Hello dushededidi,
I know what's your point.
Unfortunately, both of your options are not appropriate.
Take it easy,it's not your fault.
It is actually a practical issue.
Here is the concept,
you pay the PV at your latest prior transaction point + interest afterwards.
Meaning, practitioner won't use discount method to get the transaction price between two coupon dates.
For your convenience,
I built a model for you.
1. get PV at time 0= 1000/(1.1)
2. get interest = 1000 - PV
3. get monthly interest = interest / 12
4. get your final price = PV + monthly interest X 11
or
FV - monthly interest
The solutions should match with each other as the model demonstrated.
I hope it helps.
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