Ah, I was so not getting that this was a bond. See how important those definitions are? Many fewer abbreviations than might be imagined just are not all that standard.
With known interest rates, you should be able to find a complete alegrbraic solution for most anything. If it is the interest rate you seek, that is another story. There certainly are methods better than "guess and check", but essentially, they are still guess and check with much, much better guessing than you or I could do of the top of our heads.
I'll demonstrate a "fixed point method". This is likely to work because we have a really good idea wher the answer should be. It can't very well be as low as -5% and it's unlikely to be as high as 28%. We probably can narrow it down much more than that. Any narrowing will be beneficial.
First, we need a more convenient expression. If you happen to remember the right formula, you're in business. Personally, I never remember forumlas. I used basic principles to write them myself. In my view, this makes it much harder to select the wrong formula.
Let's define a few things:
i = the mysterious annual interest rate that we seek.
v = 1/(1+i) = the annual discounting factor associated with i
w = sqrt(v) -- since we'll need semi-annual compounding.
Having done that, we can just write down the entire structure:
P0 = $1,050 = $40(PVIFA R%,24) + $1,000(PVIF R%,24)
1050 = 40w + 40w^2 + 40w^3 + ... + 40w^24 + 1000w^24 -- Done.
This is where it comes in handy to remember everything you ever learned about geometric series.
1050 = 40w(1 + w + w^2 + ... + w^23) + 1000w^24
\(\displaystyle 1050 = 40\cdot w\cdot\frac{1-w^{24}}{1-w} + 1000\cdot w^{24}\)
If you do not know how to do that, you should practice and become proficient. It WILL save you on an exam.
Now, with a little judgment based on experience, I'm going to solve this mess for just one of the "w"s. It may not matter much, but I'm kind of fond of the one in the denominator. After a little algebra, we get:
\(\displaystyle w = 1 - 4\cdot w\cdot\frac{1-w^{24}}{105-100\cdot w^{24}}\)
Why did I do that? This is the beauty of it. All we need is a good guess to get us started. We seem to have 8% coupons, so let's just go with that.
\(\displaystyle w_{0} = \sqrt{\frac{1}{1.08}}\;=\;0.9622504486\)
I added the subscript because we're about to substitute this value into that wonderful equation I created and get a better guess.
\(\displaystyle w_{1} = 1 - 4\cdot w_{0}\cdot\frac{1-w_{0}^{24}}{105-100\cdot w_{0}^{24}}\;=\;0.9644576639\)
Wasn't that awesome? Do it again.
\(\displaystyle w_{2} = 1 - 4\cdot w_{1}\cdot\frac{1-w_{1}^{24}}{105-100\cdot w_{1}^{24}}\;=\;0.9644813361\)
That was sweet. Notice how the first four digits didn't flinch. Do it again.
\(\displaystyle w_{3} = 1 - 4\cdot w_{2}\cdot\frac{1-w_{2}^{24}}{105-100\cdot w_{2}^{24}}\;=\;0.9644816647\)
This time, the first six digits failed to change. One more time.
\(\displaystyle w_{4} = 1 - 4\cdot w_{3}\cdot\frac{1-w_{3}^{24}}{105-100\cdot w_{3}^{24}}\;=\;0.9644816692\)
Is 8 decimal places enough? If not, just do it again. One word of warning, if your initial guess isn't good enough, this may not work very well. If your crazy magic equation isn't good enough, it may not work. You do have to keep your eyes on it.
Anyway, \(\displaystyle i\;=\;\frac{1}{w_{4}^{2}}-1\;=\;0.07500886\)
There are many other methods to find this interest rate. The "fixed point" methods are kind of fun, wouldn't you say!
