Radian Group Inc. Notes to Consolidated Financial Statements - (Continued) Present Value of Expected Future Contractual Premiums —Our contractual premiums are subject to change primarily for  two reasons: (1) all of our contracts provide our counterparties with the right to terminate upon our default; and (2)  80% of the  aggregate net par outstanding of our corporate CDO transactions  (as of December 31, 2012) provide our counterparties with the  right to terminate these transactions. In determining the expected future premiums of these transactions, we adjust the  contractual premiums for such transactions to reflect the estimated fair value of those premiums based on our estimate of the  probability of our counterparties exercising this downgrade termination right and the impact it would have on the remaining  expected lifetime premium.We also cap the total estimated fair value of the contracts subject to termination such that none of  these contracts are in a derivative asset position.  As of December 31, 2012, 37% of the aggregate net par outstanding of our  corporate CDO transactions was capped in this manner . The discount rate we use to determine the present value of expected  future premiums is our CDS spread plus a risk-free rate.  This discount rate reflects the risk that we may not collect future  premiums due to our inability to satisfy our contractual obligations, which provides our counterparties the right to terminate the  contracts. Determining the Fair Premium Amount —For each corporate CDO transaction, we perform three principal steps in  determining the fair premium amount: •  first, we define a tranche on the CDX index (defined below) that equates to the risk profile of our specific transaction  (we refer to this tranche as an “equivalent-risk tranche”); •  second, we determine the fair premium amount on the equivalent-risk tranche for those market participants engaged in  trading on the CDX index (we refer to each of these participants as a “typical market participant”); and •  third, we adjust the fair premium amount for a typical market participant to account for the dif ference between the  non-performance or default risk of a typical market participant and the non-performance or default risk of a financial  guarantor of similar credit quality to us (in each case, we refer to the risk of non-performance as “non-performance  risk”). Defining the Equivalent-Risk Tranche —Direct observations of fair premium amounts for our transactions are not  available because these transactions cannot be traded or transferred pursuant to their terms and there is currently no active  market for these transactions. However , CDS on tranches of a standardized index (the “CDX index”) are widely traded and  observable and provide relevant market data for determining the fair premium amount of our transactions, as described more  fully below. The CDX index is an index based on a synthetic corporate CDO that comprises a list of corporate obligors and is  segmented into multiple tranches of synthetic senior unsecured debt of these obligors ranging from the equity tranche (i.e., the  most credit risk or first-loss position) to the most senior tranche (i.e., the least credit risk).  We refer to each of these tranches as  a “standard CDX tranche.” A tranche is defined by an attachment point and detachment point, representing the range of  portfolio losses for which the protection seller would be required to make a payment. 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.  198