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1 IMS DISCUSSION PAPR SRIS Creit Risk Assessment Consiering Variations in xposure : Application to Commitment Lines Shigeaki Fujiwara Discussion Paper No INSIU FOR MONARY AND CONOMIC SUDIS BANK OF JAPAN 2-- NIHONBASHI-HONGOKUCHO CHUO-KU, OKYO JAPAN You can ownloa this an other papers at the IMS Web site: Do not reprint or reprouce without permission.

2 NO: IMS Discussion Paper Series is circulate in orer to stimulate iscussion an comments. Views expresse in Discussion Paper Series are those of authors an o not necessarily reflect those of the Bank of Japan or the Institute for Monetary an conomic Stuies.

3 IMS Discussion Paper Series February 2008 Creit Risk Assessment Consiering Variations in xposure: Application to Commitment Lines Shigeaki Fujiwara* Abstract With the worlwie financial market confusion cause by the subprime mortgage problem an the increase in creit line contracts with relaxe covenants, there are cases where financial institutions are facing emans to provie aitional creit to securitize vehicles with heightene liquiity an creit risks. hese are typical examples emonstrating the importance of risk management consiering variations in exposure. here are also calls for incorporation of future variations in exposure into the moel for the Basel II avance internal ratings-base approach. his paper aopts commitment lines as a creit provision with variable exposure an constructs a creit risk moel whereby stochastic new borrowing eman is linke to changes in a firm s asset value. hrough simulations, the paper then consiers the interepenence among exposure at efault, probability of efault, loss given efault, expecte loss, an unexpecte loss. he paper also prepares a simple moel for the covenants, an verifies the influence of the riginess of covenants on expecte loss an other risk factors. Key Wors: Commitment lines; Probability of efault; Loss given efault; xposure at efault; xpecte loss; Unexpecte loss JL classification: G2, G32, G33 * Deputy Director an conomist, Institute for Monetary an conomic Stuies, Bank of Japan -mail: shigeaki.fujiwara@boj.or.jp he author woul like to thank Associate Professor Satoshi Yamashita of the Institute of Statistical Mathematics an the staff of the Bank of Japan for their useful comments. Views expresse in this paper are those of the author, an o not necessarily reflect the official views of the Bank of Japan.

4 . Introuction he subprime mortgage problem, which became more severe uring 2007, has triggere confusion in worlwie financial markets. It is also exerting iverse influences on financial institutions that o not irectly hol such loans. For example, some types of financial vehicles, which investe in securitize subprime mortgages, epene on ABCP for part of their funraising; as the ABCP has become ifficult to roll over, these vehicles have mae use of large sums of supplementary liquiity facilities backup lines from certain uropean an US banks. As a result of unexpecte liquiity provision, banks balance sheets are suenly expaning. In some cases, this strains the funing of the banks themselves, an the provision of creit to entities with heightene creit risk is resulting in massive losses. hese cases illustrate just how important it is for financial institutions to manage risk stemming from variations in exposure beforehan. here are also calls for the Basel II framework to manate the evaluation of creit risk consiering exposure variations. he first pillar of the Basel II accor requires the measurement of creit risk, market risk, an operational risk in the computation of capital aequacy for risk assets Basel Committee on Banking Supervision [2005a, 2005b]. Among these, banks must choose one of three methos for the measurement of creit risk: the stanarize approach, the founation internal ratings-base approach, an the avance internal ratings-base approach. Banks that choose the avance internal ratings-base approach must estimate the values of three factors that etermine the amount of losses: probability of efault PD, loss given efault LGD, an exposure at efault ad. A great eal of research has alreay been conucte on PD an LGD, an progress has been achieve in the evelopment of moels. In contrast, for ad almost all of the research substitutes exposure at the time when creit risk is evaluate, an only a few stuies such as Moral [2006], Yamashita an Yoshiba [2007], an Kupiec [2007] aress exposure variations through maturity. he SPCs known as ABCP conuits are one example. ABCP conuits use CP an other relatively short-term funraising an invest in iverse securitize proucts. Banks provie these SPCs with securitize proucts, an provie supplementary liquiity facilities when issuing ABCP.

5 ypical examples of exposure variation inclue i erivatives, 2 ii bills, housing loans, an other instruments whose principal is repai in installments incluing prepayments, etc., iii aitional loans, an iv commitment lines. Among these, aitional loans an commitment lines require a creit risk evaluation moel that assumes changes in exposure because their risk is expecte to increase via increases in exposure. specially with commitment lines, banks cannot control their exposure, because banks must passively provie new loans in response to firms execution for withrawal rights obtaine in compensation for commission fees. Accoringly, banks set covenants beforehan to prevent increases in exposure to firms with heightene creit risk. For covenants, banks use a combination of multiple conitions that can be observe an easily verifie, such as the worsening of a given financial variable beyon a given level. Recently there has been an increase in so-calle covenants light contracts with comparatively loose waiver clauses in uropean an US lening markets, an such lening has resulte in a stronger nee for risk management. Chart presents the creit limits an amount rawn own uner commitment lines in Japan. he creit limits have consistently risen since 200, an are now approaching 30 trillion yen. he rawn own amount remaine aroun two trillion yen through mi with little change, has been rising since then, an presently excees five trillion yen. 2 For example, the exposure in interest rate swaps is zero at the contract ate, but the exposure then stochastically changes to positive or negative with subsequent changes in the yiel curve, to be precise, the six-month forwar rate curve. 2

