The ways to calculate distance to default based on KMV model derivation

Transcription

1 he ays o calculae isance o efaul base on KM moel erivaion Danan Zhou,a, Fuei Li,b, Jie Gao, c School of Economics an Managemen, Chongqing Universiy of Poss an elecommunicaions, Chongqing 465, China. Absrac a anan.j.zhou@qq.com, b fuei9@gmail.com, c JessicaG4@63.com his paper escribes he basic iea of KM an re-erives he relevan mahemaical equaions in he moel, realizing he pracical applicaion of KM moel, an calculaing he efaul isance of 53 enerprises in manufacuring. A las, i provies a heoreical basis for improving he efficiency of enerprise risk managemen. Keyors KM Moel, Opion Pricing Moel, Mahemaical Derivaion,he Disance o Defaul.. Inroucion he efaul isance meho is base on he "Black-Scholes-Meron" opion pricing moel[], evelope by KM crei raing company ino he KM crei raing meho; In, KM as acquire by Mooy's, one of he orl's hree major raing agencies, forming Mooy's KM raing meho[]. KM moel regars he company's shareholers' equiy, ha is, he equiy value as a call opion base on he marke value of he enerprise's asses, ih he oal liabiliies as he execuion price. When he marke value of he enerprise asses is less han he oal liabiliies, he enerprise ill face he efaul risk of no being able o repay he loan. In he KM moel, he efaul isance DD is use o quanify he crei risk of he enerprise[3]. he greaer he efaul isance, he loer he crei risk[4]. he efaul isance is use as a crei risk measure, an is robusness is iely use in raing pracice[5]. he core of he KM crei raing meho is he efaul isance (DD) crei meric, hich is base on ynamic aa in he real marke, eermine by asse marke value, asse marke value volailiy, equiy value, equiy value volailiy, an efaul poins.. heoreical basis an moel. he basic iea of KM moel A of he enerprise asse is less han he oal eb D of he enerprise, he When he marke value enerprise ill face he risk of no being able o repay he loan, hich is calle he enerprise efaul. In he KM moel, he efaul isance DD is use o inicae he value of he asse marke value A from he poin of efaul DP,hus reflecing he possibiliy of efaul. he greaer he efaul isance, he less likely he company ill efaul. On he conrary, he smaller he efaul isance, he more likely he company ill efaul. he seps o calculae he efaul isance are as follos: Firsly, he asse marke valuea an volailiy are solve, bu hey canno be irecly obaine, hey nee o be obaine by he relaionship beeen he equiy marke value E, he equiy marke value volailiy E an he eb amoun D.he simulaneous equaions are solve as follos: 5

2 E N( ) De N( ) A A ln( ) ( r )* D () E A E N ( ) () Afer obaining he marke valuea an volailiy of he asse, he efaul isance can be obaine accoring o he efiniion of he efaul isance DD. he expression is as follos[, 6, 7]: A A DP DD A v (3) is he marke value of he asse a he ime of mauriy (ie ) of he corporae eb, bu his paper assumes ha he groh raio of he corporae asse uring he liabiliy perio is [], ha is, he A of he efaul isance is he above-menione calculaion.he poin of efaul DP is a value beeen curren liabiliies an oal liabiliies, an he mahemaical form is DP SD.5* LD. SD is he shor-erm eb a he en of he business, an LD is long-erm eb. his paper assumes ha he company only has a single ineres-free eb, he amoun of eb [8]here is DP.. he soluion of relae parameers of KM moel A, E, DP E, r,, A v () he calculaion of sock volailiy E he raiional meho of calculaing sock volailiy is o use hisorical sock price aa o solve he Si problem. he specific soluion formula is as follos: u ln( ). is he sock aily reurn rae, an S i is he sock closing price afer he en of he firs ime inerval. Accoring o he efiniion of sock volailiy, calculae he sanar eviaion of sock aily yiel, an obain aily volailiy: n S ( u u) n 6 i Si u i i i (4) Obaine he aily volailiy, an assume ha he annual sock raing ay is abou 4 ays, E S 4 he annualize volailiy is obaine. Hoever, a large number of lieraures sho ha he changes in sock reurns are clusere in volailiy[9]. he so-calle volailiy clusering means ha he flucuaion of he long-erm financial ime series in a cerain perio of ime ofen shos a siuaion of coninuous high or lo, ha is, he ranom isurbance is folloe by a large ampliue flucuaion folloe by a large flucuaion, hich is smaller. Ampliue flucuaions are

