Q:

Price of a Bond

I am working through an example in a textbook and, for the life of me, cannot figure out how they got from one step to the next. Here's the example:
A bond of 500, redeemable at par after 5 years, pays interest at 13% per year convertible semiannually. Find the price to yield an investor 8% effective per half year.
And I am supposed to use the formula: $$P=(Fr)a_{\bar n|i} + C(1+i)^{-n}$$
The only line of the example has:
$$P=32.5a_{\bar 10|.08}+500(1.08)^{-10}=449.67$$
And the whole line makes sense to me, except I am not coming to the same answer, so I know my error is in calculating $a_{\bar n|i}$. I assumed it would be $(1+\frac{.08}{2})^{10}$.
Can anyone help with what I'm doing wrong?

A:

As far as I understand, $F = 500$ is the face value, $r = 6.5\%$ (half of the annual $13\%$ since it's semiannual) is the contractual interest rate, $C = 500$ is the market value at maturity, and $i = 8\%$ is the per period desired interest rate. Then according to all the references I can find,
$$P = 500 \cdot 6.5\% \left(\frac{1-(1+8\%)^{-10}}{8\%}\right) + 500(1+8\%)^{-10}$$
which comes out to $449.67$. (Note that $500\cdot6.5\% = 32.5$.)
This means your $a_{\bar{n}|i}$ is likely defined as $a_{\bar{n}|i} = \frac{1-(1+i)^{-n}}{i}$.

