LGD Risk Resolved. Abstract

Transcription

1 LGD Risk Resolve Jon Frye (corresponing author) Senior Economist Feeral Reserve Bank of Chicago 230 South LaSalle Street Chicago, IL Michael Jacobs Jr. Senior Financial Economist Office of the Comptroller of the Currency One Inepenence Square Washington, DC The authors thank Irina Barakova, Any Feltovich, Brian Goron, Paul Huck, E Pel, Michael Pykhtin, an May Tang for comments on previous versions. Abstract This stuy provies a practical metho to anticipate systematic LGD risk. It introuces a moel in which the relation between LGD an efault is fully etermine by parameters that are alreay in han. The associate creit loss moel is a simple one that survives statistical testing. Alternative moels o not emonstrate significance an o not suggest that the propose approach misstates the istribution of creit loss ata. Any views expresse are the authors an o not necessarily represent the views of the management of the Feeral Reserve Bank of Chicago, the Feeral Reserve System, the Office of The Comptroller of the Currency or the U.S. Department of the Treasury.

2 Creit loss varies from perio to perio because both the efault rate an the loss given efault (LGD) rate vary. The efault rate has been tie to a firm's probability of efault (PD) an to correlations between factors responsible for efault. The LGD rate has prove much more ifficult to moel. Because LGD is a continuous variable, its istribution is more subtle than binary efault. Although every exposure shes light on the efault rate, only efaulte exposures she light on LGD. When a efault occurs, the resulting loss can be ifficult to measure accurately. Thus, LGD is more subtle than efault, an yet the LGD ata are fewer in number an lower in quality. Nonetheless, stuies show that the two rates vary in concert. Such systematic variation works against the lener, who can fin that an increase in the number of efaults coincies with an increase in the fraction of exposure that is lost in each efault. Leners shoul therefore anticipate systematic LGD risk in their creit portfolio loss moels. For certain institutions, regulators require such moels to account for all material risks, incluing systematic LGD risk. This paper introuces an LGD moel in which LGD an efault vary together in a particular way. The relationship is fully etermine by parameters that are alreay in han. The associate creit loss moel is a simple one. This is important, because simple moels are generally preferre unless elaborate moels emonstrate statistical significance. The alternatives trie here o not emonstrate significance an o not suggest that the propose approach misstates the istribution of creit loss ata. Therefore, the propose moel can be sai to resolve systematic LGD risk to the extent permitte by these ata. The propose LGD moel In an asymptotic single risk factor moel, 2 the conitional efault rate (cdr) an the conitional LGD rate (clgd) each epen on an unobserve creit stress factor, Z: () closs [ cdr[ * clgd[ The conitional efault rate is often assume to obey the Vasicek istribution: 3 [ PD (2) cdr[ ; ; ~ N [0, where [ is the stanar normal cumulative istribution function. The expecte value of cdr is the probability of efault. For a given value of greater values of, correlation, imply greater variance of cdr. The propose LGD rate moel uses, an expecte loss given efault (ELGD). Specifically, Altman an Karlin; Frye (2000). 2 Gory. 3 Vasicek.

3 (3) clgd [ ; ELGD, [ PD ELGD cdr ; ; ~ N [0, Equation (3) is an S curve: a strictly monotonic function of Z that is boune between 0% an 00%. The same is true of Equation (2). Therefore, greater creit stress (greater Z) implies both a greater efault rate an a greater LGD rate. This creates a connection between efault an LGD that will be explore in the next section. It woul be possible to stuy the istribution of the LGD rate by itself. However, the istribution of the creit loss rate is more important, because creit loss can cause a financial institution to fail. Therefore, we examine LGD moels in the context of the creit loss moels that they imply. In an asymptotic portfolio, the propose LGD moel implies the following creit loss moel: (4) closs [ ; ELGD, [ PD ELGD ; ~ N [0, This istribution has two parameters. Its mean is expecte loss (EL), which is the prouct of PD an ELGD. For a given value of EL, greater implies greater variance. Creit loss moels with more parameters have not been shown to significantly improve this moel when loss ata are assume to reflect an asymptotic portfolio. 4 A simpler LGD moel oes not necessarily imply a simpler creit loss moel. For example, the simplest LGD moel assumes that LGD is a constant equal to ELGD. The associate creit loss moel is: (5) closs [ ; ELGD, ELGD [ PD ; ~ N [0, This istribution has three parameters rather than two. It requires that ifferent parts of EL be hanle ifferently: PD interacts with systematic risk but ELGD is hel separate from it. Whether or not this ifferential hanling makes a statistically significant contribution is an empirical question that is investigate below. To prefigure the result, we fin no evience that this makes a significant contribution to the unerstaning of creit loss ata. Although Equation (4) an Equation (5) share the same mean, they iffer in the tails: a high percentile of Equation (4) is greater than the same percentile of Equation (5). Figure illustrates this at the 99.9 th percentile for selecte values of ELGD, an. Simply put, creit risk is greater in Equation (4) than when LGD is assume fixe as in Equation (5). Whether or not this extra risk equals the effect of LGD variation is an empirical question. In the tests that follow, Equation (4) oes not appear to either overstate or unerstate creit risk. 4 Frye (200). 2

