Finance Management

Bonds and Their Valuation-2

 Bond Valuation
• The value of any financial asset, a stock, a bond,
a lease, or even a physical asset such as an
apartment building or a piece of machinery, is
the present value of the cash flows the asset is
expected to produce.
• The cash flow of a standard coupon-bearing
bond consist of interest payments during the
bond’s life plus the amount borrowed (generally
the par value) when the bond matures.
 Bond Valuation
• For a ‘’regular’’ bond with a fixed coupon, like
Allied’s, here is the situation:
 Bond Valuation
rd = The market rate of interest on the bond, 10%. This is the discount rate used to
calculate the present value of the cash flows, which is also the bond’s price.
𝑟d is not the coupon interest rate. However, 𝑟d is equal to coupon rate at
times, especially the day the bond is issued. When the rates are equal, as in
this case, the bond sells at par.
N= The number of years before the bond matures = 15. N declines over time after
the bond has been issued; so a bond that had a maturity of 15 years when it
was issued (original maturity = 15) will have N = 14 after one year, N = 13 after
two years, and so forth. At this point, we assume that the bond pays interest
once a year, or annually; so N is measured in years.
INT = Dollars of interest paid each year = Coupon rate x Par value = 0.10x($1,000) =
$100. In calculator terminology, INT = PMT = 100. If the bond had been a
semiannual payment bond, the payment would have been $50 every six
months. The payment would have been zero if Allied has issued ‘zero coupon’
bonds, and it would have varied over time if the bond had been a ‘floater’.
M= The par, or maturity value of bond = $1,000. This amount must be paid at the
maturity.
 Bond Valuation
• We can now redraw the time line to show the
numerical values for all variables except the
bond’s value (and price, assuming an
equilibrium exists), VB :
 Bond Valuation
• The following general equation can be solved to find
the value of any bond:

𝐁𝐨𝐧𝐝′ 𝐬 𝐕𝐚𝐥𝐮𝐞 =

𝐈𝐍𝐓 𝐈𝐍𝐓 𝐈𝐍𝐓 𝐌

𝐕𝐁 = 𝟏
+ 𝟐
+ …+ 𝐍
+ 𝐍
(𝟏 + 𝐫𝐝 ) (𝟏 + 𝐫𝐝 ) 𝟏 + 𝐫𝐝 𝟏 + 𝐫𝐝

𝐍
𝐈𝐍𝐓 𝐌
=෍ 𝐭
+ 𝐍
𝟏 + 𝐫𝐝 𝟏 + 𝐫𝐝
𝐭=𝟏
 Bond Valuation
• Inserting values for the Allied bond, we have