6 Chart : Commitment line market in Japan trillion yen Creit limits Amount rawn own Source: Bank of Japan Note: Figures cover city banks, trust banks, Saitama Resona Bank, Shinsei Bank, Aozora Bank, Regional Banks, an Regional Banks II. his paper consiers commitment lines, whose market scale has expane in recent years, an aims to present an example of a creit risk moel that incorporates stochastic changes in exposure. he moel also incorporates the effect of covenants on creit risks. he paper constructs a creit risk moel whereby new loan eman changes stochastically in association with changes in a firm s asset value, i.e., a eterminant factor of its PD, an evaluates the creit risk of commitment lines uner conitions whereby there are correlations between ad, PD, an LGD. While more etaile verification base on the ata of iniviual companies is, of course, necessary to confirm the appropriateness of the moel an the parameter settings, the investigations focus on the following points using simple moel settings an simulations through which cause effect relations can easily be confirme. I. What interepenence exists among ad, PD, an LGD? II. What influence is consequently exerte on expecte loss L an unexpecte loss UL? III. How oes creit risk change with the riginess of covenants, an what factors affect its optimal settings? Major finings an quantitative examinations are as follows. I. Creit risk tens to increase with exposure to the firm whose new loan eman 3

7 rises when its asset value eclines. Positive correlation between ad an PD in the moel escribes this result. II. Covenants that prevent uncontrollable increases in exposure to poor performing firms are useful for limiting creit risk. In some cases, lax covenants can lea to greater losses, an this largely epens on the structure of how changes in a firm s asset value lea to new loan eman. III. Consequently, verifying this structure is important for controlling creit risk when proviing commitment lines an setting covenants. IV. here is a covenants level that optimizes the trae-off between the increase in loan interest earnings an the increase in L. he remainer of this paper is structure as follows. Chapter 2 presents prior stuies aressing exposure variations. Chapter 3 first presents an overview of commitment lines, an then attempts to create a moel for stochastic loan eman via commitment lines, using the financial ata of iniviual firms. By linking changes in a firm s asset value to changes in loan eman, a Merton-type structural moel, which can be generally use to evaluate creit risk, is applie. Chapter 4 evaluates the creit risk of commitment lines using a Monte Carlo simulation, an consiers the optimal covenant settings for banks. Chapter 5 conclues the paper with a summary of the finings. 2. Prior Research on xposure Variations his chapter presents an overview of three research papers that aress variations in exposure: Moral [2006], Yamashita an Yoshiba [2007], an Kupiec [2007]. Moral [2006] Moral ecomposes ad: ad = t LQ L t, where t is the exposure at time t, L is the creit limit of the commitment line a bank sol at/before t = 0, an LQ is the percentage of the commitment line use from time t to efault ate relative to the unuse commitment line at time t. Moral attempts to estimate the unknown LQ loan equivalent, an attempts to erive ad at 4

8 time. Specifically, Moral introuces the following three simplifie methos. i Simple average of the observe values We consier the LQ over the coming year. First we collect ata on efault firms whose commitment lines have similar properties, such as the firm s inustry an stan-by or revolving type see Chapter 3 for etails, an then calculates the actually observe LQ from the ad at the efault ate an exposure one year prior to efault. he LQ value is estimate to minimize the sum of the squares of the error between the observe LQ an the estimate value in other wors, the simple average of the observe LQ is taken as the LQ estimate value. ii Weighte average of the observe values While the above metho treats all of the observe ata equally, another approach is to weight the iniviual ata by importance an then calculate the estimate value. For risk management, rather than knowing the future rawn own amount by parties that are alreay close to full raw own of their commitment line, it is more important to know the future raw own by parties that still have a large amount of unuse line. In this case, the weighte average of the LQ values can be aopte as the estimate LQ. One metho for computing the weighte average is to use the square of the unuse portion as the weight. iii Minimization of the loss function he amage from wrong risk management estimates is ifferent between the cases of an overestimate an an unerestimate of the potential amounts of withrawal. One way to aress this is to incorporate a large penalty for unerestimating the rawn own amounts of commitment lines when estimating the LQ. As an application example, Moral points to the estimation of minimum capital requirements, using an asymmetrical evaluation function that assigns larger weights for unerestimating the minimum capital requirements. Assuming that there is no estimation error for PD or LGD, the estimation error for minimum capital requirements results solely from the ad estimation error. he estimate LQ is then foun by minimizing the evaluation function. 3 3 When there are estimation errors in PD or LGD, or when they are time variable, this is reflecte in 5

