### Chapter 2

```Chapter 2
Pricing of Bonds
Time Value of Money (TVM)
 The price of any security equals the PV of the
security’s expected cash flows.
 So, to price a bond we need to know:
• The size and timing of the bond’s expected cash flows.
• The required return (commensurate with the riskiness of
the cash flows).
 You must be comfortable with TVM:
• PV and FV of lump sums and annuities.
Two Important PV Formulas
 PV of a lump sum:
PV 
C
(1  y )
n
 PV of an ordinary annuity:
1

 1  (1  y ) n
PV  C 
y





Time Value
 Future Value
P n  P o (1  r )
n
where:
n = number of periods
Pn = future value n periods from now (in dollars)
Po = original principal (in dollars)
r = interest rate per period (in decimal form)
 Future Value of on Ordinary Annuity
 (1  r )  1 
Pn  A

r


n
Bond Pricing
 price = PV of all future cash flows
 to find price, you need

expected CFs
• coupon payments
• par value

yield
P 
n
t 1
C
(1  r )
t

M
(1  r )
n
Pricing A Bond
 We begin with a simple bullet bond:
•
•
•
•
Noncallable (maturity is known with certainty)
Coupons are paid every six months.
The next coupon is received exactly six months from now.
The interest rate at which the coupons can be invested is
fixed for the life of the bond.
• Principal is paid at maturity (no amortizing).
• Coupon fixed for the life of the bond.
Bond Pricing Formula
 Notation:
•
•
•
•
P = price of the bond (in \$)
n = number of periods (maturity in years  2)
C = semiannual coupon (in \$)
M = maturity value
 The bond price is:
P
C
(1  y )
n
P 

t 1

C
(1  y )
2

C
(1  y )
3


C
(1  y )
1

1

n

(1  y )
C
M
C

n
n
y

(1  y )
(1  y )
n

M
(1  y )


M

n
 (1  y )
n
Note: All inputs to the
bond pricing formula
are fixed except for y. As
y changes so does P.
Example
 Price a 20-year 10% coupon bond with a face value of
\$1,000 if the required yield on the bond is 11%.
 Formula inputs:
•
•
•
•
The coupon is: 0.10  1,000 = \$100.
The semiannual coupon, C, is: \$50.
n = 40
y = 0.055
1

1
n

(1  y )
P C
y


1


1
40


M
(1.055)
 50 

n
(1

y
)
0.055






1, 000
 8 0 2 .3 1  1 1 7 .4 6  9 1 9 .7 7

40
 (1.055)

A few good points…
 Projecting cash flows for fixed income securities is
relatively straightforward – but sometimes it may be harder,
for example:

if the issuer or the investor has the option to change the
contractual due date for the payment of the principal
(callable bonds, putable bonds)

if the coupon payment is reset periodically by a formula
based on some value or values of reference rates (floating
rate securities)

if the investor has the choice to convert or exchange the
security into common stock (convertible bond)
Pricing Zero-Coupon Bonds
 Zero-coupon bonds (zeros) are so called because
they pay no coupons (i.e., C = 0):
 They have only maturity value:
P
P
0
(1  y )

M
(1  y )
n
0
(1  y )
2

0
(1  y )
3


0
(1  y )
n

M
(1  y )
n
Example
 Price a zero that expires 15 years from today if
it’s maturity value is \$1,000 and the required
yield is 9.4%
An investor would pay
 Formula inputs:
\$1,000 in 15 years.
• M = 1,000
• n = 30
• y = 0.047
P 
M
(1  y )
n

1, 000
(1.047 )
30
 2 5 2 .1 2
Price-Yield Relationship
 A fundamental property of bond pricing is the inverse
Price
relationship between bond yield and bond price.
Yield
Price-Yield Relationship
 For a plain vanilla bond all bond pricing inputs
are fixed except yield.
 Therefore, when yields change the bond price
must change for the bond to reflect the new
required yields.
 Example: Examine the price-yield relationship
on a 7% coupon bond.
• For y < 7%, the bond sells at a premium
• For y > 7% the bond sells at a discount
• For y = 7%, the bond sells at par value
Yield
Price
5.0
1,307.45
5.5
1,218.01
6.0
1,137.65
6.5
1,065.29
7.0
1,000.00
7.5
940.95
8.0
887.42
8.5
838.80
9.0
794.53
Price-Yield Relationship
 The price-yield relationship can be summarized:
• yield < coupon rate ↔ bond price > par (premium bond)
• yield > coupon rate ↔ bond price < par (discount bond)
• yield = coupon rate ↔ bond price = par (par bond)
 Bond prices change for the following reasons:
• Discount or premium bond prices move toward par value
as the bond approaches maturity.
• Market factors – change in yields required by the market.
• Issue specific factors – a change in yield due to changes in
the credit quality of the issuer.
Example
 Suppose that you are reviewing a price sheet for
bonds and see the following prices (per \$100 par
value) reported. You observe what seem to be
several errors. Without calculating the price of each
bond, indicate which bonds seem to be reported
incorrectly and explain why.
Bond
Price
Coupon %
Yield %
U
V
W
X
Y
Z
90
96
110
105
107
100
6
9
8
0
7
6
9
8
6
5
9
6
Complications to Bond Pricing

We have assumed the following so far:
1.
