995 to 2000. is equal to the undrawn amount. The data showed an increasing trend of average LEQ for Equation 4 should be greater than zero if there is a the best rated grades (for example, AAA/AA- to A+/A-) drawing action at default. In this case, the net book bal- and a decreasing trend for the worst ones. However, the ance will increase if the denominator is positive. How- model predicted a decreasing trend with increasing rating ever, banks will constrain the further drawing, and, in grades. The authors also found an increasing relationship many cases, the relationship managers will require the between LEQ and time to default. Their results were simi- borrowers to pay down a bit of the amount in order to lar to those of Asarnow and Marker on the relationship avoid default. In this case, the numerator will be zero or between LEQ and facility ratings. negative, which reveals that banks do not allow the ad- Based on peer research, including that of Araten and ditional draw when an obligation was closely watched to Jacobs, this article argues that LEQ rises with increasing default, suggesting better risk controls and risk manage- risk ratings for the better grades and decreases for the ment on the part of banks. In this case, LEQ is truncated worse ones. This finding reflects the competition between to zero. banks and borrowers with regard to the drawdown of the The other case is current over-limits—or when the de- unused commitment. When a borrower goes into finan- nominator of equation 4 becomes zero or negative. Bank cial distress, she or he would like to use the undrawn policy usually imposes strict constraints on over-limits for commitment. On the other hand, through the use of loan commercial loans. Generally, there are two approaches to covenants, banks will protect themselves by closing the approving the applications for over-limits. If the amount unused commitment. is less than or equal to a small percentage of commitment, This article proposes a two-piecewise linear model to fit a de minimus can be made by lenders without having to go empirical LEQ. The other proposed model is for the mean to the loan approval committee. However, if the amount of LEQ on a bucket level (a cluster) based on risk ratings. for the over-limits application is greater than a percentage

The RMA Journal June 2009 51 of commitment, the loan will need to go through a formal • Utilization percentage (UT) is defined as net book bal- process known as a fresh decision. ance divided by net exposure at time t. Usually, UT has In either case, the current limit will be increased, which a negative correlation with LEQ (see the discussion in will result in a positive denominator. In this case, it is sim- Araten and Jacobs, 2001). ply assumed that the LEQ is equal to 1 for both de minimus • Remaining life (RL) of an obligation is defined as ma- and fresh decision. After the truncation and cap, the distri- turity date minus the current date for an obligation. bution of LEQ should be between 0 and 1, which satisfies If an obligation is composed of several loan notes, a the definition of the probability. Usually, the distribution weighted average maturity will be used. of LEQ is bimodal, based on how much data has been • The age of an obligation is defined as the current date truncated or capped. minus the origination date. As in discussions of the ma- The heavy tail of LEQ is another characteristic of its turity date, the origination date will be a weighted aver- distribution. As discussed above, the zero LEQ reflects age if there are multiple loans in a single obligation. banks’ strict policy on drawing revolving lines of credit • Risk rating (RR) is an important independent variable for the close-to-default obligations. It also reflects the res- of LEQ since, as discussed previously, LEQ depends on cue efforts by allowing borrowers to pay a certain amount the credit situations of borrowers. Accordingly, a model to avoid default. In order to capture those characteristics using two pieces of a linear relationship is proposed to of risk control and risk management in banks, the non- reflect the relationship between LEQ and risk rating. positive LEQ will be kept and truncated to zero. On the (The next section will discuss this topic in more detail.) other hand, LEQ is capped to 1 if it is greater than 1, • Industry index is a dummy variable defined as follows: which reflects another type of risk management practice 1, sec 1 that banks frequently use. Constraining LEQ into 0 to 1 ind = makes sense in both statistics (the property of probability) {0, sec 2 and business practice. where sector 1 may include the major portfolio in commer- cial loans such as C&I. Sector 2 may include other whole- Principles of Variable Selection on LEQ sale portfolios such as commercial real estate and leasing. It The selection of key drivers of LEQ is based on two prin- is convenient to consider sector 1 as a base reference. ciples. First, it is based on the available data in banks. The variables can be classified into continuous variables or dis- Proposed Models for LEQ crete ones based on data characteristics in model develop- If the LEQ for each individual obligation is considered ment. The variables can also be classified as raw variables as a random variable, it can be modeled using the linear and defined variables. The raw variables can be obtained regression approach. In order to capture the character- directly from a database. The defined variables are ob- istics in credit risk management as discussed previously, tained by redefining the raw variables. It would be unwise the following generic form of a two-piece linear model is to make any correlations between defined variables and suggested: raw variables. For example, it may be that the net book balance is correlated with the average of net book balance, a 5 a a a Aaa,Aa,A 0 + 1RR + ∑ i xi which is a defined variable. i=2 LEQ = b In order to examine the correlation between indepen- β β β 0 – 1RR + ∑ i yi Baa,...,C dent variables and between LEQ (a dependent variable) i=2 and independent variables, the second principle of vari-

able selection should be considered, which is based on the where ai and βi are model parameters that can be deter- Pearson correlation coefficient. A SAS code “PROC CORR” mined using statistical tools, such as SAS “PROC REG” can be used to conduct the correlation check. The null hy- with “Selection=Stepwise” (stepwise method). In 5 above,

pothesis (H0) for Pearson correlation is that the correlation the sign for risk rating (RR) is opposite in the two equa- coefficient is equal to zero. If a p-value is small (usually tions. The positive sign in the first equation indicates an taking a 0.05 significance level), the null hypothesis will increasing function of LEQ in the risk rating. The second be rejected, and therefore the two variables are correlated equation has a negative sign between the LEQ and the risk with each other. rating, which implies a decreasing relationship between This conclusion is favorable for checking the relation- the LEQ and the risk rating. The number for other inde-

ship between LEQ and its independent variables. The op- pendent variables (xi or yi) in both equations is not neces- posite criterion should be applied in order to check the sarily equal (a ≠ b) in general. relationship between the independent variables. The fol- The other criterion for the model selection is the R- lowing defined variables are suggested. square of the model. The greater the R-square, the better

