can be regarded as a risk-adjusted discount factor. Proposition 1 says that the unilateral risky valuation of the single payoff contract has a dependence on the sign of the payoff. If the payoff is positive, the risky value is equal to the risk-free value minus the discounted potential loss. Otherwise, the risky value is equal to the risk-free value.
Proposition 1 can be easily extended from one-period to multiple-periods. Suppose that a defaultable contract has m cash flows. Let the m cash flows be represented as