9 2 Yamashita an Yoshiba [2007] Yamashita an Yoshiba examine the conitions where the bank supplies aitional loans to minimize the expecte loss, an analytically evaluate how the new loan affects L an UL. 4 hey assume that a firm s asset value follows a geometric Brownian motion, an that banks can provie aitional loans at specific points in time. Defaults by the firm occur when its asset value is less than its liabilities at loan maturity. Uner these moel settings, aitional loans are mae in two opposite cases. he first is the low asset value case in which the new loan contributes to a ecline in PD. he secon is the high asset value case in which the new loan contributes to an increase in interest income. he paper also notes that when aitional loans are provie following the principle of minimizing L, UL conversely rises, an highlights the unetecte risk in the case of a bank ecision mae only on the expecte value. 3 Kupiec [2007] he Basel II formula for calculating minimum capital requirements in the first pillar takes ad an LGD as given an makes the calculation using an asymptotic single risk factor ASRF moel. On the other han, there are empirical analyses showing that PD, LGD, an ad all grow larger uring recessions. 5 Kupiec attempts to moel these positive correlations among PD, LGD, an ad. Specifically, Kupiec uses latent variables to etermine a firm s asset value, LGD an ad, using a common factor an iniviual factors that follow normal istributions. he setup establishes the correlations among PD, LGD, an ad through the common factor. Kupiec then erives the analytical solution for the portfolio loss rate uner these moel settings. However, the value of the LQ estimate. stimation methos for ad alone instea of the minimum capital requirements may be esirable to avert such bias. 4 UL is efine as the expecte loss uner stress SL minus L. See Chapter 4 for etails. 5 ad apparently increases uring recessions because funs eman rises as cash flow worsens uner poor business performance. One reason why LGD increases uring recessions is that for loans with collateral the value of the collateral eclines uring recessions, resulting in a lower recovery rate upon efault. Pykhtin [2003] notes that ignoring the correlation between PD an LGD results in an unerestimation of the creit risk. 6

10 Kupiec oes not explicitly incorporate the cause of the variation in ad such as changes in firm funs eman into the moel, an simply introuces changes in ad as a stochastic process. 3. Commitment Lines an Funs Deman Moel Outline of commitment lines he law concerning commitment line agreements 6 efines commitment lines as a contract between a bank an its corporate clients, which legally obliges the bank to exten loans to the clients upon their request up to the amount that is agree at the time of the contract within the term of valiity that is also agree at the time of the contract. he contract grants corporate clients the right to withraw any time within the term of valiity any amount up to the limit esignate in the contract. In return for the commitment, the bank receives a commitment fee. More specifically, the characteristics of commitment line contracts can be summarize as follows. 7 Corporate clients can borrow funs at any time within the creit limit. Commitment lines can be categorize into two basic types: stan-by lines of creit use as a backup for emergency cases such as when it becomes ifficult to issue CP, an revolving lines of creit for borrowings uring normal times. he loan interest rate is stipulate at the time of the contract as a specific rate or as a sprea over a reference rate such as ibor or the prime rate. Corporate clients pay a commitment fee on the creit line, or on the unuse portion of the creit line. Banks sometimes set covenants in the contracts whereby they can refuse to provie loans uner commitment lines, mainly base on the eterioration in 6 Operational since March 999. With the implementation of this law, the use of commitment line contracts starte to sprea in Japan. his law stipulates that the commissions pai by borrowers uner commitment line contracts are not regulate by the Interest Rate Restriction Law or the Capital Subscription Law. 7 his list was mostly prepare referring to Dai-Ichi Kangyo Bank International Finance Department [200]. 7

11 financial conitions of the corporate clients such as the capital ratio an the interest coverage ratio. 2 Factors etermining funs eman Many stuies have examine commitment lines, incluing Campbell [978] an Martin an Santomero [997], partly because the commitment line market evelope long ago in the US. hose stuies, however, are primarily concerne with the etermination of the fair value of commission fees an o not focus on moeling funs eman as the motivation for rawing own new loans. herefore, the simple assumptions of funs eman moels are unsuitable for the creit risk valuation moel that consiers the evelopments in ad. Specifically, many of the earlier moels cannot express the partial use of creit an raw owns before maturity, because they make the simple assumption of full use of the commitment lines or no use at all, which is totally oppose to the concept of continuous stochastic changes in the ad. o evelop such a stochastic ad, this paper begins by moeling new funs eman referring to the financial ata of iniviual firms. his paper focuses on changes in a firm s asset value as a factor etermining funs eman. In general, funs eman is expecte to increase when a firm s asset value is rising, an vice versa. However, a firm s asset value may rise when the firm improves its financial position, for example, through a reuction in borrowings. It woul be the case that a firm s asset value may also rise using ebt leverage contrary to the financial restructuring. hus, the relationship between a firm s asset value an funs eman may vary by company an across time. Moreover, funs eman oes not necessarily ecrease when a firm s asset value eclines. here may be cases where funs eman rises because of temporary cash flow tightening or back out of certain business line incurring personnel restructuring costs. Because commitment lines are often contracte for working capital, it might be more appropriate to classify funs eman into funs for working capital an funs for capital investment; however, in this paper, for simplification we o not consier the use of funs. We begin by examining the relationship between a firm s asset value an funs eman using financial ata on iniviual firms. First, we observe annual changes in 8