15
$100 M
VB = ෍ t
+ 15
1.10 1.10
t=1
• The cash flows consist of an annuity of N years
plus a lump sum payment at the end of Year
N, as reflected in equation above.
 Bond Valuation
• We could simply discount each cash flow back
to the present and sum those PVs to find the
bond’s value.
• See figure below for an example.
• However this procedure is not very efficient,
especially when the bond has many years to
maturity.
 Bond Valuation
• Therefore, we use a financial calculator
to solve the problem.
• Here is the set up:
 Bond Valuation
• In this Allied example, the bond is selling at a
price equal to its par value.
• Whenever the bond’s market, or going, rate,
rd ,is equal to its coupon rate is set at the
fixed-rate bond will sell at its par value.
• Normally, the coupon rate is set at the going
rate in the market the day a bond is issued,
causing it to sell at par initially.
 Bond Valuation
• The coupon rate remains fixed after the bond is
issued, but interest rates in the market move up
and down.
• Looking at the bond valuation equation:
𝐈𝐍𝐓 𝐈𝐍𝐓 𝐈𝐍𝐓 𝐌
𝐕𝐁 = 𝟏 + 𝟐 + …+ 𝐍+ 𝟏+𝐫 𝐍
(𝟏 + 𝐫𝐝 ) (𝟏 + 𝐫𝐝 ) 𝟏 + 𝐫𝐝 𝐝
• We see that an increase in the market interest
rate (rd ) causes the price of an outstanding bond
to fall,
• Whereas a decrease in the rate causes the bond’s
price to rise.
 Bond Valuation
• For example, if the market interest rate on
Allied’s bonds increased to 15% immediately
after it was issued, we would recalculate the
price with the new market interest rate as
follows:
 Bond Valuation/discount bond
• The bond’s price would fall to $707.63, well
below par, as a result of the increase in
interest rates.
• Whenever the going rate of interest rises
above the coupon rate, a fixed-rate bond’s
price will fall below its par value.
• This type of bond is called a discount bond.
Bond Valuation
 Bond Valuation
• On the other hand, bond prices rise when
market interest rates fall.
• For example, if the market interest rate on
Allied’s bond decreased to 5% immediately
after it was issued, we could once again
recalculate its price as follows:
 Bond Valuation/premium bond
• In this case, the price rises to $1,518.98.
• In general, whenever the going interest rate
falls below the coupon rate, a fixed-rate
bond’s price will rice above its par value.
• This type of bond is called a premium bond.
 Bond Valuation/summarize
• Here is the situation:
- 𝐫𝐝 = coupon rate, fixed-rate bond sells at par;
hence, it is par bond.
- 𝐫𝐝 > coupon rate, fixed-rate bond sells below
par; hence, it is a discount bond.
- 𝐫𝐝 < coupon rate, fixed-rate bond sells above
par; hence, it is a premium bond.
 Question
• A friend of yours invested in an outstanding
bond with a 5% annual coupon and a
remaining maturity of ten years.
• The bond has a par value of $1,000 and the
market interest rate is currently 7%.
a) How much did your friend pay for the bond?
b) Is it par, premium, or discount bond?
 Answer-a with formula
𝐁𝐨𝐧𝐝′ 𝐬 𝐕𝐚𝐥𝐮𝐞 = 𝐕𝐁

50 50 50 1,000
= 1
+ 2
+ ⋯+ 10
+ 10
(1 + 0.07) (1 + 0.07) 1 + 0.07 1 + 0.07

10
50 1,000
=෍ 10
+ 10
= $𝟖𝟓𝟗. 𝟓𝟑
1 + 0.07 1 + 0.07
t=1
Answer-a with calculator & Excel
Answer-a with Excel
 Answer-b
• Because the bond’s coupon rate (5%) is less
than the current market interest rate (7%),
• The bond is a discount bond.
• It reflects that interest rates have increased
since this bond was originally issued.
 Question

1) A bond that matures in 8 years has a par value

of $1,000 and an annual coupon payment of $70; its
market interest rate is 9%.
What is its price?
2) A bond that matures in 12 years has a par value
of $1,000 and an annual coupon rate of 10%; the market
interest rate is 8%.
What is its price?
3) which of those bonds is a discount bond, and
which is a premium bond?
Explain.
Answer-1
Answer-2
 Answer-3
• The bond which has the price of $889.30 is a
discount bond, because the bond’s coupon
rate (7%) is less than the current market
interest rate (9%).
• The other bond which has the price of
$1,150.72 is a premium bond, because the
bond’s coupon rate (10%) is higher than the
current market interest rate (8%)
 Bond Yields
• Unlike the coupon interest rate, which is fixed,
the bond’s yield varies from day to day depending
on market conditions.
• The bond’s yield gives us an estimate of the rate
of return we would earn if we purchased the
bond today and held it over its remaining life.
• If the bond is not callable, its remaining life is the
years to maturity.
• If it is callable, its remaining life is the years to
maturity if it is not called or the years to the call if
it is called.
 Yield to maturity
• Suppose you were offered a 14-year, 10%
annual coupon, 1,000 par value bond at a
price of $1,494.93.
• What rate of interest would you earn on your
investment if you bought the bond, held it to
maturity, and received the promised interest
and maturity payments?
 Yield to maturity
• This rate is called the bond’s yield to maturity
(YTM).