2.
3.
4.

Next coupon is due in six months.
Cash flows are known with certainty
We can determine the appropriate required yield.
One discount rate applies to all cash flows.
These assumptions may not be true and therefore
complicate bond pricing.
Complications to Bond Pricing:
Next Coupon Due < 6 Months
 What if the next coupon payment is less than six
months away?
 Then the accepted method for pricing bonds is:
n
P 
 (1  y )
t 1
v
C
v
(1  y )
t 1

M
(1  y ) (1  y )
v
n 1
# days betw een settlem ent and next coupo n
# days in a six-m onth period
Complications to Bond Pricing:
CFs May Not Be Known
 For a noncallable bond cash flows are known with
certainty (assuming issuer does not default)
 However, lots of bonds are callable.
 Interest rates then determine the cash flow:
• If interest rates drop low enough below the coupon rate,
the issuer will call the bond.
 Also, CFs on floaters and inverse floaters change
over time and are not known (more on this later).
Complications to Bond Pricing:
Determining Required Yield
 The required yield for a bond is: R = rf + RP
• rf is obtained from an appropriate maturity Treasury
security.
• RP should be obtained from RPs of bonds of similar risk.
• This process requires some judgement.
Complications to Bond Pricing:
Cash Flow Discount Rates
 We have assumed that all bond cash flows should
be discounted using one discount rate.
 However, usually we are facing an upward sloping
yield curve:
• So each cash flow should be discounted at a rate consistent
with the timing of its occurrence.
 In other words, we can view a bond as a package of
zero-coupon bonds:
• Each cash coupon (and principal payment) is a separate
zero-coupon bond and should be discounted at a rate
appropriate for the “maturity” of that cash flow.
Pricing Floaters
 Coupons for floaters depend on a floating reference
interest rate:
• coupon rate = floating reference rate + fixed spread (in bps)
• Since the reference rate is unpredictable so is the coupon.
 Example:
• Coupon rate = rate on 3-month T-bill + 50bps
Reference Rate
 Floaters can have restrictions on the coupon rate:
• Cap: A maximum coupon rate.
• Floor: A minimum coupon rate.
Pricing Inverse Floaters
 An inverse floater is a bond whose coupon goes up when
interest rates go down and vice versa.
 Inverse floaters can be created using a fixed-rate security
(called the collateral):
• From the collateral two bonds are created: (1) a floater, and (2) an
inverse floater.
 These bonds are created so that:
• Floater coupon + Inverse floater coupon ≤ Collateral coupon
• Floater par value + Inverse floater par value ≤ Collateral par value
 Equivalently, the bonds are structured so that the cash flows
from the collateral bond is sufficient to cover the cash flows
for the floater and inverse floater.
Inverse Floater Example
 Consider a 10-yr 15% coupon bond (7.5% every 6 months).
 Suppose \$100 million of bond is used to create two bonds:
• \$50 million par value floater and \$50 million par value inverse floater.
 Assume a 6-mo coupon reset based on the formula:
• Floater coupon rate = reference rate + 1%
• Inverse coupon rate = 14% - reference rate
 Notice: Floater coupon rate + Inverse coupon rate = 15%
• Problem: if reference rate > 14%, then inverse floater coupon rate < 0.
• Solution: put a floor on the inverse floater coupon of 0%.
• However, this means we must put a cap in the floater coupon of 15%.
 The price of floaters and inverse floaters:
• Collateral price = Floater price + Inverse floater price
Price Quotes on Bonds
 We have assumed that the face value of a bond is
\$1,000 and that is often true, but not always:
• So, when quoting bond prices, traders quote the price as a
percentage of par value.
• Example: A quote of 100 means 100% of par value.
Prices Converted into a Dollar
Price
(1)
Price Quote
(2)
Converted to a Decimal [ =
(1)/100]
97
0.9700000
(3)
Par Value
\$
10,000
(4)
Dollar Price
[ = (2) x (3)]
\$
9,700.00
85 1/2
0.8550000
100,000
85,500.00
90 1/4
0.9025000
5,000
4,512.50
80 1/8
0.8012500
10,000
8,012.50
76 5/32
0.7615625
1,000,000
761,562.50
86 11/64
0.8657588
100,000
86,172.88
100
1.0000000
50,000
50,000.00
109
1.0900000
1,000
1,090.00
103 3/4
1.0375000
100,000
103,750.00
105 3/8
1.0537500
25,000
26,343.75
103 19/32
1.0358375
1,000,000
1,035,937.50
Clean vs. Dirty Price
 Most bond trades occur between coupon payment
dates.
• Thus at settlement, the buyer must compensate the seller
for coupon interest earned since the last coupon payment.
• This amount is called accrued interest.
• The buyer pays the seller: Bond price + Accrued Interest
(often called the dirty price).
• The bond price without accrued interest is often called the
“clean price.”
Clean vs. Dirty Price
 Suppose a bond just sold for 84.34 (based on par value of
\$100) and pays a coupon of \$4 every six months.
 The bond paid the last coupon 120 days ago.
 What is the clean price? What is the dirty price?
 Clean price:
• \$84.34
 Dirty price:
• \$84.34 + 120/180(\$4) = \$87.01
```