12 total assets for a firm s asset value an total liabilities for funs eman. Book values are use for both for convenience. For total assets, we o not ajust for market value using market capitalization to calculate total assets as the sum of liabilities an share value. hat approach woul actually make the relationship between liabilities an total assets even more ifficult to grasp, as share prices are strongly influence by other omestic an foreign financial assets an especially by stock markets in other inustrialize countries, which have been becoming increasingly linke in recent years. he ata source is the Development Bank of Japan, with figures for each fiscal year from FY 999 through FY he figure covers firms in the real estate an construction sectors liste on the First Section of the okyo Stock xchange. Firms in these inustries have large changes in asset value an liabilities, an are sai to have receive their financial support from banks. Chart 2 presents the annual percent changes in total assets an total liabilities for each firm Chart 2: otal assets an total liabilities y/y % change in otal Liabilities iconstruction y/y % change in otal Assets 9

13 ii Real estate y/y % change in otal Liabilities y/y % change in otal Assets Note: A small number of observations coul not be plotte within the graph bounaries. While Chart 2 inicates a strong positive correlation between the changes in total assets an liabilities, a consierable number of observations significantly eviate from the 45 egree line. he construction inustry inclues a substantial number of firms where liabilities increase while total assets ecline, or where liabilities i not ecline as much as total assets i. Conversely, liabilities of some firms below the 45 egree line in the thir quarant ecrease more than total assets i. his might be a result of ebt-equity swaps an write-offs of ebt. Similar trens are observe in the real estate inustry, but they are not as pronounce as in construction. Next, looking at the region where total assets increase, in many cases the increase was the result of greater ebt leverage. hat tren is observe especially in the construction inustry. Conversely, there are some firms for which total assets increase by a smaller amount than their liabilities i. 3 Funs eman moel We procee to evelop the moel with the assumptions that changes in a firm s asset value influence funs eman, that banks are proviing aequate commitment lines, an that aitional funs eman over the existing loan amount will be fille using the 0

14 commitment lines without contracting new loans. First we use a Merton-type structural moel, that is, the stanar creit risk moel, to escribe changes in a firm s asset value. Specifically, a firm s asset value A t follows the geometric Brownian motion shown in equation 2: A t = µ At t At Wt,, 2 where µ an respectively represent the rift an the volatility in the firm s asset value growth rate. he only liabilities are borrowings, an the liabilities at maturity will be the borrowings at the initial time 0 plus the loans rawn on the commitment line. Default occurs if the firm s asset value falls uner total liabilities at maturity. In that case, net liabilities become the loss at efault. As for the timing of the rawing own, the perio of time until maturity is ivie into n parts, an the timing is expresse as t = t, 2 t,..., n t, where t = / n. he amount of each new loan via the commitment line, t t t t, is moele as: an t = max[ { Ratio } >α t,0], 3 t = b t D At { A 0} D At { A 0} 2 tε t t < t,2 t, 4 where {} is an inicator function with a value of when the conition insie the parentheses {} is true, an value of 0 at all other times. quation 4 for t represents the linkage of the stochastic changes in funs eman to the changes in a firm s asset value. We assume that funs eman is D At when a firm s asset value rises an D A t when a firm s asset value eclines. ven when A t = 0, liabilities may be trening up or own, so the time tren term b t is inclue. Furthermore, t is expresse as a stochastic process with the probability term 2 tε t, 2. 2 represents volatility reflecting the uncertainty of funs eman an ε t, 2 is a ranom variable following a stanar normal istribution. he error terms in equation 2 an equation 4 are inepenent of each other. quation 3 inclues a MA function because of the assumption that loans via the commitment line are not repai until maturity time. While liabilities coul actually be reuce through repayment prior to maturity, this

15 assumption seems reasonable in the following sense. In general, loan terms seem to be etermine in perios in which funing for a firm s investments an business operations are stable. Unexpecte increases in funing eman seem to be fille by the rawing own of the commitment line. hus, the moel in this paper targets creit risk valuation for a shorter perio than the perio reflecting long-term evelopments in loan amounts. When new loan eman t occurs, the entire amount is borrowe as long as it is within the line limits an the covenants are not violate. When there is conflict with the covenants, the request for new rawing owns is refuse. As for covenants, the interest coverage ratio an other financial inices are commonly use. In this paper the marketvalue basis capital ratio Ratio t, asset value liabilities/asset value, is aopte for the financial inex referre to in covenants in orer to incorporate the evelopments in into the creit risk moel. In short, commitment line loans are issue on eman only when the capital ratio excees α. he new loan makes a firm s asset value an liabilities rise by the same amount. hereafter, a firm s asset value once again follows a geometric Brownian motion. Collateral is not consiere, so the LGD becomes liabilities asset value / liabilities. Chart 3 presents an image of the moel. Chart 3: xposure variation moel image Asset value Defaults o not occur Default threshol Initial liabilities Loss amount Defaults occur t Maturity Loan eman occurs 2

16 4 Determination of D, D Section 3 presente a moel incorporating changes in a firm s asset value into stochastic evelopment in funs eman via a commitment line. In this section, we consier the specification of parameters D an D. he scatter iagram presente as Chart 4 uses the same ata as in Chart 2, on an annual change basis, an scales the ata with total assets in FY 999 = 00. he histograms show the istribution of the slopes from the origin for these scale annual change ata. In the construction an the real estate inustries, the istributions show a maximum slope of.0, regarless of whether A is positive or negative. his leas to the hypotheses that the values of D an D equal, an the presence of the stochastic term in equation 4 results in a istribution aroun the slope. Looking at the slope istribution in etail, we see that when A is positive the maximum is, but there tens to be a long tail towar the negative, especially for the real estate inustry. In contrast, when A is negative, the istribution is approximately symmetrical. With reference to these observations, this paper sets the value of D at. On the other han, the value of D is set as 0 or negative uner the assumption that firms cannot restrain liabilities when their asset value is ecreasing, an there are concerns that such firms may be force to take on aitional loans. Such a hypothesis shoul properly be subjecte to empirical analysis base on the iniviual company ata with commitment line contracts. Because such ata o not exist, this paper analyzes creit risk base on the above-state hypothesis as the first step in the investigation of creit risk moels for stochastic ad. 3

