Pricing of Bonds
Zvi Wiener
Based on Chapter 2 in Fabozzi
Bond Markets, Analysis and Strategies
Fall-02
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Time value of money
How to calculate price of a bond
Why the price of a bond changes
Relation between yield and price
Relation between coupon and price
Price changes when approaching maturity
Floaters and inverse floaters
Accrued interest and price quotes
Zvi Wiener
Fabozzi Ch 2
slide 2
Time Value of Money
present value PV = CFt/(1+r)t
Future value FV = CFt(1+r)t
Net present value NPV = sum of all PV
-PV
5
5
5
5
105
4
5
105
PV  

t
5
(
1

r
)
(
1

r
)
t 1
Zvi Wiener
Fabozzi Ch 2
slide 3
Time Value
You have $100 now and are going to deposit it
for 5 years with 6% interest.
What will be the final amount?
It depends on calculation method!
Yearly compounding: $100*1.065
Semiannual compounding: $100*1.0312
Monthly compounding: $100*1.00560
Zvi Wiener
Fabozzi Ch 2
slide 4
Periodic Rate
Annual interest rate
A
r=
=
Number of periods in a year
n
Effective Rate
n
 A
1  r   1    1  R
 n
n
Zvi Wiener
Fabozzi Ch 2
slide 5
Pricing of Bonds
T
Ct
PV  
t
t 1 (1  r )
Zero coupon bond
Zvi Wiener
100
price 
(1  r )t
Fabozzi Ch 2
slide 6
Pricing of Bonds
T
Ct
PV  
t
t 1 (1  r )
Term structure of interest rates
T
Ct
PV  
t
(
1

r
)
t 1
t
Zvi Wiener
Fabozzi Ch 2
slide 7
Yield
Yield = IRR = Internal Rate of Return
T
Ct
Price  
t
(
1

y
)
t 1
How do we know that there is a solution?
Zvi Wiener
Fabozzi Ch 2
slide 8
Example
Price calculation:
5
5
105


2
3  83.34
1.10 1.11 1.12
Yield calculation:
5
5
105


 83.34
2
3
1  y (1  y ) (1  y )
y  11.9278%
Zvi Wiener
Fabozzi Ch 2
slide 9
Price-Yield Relationship
Price and yield (of a straight bond) move in
opposite directions.
price
yield
Zvi Wiener
Fabozzi Ch 2
slide 10
General pricing formula
n
Ct
P
v
t 1
(
1

r
)
(
1

r
)
t 1
daysbetweensettlem entand nextcoupon
v
daysin six m onthsperiod
Zvi Wiener
Fabozzi Ch 2
slide 11
Accrued Interest
Accrued interest = interest due in full period*
(number of days since last coupon)/
(number of days in period between coupon
payments)
Zvi Wiener
Fabozzi Ch 2
slide 12
Day Count Convention
Actual/Actual - true number of days
30/360 - assume that there are 30 days in each
month and 360 days in a year.
Actual/360
Zvi Wiener
Fabozzi Ch 2
slide 13
Floater
The coupon rate of a floater is equal to a
reference rate plus a spread.
For example LIBOR + 50 bp.
Sometimes it has a cap or a floor.
Zvi Wiener
Fabozzi Ch 2
slide 14
Inverse Floater
Is usually created from a fixed rate security.
Floater coupon
= LIBOR + 1%
Inverse Floater coupon = 10% - LIBOR
Note that the sum is a fixed rate security.
If LIBOR>10% there is typically a floor.
Zvi Wiener
Fabozzi Ch 2
slide 15
Price Quotes and Accrued Interest
Assume that the par value of a bond is $1,000.
Price quote is in % of par + accrued interest
the accrued interest must compensate the
seller for the next coupon.
Zvi Wiener
Fabozzi Ch 2
slide 16
Home Assignment
Chapter 2
Questions 2, 3, 7, 8, 11
Zvi Wiener
Fabozzi Ch 2
slide 17
FRM-99, Question 17
Assume a semi-annual compounded rate of
8% per annum. What is the equivalent
annually compounded rate?
A. 9.2%
B. 8.16%
C. 7.45%
D. 8%
Zvi Wiener
Fabozzi Ch 2
slide 18
FRM-99, Question 17
(1 + ys/2)2 = 1 + y
(1 + 0.08/2)2 = 1.0816 ==> 8.16%
Zvi Wiener
Fabozzi Ch 2
slide 19
FRM-99, Question 28
Assume a continuously compounded interest
rate is 10% per annum. What is the equivalent
semi-annual compounded rate?
A. 10.25% per annum.
B. 9.88% per annum.
C. 9.76% per annum.
D. 10.52% per annum.
Zvi Wiener
Fabozzi Ch 2
slide 20
FRM-99, Question 28
(1 + ys/2)2 = ey
(1 + ys/2)2 = e0.1
1 + ys/2 = e0.05
ys = 2 (e0.05 - 1) = 10.25%
Zvi Wiener
Fabozzi Ch 2
slide 21
Mortgage example
You take a mortgage $100,000 for 10 years
with yearly payments and 7% interest.
What is the size of each payment?
10
x
100,000  
t
(
1

0
.
07
)
t 1
10
1
100,000  x
t
1
.
07
t 1
Zvi Wiener
Fabozzi Ch 2
x  14.2378
slide 22
Mortgage example
How much do you own bank after 3 first
payments?
7
14.2378
 76.7314

t
t 1 1.07
What is the fair value of your debt if interest
rates are 5%?
7
14.2378
 82.3849

t
t 1 1.05
Zvi Wiener
Fabozzi Ch 2
slide 23