Hi cop189!
Here are the answers to your questions.
a. In all cases, recall that the price of a bond is the discounted
value of its flow of payments. Let's assume for simplicity that all
bonds have a face value of 100.
The Treasury bill will then pay 100 in a year. Its price now is 93.46.
Since the value of this bond is the discounted value of its flow of
payments (there is only one payment, one year from now), we have that:
93.46 = 100/(1+r1)
where r1 is the spot one-year interest rate. From this equation, we get:
1+r1 = 100/93.46
r1 = 0.07 (rounded)
So the spot one-year interest rate is 7%.
In order to calculate the implied interest rate for the next year, we
use the second bond. Since the face value is 100 this bond pays 4 one
year from now and 104 (coupon + face value) two years from now. Again,
we use the bond value formula and isolate the interest rate from
there:
94.92 = 4/(1+r1) + 104/[(1+r1)(1+r2)]
We alredy know that r1=0.07, so we replace that value.
94.92 = 4/(1.07) + 104/[1.07(1+r2)]
91.18 = 104/[1.07(1+r2)]
97.56 = 104/(1+r2)
r2 = 0.066
Therefore, the implied one-year interest rate one year from now is
6.6%. This is the one-year interest rate forward for next year.
Finally, we must find the one-year interest rate forward for two years
from now. We follow the same procedure as before, using the r1 and r2
we found:
103.64 = 8/(1+r1) + 8/[(1+r1)(1+r2)] + 108/[(1+r1)(1+r2)(1+r3)]
103.64 = 8/1.07 + 8/(1.07*1.066) + 108/[1.07*1.066*(1+r3)]
89.14 = 108/[1.1406(1+r3)]
101.67 = 108/(1+r3)
r3 = 0.0622
Therefore, the implied one-year interest rate forward two years from now is 6.22%.
b. I assume here that this question is to be answered using the
previous question's data.
In order to see if there is an arbitrage opportunity here, we should
first decide whether this bond is correctly priced. Given the spot and
ofrward interest rate we found in the previous question, the value of
this bond should be:
Value = 4/1.07 + 4/(1.07*1.066) + 104/(1.07*1.066*1.0622)
Value = 3.738 + 3.506 + 85.855
Value = 93.08
Therfore, at 95.00, this bond is overpriced. So in order to take
advantage of this, you'll need to do things that involve shorting this
bond.
Notice that in the previous equation I specified the present value of
each payment of this bond (3.738, 3.506 and 85.855), which total
93.08, the "fair" value of this bond. We'll use these values to define
how we take advantage of the arbitrage opportunity.
Recall that we're shorting this bond. So we'll have to PAY $4 next
year, $4 two years from now, and $104 three years from now.
Now, after shorting this bond, we should do the following things:
1. Invest $3.738 in Treasury bills, at the spot interest rate of 7%.
This will give $4 in one year.
2. Invest $3.506 at the spot rate, and agree to invest the proceeds
(3.506*1.07=$3.751) next year at the next-year forward interest rate
(forward rate agreement). This will give you 3.751*1.066=$4 in two
years.
3. Invest $85.855 at the spot rate. Agree to invest the proceeds
(85.855*1.07=$91.864) next year at the next-year forward interest
rate. Agree to invest the proceeds of the last operation
(91.864*1.066=$97.927) in two years at the two-years-from-now forward
interest rate of 6.22%. This will give you $104 in three years.
As you can see, the result of those operations is obtaining $4 next
year, $4 in two years, and $104 in three years, which is exactly what
we need to pay the coupons and capital of the bond we shorted. We
spent in the last operations exactly $3.738 + $3.506 + $85.855 =
$93.08, while we received $95 for the bond. Therefore, we have
realized a risk-free profit of 95-93.08=$1.92.
I hope this helps! If you have any questions regarding my answer,
please don't hesitate to request a clarification. Otherwise I await
your rating and final comments.
Best wishes!
elmarto |
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