17 y/y change in otal Liabilities Chart 4: Scale total assets an total liabilities i Construction y/y change in otal Assets A is positive Frequency All samples = A is negative Frequency All samples = 0.00 < >3.0 Slope 0.00 < >3.0 Slope 4

18 ii Real estate y/y change in otal Liabilities y/y change in otal Assets A is positive A is negative 0.30 Frequency All samples = 0.30 Frequency All samples = < >3.0 slope 0.00 < >3.0 slope 4. Simulation Results he simulation aopts the following simple settings to broaly reflect the characteristics of the creit risk moel uner which ad epens on a firm s asset value. First, we break own the continuous time moel into a iscrete time moel with two perios an three points of time present, six months later, an one year later, an 5

19 assume that aitional loan eman might occur six months later in a stochastic manner. he maturity ates for the liabilities at the initial time an for the loans rawn own on the commitment line are all one year later. We now procee to calculate the values of PD, LGD expecte LGD, L, an UL through a Monte Carlo simulation uner the above assumptions. Before conucting the simulation, we set values for the parameters other than D an D in equations 2 an 4, initial asset value A 0 an initial liabilities 0, using annual ata from 979 to 2006 for nonfinancial corporations uner the Flow of Funs Statistics. First, we set A 0 = 00 an 0 = 70 base on the average capital ratio throughout the full perio. he time series of this ratio generally shows fluctuation within the range of 20%50%. We o not assume any specific scenario on the current conition in the ratio such as heightene ebt leverage, so average values are use. Next, for the parameters in equation 2, we compute the average an variance of the annual ratio series for total assets an set the values at µ = 5%, = 0%. Finally for the parameters in equation 4 we scale the 979 liabilities at 70 an set the values at b = 2, 2 = 7 base on the average an variance of the liabilities annual ifference series. Uner these parameter settings, however, the one-year PD becomes extremely low at 0.003%. 8 Because the moel characteristics can be interprete more easily from the simulation results, we assume an enterprise with a somewhat high PD an set at 20% higher than the original one of 0%. he values of the other parameters remain as state above. With these settings, the PD becomes 2.7%, which generally correspons to PDs for firms with a BB bon rating. he creit limit on the commitment line is set at 20, an interest earnings for the bank are not consiere in the creit risk analysis. he simulation is conucte 200,000 times to stabilize the calculation results. Case when D = 0 First we examine the case where the value of parameter D is zero. Chart 5 shows the simulation results for the expecte value of the new loan amount, PD, LGD, L, 8 On a global basis, this is aroun the level of PD for corporate bons with a rating of AA. 6

20 an UL. he horizontal axis of the graph shows the measurement for the covenants, that is, the capital ratio in percent. 9 UL is calculate as follows. We assume a single-factor Merton moel 0 uner the assumption that the bank s creit portfolio is well iversifie; in other wors, the amount of creit to any given borrower is sufficiently small in the overall portfolio. UL is calculate by subtracting the value of L from the value of L uner stress stresse L, hereafter, SL. We also assume a confience level of 99.9% for the common factor uner stress, with 99.9%. he correlation between the common factor an a firm s asset value is R = In Chart 5 the covenants are assume to be between 50 an 50. Covenant values of aroun the initial capital ratio of 30% or a bit lower are consiere realistic. Despite this, we bolly set the covenants over a wie range for the following reasons. First, negative values, i.e., capital eficits, are consiere because the moel oes not consier efault prior to the loan maturity uring which firms may fall into negative net worth. Covenants above 30% are also consiere to have the effect of setting strict covenants in cases when the bank oes not allow further eclines in a firm s asset value with high creit risk or other more extreme cases when the banks provie a commitment line with a high commission expecting an increase in the firm s capital. 9 In this paper, covenants refer to whether or not the capital ratio is less than α. For simplification, hereafter α is referre to as the covenants. 0 See, for example, Yamashita an Yoshiba [2007] for the single-factor Merton moels an the efinition of UL. Uner the Basel II creit risk measurement for loans to nonfinancial corporations, parameter R must be set at 0.2~0.2, so here the intermeiate value 0.8 is aopte. 7

21 Chart 5: Simulation results when D = 0 Increase in xposure LGD Covenants Covenants 3.0 % PD 0.8 LAmount Basis Covenants SL, ULAmount Basis SL 0.80 UL Covenants Covenants Note: Graphs show mean values of the istributions for each of the risk inices, conucting 200,000 simulations. 8

