THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

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1 HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo of loans, each of whch s subject to default resultng n a loss to the lender. Suppose the portfolo s fnanced partly by equty captal and partly by borrowed funds. he credt qualty of the lender's notes wll depend on the probablty that the loss on the portfolo exceeds the equty captal. o acheve a certan credt ratng of ts notes (say Aa on a ratng agency scale), the lender needs to keep the probablty of default on the notes at the level correspondng to that ratng (about.001 for the Aa qualty). It means that the equty captal allocated to the portfolo must be equal to the percentle of the dstrbuton of the portfolo loss that corresponds to the desred probablty. In addton to determnng the captal necessary to support a loan portfolo, the probablty dstrbuton of portfolo losses has a number of other applcatons. It can be used n regulatory reportng, measurng portfolo rsk, calculaton of Value-at-Rsk (VaR), portfolo optmzaton and structurng and prcng debt portfolo dervatves such as collateralzed debt oblgatons (CDO). In ths paper, we derve the dstrbuton of the portfolo loss under certan assumptons. It s shown that ths dstrbuton converges wth ncreasng portfolo sze to a lmtng type, whose analytcal form s gven here. he results of the frst two sectons of ths paper are contaned n the author s techncal notes, Vascek (1987) and (1991). For a revew of recent lterature on the subject, see, for nstance, Pykhtn and Dev (00). he lmtng dstrbuton of portfolo losses Assume that a loan defaults f the value of the borrower's assets at the loan maturty falls below the contractual value B of ts oblgatons payable. Let A be the value of the -th borrower s assets, descrbed by the process da = µ Adt + σ Adx he asset value at can be represented as * Publshed n Rsk, December 00 1

2 log A ( ) = log A + µ σ + σ X (1) 1 where X s a standard normal varable. he probablty of default of the -th loan s then where p = P[ A ( ) < B ] = P[ X < c ] = N( c ) 1 log B log A µ + σ c = σ and N s the cumulatve normal dstrbuton functon. Consder a portfolo consstng of n loans n equal dollar amounts. Let the probablty of default on any one loan be p, and assume that the asset values of the borrowng companes are correlated wth a coeffcent ρ for any two companes. We wll further assume that all loans have the same term. Let L be the gross loss (before recoveres) on the -th loan, so that L = 1 f the -th borrower defaults and L = 0 otherwse. Let L be the portfolo percentage gross loss, 1 n L n = 1 L = If the events of default on the loans n the portfolo were ndependent of each other, the portfolo loss dstrbuton would converge, by the central lmt theorem, to a normal dstrbuton as the portfolo sze ncreases. Because the defaults are not ndependent, however, the condtons of the central lmt theorem are not satsfed and L s not asymptotcally normal. It turns out, however, that the dstrbuton of the portfolo loss does converge to a lmtng form, whch we wll now proceed to derve. he varables X n Equaton (1) are jontly standard normal wth equal parwse correlatons ρ, and can therefore be represented as X = Y ρ + Z 1 ρ () where Y, Z 1, Z,, Z n are mutually ndependent standard normal varables. (hs s not an assumpton, but a property of the equcorrelated normal dstrbuton.) he varable Y can be nterpreted as a portfolo common factor, such as an economc ndex, over the nterval (0,). hen the term Y ρ s the company s exposure to the common factor and the term Z (1 ρ) represents the company specfc rsk.

3 We wll evaluate the probablty of the portfolo loss as the expectaton over the common factor Y of the condtonal probablty gven Y. hs can be nterpreted as assumng varous scenaros for the economy, determnng the probablty of a gven portfolo loss under each scenaro, and then weghtng each scenaro by ts lkelhood. When the common factor s fxed, the condtonal probablty of loss on any one loan s 1 N ( p) Y ρ p( Y ) = P[ L = 1 Y ] = N (3) 1 ρ he quantty p(y) provdes the loan default probablty under the gven scenaro. he uncondtonal default probablty p s the average of the condtonal probabltes over the scenaros. Condtonal on the value of Y, the varables L are ndependent equally dstrbuted varables wth a fnte varance. he portfolo loss condtonal on Y converges, by the law of large numbers, to ts expectaton p(y) as n. hen ( ) 1 1 P[ L x] = P[ p( Y ) x] = P[ Y p ( x)] = N p ( x) and on substtuton, the cumulatve dstrbuton functon of loan losses on a very large portfolo s n the lmt ρ N ( x) N ( p) P[ L x] = N (4) ρ hs result s gven n Vascek (1991). he convergence of the portfolo loss dstrbuton to the lmtng form above actually holds even for portfolos wth unequal weghts. Let the portfolo weghts be w 1, w,, w n wth Σw =1. he portfolo loss L n = = 1 w L condtonal on Y converges to ts expectaton p(y) whenever (and ths s a necessary and suffcent condton) n = 1 w 0 In other words, f the portfolo contans a suffcently large number of loans wthout t beng domnated by a few loans much larger than the rest, the lmtng dstrbuton provdes a good approxmaton for the portfolo loss. 3

