Hi doogieh59!
Here are the answers to your questions.
Question 1
In all cases, remember that the price of a bond is the discounted (present)
value of its stream of payments. For the sake of simplicity, assume that all
bonds have a face value of 100.
The Treasury Bill makes only one payment of 100, one year from now.
Its current price now is 93.46. Using the fact that the price of a
bond is the present value of its stream of payments, we have the
following equation:
93.46 = 100/(1+r1)
where r1 is the spot one-year interest rate. We now isolate r1 from this equation:
1+r1 = 100/93.46
r1 = 0.07
The actal value of r1 is 0.069976..., but I rounded it in order to
keep things simple. We've found, then, that the spot one-year interest
rate is 7%.
Let's call r1 the "Year 0 one-year rate" so as to avoid mixing it with
the forward rates. I'll call "Year 1 one-year rate" to the forward
rate one year from now; and "Year 2 one-year rate" to the forward rate
two years from now.
We must now find the Year 1 forward rate; here we'll use Bond #1. Once
again, we use the fact that the price of a bond is the present value
of its stream of payments. This bond pays 4 in one year (the coupon)
and 104 in two years (coupon + face value). The first payment should
be discounted at the rate r1 (Year 0 rate), while the second payment,
of 104, should be discounted using both the Year 0 rate and the Year 1
rate in the following way:
97.92 = 4/(1+r1) + 104/[(1+r1)(1+r2)]
Since r1=0.07, we replace that value in the equation:
97.92 = 4/(1.07) + 104/[1.07(1+r2)]
94.18 = 104/[1.07(1+r2)]
100.77 = 104/(1+r2)
r2 = 0.032
Therefore, this bond has embedded a Year 1 forward rate of 3.2% (I
rounded it once again).
Now we have to find the Year 2 forward rate, for which we'll use Bond
#2 and the familiar formula for pricing bonds. Note that this bond
pays 8 in one year (1st coupon), 8 in 2 years (2nd coupon) and 108 in
3 years (3rd coupon + face value). The pricing formula for this bond
then becomes:
103.64 = 8/(1+r1) + 8/[(1+r1)(1+r2)] + 108/[(1+r1)(1+r2)(1+r3)]
We replace the values of r1 and r2 we've already found:
103.64 = 8/1.07 + 8/(1.07*1.032) + 108/[1.07*1.032*(1+r3)]
88.91 = 108/[1.104(1+r3)]
98.15 = 108/(1+r3)
0.9088 = 1/(1+r3)
r3 = 0.1
Therefore, the embedded Year 2 forward rate is 10% (again, rounded).
Summing up, we've found the following embedded rates:
Year 0 (spot) rate 7%
Year 1 (forward) rate 3.2%
Year 2 (forward) rate 10%
Notice that all these are one-year interest rates. That is if we agree
today (beginning of year 0) to invest $100 at the beginning of year 2,
we should be paid $110 at the beginning of year 3.
Question 2
I think this question should use the answers found in the previous
one, otherwise it can't be answered. The answer I give here is based
on that.
Arbitrage opportunities arise when assets are incorrectly priced.
Thus, arbitrage can usually be exercised by selling (or shorting) an
asset that is overpriced; or buying an asset that is underpriced.
We now use the familiar formula for bond pricing in order to find if
the bond mentioned in this question is correctly priced.
P = 4/1.07 + 4/(1.07*1.032) + 104/(1.07*1.032*1.1)
P = 3.73 + 3.62 + 85.62
P = 92.97
At the prevailing spot and forward market interest rates, this bond is
overpriced at 95. It should be valued at 92.97. Therefore, there
exists an arbitrage opportunity, which must be exploited by selling or
shorting this bond (we're selling something that is more expensive
than it should be).
Since we'll be shorting this bond instead of buying it, bear in mind
that we'll receive $95 today (the value of the bond); and we'll have
to pay $4 one year from now (1st coupon), $4 two years from now (2nd
coupon) and $104 three years from now (3rd coupon + face value).
Notice, then, that shorting a bond is somewhat equivalent to borrowing
money.
The other "side" of the arbitrage comes from lending (investing) money
at the spot and forward market rates. These are the precise steps you
should follow:
1. Invest $3.73 in Treasury bills, at the spot rate of 7%. This will
give 3.73*1.07=$4 in one year.
2. Invest $3.62 at the spot rate. Through a forward rate agreement
(FRA), agree to invest the proceeds [3.62*1.07=$3.87] next year at
Year 1 forward rate (3.2%). Thus, you'll be paid 3.87*1.032=$4 in two
years.
3. Invest $85.62 at the spot rate. Through an FRA, agree to invest the proceeds
[85.62*1.07=$91.61] next year at the Year 1 forward rate (3.2%).
Through another FRA, agree to invest the proceeds of the previous FRA
[91.61*1.032=$94.54] in two years, at the Year 2 forward rate of 10%.
This will result in a payment to you of 94.54*1.10=$104 in three
years.
As a result of following these steps, you shall receive $4 in one
year, $4 intwo years, and $104 in 3 years. Sounds familiar? It's
exactly what you need to make the coupon and face value payments of
the bond you initially shorted.
[Actually, if you make the calculations, you'll find that the numbers
differ a little bit -such as getting $3.99 instead of $4. This is
entirely due to rounding; you can get more precise numbers by doing
the exact same calculations but keeping more decimals]
So where's the arbitrage? The cost today of the 3 steps described
above is 3.73+3.62+85.62=$92.97. However, by shorting the bond, you
also received $95 today. Therefore, a totally risk free 95-92.97=$2.03
profit per bond shorted was realized. Don't expect to find an
opportunity like this in real trading, though :-)
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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