22 Next, we explain the changes in each risk parameter. We review the simulation results in the three broa categories of covenants levels: i the stanar setting range of 0%30%; ii a very loose setting of 0% or less; an iii a strict setting of 30% an above. Increases in exposure i 0%30% he expecte value of the increase in exposure is more than six an less than eight, an the liabilities increase by about 0% from their initial value of 70. he increase is more or less constant at a level less than 0% of covenants, inicating that the existence of the covenants oes not restrict the increase in exposure in the range. ii 0% or less he covenants are not a constraint, an the new loan eman is just below eight. he new exposure figure epens on the probability of improvement in a firm s asset value A through equation 4 with D = 0. iii 30% an above t Because there are a large number of cases in the simulation where the covenants prevent new loan withrawal, the mean increase in exposure eclines as the covenants become strict. Changes in PD Uner Merton-type structural moels, the liability ratio etermines PD. In other wors, PD rises as ebt leverage is utilize. his can be explaine with equations as follows. Just before the commitment line loan is rawn own, the firm s asset value is A, the liabilities are, an the perio until maturity is. PD is given by: 2 PD = {ln µ }, 5 A 2 where is a istribution function of a normal istribution. It is clear that when the other parameters in equation 5 are fixe, PD is etermine by the liability ratio / A. 9

23 We now consier the influence of the new loan on PD. When the amount of the new loan is, the PD just after the loan is rawn becomes: 2 PD = {ln µ }. 6 A 2 his shows that PD rises falls because of the commitment line loan when the liability ratio / A rises falls from / A. Regarless of the amount of the commitment line loan, the liability ratio will increase whenever the loan is rawn in the case of net positive assets < A, an ecrease whenever the loan is rawn in the case of excess ebt > A. We now examine the changes in PD among ifferent covenants levels. i 0%30% In the case of net assets, new loans always raise PD, but borrowing is restricte as the covenants become more severe. For that reason, PD is a ecreasing function of the covenants level. ii 0% or less PD remains more or less constant at 2.9%. In a theoretical sense, in the case of excess ebt, commitment line loans reuce PD, an 2 making the covenants more severe weakens this effect, so PD is an increasing function of the covenants level, opposite to i. As iscusse above in the examination of exposure, however, the covenants is not a bining conition for new loan withrawal, so the effect of 2 is extremely small, an on Chart 5 PD is observe to be essentially flat. iii 30% an above Because the severe covenants restrict loan withrawal, PD eclines following the above-mentione effect an ultimately converges to 2.7% when no new loans are mae. Changes in LGD LGD remains constant at 0.07 regarless of the covenants level. Because LGD has a low sensitivity to the amount of new loans, this has almost no influence on LGD, even if new loans are restricte by covenants. An intuitive explanation is as follows. 20

24 LGD is calculate as the conitional expecte value A / A < ]. Here [ A, respectively represent the firm s asset value an the liabilities at maturity, an the A / istribution etermines LGD. he A / istribution in logarithmic form for the new loan amount of is given by: A ln 2 A 2 Nln µ,. 7 2 Because ln A / has little sensitivity to an the ln A / istribution is not influence significantly by, LGD remains essentially constant regarless of the covenants level. An explanation using ifferential calculations is presente as the Appenix. Changes in L L is approximately equal to the prouct of ad, PD, an LGD. Because LGD is essentially constant as state above, the change in L is roughly explaine by the changes in ad an PD. However, to be precise, because all three variables follow istributions with correlations in the creit risk moel, L is not actually etermine as the prouct of ad, PD, an LGD. hus, to calculate L, we calculate the expecte loss through the simulation. While it is inappropriate to interpret L as the prouct of the three variables, that approach is aopte here so that the interpretation is easy. i 0%30% L graually eclines, mostly from the ecline in PD. he effect of the ecrease in ad also contributes to the ecline in L. ii 0% or less Because ad an PD are essentially constant, L also remains constant at aroun 0.5. iii 30% an above he ecline in L is even gentler than that for i. his is because while ad suenly eclines, the ecline in PD is limite. PD influences L via existing exposure as well as new loan exposure. Ultimately L converges to 0.2 when no new loans are mae. 2

25 Changes in SL i 0%30% SL is a ecreasing function of the covenants. his is because if exposure is restraine by the covenants uring the loan term perio, SL can be reuce even if a stress event occurs at maturity. SL eclines by 0. from covenants 0% to 30%. his is largely greater than the 0.0 ecline in L, emonstrating that the covenants have a greater effect on SL than on L. ii 0% or less Because the increase in exposure an PD are essentially constant, SL is also flat. Because the covenants o not restrict an increase in exposure, SL takes a maximum value. iii 30% an above Similar to L, the size of the ecline in SL is graual compare with i. Furthermore, the area where the curve is convex is further to the right compare with L. his is because the ecline in ad has a stronger influence on SL than on L. Ultimately SL converges to 0.95 when there are no new loans. As for UL = SL L, because the variations in L are less than those in SL, UL has essentially the same shape as SL. 2 Cases where D is or 2 hus far we have assume that D = 0, where no loan eman arises from eclines in a firm s asset value. We now investigate the changes in risk when D is or 2 assuming increase borrowings following a ecline in a firm s asset value, which poses higher risk to banks. Here, the settings of all the parameters other than D are left unchange to observe the influence from changes in D. Chart 6 presents the simulation results. 22

