Managing Credit Risk in Corporate Bond Portfolios : A Practitioner's ...

72 MANAGING CREDIT RISK IN CORPORATE BOND PORTFOLIOSIn equation (5.4), one takes the logarithm on both sides of the inequalitybecause doing so does not change the probabilities. Rearranging theterms in equation (5.4), one can represent the probability of default forthe firm asPD prob c z T ln(A 0 F) A 0.5 2 ABTd A 2T(5.5)Because z T is a normally distributed random variable, the probability ofdefault can be represented aswherePD N(D) (5.6)D ln(A 0F) A 0.5 2 ABT A 2TDN(D) 1 exp(0.5x 2 )dx22pq(5.7)(5.8)In equation (5.7), D represents the distance to default, which is the distancebetween the logarithm of the expected asset value at maturity and the logarithmof the default point normalized by the asset volatility.Although Merton’s framework for determining the probability ofdefault for issuers is rather simple, applying this directly in practice runsinto difficulties. This is because firms seldom issue zero-coupon bonds andusually have multiple liabilities. Furthermore, firms in distress may be ableto draw on lines of credit to honor coupon and principal payments, resultingin a maturity transformation of their liabilities.To resolve these difficulties, the KMV Corporation suggested somemodifications to Merton’s framework to make the default probability estimatemeaningful in a practical setting 1 (KMV refers to the probability ofdefault as the expected default frequency, or EDF). For instance, rather thanusing the face value of the debt to denote the default point, KMV suggestsusing the sum of the short-term liabilities (coupon and principal paymentsdue in less than 1 year) and one half of the long-term liabilities. This choiceis based on the empirical evidence that firms default when their asset valuereaches a level between the value of total liabilities and the value of short-termliabilities. Furthermore, because the asset returns of the firms may in practice