Complex CDS valuation model The term structure of default: PDs cumulative, marginal and seen from today The term structure of PD (probability of default) probabilities that default occurs – at various dates in the future. - probability of default for each future period t as seen from today (t=0), where the superscript zero indicates today’s view. Consider a pool of 100 issuers and suppose that you expect to see two defaults within the second year from today, then =2%. So is the probability to default between January 1 of year t and December 31 of year t as expected today. - cumulative PD over t years. Suppose you expect a time-constant of 2%, then the probability to default over two years, , is 4%. In general, = + (1) marginal PD in year t. It is the probability to default during year t conditional on having - survived the beginning of year t: = / (1 - ) (2) Combining both equations, we obtain an expression for the cumulative PD in terms of the marginal PD: = + *(1 - ) (3) Finally, the marginal default probability in terms of the cumulative default probability: = ( - ) / (1 - ) (4) To clarify all these formulas, consider Figure 1, where is PD as seen from today, colored by dotted line Figure 1. The term structure of defaults. Calculation of the risk-neutral default probabilities.
Complex CDS valuation model The term structure of default: PDs cumulative, marginal and seen from today The term structure of PD (probability of default) – probabilities that default occurs at various dates in the future. - probability of default for each future period t as seen from today (t=0), where the superscript zero indicates today’s view. Consider a pool of 100 issuers and suppose that you expect to see two defaults within the second year from today, then =2%. So is the probability to default between January 1 of year t and December 31 of year t as expected today. - cumulative PD over t years. Suppose you expect a time-constant of 2%, then the probability to default over two years, , is 4%. In general, = + (1) - marginal PD in year t. It is the probability to default during year t conditional on having survived the beginning of year t: = / (1 - ) (2) Combining both equations, we obtain an expression for the cumulative PD in terms of the marginal PD: = + *(1 - ) (3) Finally, the marginal default probability in terms of the cumulative default probability: = ( - ) / (1 - ) (4) To clarify all these formulas, consider Figure 1, where is PD as seen from today, colored by dotted line Figure 1. The term structure of defaults. Calculation of the risk-neutral default probabilities.