26 Chart 6: Simulation results when the value of D is change Increase in xposure LGD LGDD-=0 LGDD-=- LGDD-= D-=0 D-=- D-= Covenants Covenants % PD LAmount Basis PDD-=0 PDD-=- PDD-= LD-=0 LD-=- LD-= Covenants Covenants ULAmount Basis ULD-=0 ULD-=- ULD-= Covenants 23

27 Increases in exposure When D is ecrease from 0 to to 2, the increase in exposure rises because greater funs eman arises when a firm s asset value eclines. In the case of no upper limit to the commitment line, the increase in exposure becomes a linear function of D from the settings of equation 4. In Chart 6, on the other han, the increase in the case of D s shifts from to 2 is less than in the case from 0 to. his is because in the simulation there are cases where new loans are restricte by the limit of the commitment line, an this reuces the average ad. When the covenants are set within the range of 0% 30%, the area where the increase in exposure begins to ecelerate is aroun 20% when D is 0 an aroun 0% when D is. his inicates that when D = an the covenants are loose, there is a large risk that the bank will not be able to restrain increase new loan eman. In contrast, when the covenants level approaches 30% exposure is almost the same regarless of the value of D, because the loan eman is suppresse by the covenants when a firm s asset value eclines. his shows that the relationship between covenants an increases in exposure can change greatly epening on the value of D, an suggests the importance of setting the covenants at an appropriate level. Changes in PD PD rises as for D becomes smaller. he maximum ifferential of PDs between 0 an 2 D is nearly %. PD grows larger as D eclines for the following reasons. New loan eman arises irrespective of whether a firm s asset value increases or ecreases six months later, but firms hol positive net assets in many cases. In this case, the liabilities ratio that etermines PD rises along with the increase in new loans. As the negative value of D eclines, the increases in the new loan an the liabilities ratio become greater. For that reason, PD rises because of eclines in D. When D is or 2, PD suenly rises as the covenants level ecreases from 20% to 0%. Because ad an PD have a strong positive correlation uner a relaxe covenants setting, creit risk increases significantly. When the covenants are set higher than aroun 25%, however, PD becomes constant regarless of the value of D because an increase in exposure ue to a ecline in the firm s asset value is restricte. 24

28 Changes in LGD LGD remains nearly constant at 0.07 regarless of the value of D an the covenants. When the covenants are lax, intuitively LGD woul be expecte to rise when large funs eman arises from eclines in a firm s asset value. However, as long as the assumptions that assets an liabilities rise equally from new loans an that the geometric Brownian motion parameter in equation 2 remains constant hol true, uner the framework of the Merton-type structural moel, LGD is barely influence by the covenants setting, similar to the case of when D = 0. See the Appenix for the reason. he gap between simulation results an intuition suggests the possibility that these two assumptions may not hol. 2 Changes in L an UL When the covenants are 30% or less, ad an PD both rise from the ecline in D, an thus L an UL both increase significantly. In particular, UL more easily rises because SL expans nonlinearly against the change in D. In Chart 6, SL is only restricte because the limit of the commitment line is set at 20, an there is great potential creit risk uner easy commitment line provision an lax covenants especially when D is negative. 3 Optimal covenants setting Up to this point, because the analysis has assume no loan interest income, banks are able to reuce L always positive by setting strict covenants. When interest earnings are consiere for valuation of expecte loss, however, another argument emerges; the optimal covenants level exists to maximize the total return compose of i higher 2 For example, even when a firm has worsening performance, at the moment of new loan withrawal, it remains as cash in the firm s balance sheet. Consequently, the assumption that assets an liabilities rise equally when new loans are rawn is feasible. In reality, however, it is entirely possible that the asset may eteriorate from the moment this cash is use for the firm s activities. In that case the firm s asset value oes not increase by the amount of the new loan. hen, LGD increases because the numerator of A / in equation 7 ecreases. Over a somewhat longer timeframe, it is also possible that the cash borrowe by firms with worsene performance may not be use effectively. In that case, the growth rate of the firm s asset value, µ, may be sai to have ecline uring the perio. hus, a possible approach to avert this kin of problem is to set µ as a function of the firm s asset value. 25

29 interest earnings from the increase in loan amount an ii higher expecte loss at efault because of the increase in loan amount. In this section we examine this trae-off assuming that banks act to maximize their expecte revenues incluing the expecte loss. 3 We now examine the optimal covenants level using simulations. he simulation settings are as follows. Banks provie new loans at a previously etermine interest rate r. he bank s funing cost is r f. he initial liability amount 0 an the new loan amount are state at face value. In other wors, the amounts that the firm actually borrows are iscounte by lening rates. 4 he simulation parameter settings are the same as for the case D = 0, an r is set at 3%. Simulations are conucte for the two cases when r f is % an 2%. Uner the above settings, the expecte revenues π are: r r r r τ f f π = 0 e [ ] e [ 0 A ], 8 where τ is the time from the raw own of the commitment line to maturity. he simulation results are presente in Chart 7. Chart 7: Optimal covenants xpecte Revenue Funing rate of % left scale Funing rate of 2% right scale Covenants Another approach woul be to consier the profit risk trae-off for risk calculate using UL. For simplification, we assume risk-neutral banks. 4 hese settings are to simplify the expression of revenues as efine in equation 8, an whether the funing amount is measure on a iscounte basis or face value basis is an unimportant issue. 26

