The document discusses bonds, including types of bonds and how bond prices and yields are calculated. It covers zero-coupon bonds and coupon bonds. For zero-coupon bonds, it shows the formulas to calculate price from yield and yield from price for terms less than and greater than one year. For coupon bonds, it explains how coupon payments are determined based on the coupon rate, payment frequency, and time to maturity. The purpose is to explain how to compute bond prices, yields, and related calculations.

1. Bonds
Prices
and
Yields

2. Learning
Objec-ves
¨ Types
of
bonds
and
bond
parameters
¨ The
concept
of
yield
¨ Bond
price
computed
as
discounted
expected
cash
flow
¤ Applica-ons
n Compute
bond
yield
from
a
known
price
n Compute
bond
price
from
a
known
yield
n Graph
the
price
v.
yield
n Compute
bond
price
when
yield
is
not
known
n Graph
price
v.
Time
to
maturity
n Compute
mortgage
payments
¨ Bond
price
quotes
2

3. Bonds
¨ Corpora-ons
and
government
en--es
can
raise
capital
by
selling
bonds
¤ Long
term
liability
(accoun-ng)
¤ Debt
capital
(finance)
¨ The
bond
has
¤ Principal,
par,
or
face
value:
F
¤ Price:
P
¤ Yield:
y
(actually
“yield
to
maturity”
and
the
discount
rate)
¤ Maturity
date,
-me
to
maturity,
term,
or
tenor:
T
n Date
at
which
the
bond
principal,
F,
is
returned
to
investors
¤ In
the
case
of
a
coupon
bond
(as
opposed
to
a
zero
coupon
bond)
n Coupon
rate:
c
(annual,
simple,
nominal
rate)
n Annual
payment
frequency:
m;
or
period
Δt
n In
the
U.S.
semiannual
coupons
is
typical:
m
=
2
or
Δt
=
.5
3

4. Zero
Coupon
Bonds
¨ ZCBs
do
not
pay
a
coupon
¨ The
return
and
‘yield’
(rate)
is
due
to
the
purchase
price
at
a
discount
to
face
value
¨ U.S.
Treasury
bills
(T
–
bills)
are
zero
coupon
bonds
¤ Time-­‐to-­‐maturity
at
issue
is
4,
13,
26,
52
weeks
¤ Face
value
$100
to
$5,000,000
¨ A
ZCB
yield
is
the
interest
rate,
and
the
discount
rate
denoted
z
4
F
P
t=0
t=T

5. Zero
Coupon
Bond
¨ For
T
≤
1
year:
where
z
is
the
annual
simple
rate
or
yield
¨ For
T
>
1
year
where
z
is
the
annualized
effec-ve
rate
or
yield
If
a
bond
has
a
term
of
a
year
or
less,
simple
interest
is
used,
otherwise
compound
annual
interest
is
used
by
conven-on
T)z(1
F
P
⋅+
=
T
z)(1
F
P
+
=
5
F
P
t=0
t=T

6. Zero
Coupon
Bond
Example
¨ The
face
value
is
$1000,
the
market
price
is
$850,
and
the
-me
to
maturity
is
3.5
years.
What
is
the
annualized
yield
?
¨ The
face
value
is
$1000,
the
market
price
is
$975,
and
the
-me-­‐to-­‐
maturity
is
0.5
years.
What
is
the
annualized
yield?
$850 =
$1000
(1+z)3.5
z =
$1000
$850
!
"
#
$
%
&
1
3.5
−1= 4.753%
$975=
$1000
(1+0.5⋅z)
z =2⋅
$1000
$975
−1
#
$
%
&
'
(=5.128%
6
T)z(1
F
P
⋅+
=
T
z)(1
F
P
+
=

7. Coupon
Bond
P
=
current
price
C
=
coupon
payment
F
=
face
or
par
value
t=0.0
t=Δt
t=2·∙Δt
t=M·∙Δt=T
i=0
i=1
i=2
i=M
t0=0.0
t1=Δt
t2=2Δt
tM=
M·∙Δt
=T
7

8. Coupon
Payment
¨ Bond
coupon
cash
flows,
C,
are
defined
by
a
nominal,
simple
coupon
rate,
c,
and
a
compounding
frequency
per
year,
m,
or
coupon
period
measured
in
years,
Δt
¨ The
total
cash
flow
at
-me
ti,
CFi,
is
defined
as:
CFi
$=$C$$$$$$$$for$$i<M
CFM
$=$C$+$F
8
$8.125
C
.5t
1000$F
%625.1c
example
tFc
C
=
=Δ
=
=
Δ⋅⋅=
%y1
2
y
1
%632.11
2
1.625%
1
y
rate,
coupon
Effective
2
2
=−⎟
⎠
⎞
⎜
⎝
⎛
+
=−⎟
⎠
⎞
⎜
⎝
⎛
+
T=num
of
years
(floa-ng)
N=num
of
years
(integer)
m=periods
per
year
In
this
course,
generally
M=NŸm
360=
30
Ÿ12

