So I was under the impression that the discount factor $v$ is always annualized. But as I work on this problem for an $n$-year bond, it seems not to be the case, as they arrive at this conclusion for the present value of some semiannual coupons (sorry not sure how to do angle-n notation):
$a_{2n}=\frac{1-v^{2n}}{i^{(2)}}$
So the $n$-year bond has $2n$ payment/conversion periods, which is fine, but doesn't $v^{2n}$ imply that $v$ is a semiannual discount factor, not an annual one? It seems to me the line of thinking goes more like this...
$a_{2n}=\frac{1-(1+\frac{i^{(2)}}{2})^{-2n}}{i^{(2)}} = \frac{1-(1+i)^{-n}}{i^{(2)}} = \frac{1-v^{n}}{i^{(2)}}$
On the other hand, the problem defines $i$ to be the semiannual yield rate, which is a bit unusual for these problems where $i$ is almost always the effective annual rate. So is the period of $v$ to be understood in terms of how we define $i$? And if so, can I take that to the bank, so to speak? Works every time?
Thanks for your help!
After some investigation, it appears that $v$ can indeed be a semiannual discount factor, and that it depends on the period defined for $i$. This problem defines $i$ to be a semiannual rate, and so $v$ follows suit.
This may be unique to this particular type of problem on Exam FM.