30 he expecte revenues are positive when there is a sprea to some extent between the lening rate an funing rate. he optimal covenants level is 30% when the funing rate is 2% an 25% when the funing rate is %, which suggests that optimal covenants are not lower than 30% of the initial capital ratio. When the funing rate is %, the volume effect of the increase exposure excees the negative influence of increase creit risk, an therefore the optimal covenants value becomes smaller than in the case of the 2% funing rate. 5. Summary his paper has constructe a creit risk moel for stochastic changes in exposure an applie it to evaluate creit risk of commitment lines. Loans uner commitment lines make it ifficult for banks to manage risk appropriately, once such contracts are conclue an firms have an option to withraw new loans. For that reason, the covenant conitions are important for risk management. We consier a moel in which ad, PD, an LGD are mutually interepenent an a firm s funs eman is explicitly linke to changes in its asset value. We apply the moel to commitment lines, an calculate L, UL, an other risk measurements. We also assume various egrees of covenants from relaxe to severe to examine the changes in ad, PD, LGD, L, an UL an investigate their interepenence in simulations. he following points were verifie quantitatively through the simulations. I. Creit risk tens to increase with exposure to the firm whose new loan eman rises when its asset value eclines. A positive correlation between ad an PD in the moel escribes this result. II. Covenants that prevent uncontrollable increases in exposure to poor performing firms are useful for limiting creit risk. In some cases, lax covenants can lea to greater losses, an this largely epens on the structure of how changes in a firm s asset value leas to new loan eman. III. Consequently, verifying this structure is important for controlling creit risk when proviing commitment lines an setting covenants. IV. here is a covenants level that optimizes the trae-off between the increase in 27

31 loan interest earnings an the increase in L. Because iniviual companies commitment line ata are not available, the examination of the suitability of the moel an parameter settings in this paper is insufficient. Moreover, this paper aopts several simplifie assumptions to unerstan the property of the moel that generates mutual epenence among ad, PD, an LGD. he following types of moel extensions coul be consiere. hese are left as future issues, along with the verification on the moel base on empirical ata. I. he funs eman moel sets D an D as constants, but it woul enhance moel applicability to make these parameters either functions of the firm s asset value or stochastic variables. II. he moel assumes that new loan eman always occurs at a specific point, but the timing an the presence or absence of loan eman coul also be incorporate into the moel. III. he moel coul accommoate funs prepayment prior to maturity instea of having all refune at maturity. IV. Funs eman is given as exogenous, but the moel coul be extene to make it an enogenous variable, for example, assuming the eman is etermine to maximize shareholer value. 28

32 29 Appenix: Changes in LGD from commitment line loans he LGD for a new loan with a face value of is given by the following equation: exp A LGD = µ, A- where } 2 {ln 2 A µ =. We now calculate the first erivative to examine the sensitivity of LGD to the amount of the new loan with a face value of : }'] }{ exp { exp [ 2 2 A LGD = µ µ A-2 he numerator of equation A-2 is moifie as follows: }, exp { exp exp exp exp } }{ exp { exp 2 A A A A A A A A A A µ φ µ φ µ φ µ µ φ µ µ = = A-3 where φ is the ensity function of the stanar normal istribution. From equation A-3, the enominator of LGD / is far larger than the numerator, so a change in leas to a small change in LGD. he following chart shows the changes in LGD when new loans are mae uner various firms asset value conitions. he parameter settings are the same as those for the case 0 = D in Chapter 4. he chart shows little change in LGD regarless of the new loan amount. In other wors, there is little influence on LGD even when new lening is restraine by covenants.

33 Reference Figure: Changes in LGD from commitment line new loans LGD Asset value 60 Asset value 70 Asset value 80 Asset value 90 Asset value New Loan Deman 30

34 Dai-Ichi Kangyo Bank International Finance Department 200. Hojin Yushiwaku Settei to Yushi orihiki Corporate Creit Line Setting an Financial ransactions BSI ucation Japanese. Yamashita, Satoshi an oshinao Yoshiba Analytical solutions for expecte an unexpecte losses with an aitional loan, Bank of Japan IMS Discussion Paper Series , Martin, J. S. an Santomero, A. M., Investment opportunities an corporate eman for lines of creit, Journal of Banking an Finance, 2, 997, pp Basel Committee on Banking Supervision BCBS, International Convergence of Capital Measurement an Capital Stanars, Basel Committee Publications No.07 November 2005a. Available at Basel Committee on Banking Supervision BCBS, An xplanatory Note on the Basel II IRB Risk Weight Functions, July 2005b. Available at Campbell,.S., A moel of the market for lines of creit, Journal of finance, 33, 978, pp Kupiec, P. H., A Generalize Single Common Factor Moel of Portfolio Creit Risk, presente at 7th Annual Derivatives Securities an Risk Management Conference, Feeral Deposit Insurance Corporation s Center for Financial Research, Moral, G., AD stimates for Facilities with xplicit Limits, in he Basel II Risk Parameters, ngelmann, B. an R. Rauhmeier es., Springer, 2006, pp Pykhtin, M., Unexpecte Recovery Risk, Risk, 68, 2003, pp