9. Coupon
Bond
Yield
¨ Yield
to
maturity
is
the
actual
yield
achieved
for
a
coupon
bond
if
¤ The
bond
is
held
to
maturity,
and
¤ Each
coupon
payment
is
reinvested
at
a
rate
of
return
of
y
through
-me
T
n The
risk
that
coupons
cannot
be
reinvented
at
a
rate
greater
than
or
equal
to
y
due
to
market
condi-ons
is
called
“reinvestment
risk”
¨ The
yield
to
maturity
is
the
investor’s
expected
return
on
investment
and
is
thus
the
issuer’s
rate
cost
¤ It’s
the
issuer’s
cost
of
debt,
kD,
for
the
bond
¨ The
yield
reflects
both
the
-me
value
of
money
and
the
credit
worthiness
of
the
borrower
¤ The
expected
variance
in
the
cash
flow
is
reflected
in
the
yield
9

10. Bond
Price
¨ The
discount
rate
y
is
the
yield
to
maturity
or
simply
the
yield
on
a
coupon
bond
¨ It’s
an
internal
rate
of
return
that
sets
the
discounted
cash
flow
on
the
right
hand
side
to
the
market
price
of
the
bond,
P,
on
the
lek
hand
side
∑= +
=
M
1i
t
i
i
)y(1
CF
P∑=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
M
1i
i
i
m
y
1
FC
P
10
y
is
the
nominal
annual
yield
to
maturity
in
this
formula
with
integer
periods
y
is
effec8ve
annual
yield
to
maturity
in
this
formula
with
discrete
real
-me
periods

11. ¨ For
a
frac-onal
ini-al
coupon
period:
t1
<
∆t
Frac-onal
Ini-al
Time
Period
For
a
bond
with
semi-­‐annual
coupons,
assume
that
the
next
coupon
payment
is
in
3
months.
The
coupon
payments
occur
at
t0=0.0,
t1=0.25,
t2=0.75,
t3=1.25,
t4
=
1.75,
…
i=0
i=1
i=2
i=M
t0=0.0
t1
t2=t1+Δt
tM=
T
C
=
coupon
payment
F
=
face
or
par
value
11

12. Zero
Coupon
Bonds
Again
¨ A
bond
dealer
can
split
a
coupon
bond
into
ZCBs
¤ one
for
the
principal
and
¤ one
for
each
coupon
¤ This
is
called
‘stripping’
the
bond
¨ The
advantage
of
a
ZCB
is
that
there
is
no
reinvestment
risk
¨ For
a
ZCB,
the
yield,
y,
is
the
zero
coupon
rate
denoted
as
z
12

13. Bond
Equa-on
Applica-ons
¨ Find
the
yield-­‐to-­‐maturity,
y,
from
a
known
market
price,
P
¤ Solve
for
y
(nominal,
y,
or
effec-ve,
y
‘bar’)
¤ Solve
for
the
roots
of
a
nonlinear
equa-on
n In
this
course
use
Excel
Goal
Seek
¤ Example:
Compute
both
the
effec8ve
and
nominal
yield
for
a
bond
with
$1000
face
value,
current
market
price
of
$800,
coupon
rate
of
7%
paid
semiannually,
and
4.5
years
to
maturity.
∑= +
=
M
1i
t
i
i
)y(1
CF
P∑=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
M
1i
i
i
m
y
1
FC
P
13

14. Bond
Equa-on
Applica-ons
$1,000 F
7.00% c nominal
13.434% y effective
t CF DF DCF
0 $0 $0.00
0.5 $35 0.939 $32.86
1 $35 0.882 $30.85
1.5 $35 0.828 $28.97
2 $35 0.777 $27.20
2.5 $35 0.730 $25.54
3 $35 0.685 $23.98
3.5 $35 0.643 $22.51
4 $35 0.604 $21.14
4.5 $1,035 0.567 $586.94
Sum $1,315 P $800.00
13.011% y nominal
t i
CF DF DCF
0 0 $0 $0.00
0.5 1 $35 0.939 $32.86
1 2 $35 0.882 $30.85
1.5 3 $35 0.828 $28.97
2 4 $35 0.777 $27.20
2.5 5 $35 0.730 $25.54
3 6 $35 0.685 $23.98
3.5 7 $35 0.643 $22.51
4 8 $35 0.604 $21.14
4.5 9 $1,035 0.567 $586.94
Sum $1,315 P $800.00
∑= +
=
M
1i
t
i
i
)y(1
CF
P ∑=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
M
1i
i
i
m
y
1
FC
P
14