4 Ratio 4.0 Figure. Equation (4) / Equation (5) with = PD = 5%, Rho = 20% PD = %, Rho = 20% PD = 5%, Rho = 5% PD = %, Rho = 5% % 20% 40% 60% 80% 00% ELGD Previous research into systematic LGD risk introuces parameters that calibrate the response of LGD to observe or unobserve factors. 5 Depening on their values, these parameters can prouce an LGD response that is greater than or less than exhibite by Equation (3). Whether or not this contributes to unerstaning creit loss is an empirical question that is investigate below. We employ several specifications an fin that none of them emonstrate statistical significance. The LGD moel introuce in Equation (3) prouces the particularly simple creit loss istribution of Equation (4). This istribution isplays greater creit risk than when LGD is assume fixe. The tests that follow o not reveal that Equation (4), restate for a finite portfolio, misstates of the istribution of a long history of creit loss ata. The connection between efault an LGD Although the istribution of creit loss is the primary focus of stuy, this section motivates the statistical tests with a visual comparison of efault rates an LGD rates. We first invert Equation (2): (6) [ cdr [ PD Substituting this into the numerator of Equation (3) prouces the relation between the conitional efault rate an the conitional LGD rate: 5 For example, Pykhtin (2003) introuces three LGD parameters. 3

5 Conitional LGD Rate (7) clgd[ cdr; ELGD, [ PD ELGD cdr [ cdr [ PD Separating the ranom variables from the parameters prouces the following: (8) [ PD ELGD [ PD [ cdr clgd [ cdr Thus, the relation between the conitional efault rate an the conitional LGD rate epens on a single constant. For selecte values of the constant, Figure 2 illustrates the positive relation between the two variables. 00% Figure 2. Conitional Default an LGD Rates 80% 60% 40% 20% 0% 0% 20% 40% 60% 80% 00% Conitional Default Rate All of the ata from an asymptotic portfolio fall on one line rawn from the infinite set of lines that is represente in Figure 2. Two points in particular are highlighte. These points reflect a portfolio that in one year can prouce rates of efault an LGD equal to % an 2%, an that in another year can prouce rates of 20% an 42%. 6 As conitions change, a twenty-fol increase in the efault rate coincies with a two-fol increase of the LGD rate. This egree of LGD variation oes not seem unrealistic, an this invites comparison of the moel to historical ata. Historical ata come from a finite portfolio; therefore, they scatter for ranom reasons. Realie rates of efault an LGD are not always equal to their conitional expectations. In aition, the 6 This portfolio might have (PD = 5%, ELGD = 3.6%, = 0%), or (PD = 5%, ELGD = 4.%, = 0%), or any other combination of values that make the right-han sie of Equation (8) equal to

