Price of a Bond

Question

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?

Answer 1

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 $a_{\bar{n}|i} = \frac{1-(1+i)^{-n}}{i}$.

Answer 2

It is not that simple as $(1+i/2)^{10}$. Since the bond pays interest $Fr$ semiannually, the price should be based on the present value of those payments too. The notation $a_{\bar{n}|i}$ is for the present value of annuity immediate, which is this case. The value $a_{\bar{10}|i}$ is for $v=1/(1+i)$, $$ \begin{align*} a_{\bar{10}|i}&=\sum_{k=1}^{10} v^{k}\\ &=v\frac{1-v^{10}}{1-v}\\ &=\frac{1-v^{10}}i\\ &=\frac{1-(1.08)^{-10}}{.08}. \end{align*} $$

It is also said in the problem that "effective semiannually 8%", so what we use is $i=8\%$ semiannually.

For viewing the financial formulas: refer to this: https://en.wikibooks.org/wiki/Financial_Math_FM/Formulas