Bond Valuation: Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

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Bond Valuation
Debashis Saha, Assistant Professor, F & B, Jahangirnagar

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The Basic Structure of Bonds
 What is a bond?
 In its broadest sense, a bond is any debt instrument that promises a
fixed income stream to the holder
 A contract under which a borrower promises to pay interest and
principal on specific dates to the holders of the bond
 Fixed income securities are often classified according to maturity, as
follows:
 Less than one year – Bills or “Paper”
 1 year < Maturity < 7 years – Notes
 < 7 years – Bonds
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 A typical bond has the following characteristics:

 A fixed face or par value, paid to the holder of the bond, at maturity
 A fixed coupon, which specifies the interest payable over the life of the bond
 Coupons are usually paid either annually or semi-annually
 A fixed maturity date

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 Bonds may be either:

 Bearer bonds
-Debt security issued by a business entity, such as a corporation, or by a
government. not registered, no records are kept of the owner
 Registered bonds
-Whose owner is registered with the bond’s issuer.
 Bond indenture - the contract between the issuer of the bond and the investors
who hold it
 The market price of a bond is equal to the present value of the payments
promised by the bond
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Cash Flow Pattern for a Traditional Coupon-Paying
FIGURE 6-1
Bond
0 1 2 3 … n
I I I I I
I = interest payments, and F = principal repayment

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Cash Flow Pattern of a Bond
0 1 2 3 4 n
Purchase Coupon Coupon Coupon Coupon Coupon +

Price Face Value
Cash Outflows Cash Inflows

to the Investor to the Investor
The Purchase Price or Market Price of a bond is simply the present

value of the cash inflows, discounted at the bond’s yield-to-maturity

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 Bond indenture is the contract between the issuer and the holder.
It specifies:
 Details regarding payment terms
 Collateral
 Positive and negative covenants
 Par or face value (usually increments of $1,000)
 Bond pricing – usually shown as the price per $100 of par value,
which is equal to the percentage of the bond’s face value

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 Term-to-maturity – the time remaining to the bond’s maturity date

 Coupon rate – the annual percentage interest paid on the bond’s face
value; If the coupon is paid twice a year, divide the annual coupon by
two
 Example: A $1,000 bond with an 8% coupon rate will have an $80 coupon if
paid annually or a $40 coupon if paid semi-annually

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Types
 Fixed rate bonds have a coupon that remains constant throughout the
life of the bond. A variation are stepped-coupon bonds, whose coupon
increases during the life of the bond.
 Floating rate notes (FRNs, floaters) have a variable coupon that is linked
to a reference rate of interest, such as LIBOR or Euribor. For example the
coupon may be defined as three month USD LIBOR + 0.20%. The coupon
rate is recalculated periodically, typically every one or three months.
 Zero-coupon bonds (zeros) pay no regular interest. They are issued at a
substantial discount to par value, so that the interest is effectively rolled
up to maturity (and usually taxed as such). The bondholder receives the
full principal amount on the redemption date.

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 High-yield bonds (junk bonds) are bonds that are rated below investment
grade by the credit rating agencies. As these bonds are more risky than
investment grade bonds, investors expect to earn a higher yield.
 Convertible bonds let a bondholder exchange a bond to a number of
shares of the issuer's common stock. These are known as hybrid securities,
because they combine equity and debt features.
 Exchangeable bonds allows for exchange to shares of a corporation other
than the issuer.

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 Asset-backed securities are bonds whose interest and principal payments are
backed by underlying cash flows from other assets. Examples of asset-backed
securities are mortgage-backed securities (MBS's),
 Subordinated bonds are those that have a lower priority than other bonds of the
issuer in case of liquidation. In case of bankruptcy, there is a hierarchy of creditors.
 A government bond, also called Treasury bond, is issued by a national government
and is not exposed to default risk. It is characterized as the safest bond, with the
lowest interest rate. A treasury bond is backed by the “full faith and credit” of the
relevant government.
 Municipal bond is a bond issued by a state, U.S. Territory, city, local government, or
their agencies. Interest income received by holders of municipal bonds is often
exempt from the federal income tax and from the income tax of the state in which
they are issued, although municipal bonds issued for certain purposes may not be
tax exempt.
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Security and Protective Provisions
 Mortgage bonds – secured by real assets

 Debentures – either unsecured or secured with a floating charge over the
firm’s assets
 Collateral trust bonds – secured by a pledge of financial assets, such as
common stock, other bonds or treasury bills
 Equipment trust certificates – secured by a pledge of equipment, such as
railway rolling stock