6 LGD Rate ata presente in this section o not control for composition; therefore, portfolio average PD or ELGD might change from year to year. For these reasons an others, a graphical comparison is illustrative but not ecisive. The effects of obligor rating, ebt seniority, an the ranom ispersion of efault an LGD rates are controlle for in the statistical tests that follow. Figure 3 isplays two sets of historical ata. The blue (upper) ata points are those provie by Dr. Ewar Altman. 7 These ata have an average annual efault rate of 4.59% an average annual loss of 2.99%; therefore, the average LGD of an exposure is 2.99%/4.59% = 65.2%. Years with above-average efault ten to have above-average LGD. Figure 3. Data an Equation (7) 80% 70% 60% 50% 40% 30% Altman ata clgd[pd=4.59%, ELGD=65.2%, rho=0% 20% MURD ata clgd[pd=4.54%, ELGD=42.6%, rho=0% 0% 0% 2% 4% 6% 8% 0% 2% 4% Default Rate The blue (upper) line is Equation (7) with PD = 4.59%, ELGD = 65.2%, an = 0%. Other choices of less than 30% or so prouce nonintersecting lines that are slightly above or slightly below the line that is illustrate; as Equation (8) suggests, the value of is much less important than PD an ELGD. The three parameters together etermine the relationship between efault an LGD. The blue line has the character of a regression line, an yet no regression has been performe. Instea, one line has been selecte from the set of lines represente in Figure 2 an compare to the Altman ata. In Figure 3, the re (lower) ata points are provie by Mooy s Ultimate Recovery Database (MURD). 8 The average annual efault rate is 4.54% an the average annual loss rate is.94%; therefore, average LGD is 42.6%. The re line is Equation (7) with PD = 4.54%, ELGD = 42.6%, an = 0%. Again, the slope of the line appears not inconsistent with the slope of the ata swarm, even though the slope of the line comes only from ELGD an. 7 Altman an Karlin. 8 Cantor et al. 5

7 A significant strength of the propose LGD moel is that it makes a preiction about the systematic efault/lgd relation base on average efault an average LGD. This preiction resembles a regression line for each of two ata sets. Whether or not the preiction can be reliably improve is an empirical matter. The next section evelops alternative LGD moels that can be use to test potential improvements. Alternatives for testing In the tests to be conucte, the null hypothesis is a creit loss moel base on the propose LGD moel. Each of the five alternatives evelope here introuce an aitional LGD sensitivity parameter: a, b, c,, or e, respectively Each assumes that the conitional efault rate has the Vasicek istribution with correlation equal to an that LGD an efault both epen on the same risk factor. In each alternative, the expecte value of the loss istribution equals the prouct of the parameters PD an ELGD. Alternative A generalies the fixe-lgd moel by placing a power of ELGD outsie the cdr[ function: (9) clossa [ ; ELGD,, a ELGD a [ PD ELGD a The conitional LGD function in Alternative A is: (0) clgda[ ; ELGD,, a clossa cdr[ ; Figure 4 illustrates Alternative A for a = 2,, 0, an -5; throughout, PD = 0%, ELGD = 30%, an = 0%. This shows that Alternative A can represent LGD risk that is greater than, equal to, or less than the LGD risk in Equation (3). 6

8 Conitional LGD Rate 50% Figure 4: Default an LGD with Alternative A 40% 30% 20% 0% 0% a = 2: negative LGD risk a = : fixe LGD a = 0: null hypothesis a = -5: high LGD risk 0% 0% 20% 30% Conitional Default Rate Alternative B places a power of PD outsie the cdr[ function. If parameter b is set to.0, Alternative B becomes a moel in which the efault rate always equals PD an all creit risk stems from the istribution of LGD: () clgdb[ ; ELGD,, b PD b [ PD b ELGD / cdr[ ; If b is set such that PD b = ELGD a, Alternative B becomes ientical to Alternative A; therefore, Alternative A an Alternative B are ientical when only a single portfolio is moele. The assumption that b is uniform across several portfolios prouces ifferent results from the assumption that a is uniform across portfolios. Alternative C provies a ifferent possible linkage between portfolios by placing a power of EL outsie the cdr[ function: (2) clgdc[ ; ELGD,, c ( PD ELGD) c [( PD ELGD) cdr[ ; c In a single portfolio, Alternative C is equivalent to Alternative A if EL c = ELGD a. Alternatives A, B, an C specify a creit loss moel an use it to infer an LGD moel. Alternative D specifies the LGD moel irectly. This has an effect on the expecte value of the loss istribution that is remove with a multiplicative constant. Alternative D generalies Equation (3) by allowing correlation to take the value : 7