15. Bond
Equa-on
Applica-ons
¤ Convert
the
nominal
yield
to
the
effec-ve
yield
¨ Find
market
price
from
a
known
yield
¤ For
the
bond
in
the
last
example,
what
is
the
price?
n Given
an
effec8ve
annual
yield
of
12%
or
n A
nominal
annual
yield
of
12%
1
2
y
1y
1
2
%011.13
1%434.13
2
2
−⎟
⎠
⎞
⎜
⎝
⎛
+=
−⎟
⎠
⎞
⎜
⎝
⎛
+=
15

16. Bond
Equa-on
Applica-ons
$1,000 F
7.00% c nominal
12.000% y effective
t CF DF DCF
0 $0 $0.00
0.5 $35 0.945 $33.07
1 $35 0.893 $31.25
1.5 $35 0.844 $29.53
2 $35 0.797 $27.90
2.5 $35 0.753 $26.36
3 $35 0.712 $24.91
3.5 $35 0.673 $23.54
4 $35 0.636 $22.24
4.5 $1,035 0.601 $621.53
Sum $1,315 P $840.34
∑= +
=
M
1i
t
i
i
)y(1
CF
P
12.000% y nominal
t i
CF DF DCF
0 0 $0 $0.00
0.5 1 $35 0.943 $33.02
1 2 $35 0.890 $31.15
1.5 3 $35 0.840 $29.39
2 4 $35 0.792 $27.72
2.5 5 $35 0.747 $26.15
3 6 $35 0.705 $24.67
3.5 7 $35 0.665 $23.28
4 8 $35 0.627 $21.96
4.5 9 $1,035 0.592 $612.61
Sum $1,315 P $829.96
∑=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
M
1i
i
i
m
y
1
FC
P
16

17. Bond
Equa-on
Applica-ons
¨ For
the
bond
with
a
12%
effec-ve
yield
and
price
$840.34
at
-me
0,
here’s
a
plot
of
price
as
-me
progress
from
0
to
4.5
years
assuming
a
constant
yield
of
12%
$825
$850
$875
$900
$925
$950
$975
$1,000
$1,025
$1,050
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Time
Price17

18. Yield
Curve
¨ A
graph
can
be
developed
that
plots
various
(annualized)
yields-­‐to-­‐maturity
against
-mes-­‐to-­‐maturity
for
bonds
from
the
same
issuing
en-ty
or
from
issuing
en--es
with
the
same
risk
y
0
T
18
Bloomberg
Animated
yield
curve

19. Yield
Curve
¨ The
most
useful
version
of
the
yield
curve
plots
zero
coupon
bond
yields
against
-mes-­‐to-­‐maturity
¨ A
zero
coupon
yield
curve
depicts
pure
interest
rates
with
no
ambiguity
in
the
risk
associated
with
the
yield
¨ Thus
we
can
use
ZCB
yield
curve
to
plot
the
interest
rate
with
respect
to
-me-­‐to-­‐maturity,
or
holding
period
¨ Yields
on
zero
coupon
U.S.
Treasury
debt
are
risk
free
¨ The
addi-onal
yield
for
non-­‐U.S.
Treasury
debt
is
due
largely
to
credit
risk
and
is
called
a
risk
premium
or
a
credit
spread
¨ Moody’s
and
Standard
&
Poor’s
evaluate
bond
credit
risk
via
analysis
of
the
issuer’s
financial
posi-on
and
assign
risk
levels
such
as
AA
that
translate
into
credit
risk
premiums
19

20. Corporate
Credit
Ra-ng
From Investopedia
20
AAA companies

21. Yield
Curve
¨ A
corporate
bond,
say
a
AA
rated
bond,
might
have
the
following
yield
curve
y
0
T
AA
Corporate
Bond
Yield
Curve
U.S.
Treasury
Debt
Yield
Curve
risk
premium
and
credit
spread
21

22. Other
Types
of
Risk
¨ Another
type
of
risk
known
as
liquidity
risk
¨ Results
from
a
bond
having
few
buyers
and
sellers
¤ The
issue
is
“illiquid”
¨ Newer
U.S.
Treasuries
(referred
to
as
“on
the
run”)
can
trade
with
a
liquidity
premium
compared
with
old
U.S.
Treasuries
(referred
to
as
“off
the
run”)
y
=
f(risk
free
-me
value
of
money,
credit
risk,
liquidity
risk)
22

