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). MARKET VALUE
 You must be comfortable with TVM:
• PV and FV of lump sums and annuities.
• Your text has a good review of the TVM concepts needed
for this course.
Two Important PV Formulas
 PV of a lump sum:
CF
PV 
(1  r ) n
 PV of an annuity (Formula 2.5, where CF = A):
1

1  (1  r ) n
PV  CF 
r





Pricing A Bond
 We begin with a simple bullet bond:
•
•
•
•
Non-callable (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 (Formulas 2.6, 2.7, 2.8):
C
C
C
P



(1  r ) (1  r ) 2 (1  r )3
C
M


(1  r ) n (1  r ) n Note: All inputs to the
bond pricing formula
1 

are fixed except for r. As
1

n
 (1  r )n 
r changes so does P.
C
M
M
P

C

n
n
(1  r )
r

 (1  r ) n
t 1 (1  r )
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
r = 0.055
1

1

 (1  r )
PC
r


1


1


 (1.055) 40
M
 50 

n
(1

r
)

 0.055




1, 000
 802.31  117.46  919.77

40
 (1.055)

Pricing Zero-Coupon Bonds
 Zero-coupons bonds (zeros) are so called because
they pay no coupons (i.e., C = 0):
 They have only maturity value:
0
0
0
P



2
3
(1  r ) (1  r ) (1  r )
M
P
(1  r ) n
0
M


n
(1  r ) (1  r ) 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:
• M = 1,000
• n = 30
• r = 0.047
P
M
1, 000
 252.12

n
30
(1  r )
(1.047)
$252.12 today and receive
$1,000 in 15 years.
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 r < 7%, the bond sells at a premium
• For r > 7% the bond sells at a discount
• For r = 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. (Time Passes)
• 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. (Credit Spreads)
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 determining 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
t 1
v
C
M

(1  r )v (1  r )t 1 (1  r )v (1  r ) n 1
# days between settlement and next coupon
# days in a six-month period
Complications to Bond Pricing:
CFs May Not Be Known
 For a non-callable bond cash flows are known with
certainty (assuming issuer does not default)
 However, most 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 (Risk Premium) 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
Spread
 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
(pg. 30 text)
 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.
Price Quotes on Bonds
 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
pg 31
 Suppose a bond just sold for 87.01 (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:
• $87.01 – (120/180)($4) = $84.34
 Dirty price:
• $87.01
Clean vs. Dirty Bonds example
A US bond has a coupon rate of 7.2% and pays 4 times a year, on the 15th of January,
April, July, and October. It uses the 30/360 US day count convention.
A trade for 1,000 par value of the bond settles on January 25th. The prior coupon date
was January 15th. The accrued interest reflects ten days' interest, or $2.00 (7.2% of
1,000 * (10 days/360 days)).
The full (Dirty) value of these bonds is set by the market at $985.50
Bond Pricing Example
Term
Value
Par value
1,000.00
Full market value
$985.50
Dirty price
98.55
Accrued interest
$2.00
Flat market value
Clean price
$983.50
98.35