3 folloe by small flucuaions, a phenomenon knon as volailiy clusering. Hoever, he above meho has obaine a phenomenon ha he sock volailiy an he acual siuaion are no consisen. herefore, he GARCH (,) moel is use o calculae he sock volailiy. In his case, he volailiy obaine is closer realiy. he concree expression of he GARCH(,) moel is as follos: Mean equaion: y c, N(, ) (5) Replace ih L Coniional variance equaion: (6), o become he folloing equaion: 7 L (7) :he preice variance of he -h perio is calle he GARCH iem; : he square of he resiual of he -h perio, he ranom isurbance erm, is calle he ARCH erm. L : Long-erm average variance;,,, :Weighs are greaer han, an GARCH(,) is calle generalize auoregressive coniional heeroskeasiciy. In his case, he sock aily volailiy is no a fixe consan, bu varies from ime o ime. he firs "" in brackes represens he number of iems in he GARCH iem, he GARCH iem is he influence of he previous preicion variance on he curren variance; he secon "" represens he number of iems in he ARCH iem, an he ARCH iem is calle he ranom isurbance iem, hich is a amoun of change over ime is measure by he volailiy of he previous perio. he meaning of "auoregressive" here is he inroucion of values of some orer of he explanaory variables such as. As an explanaory variable, explaining he change of he variable ogeher ih he ranom isurbance erm. ha is, he regression ha inrouces he regression variable iself is calle auoregression. Ierovariance refers o he regression moel, he variance beeen he ranom isurbances is ifferen. So he consrucion of he GARCH moel is o eliminae his heerosceasiciy as much as possible. Suppose ha i obeys a mean of an he variance is -, an his variance is a linear combinaion of he square of he lag of is ranom isurbance erm, calle he coniional variance. Afer fiing he GARCH(,) moel ih he enerprise aa, he aily volailiy of he company's sock reurn rae is obaine, assuming ha he raing ay of he year is 4 ays, an hen convere ino he annualize volailiy of he calculaion base ae. () Calculaion of equiy marke value I shoul be noe ha hen calculaing he equiy marke value of lise companies, if he price ifference beeen he raable shares an he non-raable shares cause by he spli of he lise companies in China is no consiere, he marke price of he raable shares is muliplie by he oal share capial o esimae he equiy of he company. he marke value may uneresimae he crei risk of he enerprise; in he special marke environmen in hich Chinese lise companies are locae, he seing of he efaul poin coefficien may affec he moel preicion abiliy. herefore, combine ih he specific siuaion of China's sock marke, he value of he equiy marke is compose of o pars, namely he marke value of he raable shares an he marke value of he non-raable shares. he specific calculaion expression is as follos: Equiy marke value = marke value of raable shares + marke value of non-raable shares = average closing price of sock eekly in he curren ay of he benchmark ay * he number of raable shares+he number of non-raable shares * ne asses /per share

4 (3) he poin of efaul DP, r an he poin of efaul DP SD a* LD, is he coefficien, hich reflecs he influence of he long-erm liabiliies of he enerprise in he efaul of he enerprise. he larger a is, he higher he proporion of long-erm ebs in corporae efauls. A large number of lieraures[, ]have verifie ha.5 can reflec he rue efaul of he company o a greaer exen, so his paper akes.5. is he annual reurn on asses, i is also calle a risk-free rae. his paper ses he one-year ime eposi rae announce by he People's Bank of China on he base ae. is quarer. r 3. Derivaion of heoreical formulas relae o KM moel () Black-Scholes opion pricing heory a KM moel is base on he Black-Scholes opion [] pricing heory. he opions are ivie ino European opions an American opions. he famous scholars Black an Scholes s suy are base on European opions, an he rules can only be exercise on he expiraion ae. he basic assumpions of Black-Scholes opion pricing heory are as follos: ) he change in sock price is subjec o a lognormal isribuion; ) he risk-free rae is fixe hroughou he exercise perio; 3) Assume ha here is no ransacion cos in he securiies ransacion an here is no ivien isribuion. Base on he above assumpions, Black an Scholes propose a pricing moel for bullish (pu) opions as follos: S Call( S, K, r,, δ) SN( ) Ke N( ) ln( ) ( ) Pu( S, K, r,, δ) Ke N( ) SN( ) S K δ r δ δ (8) is he price of he sock a he ime; K is he srike price of he opion; r is he risk-free ineres rae; r is he volailiy of he sock's reurn; is he mauriy ae of he hel opion. ()Iō's lemma G is a funcion of an, an mus saisfy he folloing equaion: G G G G G ( * ) Z (9) ( ) Z (3) Derivaion of he propery of lognormal isribuion ) Meaning: Assume ha he value of financial asses obeys a lognormal isribuion, ha is ln obeys a normal isribuion. ) Derivaion: ln N[ln ( ), ] Le G ln, G is a funcion of an, an G G G,,. herefore, base on formula Iō's lemma, he funcion G ih an obeys: 8