9 8 (3) PD PD ELGD PD PD ELGD PD PD ELGD ELGD PD clgd [ / [ [ [ [ [,,, ; [ where [ symbolies the stanar normal probability ensity function. If >, LGD respons to Z more strongly than in the null hypothesis, an if < it respons less strongly If = 0, Alternative D becomes the fixe-lgd moel. By necessity, 0; therefore, Alternative D cannot represent negative epenence between LGD an efault. Although the formulas for Alternative D an Alternative A are quite ifferent, they give similar results. Experimentation reveals that for given values of an, there is a value of a that makes Alternative A nearly equal to Alternative D for all values of PD an ELGD. Therefore, even when Alternative D is applie across portfolios that have ifferent values of PD an ELGD, its results are nearly the same as Alternative A. As a consequence, Alternative D is not employe in the statistical tests. Alternative E asserts that the loss istribution has two parameters, but that the correlation relevant for loss is ifferent from the correlation relevant for efault. The effect is that numerator an enominator of clgde iffer in correlation. The conitional LGD function is as follows: (4), ; [ / [,,, ; [ PD cdr e e PD ELGD e ELGD PD clgde Alternative E is istinct from the other alternatives in that it alone oes not contain the fixe- LGD moel as a special case. Table summaries the egree of LGD risk for each of the alternatives.

10 Table. Sensitivities of LGD to efault in five alternative moels Degree of LGD risk Alt. A Alt. B Alt. C Alt. D Alt. E Greater than null hypothesis a < 0 b < 0 c < 0 > e > Equal to null hypothesis a = 0 b = 0 c = 0 = e = Less than null hypothesis a > 0 b > 0 c > 0 < e < Zero; LGD is fixe at ELGD a = b = Log[ELGD/Log[PD c = Log[ELGD/Log[EL = LGD negatively relate to efault a > b > Log[ELGD/Log[PD c > Log[ELGD/Log[EL This section introuces five alternative LGD functions. Each contains an extra parameter that can allow for greater or less LGD sensitivity than the comparatively restrictive null hypothesis. Creit loss in a finite portfolio A creit loss ata set might have only a hanful of efaults in a year. To analye such ata requires the istribution of loss in a finite portfolio. The erivation begins with finite-portfolio istributions of efault an LGD conitione on Z. The prouct of these is the joint istribution of efault an LGD. This is transforme to a joint istribution of efault an loss. The marginal istribution of loss is foun by summing over the number of efaults an removing the conitioning on Z. Essentially, Equation (4) is the istribution of creit loss in the asymptotic portfolio, an this section prouces the istribution when the portfolio is finite. We begin by conitioning on Z an recogniing two cases. The first case is that the number of efaults, D, equals 0. This occurs with probability equal to ( - cdr[; ) N, where N is the number of firms. With this probability mass, creit loss equals ero. The secon case, conitione on both D = > 0 an on Z, prouces a istribution of the portfolio average LGD rate. The istribution is assume to be normal with variance 2 /D: LGD clgd (5) f [ LGD LGDD, / / where LGD symbolies the portfolio average LGD rate an clgd stans for one of the LGD functions. The Central Limit Theorem implies normality when the number of efaults is large, but this is selom the case. More practically, normality allows LGD outsie the range [0,. This is important because the ata inclues portfolio-years where average LGD is negative. The conitional istribution of the number of efaults is assume Binomial with parameters N an cdr[. The joint istribution of D an LGD is then the prouct: (6) f D, LGD [, LGD cdr[ ; ( cdr[ ; ) 9 N N LGD / / clgd