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Security and Protective Provisions
 Covenants
 Positive covenants – things the firm agrees to do
 Supply periodic financial statements

 Maintain certain ratios
 Negative covenants – things the firm agrees not to do
 Restricts the amount of debt the firm can take on

 Prevents the firm from acquiring or disposing of assets

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More Bond Features
 Call feature – allows the issuer to redeem or pay off the bond prior to
maturity, usually at a premium
 Retractable bonds – allows the holder to sell the bonds back to the issuer
before maturity
 Extendible bonds – allows the holder to extend the maturity of the bond
 Sinking funds – funds set aside by the issuer to ensure the firm is able to
redeem the bond at maturity
 Convertible bonds – can be converted into common stock at a pre-
determined conversion price

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Bond Valuation
 The value of a bond is a function of:

 Par value
 Term to maturity
 Coupon rate
 Investor’s required rate of return (discount
rate is also known as the bond’s yield to
maturity)
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Bond Value
General Formula
 1 
1 − ( 1 + k )n
[ 6-1]
 1
B = I  b
+F
 kb  ( 1 + k b )n
 
Where:
I = interest (or coupon ) payments
kb = the bond discount rate (or market rate)
n = the term to maturity
F = Face (or par) value of the bond
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Bond Valuation: Example
 What is the market price of a ten-year, $1,000 bond

with a 5% coupon, if the bond’s yield-to-maturity is 6%?
1 − (1 + kb )− n  F Calculator Approach:
B=I + 1,000 FV
 ( + b)
n
 k b 1 k 50 PMT
10 N
1 − (1.06 )−10  1, 000 I/Y 6
= 50  +
 (1.06 )
10 CPT PV 926.40
 0.06
= $926.40
Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

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Factors Affecting Bond Prices
Bond Price-Yield Curve
When interest rates increase, bond prices fall
FIGURE 6-2
Price
($)
Market Yield (%)

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 The relationship between the coupon rate and the bond’s

yield-to-maturity (YTM) determines if the bond will sell at a
premium, at a discount, or at par
If Then Bond Sells at a:
Coupon < YTM Market < Face Discount
Coupon = YTM Market = Face Par
Coupon > YTM Market > Face Premium

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Bond Valuation: Semi-Annual Coupons
 So far, we have assumed that all bonds have annual pay coupons. While
this is true for many Eurobonds, it is not true for most domestic bond issues,
which have coupons that are paid semi-annually
 To adjust for semi-annual coupons, we must make three changes:
 Size of the coupon payment (divide by 2)
 Number of periods (multiply by 2)
 Yield-to-maturity (divide by 2)

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Bond Valuation: Semi-Annual Coupons
For example, suppose you want to value a five-year, $10,000 Government of

Canada bond with a 4% coupon, paid twice a year, given a YTM of 6%.
  kb −2 n  Calculator Approach:
1 − 1 +  
I   2  F 10,000 FV
B= + 400 ÷ 2 =
2 kb   kb  2 n PMT
  1 +  5x2= N
 2   2
6 ÷ 2 = I/Y
  .06 −2 x 5  CPT PV 926.40
1 − 1 +  
400   2   10, 000
= +
2  0.06   .06 2 x 5
 2  1 + 
   2 
= $9,146.98

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 There are three factors that affect the price volatility of a bond
 Yield to maturity
 Time to maturity
 Size of coupon

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Yield to Maturity (YTM)
 Yield to maturity (YTM) measures the annual return an investor would receive if he or she held a
particular bond until maturity. It is essentially the internal rate of return on a bond and it equates
the present value of bond future cash flows to its current market price.
 To understand YTM, one must first understand that the price of a bond is equal to the present
value of its future cash flows, as shown in the following formula:
 B0 = the bond price,
 C = the annual coupon payment,
 F = the face value of the bond,
 YTM = the yield to maturity on the bond, and
 t = the number of years remaining until maturity.

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 The price of a bond is $920 with a face value of $1000 which is the face
value of many bonds. Assume that the annual coupons are $100, which is
a 10% coupon rate, and that there are 10 years remaining until maturity.
 11.25%

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 Find the yield to maturity on a semiannual coupon bond with a face value
of $1000, a 10% coupon rate, and 15 years remaining until maturity given
that the bond price is $862.35.

