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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 !

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