52 June 2009 The RMA Journal Table 1 modeled results are also given in Table 1. The kernel func- An Illustrated Average LEQ in Risk-Rating Grade tion 6 fits well with the data, especially for grades 2 to 7.

RR 1 2 3 4 5 6 7 8 9 Conclusions The empirical credit conversion factor (CCF) and loan LEQ 0.001 0.05 0.23 0.19 0.11 0.08 0.05 0.03 0.02 equivalency factor (LEQ) have been discussed for both on-balance-sheet and off-balance-sheet exposures. The LEQ 0.010 0.05 0.23 0.16 0.11 0.08 0.05 0.04 0.03 major conclusion is that in order to calculate LEQ for off- Model balance-sheet items, it is necessary to conduct truncating and capping techniques, an approach that guarantees LEQ the interpretation of modeled LEQ on raw data. Using two declining by a reasonable range (from 0 to 1) and reflects pieces of the linear equation can improve R-square if raw the bank’s credit risk management strategies. The distribu- data shows two different linear trends of LEQ. The cut- tion of LEQ relies very much on such strategies. point of risk rating between the two pieces of the model The principles of variable selection for LEQ have been depends on the data, which reflects the practice of credit established. Two model methodologies are proposed risk management. Equation 5 uses Moody’s risk-rating based on different levels of LEQ—the individual level scale as an illustrated example for cut-point. and the bucket level. A two-piecewise linear relationship Finally, the variables selected using both the correlation between LEQ and risk rating has been developed in indi- coefficient and the linear regression should be the same. vidual levels of LEQ, and a nonlinear model with a kernel Two methods can be used for comparison to determine function form has been proposed to model average LEQ the final independent variables of LEQ. in the bucket level. Generally, if a borrower is in financial The LEQ can also be clustered into buckets that are distress or in a situation of deteriorated credit, he or she usually measured by rating grade and other factors. A will draw more from the unused commitment. Meanwhile, two-dimensional bucket is more practical and intuitive. according to established loan covenants, banks will forbid For example, two factors, rating grade and time to default, the borrower from withdrawing the unused funds. This have been used by Araten and Jacobs (2001). It is also competition between banks and borrowers results in two practical to use risk-rating grade and industry sector for a pieces of the LEQ in risk rating at an individual level and a bucket, which makes more business sense. nonlinear bell-shape mean of the LEQ at a bucket level. The volatility of LEQ will be reduced by carefully se- A bimodal distribution presents difficulties for model lecting the bucket and considering an average LEQ in development using the linear regression approach, which a bucket. In this case, the variance of LEQ is separated will result in a low R-square of the model. The model built into the variance between the buckets and within the on the bucket level for the mean of LEQ performs much buckets. Such diversification of LEQ variance can reduce better than the model built on the individual level. v the volatility of LEQ, which is usually very high due to a bimodal distribution of LEQ. The following example il- •• lustrates such a methodology. The risk rating is based on Kelin Pan, Ph.D., has worked at the University of Western Ontario, the University of Moody’s rating scale from Aaa to C, which corresponds British Columbia, and Citizens Bank as adjunct professor, lecturer, and senior quantitative to grade numbers 1 to 9 (Aaa, Aa, A, Baa, Ba, B, Caa, Ca, analyst, respectively. The author of numerous publications, he is now working in the area and C). of financial risk management. The views expressed in this paper are those of the author In Table 1, the cut-point between the two pieces of LEQ and do not necessarily reflect the opinions of Citizens Bank. Contact him by e-mail at is set at rating grade 3 (grade A). In each piece, it is obvi- [email protected]. ous that no linear relationship exists between LEQ and risk rating, which is commonly encountered in practice. References For capturing the nonlinear relationship between LEQ Araten, Michel, and Michael Jacobs Jr., “Loan Equivalents for Revolving and RR, the following kernel function is suggested, Credits and Advised Lines,” The RMA Journal, May 2001, pp. 34–39. Asarnow, Elliot, and James Marker, “Historical Performance of the

6 LEQ = A exp[– B x – x3 ], U.S. Corporate Loan Market: 1988–1993,” The Journal of Commercial Lending, Spring 1995, pp. 13–32. where A and B are model parameters, and x and x3 corre- Basel Committee on Banking Supervision, “International Convergence spond to RR and RR=3, respectively. Two parameters can be of Capital Measurement and Capital Standards,” Item 474, June 2006, determined using the data points in Table 1. After a simple p. 104. Available at http://www.bis.org/publ/bcbs128.pdf. computation, we have A=0.23, B=Ln(0.23/0.05)=1.526 for RR≤3, and B=[Ln(0.23/0.11)]/2=0.3688 for RR>3. The Write to [email protected].

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