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 Bond prices go down when the YTM goes up
 Bond prices go up when the YTM goes down
 Look at the graph on the next slide. It shows how the price of a 25 year,
10% coupon bond changes as the bond’s YTM varies from 1% to 30%
 Note that the graph is not linear – instead it is said to be convex to the
origin

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Price and Yield: 25 Year Bond, 10% Coupon
Price/Yield Relationship
Price per $100 of Face 350

300
250
Value
200
150
100
50
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Percent YTM

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Bond Convexity
 The convexity of the price/YTM graph reveals two important insights:

 The price rise due to a fall in YTM is greater than the price decline due to a rise in
YTM, given an identical change in the YTM
 For a given change in YTM, bond prices will change more when interest rates
are low than when they are high

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 Time to maturity
 Long bonds have greater price volatility than short bonds
 The longer the bond, the longer the period for which the cash flows are fixed
 Size of coupon
 Low coupon bonds have greater price volatility than high coupon bonds
 High coupons act like a stabilizing device, since a greater proportion of the
bond’s total cash flows occur closer to today & are therefore less affected
by a change in YTM

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Interest Rate Risk & Duration
 The sensitivity of bond prices to changes in interest rates is a measure of

the bond’s interest rate risk
 A bond’s interest rate risk is affected by:
 Term to maturity
 Size of coupon
 These three factors are all captured in one number called duration

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Duration
 Duration is a measure of interest rate risk

 The higher the duration, the more sensitive the bond is to changes in
interest rates
 A bond’s duration will be higher if its:
 YTM is lower
 Term to maturity is longer
 Coupon is lower

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Bond Quotations
Issuer Coupon Maturity Price Yield
Canada 5.500 2009-Jun-01 103.79 4.16

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Bond Yields
 Yield-to-maturity (YTM) – the discount rate used to evaluate bonds

 The yield earned by a bond investor who:
 Purchases the bond at the current market price
 Held the bond to maturity
 Reinv ested all of the coupons at the YTM
 Is the bond’s Internal Rate of Return (IRR)

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Current Yield
 The current yield is the yield on the bond’s current market price
provided by the annual coupon
 It is not a true measure of the return to the bondholder because it does
not consider potential capital gain or capital losses based on the
relationship between the purchase price of the bond and it’s par value.
[ 6-4]
Annual interest
CY =
B

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Current Yield
Example
 The current yield is the yield on the bond’s current market price
provided by the annual coupon
 Example: If a bond has a 5.5% annual pay coupon and the
current market price of the bond is $1,050, the current yield is:
Annual Coupon
Current Yield =
Current Market Price
55
=
1, 050
= 5.24%

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Zero Coupon Bonds
 A zero coupon bond is a bond issued at a discount that matures at par or

face value
 A zero coupon bond has no reinvestment rate risk, since there are no
coupons to be reinvested
 To calculate the price of a zero coupon bond, solve for the PV of the face
amount

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Zero Coupon Bonds
 Example: What is the market price of a $50,000 zero coupon bond

with 25 years to maturity that is currently yielding 6%?
F
B=
(1 + kb )
n
50, 000
=
(1.06 )
25
= $11, 649.93

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Floating Rate & Real Return Bonds
 Floating rate bonds have a coupon that floats with some reference rate,
such as the yield on T bills
 Because the coupon floats, the market price will typically be close to the bond’s
face value
 Real return bonds are issued by the Government of Canada to protect

investors against unexpected inflation
 Each period, the face value of the bond is grossed up by the inflation rate. The
coupon is then paid on the grossed up face value.

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Callable Bond
 A callable bond gives the borrower (issuer) the right to pay back the obligation
to the lender (bondholder) before the stated maturity date.
 A callable bond (also called a "redeemable bond") is a bond with an
embedded call option. If the issuer agrees to pay more than the face value
amount of the bond when called, the excess of the payment over the face
amount is the "call premium". In most cases, the call price is greater than the
par (or issue) price.

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 Company ABC decides to borrow $10 million in the bond market. The bond's coupon
rate is 8%. Company analysts believe interest rates will go down during the 7 year term
of the bonds. To take advantage of lower rates in the future, ABC issues callable bonds.
 Under the terms of the bonds (the "indenture"), ABC has the option to call the bonds
(meaning, pay them back) any time after year 3. However, if ABC decides to exercise its
right to call, it needs to pay bondholders $102 for every $100 of principal.
 Let's assume that in year 4, interest rates fall to 6%. ABC exercises its right to redeem the
bonds. It borrows money from a bank at 6% and pays back the 8% bonds.
 Even though ABC had to spend $10.2 million to pay back its current bondholders, it will
benefit going forward because future interest payments will be only $612,000 per year
($10,200,000 * 6%) vs. $800,000 per year ($10,000,000*8%).