11 Density We efine loss per unit of exposure an pass from variables (D, LGD ) to variables (D, Loss) with the following transformation: (7) Loss D LGD / N; D D The Jacobian eterminant is N/D. The transforme joint istribution is then: (8) f D, Loss [, Loss N cdr[ ; ( cdr[ ; ) N N N Loss / / / clgd Summing this over, combining the two cases, an removing the conitioning on Z prouces the istribution of creit loss in the finite portfolio. The istribution epens on N, ELGD, an, an an aitional LGD sensitivity parameter (a, b, c, or e) when one of the alternatives is being use: (9) f Loss [ Loss I[ Loss 0 I[ Loss 0 [ ( cdr[ ; ) [ N fd, Loss [, Loss N Figure 5 compares this to the istribution of loss in the asymptotic portfolio, Equation (4). Each istribution has EL = 5% an = 5%. The finite portfolio has ten exposures with PD = 0%. With probability 43% there are no efaults an no loss, so the area uner this istribution is Using = % prouces the istinct humps for one, two, an three efaults. The hump for one efault is centere at less than 5% loss, while the hump for three efaults is centere at greater than 5% loss. Thus, Figure 5 illustrates the particular way that the null hypothesis links the LGD rate an the efault rate Figure 5. Distributions of loss for asymptotic an finite portfolios Asymptotic portfolio: PD = 0% ELGD = 50% = 5% Portfolio with 0 loans: PD = 0%, ELGD = 50%, = 5%, = % 5 0 0% 5% 0% 5% 20% Creit Loss Rate 0

12 Uner the usual statistical assumptions the parameters are stable over time an Z is inepenent each year the log of the likelihoo function of the sequence of loss rates is then: (20) LnLLoss[ Loss, Loss 2,..., LossT Log[ floss[ Loss t T t Data The ata use are from Mooy's Corporate Default Rate Service. 9 An exposure "cell" the intersection of a rating grae an a seniority class is assume to be a homogenous portfolio of statistically ientical exposures such as calle for in the loss moels. We efine efault only when the loss is observe. By contrast, stuies of efault in isolation can inclue efaults that prouce unobserve loss. We refer to this less-restrictive efinition as nominal efault an note that it prouces greater efault rates, as shown below. We elimit the ata set in several ways. To have notche ratings available at the outset, the ata sample begins with 983. To align with the assumption of a single risk factor, firms must be classifie as inustrial, public utility, or transportation firms heaquartere in the US. Ratings are taken to be Mooy's "senior" ratings of firms, which usually correspons to the rating of the firm s long-term senior unsecure ebt if such exists. To focus on cells that have aequate numbers of efaults, we analye firms rate Baa3 or lower. We group the ratings C, Ca, Caa, Caa, Caa2, an Caa3 into a single grae we esignate "C". This prouces five obligor rating graes altogether: Ba3, B, B2, B3, an C. Again to align with the assumption of a single risk factor, ebt issues must be ollar enominate, intene for the U.S. market, an not guarantee or otherwise backe. We efine five seniority classes: Senior Secure Loans (Senior Secure instruments with Debt Class "Bank Creit Facilities") Senior Secure Bons (Senior Secure instruments with Debt Class "Equipment Trusts", "First Mortgage Bons", or "Regular Bons/Debentures") Senior Unsecure Bons ("Regular Bons/Debentures" or "Meium Term Notes") Senior Suborinate Bons ("Regular Bons/Debentures") Suborinate Bons ("Regular Bons/Debentures"). This exclues convertible bons, preferre stock, an certain other instruments. A firm is expose in a cell-year if on January st the firm itself has one of the five obligor ratings, it is not currently in efault, an it has a rate issue in the seniority class. A firm efaults if one or more post-efault prices are observe. LGD is par minus the average of such prices. The efault rate in the cell-year is the number of LGD's ivie by the number of firms that are 9 Mooy s.

13 expose, an the loss rate is the sum of the LGD's ivie by the number of firms that are expose. Testing cell-by-cell This section presents tests performe one cell at a time. To prefigure the results, none of the alternatives prouce a significant improvement over the null hypothesis. We fin no evience that the null hypothesis seriously misstates LGD risk. Risk managers often use averages to estimate creit moel parameters. We follow that practice rather than introucing more sophisticate estimation techniques. In each cell, the estimate of PD equals the average annual efault rate. The estimate of EL equals the average annual loss rate, an ELGD equals EL / PD. The estimate of equals the average stanar eviation of LGD taken in any cell-year that has at least two efaults; this overall average is 20.30%. Risk managers take estimates of from various sources incluing vene moels, asset or equity return correlations, creit efault swaps, regulatory authorities, an inferences from acaemic stuies. It is simpler for us to perform maximum likelihoo estimation by maximiing the following expression of : (2) LnL [, T 2, t..., T Log, N, N,..., N, PD 2 [ cdr[ ; T ^ ^ t ( cdr[ ; ) ^ Nt t Nt t The next section consiers the effect of other values of correlation an fins that the null hypothesis hols up over a broa range of possible values of. Given estimates of ELGD,, an, each of the alternatives call for the estimation of one aitional parameter. This parameter is estimate by maximiing the likelihoo of Equation (20). Table 2 shows summary statistics, parameter estimates, an test statistics for each cell. The test statistics are state as the log likelihoo using the alternative less the log likelihoo using the null hypothesis. Twice this ifference has the asymptotic chi-square istribution with one egree of freeom. Differences greater than the 5% critical value (.92) are note in bol face. The test statistics for Alternatives B an C woul be ientical to those presente for Alternative A, because only a single cell at a time is subject to a test. 2