4 Propertes of the loss dstrbuton he portfolo loss dstrbuton gven by the cumulatve dstrbuton functon ρ N ( x) N ( p) F( x; p, ρ ) = N ρ s a contnuous dstrbuton concentrated on the nterval 0 x 1. It forms a twoparameter famly wth the parameters 0 < p, ρ < 1. When ρ 0, t converges to a one-pont dstrbuton concentrated at L = p. When ρ 1, t converges to a zero-one dstrbuton wth probabltes p and 1 p, respectvely. When p 0 or p 1, the dstrbuton becomes concentrated at L = 0 or L = 1, respectvely. he dstrbuton possesses a symmetry property F( x; p, ρ ) = 1 F(1 x;1 p, ρ ) he loss dstrbuton has the densty ( ) 1 ( ) 1 ρ 1 f ( x; p, ρ ) = exp 1 ρ N ( x) N ( p) + N ( x) ρ ρ whch s unmodal wth the mode at Lmode 1 ρ 1 ρ 1 = N N ( p) when ρ < ½, monotone when ρ = ½, and U-shaped when ρ > ½. he mean of the dstrbuton s EL = p and the varance s ( ) s = Var L = N N ( p), N ( p), ρ p 1 1 where N s the bvarate cumulatve normal dstrbuton functon. he nverse of ths dstrbuton, that s, the α-percentle value of L, s gven by L = F( α ;1 α p,1 ρ ) he portfolo loss dstrbuton s hghly skewed and leptokurtc. able 1 lsts the values of the α-percentle L α expressed as the number of standard devatons from the mean, for several values of the parameters. he α-percentles of the standard normal dstrbuton are shown for comparson. (5) 4

5 able 1. Values of (L α p)/s for the portfolo loss dstrbuton p ρ α =.9 α =.99 α =.999 α = Normal hese values manfest the extreme non-normalty of the loss dstrbuton. Suppose a lender holds a large portfolo of loans to frms whose parwse asset correlaton s ρ =.4 and whose probablty of default s p =.01. he portfolo expected loss s EL =.01 and the standard devaton s s =.077. If the lender wshes to hold the probablty of default on hs notes at 1 α =.001, he wll need enough captal to cover 11.0 tmes the portfolo standard devaton. If the loss dstrbuton were normal, 3.1 tmes the standard devaton would suffce. he rsk-neutral dstrbuton he portfolo loss dstrbuton gven by Equaton (4) s the actual probablty dstrbuton. hs s the dstrbuton from whch to calculate the probablty of a loss of a certan magntude for the purposes of determnng the necessary captal or of calculatng VaR. hs s also the dstrbuton to be used n structurng collateralzed debt oblgatons, that s, n calculatng the probablty of loss and the expected loss for a gven tranche. For the purposes of prcng the tranches, however, t s necessary to use the rsk-neutral probablty dstrbuton. he rsk-neutral dstrbuton s calculated n the same way as above, except that the default probabltes are evaluated under the rsk-neutral measure P *, 1 * * log B log A r + σ p = P [ A( ) < B] = N σ where r s the rsk-free rate. he rsk-neutral probablty s related to the actual probablty of default by the equaton 5

6 ( M ) = N N ( ) + λρ (6) * 1 p p where ρ M s the correlaton of the frm asset value wth the market and λ = (µ M r)/σ M s the market prce of rsk. he rsk-neutral portfolo loss dstrbuton s then gven by 1 1 * * 1 ρ N ( x) N ( p ) P [ L x] = N (7) ρ hus, a dervatve securty (such as a CDO tranche wrtten aganst the portfolo) that pays at tme an amount C(L) contngent on the portfolo loss s valued at V = e C L r * E ( ) where the expectaton s taken wth respect to the dstrbuton (7). For nstance, a default protecton for losses n excess of L 0 s prced at ( ( )) V = e E ( L L ) = e p N N ( p ), N ( L ), 1 ρ r * r * 1 * he portfolo market value So far, we have dscussed the loss due to loan defaults. Now suppose that the maturty date of the loan s past the date H for whch the portfolo value s consdered (the horzon date). If the credt qualty of a borrower deterorates, the value of the loan wll declne, resultng n a loss (ths s often referred to as the loss due to credt mgraton ). We wll nvestgate the dstrbuton of the loss resultng from changes n the marked-to-market portfolo value. he value of the debt at tme 0 s the expected present value of the loan payments under the rsk-neutral measure, D = e Gp r * (1 ) where G s the loss gven default and p * s the rsk-neutral probablty of default. At tme H, the value of the loan s 1 r( H ) log B log A( H ) r( H ) + σ ( H) D( H ) = e 1 G N σ H Defne the loan loss L at tme H as the dfference between the rskless value and the market value of the loan at H, r( H ) L e = D H ( ) 6