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 It's extremely important for investors to realize the presence of an embedded call option in a bond affects the value
of the bond.
 A callable bond is worth less to an investor than a noncallable bond because the company issuing the bond has the
power to redeem it and deprive the bondholder of the additional interest payments he'd be entitled to if the bond
was held to maturity.
 From the company's perspective, having the ability to call the bonds adds value because the company is given the
flexibility to adjust its financing costs downward if interest rates decline.
 Typically, bonds are called when interest rates fall so dramatically, the issuer can save money by floating new bonds
at lower rates. If by the time of the call date interest rates have significantly dropped, the issuer is motivated to call
the bonds because doing so will allow it to refinance its debt at a cheaper level. From another perspective, the
issuer is incentivized to buy bonds back at par value, because as interest rates go down, the price of the bonds goes
up.
 Callable bonds are attractive to investors because they usually offer higher coupon rates than non-callable bonds.
But as always, in return for this investment advantage comes greater risk.
 If interest rates drop, the bond's issuer will be strongly motivated to save money by replaying it callable bonds and
issuing new ones at lower coupon rates. In these circumstances, the investor that holds the bonds will see his interest
payments stop and obtain his principal early. If the investor then reinvests this principal in bonds again, chances are
that he will be forced to accept a lower coupon rate that is in line with the prevailing (and lower) interest rates
(called "interest rate risk").
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Callable bonds
 Bonds that may be repurchased by the issuer at a specified call

price during the call period
 A call usually occurs after a fall in market interest rates that allows
issuers to refinance outstanding debt with new bonds.
 Generally, the call price is above the bond’s face value. The
difference between the call price and the face value is the call
premium
 Bonds are not usually callable during the first few years of a bond’s
life. During this period the bond is said to be call-protected.
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 Investors are typically interested in knowing what the yield

will be if the bond is called by the issuer at the first possible
date. This is called yield to call (YTC).
 Suppose that we have a 3-year, $1,000 par value, 6%
semiannual coupon bond. We observe that the value of
the bond is $852.48. The first call price is $1,060 in 2 years.
Find YTC.
 N= 4, FV = 1060 PMT=30 PV = -852.48
 I/YR = 8.85*2= 17.707%

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More on Bond Prices
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 
C  1 + FV
Bond price = 1−
YTM 

1 + YTM
2
( )
2M 

(
1 + YTM
2
)2M
Now assume a bond has 25 years to maturity, a 9% coupon, and the YTM is 8%.
What is the price? Is the bond selling at premium or discount?
 
90  1 + 1000
Bond price = 1− = $1,107.41
.08 
 1 + .08
2
( )
50 
 (
1 + .08
2
)
50
Now assume the same bond has a YTM of 10%. (9% coupon & 25 years to maturity)
What is the price? Is the bond selling at premium or discount?
 
90 
Debashis Saha, Assistant Professor, F & B, Jahangirnagar 1 + 1000
University Bond price = 1− = $908.72
 50  50
More on Bond Prices (cont’d)
45
Now assume the same bond has 5 years to maturity (9% coupon
& YTM of 8%) What is the price? Is the bond selling at
premium or discount?
 
90  1 + 1000
Bond price = 1− = $1,040.55
.08 
 1 + .08
2 
(
10 
 1 + .08)2
10
( )
Now assume the same bond has a YTM of 10%. (9% coupon &
5 years to maturity) What is the price? Is the bond selling at
premium or discount?
 
90  1 + 1000
= − = $961.39
( ) ( )
Bond price 1
.10  10  10
University
 1 + .10  1 + .10
2  2
More on Bond Prices (cont’d)
46
Where does this leave us? We found:

Coupon Years YTM Price
9% 25 8% $1,107
9% 25 10% $ 908
9% 5 8% $1,040
9% 5 10% $ 961
$1,150
$1,100 25 years
$1,050
$1,000 5 years
$950
$900
University 8% 9% 10% 11%
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 Decreasing yields cause bond prices to rise, but long-term bonds

increase more than short-term. Similarly, increasing yields cause
long-term bonds to decrease in price more than short-term
bonds.

