A company plans to issue $5 million of perpetual bonds. The face value
of each bond is $1000. The annual coupon on the bonds is 12 percent.
Market interest rates on one year bonds are 11 percent. With equal
probability, the long term market interest rate will be either 14
percent or 7 percent next year. Assume investors are risk-neutral. If
the company bonds are noncallable, what is the price of the bonds? If
the bonds are callable one year from today at $1,450, will their price
be greater than or less than the price computed in the first question?
Why?

Hi cop189!!

If the company bonds are noncallable, what is the price of the bonds?

If the bonds are noncallable, the present value of the bond?s payments
is the price of the bond. Then the price of the bond depends on the
interest rate which prevails in the market.
The bond pays an annual coupon equal to its coupon rate times its face
value, in this case:
$1000*0.12 = $120 


If the interest rate in one year will be 14%, the value of the bond in
one year will be:
$120 + $120/0.14 = $977.14

Discounting this amount by the prevailing market interest rate gives
the current price of the bond for the 14% rate case:

P(14%) = $977.14/1.11 = $880.31



If the interest rate in one year will be 7%, the value of the bond in
one year will be:
$120 + $120/0.07 = $1,834.29

Discounting this amount by the prevailing market interest rate gives
the current price of the bond for the 7% rate case:

P(7%) = $1,834.29/1.11 = $1,652.51


Note that we don't know the interest rate of the next year, but we
know each interest rate's probability. Investors are risk-neutral,
then the value of the bond is the weighted sum (for probabilities) of
the value of the bond in each scenario:
P = 0.5*$880.31 + 0.5*$1,652.51 = $1,266.41


If the bonds are noncallable, the current price of the bonds is $1,266.41

                -----------------------

If the bonds are callable one year from today at $1,450, will their price
be greater than or less than the price computed in the first question? Why?

Callable bonds are bonds which the firm can redeem at a stated price
prior to its maturity date. If the firm calls the bonds, they pay the
stated price to the bond holders and retire it before its original
maturity date. The firm will only call the bonds if it is in its
interests to do so.


Again we have two possible scenarios.

-. Scenario 1 or 14% interest rate scenario:

The price of the bond in one year will depend on the interest rate
which prevails in the market.
The bond pays an annual coupon equal to its coupon rate times its face
value, in this case:
$1000*0.12 = $120 
If the interest rate in one year will be 14%, the value of the bond in
one year will be:
$120 + $120/0.14 = $977.14

The above result can be expressed in the following statement:
From the next year the company will pay to the bond holders infinite
payments that will have a present value of $977.14

The above is clearly preferable to pay next year $1,450 of bond redemption.
Then the company will not redeem the bond in this case (because
actually it is financing its debt at a rate less than 14%), then the
value of the bond will remain at $977.14 in one year.
Discounting this amount by the prevailing market interest rate gives
the current price of the callable bond for the 14% rate case:

P(14%) = $977.14/1.11 = $880.31



-. Scenario 2 or 7% interest rate scenario:

The price of the bond in one year will depend on the interest rate
which prevails in the market.
The bond pays an annual coupon equal to its coupon rate times its face
value, in this case:
$1000*0.12 = $120 
If the interest rate in one year will be 7%, the value of the bond in
one year will be:
$120 + $120/0.07 = $1,834.29

The above result can be expressed in the following statement:
From the next year the company will pay to the bond holders infinite
payments that will have a present value of $1,834.29

The above is clearly NOT preferable to pay next year $1,450 of bond redemption.
Then the company will redeem the bond in this case (because actually
it is financing its debt at a rate greater than 7%), then the value of
the bond will be $1,450.00 in one year.
Discounting this amount by the prevailing market interest rate gives
the current price of the callable bond for the 7% rate case:

P(7%) = $1,450.00/1.11 = $1,306.31


Again we don't know the interest rate of the next year, but we know
each interest rate's probability. Investors are risk-neutral, then the
value of the bond is the weighted sum (for probabilities) of the value
of the bond in each scenario:

P = 0.5*$880.31 + 0.5*$1,306.31 = $1,093.31

If the bonds are callable, the current price of the bonds is $1,093.31
, less than the price computed in the first question ($1266.41).

Why?
A callable bond is sold for less than an otherwise identical ordinary
bond. This is because the buyer of the bond is giving up something:
the right
to hold this bond until maturity under certain conditions. In this
case if the bond price rises above $1,450, the company will call it,
and with this action wealth will transferred from the bondholders to
the shareholders. Thus, the buyer is only willing to pay less for the
callable bonds.

----------------------------------------------------------

I hope that this helps you. Feel free to request for a clarification
if you find something unclear. I will be glad to clarify the answer
and/or give you further assistance on this topic if you need it.

Best regards.
livioflores-ga

Well, there's no polite way to do this, but the answer above looks incorrect to me.

First Question: If the company bonds are noncallable, what is the
price of the bonds?

Livioflores is correct that in this case, the present value (PV) of
all payments = price. However, he mis-calculated the PV. This bond is
a perpetuity; it pays interest forever. Finding the PV of a perpetuity
is easy: [cash flow] / [current interest rate]. This C/r formula is
the mathematical simplification of the standard, drawn-out PV
cacluation.

In this case, C/r = $120 / 0.11 = $1090.91. Expected future interest
rates do not matter, just like they do not matter any other time
you're cacluating the PV of a stream of cash flows.

I located some 'offical' notes to back this up. Go to
http://tinyurl.com/52zms for academic notes. It's PDF from
Northwestern U. Look on page 25.


Second Question: If the bonds are callable one year from today at
$1,450, will their price be greater than or less than the price
computed in the first question?

Generally, you can answer any question like this (a 'normal' security
bundled with some sort of option) by figuring out what the value of
the option would be by itself, then adding/subtracting that value as
necessary from the original price.

In this case, the company is selling you one thing:
A. plain-vanillia bond as before (PV = $1090.91)

And you are selling the company one thing:
B. call option with a strike price of $1450 and maturity of one year.
Think about this one... they have the right, but not the obligation,
to buy the bond from you in one year. This is the definition of a call
option. A call option can never have a negative value, so the company
would ALWAYS have to pay you some money in exchange for this option.

(A) is a $1090.91 payment from you to them
(B) is an unknown payment from them to you

A + B < $1090.91, so the price of the callable bond is less than the
price of the non-callable bond.

If you're really interested, you can actually calculate the value of
(B), assuming you have all of the information, but your question
didn't indicate that this was necessary.