Step 2: Solve for Bond 2's Expected Price
Now this is where it gets interesting.
We used the coupon rates to estimate what percentage of our money
should be invested in Bond 1 and Bond 3 in order to equal the rate of
Bond 2. We were able to do this because we knew all three coupon
rates. However, the basis of this question implies that the current
price of Bond 2 is not what it should be, due to the "callable
feature." And in fact, if you look at the price of Bond 2, you
could probable deduce that something was affecting the price since it
is lower than both Bond 1 and Bond 3 despite having a coupon rate in
the middle of the two other bonds.
So since we now know what percentage of
our investments in Bond 1 and Bond 3 will equal Bond 2's coupon rate,
we can extrapolate this information and apply it to the bond price in
order to estimate what the price of Bond 2 should be without the callable feature. Essentially, we
are wanting to "pay" for Bond 2's price without buying Bond 2, which we
can do by purchasing a portion of the cheaper Bond 1 and a portion of
the more expensive Bond 3 (we already solved for these portions using
the coupon rates).
Again, our basic assumption is that the
prices of Bond 1 and Bond 3 are correct, but that the price of Bond 2
is not what it should be because it is affected by the callable
feature. The formula we use looks basically the same as the previous
Price 2 = (Price 1 * 0.4516) + (Price 3
Note: As mentioned previously, it
matters that we are using decimals for our "weighing"
percentages because we are using them to "weigh" the
prices. You will get an error if you try to use 45.16 instead of
0.4516. Technically, our prices are in percentage form too, but
because we are not using the price to weigh anything, it does not
affect our answer to use the percentage version instead of decimals. So we can use
112.78 instead of the appropriate decimal of 1.1278 (112.78%).
Now, simply plug in the prices we know
are unaffected by a call feature, which is the prices for Bonds 1 and
3, and then solve for what the price of Bond 2 should be. Always use the Ask prices when using the price of the bond. The Ask
price is always the price an investor will pay to purchase the bond
(it is the dealer's asking price).
Price 2 = (112.78
* 0.4516) + (114.61 *
Price 2 = (50.93) + (62.85)
Price 2 = 113.78
The price of Bond 2 should be 113.78
(or 113.78% of the face value) – this makes sense since the price
of Bond 1 is 112.78 and the price of Bond 3 is 114.61. However, the
actual price of Bond 2 is 109.65.
The difference between what the price of Bond 2 should be (113.78)
and the actual price of Bond 2 (109.65) is the implied value of the
– 109.65 = 4.13
that the prices are actually percentages of the face value, so 4.13
is actually 4.13% of the bond's face. Typically we assume bond values to
be $1000, so the dollar value of the call feature is $41.30.
You now know how to solve this difficult problem!
Deriving the Formula for Future Value
with an Annuity
Business Math Articles