14 Averages Table 2. Basic statistics, parameter estimates, an test statistics by cell Senior Senior Senior Senior Suborinate Secure Loans Secure Bons Unsecure Bons Suborinate Bons Bons Averages EL D 0.2% 4 0.7% 3 0.4% 6 0.8% 9 0.9% % 0 PD N 0.6% 66 2.% % 703.2% 525.5% 874.% 579 ELGD D Years 42% 3 33% 3 49% 4 63% 6 64% 8 55% 5 Ba3 N Years 7.6% 4.0% % % % 2.8% 23 NomPD NomD 0.7% 5 2.% 3.2% 9.2% 9.7% 3.3% a LnL e LnL 20.4% % % % % % 0.24 EL D 0.2% 9 0.2% 2.0% 3.4% 22.3% % 7 PD N 0.8% % 205.8% 757.9% % 756.5% 792 ELGD D Years 28% 5 29% 2 53% 0 74% 0 5% 0 54% 7 B N Years 4.4% 4.0% 27.0% % % % 24 NomPD NomD.8% % 3 2.3% 7 2.0% % 45 2.% 23 a LnL e LnL 2.3% % % % % % 0.08 EL D 0.4% % 4 2.5% 45 2.% % 35.5% 29 PD N.2% % 68 4.% % % % 780 ELGD D Years 36% 0 43% 4 60% 4 69% 8 57% 55% 9 B2 N Years 5.0% % 26.6% % 2 2.% % 23 NomPD NomD 3.0% 6 6.% 5 4.9% 5 3.2% 40 6.% % 39 a LnL e LnL 6.9% % % % % % 0.39 EL D 0.4% 9 2.9% 3.5% % % % 33 PD N.4% % % % % % 632 ELGD D Years 29% 6 40% 8 52% 7 40% 4 64% 3 48% 2 B3 N Years 2.3% % % % 22.7% 2 8.0% 22 NomPD NomD 4.% 47 8.% 3 8.5% % % % 45 a LnL e LnL 3.% % % %.8 9.9% 0.4.4%.0 EL D 2.% 88 5.% % % % 2 6.0% 66 PD N 5.6% % 449.5% % % % 558 ELGD D Years 38% 0 52% 7 60% 2 76% 4 73% 6 59% 4 C N Years 6.9% 4 2.3% % % 6.2% 9 2.9% 2 NomPD NomD 23.2% % % % % 8 8.3% 95 a LnL e LnL 23.3% % % % % % 0.84 EL D 0.6% % 4 3.0% % % 34 2.% 3 PD N.8% % % % % % 668 ELGD D Years 35% 7 47% 7 57% 3 65% 0 6% 0 55% 9 N Years 2.8% 4 9.5% 26 2.% % % 22.8% 22 NomPD NomD 5.9% % % % % % 43 a LnL e LnL 3.2% % 0.56.% % % % 0.5 Key to Table 2: EL, an : Estimates as iscusse in the text; ELGD = EL / PD. D: The number of efaults in the cell, counting within all 27 years. N: The number of firm-years of exposure in the cell, counting within all 27 years. D Years: The number of years that have at least one efault. N Years: The number of years that have at least one firm expose. NomD: The number of nominal efaults (incluing where the resulting loss is unknown). NomPD: Average of annual nominal efault rates. a, e: MLEs of the parameters in Alternatives A an E. LnL: the pick-ups in LnL Loss provie by Alternative A or E relative to the null hypothesis. Statistical significance at the 5% level is inicate in bol. Along the right an bottom margins, average EL, an are weighte by N; other averages are unweighte. 3