5 G G G G G ( * ) Z ( ) Z () his shos ha G ln is a generalize Wiener process ha saisfies a rif rae an a variance rae. ha is, -he change beeen he imes obeys he mean ( ) an he normal isribuion of variance is.hen ln ln N[( ), ],go i ln N[ln ( ), ]. is he annual rae of reurn on he value of financial asses; is he volailiy of he reurn on financial asses. (4) Black-Scholes-Meron Deb Pricing Moel ) hough he ebor's shareholer's equiy is regare as a kin of he caller's asse marke value, an he ebor's oal liabiliies (his aricle assumes no ineres) D as he srike price of he execuion price []. Base on he risk-neural coniion, he opion price of a European call opion is he value of is expece value iscoune a a risk-free rae. hen he value of he ebor's sock a he ln momen is he value of is expece value E[max( D,)] iscoune a he risk-free rae. ) Derivaion of Black-Scholes-Meron Deb Pricing Moel here are o mehos for eriving he Black-Scholes-Meron eb pricing moel: solving ifferenial equaions an eriving resuls base on risk-neural pricing. his paper is base on he preconiions of risk neural pricing. Derivaion prerequisies:. Base on risk neural coniions;. Assume ha he expece rae of reurn of he ebor s financial asses is consisen ih he risk-free rae; 3. he company has no ivien isribuion uring he perio; 4. he eb is a single ineres-free liabiliy an he borroing perio is assume o be year; E N( ) De N( ) ln( ) ( r )* D Explanaion of he symbol: E is he ebor's shareholer's equiy; is he asse marke value of he ebor a he momen ; D is he ebor s poin of efaul; r is a risk-free rae; () is he variance of he logarihm of he annual reurn on asses calculae by coninuous compoun ineres; is he ime from he expiraion of he loan; 9

6 Uner risk-neural coniions, Black-Scholes-Meron eb pricing moel reas he ebor's sock value as a call opion ih he oal marke liabiliy as he execuion price, ih he oal marke value of he eb D as he arge. hen, a he momen, he sock value E is he presen value afer iscouning is expecaion a a risk-free rae r. herefore, a he momen expression of he sock value E e E[max( D,)],, is sock value is E e E[max( D,)]. Proof () is he E of he company a he momen is he value of he asse marke a he momen.. Namely: he proof process is as follos: Assume ha he marke value of he asse is subjec o he mean m an he lognormal isribuion of variance, ie ln N( m, ). he properies of he lognormal isribuion are: ln m le, N(,),hen he probabiliy ensiy funcion of is A he same ime, e se momen g ( ), an he upper an loer limis of he inegral of limis of he inegral of. mln.sanarizaion, h( ) e. as he probabiliy ensiy funcion of he asse marke value ln m ( D, ),ln (ln D, ),ln m, D(, ) Prove E E[max( D,)] D ( D) g( ) ln Dm ln Dm ln Dm ln Dm m ( e D) h( ) m e h( ) Dh( ) m ln Dm ln Dm m e e D e ln D m e D[ N( )] ln Dm e ( ) m m ( ) ln Dm ln Dm m ln D DN( ) ln ln D e e ( DN ) ln( / D) m e N( ) ( DN ) ln( / ) ln D m Dm e [ N( )] DN( ) ln( / ) m ln D m D e N( ) DN( ) ln( m / D) ln( / D) e N( ) DN( ) Also, by he naure of he lognormal isribuion, ln N(ln ( - ), ) a he A are replace by he upper an loer

