Our corporate CDO transactions possess similar structural features to the standard CDX tranches, but often dif fer with  respect to the referenced corporate entities, the term, the attachment points and the detachment points.  Therefore, in order to  determine the equivalent-risk tranche for each of our corporate CDO transactions, we determine the attachment and detachment  points on the CDX index that have comparable estimated probabilities of loss as the attachment and detachment points in our  transactions. We begin by performing a simulation analysis of referenced entity defaults in our transactions to determine the  probability of portfolio losses exceeding our attachment and detachment points.  The referenced entity defaults are primarily  determined based on the following inputs: the market observed CDS credit spreads of the referenced corporate entities, the  correlations between each of the referenced corporate entities and the term of the transaction. For each referenced corporate entity in our corporate CDO transactions, the CDS spreads associated with the term of our  transactions (“credit curve”) define the estimated expected loss for each entity (as applied in a market standard approach known  as “risk neutral” modeling). The credit curves on individual referenced entities are generally observable.  The expected  cumulative loss for the portfolio of referenced entities associated with each of our transactions is the sum of the expected losses  of these individual referenced entities.  With respect to the correlation of losses across the underlying reference entities, two  obligors belonging to the same industry or located in the same geographical region are assumed to have a higher probability of  defaulting together (i.e., they are more correlated).  An increase in the correlations between the referenced entities generally  causes a higher expected loss for the portfolio associated with our transactions.  The estimated correlation factors that we use are  derived internally based on observable third-party inputs that are based on historical data.  The impact of our correlation assumptions currently does not have a material ef fect on our fair premium estimates in light  of the significant impact of our non-performance risk adjustment as described below . Once we have established the probability of portfolio losses exceeding the attachment and detachment points in our  transactions, we then use the same simulation method to locate the attachment and detachment points on the CDX index with  comparable probabilities. These equivalent attachment and detachment points define the equivalent-risk tranche on the CDX  index that we use to determine fair premium amounts. Determining the Typical Fair Premium Amount —The equivalent-risk tranches for our corporate CDO transactions often  are not identical to any standard CDX tranches.  As a result, fair premium amounts generally are not directly observable from  the CDX index for the equivalent-risk tranche and must be separately determined.  We make this determination through an  interpolation in which we use the observed premium rates on the standard CDX tranches that most closely match our  equivalent-risk tranche to derive the typical fair premium amount for the equivalent-risk tranche. Non-Performance Risk Adjustment on Corporate CDOs —The typical fair premium amount estimated for the equivalent- risk tranche represents the fair premium amount for a typical market participant—not Radian.  Accordingly, the final step in our  fair value estimation is to convert this typical fair premium amount into a fair premium amount for a financial guarantor of  similar credit quality to us. A typical market participant is contractually bound by a requirement that collateral be posted  regularly to minimize the impact of that participant’ s default or non-performance.  This collateral posting feature makes these  transactions less risky to the protection buyer , and therefore, priced dif ferently. None of our contracts require us to post  collateral with our counterparties, which exposes our counterparties fully to our non-performance risk.  We make an adjustment  to the typical fair premium amount to account for both this contractual dif ference, as well as for the market’ s perception of our  default probability, which is observable through our CDS spread.  The amount of the non-performance risk adjustment is computed based, in part, on the expected claim payment by  Radian. To estimate this expected payment, we first determine the expected claim payment of a typical market participant by  using a risk-neutral modeling approach. A significant underlying assumption of the risk-neutral model approach that we use is  that the typical fair premium amount is equal to the present value of expected claim payments from a typical market  participant. Expected claim payments on a transaction are based on the expected loss on that transaction (also determined using  the risk-neutral modeling approach). Radian’ s expected claim payment is calculated based on the correlation between the  default probability of the transaction and our default probability . The default probability of Radian is determined from the  observed Radian Group CDS spread and the default probability of the transaction is determined as described above under  “Defining the Equivalent-Risk Tranche.” The present value of Radian’ s expected claim payments is discounted using a risk-free  interest rate, as the expected claim payments have already been risk-adjusted. The reduction in our fair premium amount related to our non-performance risk is limited to a minimum fair premium  amount, which is determined based on our estimate of the minimum fair premium that a market participant would require to  assume the risks of our obligations.  Approximately 32% of our corporate CDO contracts as of  December 31, 2012 are subject to  this minimum fair premium. Our non-performance risk adjustment currently results in a material reduction of our typical fair  premium amounts, which in turn has a positive impact on the fair value of these derivatives.  151