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Here is my understanding of your question, I might have oversimplified your problem, and made some hypothesis that were not yours. Assuming we are talking about a bond paying $1$ each day until $T$ or default event if occuring before $T$. Let's write the risky coupon bond payment in a continous time manner: $$\int_0^T \mathbb{1}_{\tau>t} dt$$ with ...

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There are different measures and interpretations of duration. One, as has been pointed out already, has a formula weighting coupons and final contractual cashflow. Other definitions of duration take a broader perspective and relate it to the interest rate sensitivity of the security and not to a particular formula. These go by names such as effective or ...

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Ditto to Larasing. Any bond's duration is just a matter of coupon, price, discount rate. Credit risk does not factor into this equation.

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