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Yield to Call
 Many bonds, especially those issued by corporations, are callable. This means that the issuer of
the bond can redeem the bond prior to maturity by paying the call price, which is greater than
the face value of the bond, to the bondholder. Often, callable bonds cannot be called until 5
or 10 years after they were issued. When this is the case, the bonds are said to be call protected.
The date when the bonds can be called is refered to as the call date.
 The yield to call is the rate of return that an investor would earn if he bought a callable bond at
its current market price and held it until the call date given that the bond was called on the call
date. It represents the discount rate which equates the discounted value of a bond's future cash
flows to its current market price given that the bond is called on the call date. This is illustrated by
the following equation:

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 B0 = the bond price,

 C = the annual coupon payment,
 CP = the call price,
 YTC = the yield to call on the bond, and
 CD = the number of years remaining until the call date.

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 Find the yield to call on a semiannual coupon bond with a face value of
$1000, a 10% coupon rate, 15 years remaining until maturity given that the
bond price is $1175 and it can be called 5 years from now at a call price
of $1100.

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 Bonds are long-term debt securities that are issued by corporations and gov ernment entities. Purchasers of bonds
receiv e periodic interest payments, called coupon payments, until maturity at which time they receiv e the face v alue
of the bond and the last coupon payment. Most bonds pay interest semiannually. The Bond Indenture or Loan Contract
specifies the features of the bond issue. The following terms are used to describe bonds.
 Par or Face Value
 The par or face v alue of a bond is the amount of money that is paid to the bondholders at maturity. For most bonds the
amount is $1000. It also generally represents the amount of money borrowed by the bond issuer.
 Coupon Rate
 The coupon rate, which is generally fixed, determines the periodic coupon or interest payments. It is expressed as a
percentage of the bond's face v alue. It also represents the interest cost of the bond issue to the issuer.
 Coupon Payments
 The coupon payments represent the periodic interest payments from the bond issuer to the bondholder. The annual
coupon payment is calculated be multiplying the coupon rate by the bond's face v alue. Since most bonds pay interest
semiannually, generally one half of the annual coupon is paid to the bondholders ev ery six months.
 Maturity Date
 The maturity date represents the date on which the bond matures, i.e., the date on which the face v alue is repaid. The
last coupon payment is also paid on the maturity date.
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 Original Maturity
 The time remaining until the maturity date when the bond was issued.
 Remaining Maturity
 The time currently remaining until the maturity date.
 Call Date
 For bonds which are callable, i.e., bonds which can be redeemed by the issuer prior to
maturity, the call date represents the date at which the bond can be called.
 Call Price
 The amount of money the issuer has to pay to call a callable bond. When a bond first
becomes callable, i.e., on the call date, the call price is often set to equal the face value
plus one year's interest.

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 Required Return
 The rate of return that investors currently require on a bond.
 Yield to Maturity
 The rate of return that an investor would earn if he bought the bond at its current
market price and held it until maturity. Alternatively, it represents the discount rate
which equates the discounted value of a bond's future cash flows to its current market
price.
 Yield to Call
 The rate of return that an investor would earn if he bought a callable bond at its
current market price and held it until the call date given that the bond was called on
the call date.

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1

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 A typical bond has the following characteristics:

 A fixed maturity date

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 Bonds may be either:

I = interest payments, and F = principal repayment

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Purchase Coupon Coupon Coupon Coupon Coupon +

Cash Outflows Cash Inflows

The Purchase Price or Market Price of a bond is simply the present

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 Term-to-maturity – the time remaining to the bond’s maturity date

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 Mortgage bonds – secured by real assets

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 Supply periodic financial statements

 Restricts the amount of debt the firm can take on

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 The value of a bond is a function of:

 What is the market price of a ten-year, $1,000 bond

Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

When interest rates increase, bond prices fall

Market Yield (%)

Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

 The relationship between the coupon rate and the bond’s

If Then Bond Sells at a:

Coupon < YTM Market < Face Discount

Coupon = YTM Market = Face Par

Coupon > YTM Market > Face Premium

Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

For example, suppose you want to value a five-year, $10,000 Government of

Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Price per $100 of Face 350

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 The convexity of the price/YTM graph reveals two important insights:

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 The sensitivity of bond prices to changes in interest rates is a measure of

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

 Duration is a measure of interest rate risk

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Issuer Coupon Maturity Price Yield

Canada 5.500 2009-Jun-01 103.79 4.16

Debashis Saha, Assistant Professor, F & B, Jahangirnagar University

 Yield-to-maturity (YTM) – the discount rate used to evaluate bonds

Debashis Saha, Assistant Professor, F & B, Jahangirnagar

Debashis Saha, Assistant Professor, F & B, Jahangirnagar University