I am working on a specific problem regarding price of bonds and it is the following.
A 10% bond with face amount $F=100$ is callable on any coupon date from $t=15.5$ years after issue up to the maturity date which is when $t=20$ yrs from issue. Find the price $P$ of bond to yield a minimum nominal annual rate of $i^{(2)}=12\%,10\%, \text{or} \ 8\%$
I understand that the present value of these types of bonds can be calculated as
$$P=Cv^n_i + Fra_{\overline{n}\rceil i}$$
I was able to solve in the case where $i^{(2)}=12\% \ \text{or} \ 10\% $, but no matter what I tried to do I cannot seem to get the correct value for $8\%$.
According to the book the price must be $120.55$.
I also understand that when a bond is purchased as a discount, the minimum yield occurs at the latest date and when it is a premium it occurs at the earliest.
This is especially true when the face value $F$ equals the redemption value $C$, but in this case $F \ne C$ so although generally I would assume the same rules apply, I checked the earliest and the latest dates... which still did not give me the same value.
I get
$$P=110(1.05)^{(-40)}+5a_{\overline{40}\rceil .04} \approx 114.59$$
So, it would help me a lot if anyone could explain to me the following.
1), How is it that the yield rate is higher than the coupon rate and yet, the minimum price occurred at the latest date rather than the earliest?
2), I do not think a 6 difference in negligible... is there something I did wrong in my calculation? Is it possible that I should be picking a date between $n=30 \ \text{and} \ 40$
By the premium/discount formula, which is
$ P = C + (Fr - iC)a_{n|i}$
you can see that, if $ Fr -iC >0 $, the bond is selling at premium so the earliest redemption date is the most favorable for the issuer. (Because he would like to stop repaying the premium via the coupon payments as soon as possible.)
For $ i^{(2)} = 0.08 $, the bond is effectively selling at premium. ( assuming C = 110, as in your computation )
The bond is callable from the 31th coupon, and the price of the bond for $ 31 <= n <= 40 $ is
$P = 110 + ( 5 - 4.4)a_{n|0.04} $
As you can see, the minimum price clearly occurs at $ n = 31 $.
For $ n = 31 : P = 120.55$
For $ n = 40 : P = 121.88$