𝐈𝐍𝐓 𝐈𝐍𝐓 𝐈𝐍𝐓 𝐌

𝐕𝐁 = 𝟏
+ 𝟐
+ …+ 𝐍
+ 𝐍
(𝟏 + 𝐫𝐝 ) (𝟏 + 𝐫𝐝 ) 𝟏 + 𝐫𝐝 𝟏 + 𝐫𝐝

$𝟏𝟎𝟎 $𝟏𝟎𝟎 $𝟏, 𝟎𝟎𝟎

$𝟏, 𝟒𝟗𝟒. 𝟗𝟑 = 𝟏
+⋯+ 𝟏𝟒
+
(𝟏 + 𝐫𝐝 ) 𝟏 + 𝐫𝐝 𝟏 + 𝐫𝐝 𝟏𝟒
 Yield to maturity
• We can substitute values for rd until you find a
value that ‘’works’’ and force the sum of the
PVs in the equation to equal $1,494.93.
• However finding rd = YTM by trial and error
would be a tedious, time consuming project.
Here is the setup:
Yield to maturity
 Question
• You have just purchased an outstanding 15-
year bond with a par value of $1,000 for
$1,145.68.
• İts annual coupon payment is $75.
• What is the bond’s yield to maturity.
Answer to bond’s yield to maturity
 Yield to maturity
• Yield to maturity can also be viewed as the
bond’s promised rate of return,
• Which is the return the investors will receive if
all of the promised payments are made.
• However the yield to maturity equals to
expected rate of return only when
(1) the probability of default is zero and
(2) the bond can not be called.
 Yield to maturity
• If there is some default risk or the bond may
be called, there is some chance that the
promised payments to maturity will not be
received,
• in which case the calculated yield to maturity
will exceed the expected return.
 Yield to maturity
• Note also that a bond’s calculated yield to
maturity changes whenever the interest rates
in the economy change.
• An investor who purchases a bond and hold it
until it matures will receive the YTM that
existed on the purchase date.
• But the bond’s calculated YTM will change
frequently between the purchase date, and
the maturity date.
 Yield to call
• If you purchase a bond that is callable, an the
company calls it, you do not have the option
to holding it to maturity.
• Therefore, the yield to maturity would not be
earned.
 Yield to call
• For example, if Allied’s 10% coupon bonds
were callable and interest rate fell from 10%
to 5%, the company could call in the 10%
bonds, replace them with 5% bonds, and save
$100-$50=$50 interest per bond per year.
• This would be beneficial to the company but
not to its bondholders.
 Yield to call
• If current rates are well below an outstanding bond’s
coupon rate, a callable bond is likely to be called.
• So, the investors will estimate its most likely rate of
return as the yield to call (YTC) rather than the yield
to maturity.
• To calculate the YTC, we modify the ‘Bond’s Value’
equation, using years to call as N and the call price
rather than the maturity value as the ending
payment.
 Yield to call
• Here is the modified equation:

