Hi bluesteel!
Here are the answers to your questions.
Question 1
We can calculate the implied 1.5 year and 2 year interest rate by
using the fact that the value of a bond should be the discounted value
of the coupons plus capital payments. We will use the six-month and
one-year sopt rates to discount the first two coupon payments, while
the rate at which the third coupon is discounted will give us the
implied 1.5 year rate. Let's see how we calculate this. I assume here
that the face value of the bond is 100. I'm also assuming here that
the rate you provide for the six months is not annualized; that is,
100 today become 101.75 in six months.
101.3518 = 3/(1.0175) + 3/(1.0195) + 103/(1+r)
The left-hand side of this equation is the value of the bond, while
the right-hand side is the present value of its payments. It's clear
now that the value of 'r' will give us the implied 1.5 year interest
rate (because in 1.5 years from now, we will receive the 103, 3 from
the coupon and 100 from the face value of the bond). So we have:
101.3518 = 3/(1.0175) + 3/(1.0195) + 103/(1+r)
95.4607 = 103/(1+r)
1+r = 103/95.4607
1+r = 1.0789
r = 0.0789
So the implied 1.5 year interest rate is 7.89%.
Now, in order to calculate the implied 2-year interest rate, we make
use of the second bond, but the process is esentially the same:
104.5764 = 4.5/(1.0175) + 4.5/(1.0195) + 4.5/(1.0789) + 104.5/(1+r)
Using basic algebra just as before, we get that r=0.1412, so the
2-year implied interest rate is 14.12%.
Note again, that both results I gave are not annualized; you can
easily annualize them.
- What is the six month forward rate 1.5 years from now?
We know that the 1.5 year rate is 7.89% and the 2-year rate is 14.12%.
The forward rate is derived in the following way. If you have 100
today, you wil have 107.89 in 1.5 year, and 114.12 in 2 years.
Equivalently, this means that investing 107.89 in 1.5 years will yield
114.12 six months after that. Therefore, the forward six-month rate in
1.5 years is basically 114.12/107.89=1.0577, so the forward rate is
5.77%, again, not annualized.
Question 2
I will assume here that the 6% rate is the rate at which EACH coupon
pays and not the annual rate (in the latter case, each coupon would
pay 3%, because they pay twice a year). The calculation of the
duration of a bond is done with a formula you can find in the
following page. You will also find a good explanation of the
interpretation of duration here.
Advanced Bond Tutorial: Duration
http://www.investopedia.com/university/advancedbond/advancedbond5.asp
[go to the part called "Macaulay Duration"]
As you can see from the formula explained there, we first need to know
the implied interest rate (which in the previous link is called
"required yield"). We calculate this again using the fact that the
value of a bond is the present value of its flow of payments:
104.375 = 6/(1+r) + 6/(1+r)^2 + ... + 106/(1+r)^6
You can use any financial calculator in order to get this value. The
value of r that solves this equation is then approximately r=0.0514.
Therefore, the six-month implied interest rate is 5.14%. Now we just
plug the values of this problem in the formula provided in the
previous link to comput ethe bond's duration:
Duration =
1*6/(1+r) + 2*6/(1+r)^2 + 3*6/(1+r)^3 + ... + 6*6/(1+r)^6 + 6*100/(1+r)^6
-------------------------------------------------------------------------
104.375
where r=0.0514, just as we found before. Bear in mind that since we're
using six-month coupon payments and the six-month interest rate, we'll
get the answer in six-month units. This means that if we get, for
example, that the duration is 2, then the duration will be 1 year (2
six-month units). The above equation gives:
Duration = 5.22
Therefore, the duration of this bond is 5.22/2 = 2.61 years.
Question 3
The bond should not be converted at the moment. The reason for this is
quite simple. We currently own a bond valued at $985. If we convert
it, we'll get 20 shares valued at $48.5, so we will get equity valued
at 20*48.5=$970. Therefore, since the value of the equity we would get
in exchange for the bond is less than the value of the bond, there is
no economic reason to convert it at the moment. You can find further
information regarding bonds convertible to equity at thye following
link:
Convertible Bonds
http://www.finpipe.com/bndcnvrt.htm
Google search terms
convertible bond "conversion rartio"
://www.google.com/search?hl=en&q=convertible+bond+%22conversion+ratio%22
finance bond duration formula
://www.google.com/search?hl=en&lr=&q=finance+bond+duration+formula
I hope this helps! If you have any doubts regarding my answer, please
don't hesitate to request clarification before rating it; otherwise I
await your rating and final comments.
Cheers!
elmarto |
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