# Forward contract on a defaultable coupon bearing bond

Notations :

$$P(t,T)$$ : the $$t$$-price of a coupon bearing bond paying coupons $$C_i$$ at $$T_i$$ maturing at $$T$$

$$B(t,T)$$ : the $$t$$-price of a non defaultable zero coupon bond paying 1 at $$T$$

$$P_r(t,T)$$ : the $$t$$-price of a (risky bond) defaultable coupon bearing bond paying coupons $$C_i$$ at $$T_i$$ if no default, else paying a fixed recovery $$R$$ at time of default $$\tau$$ and maturing at $$T$$

According to Price a forward contract on a zero-coupon bond

The forward price of a forward contract on a standard coupon bearing bond settling at $$t_1$$ could be expressed as :

$$F(t_1,T)= \mathbb{E}^{t_1}\left ( P(t_1,T) \right | \mathbb{F}_t)= \frac{\sum_{T_i\geq t_1}C_iB(t,T_i)+B(t,T)}{B(t,t_1)}= \frac{P(t,T)-\sum_{T_i\leq t_1}C_iB(t,T_i)}{B(t,t_1)}$$

Hence we could write Forward Bond($$t_1$$) + past coupons($$t_1$$) = standard bond($$t$$). (*)

Likewise, how to establish the forward expression for $$P_r (t_1, T)$$? does an expression similar to (*) hold? I find it difficult to account for the defaultability of the risky bond.

Thanks !