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

100 MANAGING CREDIT RISK IN CORPORATE BOND PORTFOLIOSMaking use of equations (6.10) and (6.12), one can rewrite equation (6.17)as follows:E(/ i/ k) NE i NE k ( PDi PDk ik PD i PD k ) LD i LD k(6.18)Incorporating equation (6.18) into (6.5) and simplifying gives the followingrelation between loss correlation and default correlation when i k: / ik NE i NE k PDi PDk LD i LD kUL i UL k ik(6.19)Under the assumption that recovery rates for bonds issued by two differentobligors are independent, equation (6.19) indicates that loss correlation islower than default correlation. In the general case, loss correlation can beeither lower or higher than default correlation depending on the level ofcorrelation between recovery rates for different obligors. The assumption / ik ik implicitly postulates that recovery rates for different obligors arepositively correlated as follows: r ik ik (UL% i UL% k PDi PDk LD i LD k ) RRi RRk ( ik PDi PDk PD i PD k )(6.20)In equation (6.20), the variable UL% is the unexpected loss as a percentageof exposure, which is given by UL% UL/NE.Estimating Default CorrelationI mentioned that estimation of the loss correlation between obligors is usuallydone through indirect methods. This is also true in the case of defaultcorrelation. Because knowledge of default correlation allows one to computeloss correlation and therefore unexpected portfolio loss, I focus onhow default correlation can be estimated. The standard technique for estimatingdefault correlation is based on the latent variable approach. In suchan approach, default of an obligor is assumed to occur if a latent variablethat is considered to play a role in the firm’s default falls below a certainthreshold value. Correlation between the latent variables of different obligorsis then used to infer the default correlation between obligors.The latent variable that is used in practice is the asset returns of theobligor. The motivation for using asset return as the latent variable is that inMerton’s model, a firm’s default is driven by changes in its asset value. As aresult, the correlation between the asset returns of two obligors can be usedto compute the default correlation between them. In practice, one uses thecorrelation between asset returns for two obligors to estimate the probability