15 Along the bottom an on the right of Table 2 are averages. The overall averages at the bottom right corner contain the most important fact about creit ata: they are few in number. The average cell has only 3 efaults, which is about one per year. Since efaults cluster in time, the average cell has efaults in only 9 years, an only these years can she light on the connection between efault an LGD. Not only are the ata few in number, they have a low signal-to-noise ratio: the ranom variation of LGD, measure by = 20.30%, is material compare to the magnitue of the systematic effect an the number of LGDs that are observe. The estimates of, a, an e tell about the egree of LGD risk. In about half the cells, a > 0 an e < (suggesting overstatement by the null) an in the other half, a < 0 an e > (suggesting unerstatement). A pattern like this is expecte if the null hypothesis correctly states the egree of systematic LGD risk. Two cells prouce pickups of log likelihoo greater than the critical value of.92, but the parameter estimates point in opposite irections. Specifically, estimate a is greater than ero in Cell B3-Senior Secure Loans, but it is less than ero in Cell C-Senior Suborinate Bons. Even if the null hypothesis is true, a few such cases of nominal significance woul be expecte because twenty-five separate tests are being performe. This section performs statistical tests of the null hypothesis one cell at a time. Two cells prouce nominal significance, consistent with the expectation that test results are ranom to some egree. There is no goo evience that the null hypothesis either overstates or unerstates LGD risk on average. Testing cells in parallel This section presents tests involving several cells at once. This allows each cell to contribute information about the systematic risk factor. We begin by analying the five cells of loans. Counting within fourteen years of rate loan exposure an five graes, there are 6,20 firmyears of exposure in all. We estimate = 8.5% by maximiing the following likelihoo: (22) LnL s [ ;{ ^ ^ t, i Nt, i t, i t, i Log [ cdr[ ; PD, ( [ ;, ) i cdr PDi t 996 i t, i, 2,...},{ N i, N 2,...} i N The average of the stanar eviation of LGD, taken across all cell-years of loans, provies the estimate = 23.3%. The estimates of EL an PD are taken from Table. The following likelihoo is then a function of only the parameter a, b, c, or e: floss [ Loss t, i i t996 i (23) LnLLosses[{ Loss, Loss 2,...} i Log[ [ 4

16 Table 3 shows the resulting estimates an the pickups of LnL. The estimates of a, b, an c are greater than ero, an the estimate of e is less than the value of. This pattern suggests slightly less LGD risk than preicte by the null hypothesis, however, none of the alternatives comes close to the statistically significant pickup of LnL >.92. Table 3. Testing cells in parallel Loans only; = 23.3%, = 8.5% Parameter Estimate LnL a b c e Bons only; = 9.7%, = 8.05% Parameter Estimate LnL a b c e Loans an bons; = 20.3%, = 9.0% Parameter Estimate LnL a b c e Turning to the twenty cells of bons exclusive of loans, some firms have bons in ifferent seniority classes in the same year. Of the total of 0,585 firm-years of bon exposure, 9.0% have exposure in two classes, 0.4% have exposure in three classes, an 0.% have exposure in all four classes. This creates an intricate epenence between cells rather than inepenence. Assuming that this egree of epenence oes not invaliate the main result, Table 3 shows that a, b, an c are less than ero, an e is greater than. This pattern suggests greater, but not significantly greater, LGD risk than preicte by the null hypothesis. When all loans an all bons are analye at once, 6.0% of firm-years have exposure to two or more classes. Again, all parameters in Table 3 point to slightly greater LGD risk than preicte by the null hypothesis, but the ifference is far from statistically significant. The previous tests use a maximum likelihoo estimate of. A ifferent value of might be assume, an this might affect the results of a test. Figure 6 assumes a range of values of among loans from 0% to 50%. Two values of the log likelihoo are shown. The lower line is LnL uner the null hypothesis, an the upper line is LnL when maximie uner Alternative A. Alternative A achieves statistical significance only if correlation is assume to be less than 4.8% or greater than 45.4%. Otherwise, Alternative A oes not provie a significant improvement. 5

