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

148 MANAGING CREDIT RISK IN CORPORATE BOND PORTFOLIOSthe joint migration probability h B2,Aaa given by equation (8.11). One can followthis procedure to compute the joint migration probabilities h ik to anyof 324 discrete states for a two-obligor portfolio.The next step in the process is to compute the credit loss g ik associatedwith each state of the discrete joint probability distribution. Here, g ikdenotes the credit loss due to the rating migration of obligor 1 from state uto state i and of obligor 2 from state to state k. Because this credit loss isnot dependent on the asset return distribution, one can compute this usingequation (6.42) in Chapter 6.Once the joint probabilities of being in each of the states and the correspondingcredit losses have been determined, it is fairly simple to compute theexpected value E(/ 1/ 2) of the joint distribution of credit loss using equation(6.43) in Chapter 6. Inserting this value into equation (6.5) in Chapter 6allows one to compute the loss correlation between the two obligors underthe migration mode when the joint distribution of asset returns is bivariatet distributed. Once the loss correlation between the obligors is determined,computing the unexpected loss of the portfolio is straightforward.Again it is instructive to compare the expected and the unexpected lossof the two-bond portfolio when asset returns are assumed to be t distributed.The various credit risk measures computed using the analytical approachpresented here for the two-bond portfolio example considered in Exhibit 6.1in Chapter 6 with 30 percent asset return correlation are as follows:The loss correlation / ik under the migration mode is 0.14905.The expected portfolio loss EL P under the migration mode is $4,740.The unexpected portfolio loss UL P under the migration mode is$32,770.As one might expect, the unexpected loss of the portfolio increases due toan increase in the loss correlation between the obligors.To provide further comparisons for the credit risk measures obtainedwhen the joint normality of asset returns is relaxed, Exhibit 8.3 shows therisk measures computed for the 23-bond portfolio considered in Exhibit 6.6in Chapter 6 assuming a multivariate t distribution for asset returns. In computingthe expected and unexpected losses for the portfolio, the indicativeEXHIBIT 8.3 Portfolio Credit Risk Using Indicative Asset ReturnCorrelation MatrixDescription EL P (mn $) UL P (mn $) %EL P (bp) %UL P (bp)Under default mode 0.660 3.783 13.8 79.4Under migration mode 1.622 5.034 34.0 105.6