(16c)
Proof: See the appendix.
We may think of as the risk-adjusted discount factor. Proposition 3 tells us that the bilateral risky price of a single-payment contract can be expressed as the present value of the payoff discounted by a risk-adjusted discount factor that has a switching-type dependence on the sign of the payoff.
Using a similar derivation as in Proposition 2, we can easily extend Proposition 3 from one-period to multiple-periods. Suppose that a defaultable contract has m cash flows. Let the m cash flows be represented as with payment dates , where i = 1,…,m . Each cash flow may be positive or negative. The bilateral risky value of the multiple-payment contract is given by
Proposition 4: The bilateral risky value of the multiple-payment contract is given by