17 Log Likelihoo State positively, the null hypothesis appears robust respect to the uncertainty in an estimate of correlation, even if the egree of uncertainty is quite substantial. 78 Figure 6. Log likelihoos for loans at assume values of 76 Max LnL Alt. A. LnL Null Hypothesis % 4.80% 0% 20% 30% 40% 45.4% 50% Assume value of rho The tests of this section employ all loans, all bons, or both loans an bons taken together. No evience is foun to suggest that the null hypothesis seriously misstates LGD risk. Applications an incentives Practical creit portfolio loss moels have multiple risk factors rather than one. A linear compoun of the factors provies the total systematic contribution. If we interpret Z as a multivariate stanar normal vector an a as a vector of coefficients, then in the multi factor case: [ PD a' (24) cdr [ ; a, ; ~ N[0, a' a an (25) clgd [ ; ELGD, a, [ PD ELGD a' a' a cdr ; a, Each formula is strictly monotonic in a Z, so as before there is a strictly monotonic relation between cdr an clgd. Most important, the value of clgd is nonranom given the elements of vector Z, so the propose moel can be reaily applie in practical situations. 6

18 99.9 percentile LGD - ELGD If a risk moel attributes capital to creit exposures, it can have an effect on business incentives. Figure 7 shows the 99.9 th percentile of clgd as a function of ELGD. For example, an exposure with PD = 0%, ELGD = 20%, = 2.% is foun on the top line; clgd is (20% + 6%) =36%. Figure percentile LGD less ELGD 20% PD = 0%, Rho = 2.% 8% PD = 3%, Rho = 4.7% 6% PD = %, Rho = 9.3% PD = 0.3%, Rho = 22.3% 4% PD = 0.%, Rho = 23.4% 2% PD = 0.03%, Rho = 23.8% 0% 8% 6% 4% 2% 0% 0% 20% 40% ELGD 60% 80% 00% A loan with PD = 0% but a ifferent ELGD woul also be foun on the top line. For example, if ELGD were equal to 0%, then clgd is (0% + 2%) = 22%, about 6% of its initial value of 36%. In the absence of systematic LGD risk, this proportion woul be 50%. This illustrates that in the propose approach, risk is less sensitive to the value of ELGD than is the case when LGD is assume to have no systematic risk. A loan with a ifferent PD woul be foun on a ifferent line. For example, if PD = % an ELGD = 20%, then clgd is (20% + 2%) = 32%, which is less than the initial value of 36%. This gives the lener a benefit on the clgd sie in aition to the benefit on the cdr sie. PD affects both cdr an clgd. Risk is more sensitive to the value of PD than is the case when LGD is assume to have no systematic risk. Net, if correlation is invariant, only a change of EL prouces a change of risk. The incentive to reuce PD by a given proportion is equal to the incentive to reuce ELGD by that same proportion. 7

19 Conclusion This stuy introuces a moel that allows LGD to vary with the efault rate. The moel uses only parameters that are alreay part of existing creit loss moels: ELGD, an correlation. This has a practical benefit an a scientific benefit. The practical benefit is that the moel can be implemente without estimating new parameters. The relation between efault an LGD is etermine by parameters that are alreay in han. The scientific benefit is a particularly simple creit loss moel. This can serve as the null hypothesis in rigorous statistical tests. Testing it against several alternatives, we fin no evience that the propose approach seriously misstates systematic LGD risk or the istribution of a long history of creit loss ata. 8

20 References Altman, E. I., an B. Karlin, 200, Special report on efaults an returns in the high-yiel bon an istresse ebt market: The year 2009 in review an outlook, NYU Solomon Center Report, February. Cantor, R., Emery, K., Keisman, D., Ou, S., Mooy's Ultimate Recovery Database: Special report. Mooy s Investor Service, April. Frye, J., 2000, Depressing recoveries, Risk 08- (November), 200, Moest means, Risk (January) Gory, M., 2003, A risk-factor moel founation for ratings-base bank capital rules, Journal of Financial Intermeiation, Volume 2, Issue 3, July 2003, Pages Gupton, G., Finger, C., an Bhatia, M., 997, CreitMetrics Technical Document Mooy's Corporate Default Rate Service, 200 Pykhtin, M., 2003, Unexpecte Recovery Risk, Risk (August), pp Vasicek, O. A., 984, Creit valuation, KMV TM Corporation, March. 9