N
INT Call price
Price of bond = ෍ t
+
(1 + rd ) (1 + rd )N
t=1

• N is the number of the years until the company can call

the bond.
• Call price is the price the company must pay in order to
call the bond (it is often set equal to the par value plus
one year’s interest).
• And 𝐫𝐝 is the YTC.
 Yield to call
• To illustrate, suppose Allied’s had a deferred call
provision that permitted to the company.
• If it desired, to call them 10 years after their issue
date at a price of $1,100.
• Suppose further that interest rates had fallen and
that 1 year after issuance, the going interest rate
had declined, causing their price to rise to
$1,494.93.
• Here is the time line and the setup for finding the
bonds’ YTC with a financial calculator below:
 Yield to call
time line & calculator
 Yield to call
• The YTC is 4.21%.
• This is the return you would earn if you
bought an Allied bond at a price of $1,494.93
and it was called nine years from now.
• It could not be called until ten years after
issuance.
• One year has gone by, so there are 9 years left
until the first call date.
 Example
• You have just purchased an outstanding 15-year bond
with a par value of $1,000 for $1,145.68.
• Its annual coupon payment is $75.
• We calculated the YTM of this bond as 6% before.
• Assume that this bond is callable in seven years at a
price of$1,075.
• What is the bond’s YTC?
• If the yield curve remains flat at its current level during
this time period, would you expect to earn the YTM or
YTC?
 Answer
• Using a financial calculator, we can determine
that the bond’s YTC is 5.81%.
 Answer/Excel
• Here we find the bond’s YTC is equal to 5.81%.
 Yield to call
• A company is more likely to call its bonds if they are
able to replace their current high-coupon debt with
cheaper financing.
• Broadly speaking, a bond is more likely to be called if
its price is above par.
• Because a price above par means that the going
market interest rate (the yield to maturity) is less than
the coupon rate.
• So, do you think Allied will call its 10% bonds when
they become callable?
• Allied’s action will depend on what the going interest
rate is when they become callable?
 Yield to call
• If the going interest rate remains 𝐫𝐝 = 5%, Allied could
save 10% − 5% = 5%, or $50 per bond per year.
• So, it would call the 10% bonds and replace them with
a new 5% issue.
• There would be some cost to the company to refund
the bonds.
• But because the interest savings would most likely be
worth the cost; Allied would probably refund them.
• Therefore, you should expect to earn the YTC=4.21%
rather than YTM=5% if you purchased the bond under
the indicated conditions.
 Example
• Halley Enterprises’ bonds currently sell for
$975.
• They have 7-year maturity, an annual coupon
of $90, and a par value of $1,000.
• What is their yield to maturity?

• The answer is 9.51%

Answer with Excel
 Example
• The Henderson Company’s bonds recently sell
for $1,275.
• They pay a $120 annual coupon, have a 20-
year maturity, and a par value of $1,000, but
they can be called in 5 years at $1,120.
• What are a) their YTM, b) their YTC, and c) if
the yield curve remained flat which rate would
investors expect to earn?
a) YTM is 8.99%
b) YTC is 7.31%, and c) YTC
 Bonds with Semiannual Coupons
• Although some bonds pay interest annually,
the vast majority actually make payments
semiannually.
• To evaluate semiannual payment bonds, we
must modify the valuation model equation as
follows:
Evaluating the Bonds with Semiannual
Coupons
1. Divide the annual coupon interest payment
by 2 to determine the dollars of interest paid
each six months.
2. Multiply the years to maturity, N, by 2 to
determine the number of semiannual
periods.
3. Divide the nominal (quoted) interest rate, 𝐫𝐝 ,
by 2 to determine the periodic (semiannual)
interest rate.
Evaluating the Bonds with Semiannual
Coupons
• On a time line, there would be twice as many
payments, but each would be half as large as with
an annual payment bond.
• Making the indicated changes results in the
following equation for finding an a semiannual
bond’s value:

2N
INT/2 M
VB = ෍ t
+
(1 + rd /2) (1 + rd /2)2N
t=1
 Example
• Assume that Allied Food’s 15-year bonds as
discussed before (section 7-3 in textbook) pay
$50 of interest each six months rather than
$100 at the end of each year.
• Thus, each interest payments is only half as
large but there are twice as many of them.
• We would describe the coupon rate as «10%
with semiannual payments».
 Example
• When the going (nominal) interest rate is rd =
5% with semiannual compounding, the value
of a 15-year, 10% semiannual coupon bond
that pays $50 interest every six months as
follows:
 Example
• Alternatively, when we know the price of a
semiannual bond, we can easily back out the
bond’s nominal yield to maturity.
• If you were told that a 15-year bond with a 10%
semiannual coupon was selling for $1,523.26, you
could solve for the bond’s periodic interest rate as
follows;