7 Which is go m ln ( - ) ; E e E[max( D,)] ln( / D) ln( / D) m e [ e N( ) DN( ) ln ( - ) e [ e N( ) DN( )] ln e N e DN ( ) ( ) N( ) De N( ) his is he Black-Scholes-Meron eb pricing moel an he resuls are as follos: E N( ) De N( ) Which ^ ln( ) ( r )* D, 3)he erivaion of he relaionship beeen he raio of he volailiy of sock value E E an o he volailiy of he marke value of corporae asses is he elasiciy of changes in he value of he company's equiy o changes in he value of he company's asses []. / E / E E E E ( ) by E N De N E go N ( ) E an ( ) N ( ) E so E E ( ) ( ) N ( ) 4. An applicaion o calculae he isance o efaul his paper selecs he aa of 53 lise companies in he manufacuring inusry in as a case o calculae he efaul isance. In his paper, he original inpu inicaors for calculaing he efaul isance of 53 lise companies are given as shon in able- (ake as an example), he inpu inex iems in ables- an he inpu parameers for calculaing he efaul isance is consisen ih.-., he resuls of he isance o efaul of 53 lise companies is shon in able 3: able Inpu inicaor aa for calculaing DD

8 Coe SD LD Circulaing A shares Limie sale of A shares Weekly average closing price BPS DP E 49.SZ.56E E E+9 4.9E+9 5.SZ.5E+9.77E E E+9 7.E+9 59.SZ.59E+ 7.85E+9.E E+ 5.4E+9 6.SZ 5.9E+9.63E+9.6E E+9.6E+.SZ 3.69E+.6E+ 8.4E E E+ E+ 4.SZ 4.E E E+8.76E E+9.7E+ 4.SZ.5E+.44E+.E+9.35E E+.E+ 44.SZ 4.8E E E+9.8E+9 43.SZ 7.3E E E+8.E+ 45.SZ.E+ 9.78E+9.6E E+ 4.6E SZ.6E E+8 5.9E SZ.3E E E SZ.5E E E SZ.43E E E+8.3E SZ 4.78E E E+8.46E SZ E SZ.77E E+8 4.3E SZ.9E E+8.9E SZ E SZ E+8 able Ra aa of he closing price of he GARCH (,) moel Coe SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ

9 3345.SZ SZ SZ SZ Conclusion able 3 he resuls of he isance o efaul Coe DD 49.SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ SZ he isance o efaul has goo sabiliy[5] in quanifying corporae crei risk, an he bigger he efaul isance, he loer he crei risk of he enerprise. On he conrary, he smaller he efaul isance, he higher he crei risk of he enerprise. hrough he mahemaical erivaion of he KM moel relae equaions, he unersaning of he KM moel is realize, hich is very helpful for calculaing he efaul isance. If in he acual managemen of he enerprise, he role of he isance o efaul in quanifying he crei risk of he enerprise can be ell consiere, an he goal of improving he managemen efficiency of he enerprise can be achieve. References [] Black F S M. he Pricing of Opions an Corporae Liabiliies[J]. Journal of Poliical Economy. 973, 3(8): [] AO. Crei aluaion[j]. Crei aluaion[m]// Finance, Economics an Mahemaics. John Wiley & Sons, Inc, 5. [3] Bharah S, Shumay. Forecasing Defaul ih he Meron Disance o Defaul Moel[J]. Revie of Financial Suies. 8, (3): [4] Yan Haifeng. Research on Crei Risk of Chinese Lise Companies Base on KM Moel[J]. Inusrial Economics Research. 9, 3: 4-. 3

10 [5] Jessen C, Lano D. Robusness of isance-o-efaul[j]. Journal of Banking & Finance. 5, 5: [6] Langohr H, Langohr P. he raing agencies an heir crei raings[m]// he Raing Agencies an heir Crei Raings: Wha hey Are, Ho hey Work, an Why hey are Relevan. 8. [7] Doumpos M, Niklis D, Zopouniis C, e al. Combining accouning aa an a srucural moel for preicing crei raings: Empirical evience from European lise firms[j]. Journal of Banking & Finance. 5, 5: [8] an Jiujun. Financial Early Warning Moel of Lise Companies ih Financial Inicaors an Defaul Disance[J]. Sysems Engineering. 5(9): 5-. [9] Liu Yingchun, Liu Wei. Research on KM Crei Risk Measuremen Base on GARCH olailiy Moel[J]. Journal of Dongbei Universiy of Finance an Economics. (3): [] Zhang Ling. Applicaion of KM Moel in Crei Risk Assessmen of Lise Companies[J]. Sysems Engineering. 4, (): [] Ma Ruoei, Zhang Wei, Bai Yukun. Improvemen of Dynamic Defaul Probabiliy KM Moel of China's Lise Companies[J]. Sysems Engineering. 4(): [] Lu Wei, Zhao Hengbiao, Fang Zhaoben, e al. Applicaion of KM Moel in Corporae alue Evaluaion [J]. Managemen Science. 3(3):