7 hs defnton of loss s chosen purely for convenence. If the loss s defned n a dfferent way (for nstance, as the dfference between the accrued value and the market value), t wll only result n a shft of the portfolo loss dstrbuton by a locaton parameter. he loss on the -th loan can be wrtten as H L = a N b X H H where a Ge r ( H ) =, 1 N ( ) b = p + λρ M H and the standard normal varables X defned over the horzon H by log A ( H ) = log A + µ H σ H + σ H X 1 are subject to Equaton (). Let L be the market value loss at tme H of a loan portfolo wth weghts w. he condtonal mean of L gven Y can be calculated as ρh µ ( Y ) = E( L Y ) = a N b Y ρh ρh he losses condtonal on the factor Y are ndependent, and therefore the portfolo loss L condtonal on Y converges to ts mean value E(L Y) = µ(y) as Σw. he lmtng dstrbuton of L s then P[ ] P[ ( ) ] x ρh L x = µ Y x = F ; N( b), (8) a We see that the lmtng dstrbuton of the portfolo loss s of the same type (5) whether the loss s defned as the declne n the market value or the realzed loss at maturty. In fact, the results of the secton on the dstrbuton of loss due to default are just a specal case of ths secton for = H. he rsk-neutral dstrbuton for the loss due to market value change s gven by x ρh = a * * P [ L x] F ; p, (9) 7

8 Adjustment for granularty Equaton (8) reles on the convergence of the portfolo loss L gven Y to ts mean value µ(y), whch means that the condtonal varance Var(L Y) 0. When the portfolo s not suffcently large for the law of large numbers to take hold, we need to take nto account the non-zero value of Var(L Y). Consder a portfolo of unform credts wth weghts w 1, w,, w n and put n δ = = 1 w he condtonal varance of the portfolo loss L gven Y s where (1 ρ) H Var( L Y ) = δa N ( U, U, ) N ( U ) ρh ρh U = b Y ρh ρh he uncondtonal mean and varance of the portfolo loss are EL = an(b) and Var L = E Var( L Y ) + Var E( L Y ) H ρh (10) = δ a N ( b, b, ) + (1 δ) a N ( b, b, ) a N ( b) akng the frst two terms n the tetrachorc expanson of the bvarate normal dstrbuton functon N (x,x,ρ) = N (x) + ρn (x), where n s the normal densty functon, we have approxmately H ρh L = δ a b + δ a b H = a N ( b, b,( ρ + δ(1 ρ)) ) a N ( b) Var n ( ) (1 ) n ( ) Approxmatng the loan loss dstrbuton by the dstrbuton (5) wth the same mean and varance, we get P[ ] x H L x = F ; N( b),( ρ + δ(1 ρ)) (11) a hs expresson s n fact exact for both extremes n, δ = 0 and n = 1, δ = 1. Equaton (11) provdes an adjustment for the granularty of the portfolo. In partcular, the fnte portfolo adjustment to the dstrbuton of the gross loss at the maturty date s obtaned by puttng H =, a = 1 to yeld ( ) P[ L x] = F x; p, ρ + δ(1 ρ ) (1) 8

9 Summary We have shown that the dstrbuton of the loan portfolo loss converges, wth ncreasng portfolo sze, to the lmtng type gven by Equaton (5). It means that ths dstrbuton can be used to represent the loan loss behavor of large portfolos. he loan loss can be a realzed loss on loans maturng pror to the horzon date, or a market value defcency on loans whose term s longer than the horzon perod. he lmtng probablty dstrbuton of portfolo losses has been derved under the assumpton that all loans n the portfolo have the same maturty, the same probablty of default, and the same parwse correlaton of the borrower assets. Curously, however, computer smulatons show that the famly (5) appears to provde a reasonably good ft to the tal of the loss dstrbuton for more general portfolos. o llustrate ths pont, Fgure gves the results of Monte Carlo smulatons 1 of an actual bank portfolo. he portfolo conssted of 479 loans n amounts rangng from.000% to 8.7%, wth δ =.039. he maturtes ranged from 6 months to 6 years and the default probabltes from.000 to.064. he loss gven default averaged.54. he asset returns were generated wth fourteen common factors. Plotted s the smulated cumulatve dstrbuton functon of the loss n one year (dots) and the ftted lmtng dstrbuton functon (sold lne). References Pykhtn M and A Dev, 00, Credt Rsk n Asset Securtsatons: An Analytcal Model, Rsk May, pages S16-S0 Vascek O, 1987, Probablty of Loss on Loan Portfolo, KMV Corporaton (avalable at kmv.com) Vascek O, 1991, Lmtng Loan Loss Probablty Dstrbuton, KMV Corporaton (avalable at kmv.com) 1 he author s ndebted to Dr. Ym Lee for the computer smulatons. 9

10 Fgures Fgure 1. Portfolo loss dstrbuton ( p =.0, rho =.1) Portfolo loss Fgure. Smulated Loss Dstrbuton for an Actual Portfolo Cumulatve Probablty %.00% 4.00% 6.00% 8.00% 10.00% Portfolo Loss 10