23. Reinvestment
Risk
¨ Consider
a
$1000
bond
with
a
coupon
rate
of
10%
paid
annually
for
10
years.
Ini-ally,
the
yield
is
11%,
the
price
is
$941.11,
and
the
yield
curve
is
flat.
Prior
to
the
payment
of
the
next
coupon,
we
consider
three
scenarios
1. the
yield
curve
shiks
parallel
down
to
9%
2. the
yield
curve
remains
flat
at
11%
3. the
yield
curve
shiks
parallel
up
to
12%
What
are
the
actual
yields?
$1,000 F
10.00% c nominal Year CF DF DCF 9% 11% 12%
11.00% y nominal 1 100$ 0.9009 90.09$ 217.19$ 255.80$ 277.31$
2 100$ 0.8116 81.16$ 199.26$ 230.45$ 247.60$
3 100$ 0.7312 73.12$ 182.80$ 207.62$ 221.07$
4 100$ 0.6587 65.87$ 167.71$ 187.04$ 197.38$
5 100$ 0.5935 59.35$ 153.86$ 168.51$ 176.23$
6 100$ 0.5346 53.46$ 141.16$ 151.81$ 157.35$
7 100$ 0.4817 48.17$ 129.50$ 136.76$ 140.49$
8 100$ 0.4339 43.39$ 118.81$ 123.21$ 125.44$
9 100$ 0.3909 39.09$ 109.00$ 111.00$ 112.00$
10 1,100$ 0.3522 387.40$ 1,100.00$ 1,100.00$ 1,100.00$
Sum 941.11$ 2,519.29$ 2,672.20$ 2,754.87$
Yield To Maturity 10.35% 11.00% 11.34%
Bond Price Calculation Future Value of Coupon Reinvestment
23

24. Bond
Price
Quotes
¨ Dirty
and
clean
prices
¤ Dirty
price
n Price
from
DCF
formula
n Transac-on
price
n When
the
seller
sells
at
this
price
she
gets
the
prorated
share
(accumulated
interest)
of
the
next
coupon
¤ Clean
price
n Price
quoted
by
bond
dealer
n Excludes
accumulated
interest
on
next
coupon
payment
¤ Clean
price
=
Dirty
price
–
accumulated
interest
n Accumulated
as
simple
interest
using
applicable
day
count
conven-on
24

25. Clean
and
Dirty
Bond
Prices
Bond
purchased
just
aker
its
8/15/2008
coupon
Bond
purchased
6/12/2009
From
8/15/08
to
8/15/09
is
365
days From
8/15/08
to
6/12/09
is
301
days
or
.825
yrs
From
6/12/09
to
8/15/09
is
64
days
or
.175
yrs
Payment
Date
t CF DF DCF
15-­‐Aug-­‐08 0 $0 1.000 $0.00
15-­‐Aug-­‐09 1 $5 0.962 $4.81
15-­‐Aug-­‐10 2 $5 0.925 $4.62
15-­‐Aug-­‐11 3 $5 0.889 $4.44
15-­‐Aug-­‐12 4 $5 0.855 $4.27
15-­‐Aug-­‐13 5 $5 0.822 $4.11
15-­‐Aug-­‐14 6 $105 0.790 $82.98
Sum $130.00 P $105.24
Payment
Date
t CF DF DCF
12-­‐Jun-­‐09 0 $0 1.000 $0.00
15-­‐Aug-­‐09 0.175 $5 0.993 $4.97
15-­‐Aug-­‐10 1.175 $5 0.955 $4.77
15-­‐Aug-­‐11 2.175 $5 0.918 $4.59
15-­‐Aug-­‐12 3.175 $5 0.883 $4.41
15-­‐Aug-­‐13 4.175 $5 0.849 $4.24
15-­‐Aug-­‐14 5.175 $105 0.816 $85.71
Sum $130.00 P $108.70
F=$100,
c=5%,
y=4%,
m=1
25

26. $100
$101
$102
$103
$104
$105
$106
$107
$108
$109
$110
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Time
PriceDirty
and
Clean
Prices
15
Aug
2008
Clean
price
@
6/12/09
=
Dirty
price
–
accumulated
interest
=
$108.70
–
$5
·∙
301
/
365
=
$108.70
-­‐
$4.12
=
$104.58
15
Aug
2009
26
12
June
2009

27. Bond
Price
Quotes
¨ Bid
and
ask
prices
¤ The
clean
price
is
quoted
for
bid
and
ask
prices
n The
dealer
will
buy
a
bond
at
the
bid
price
n The
dealer
will
sell
a
bond
at
the
ask
(offer)
price
¨ Bond
prices
are
quoted
rela-ve
to
100
regardless
of
actual
par
value
¤ Price
is
quoted
as
a
percent
of
par
¨ Example
27

28. Plot
price
v.
yields
to
maturity
$700
$800
$900
$1,000
$1,100
$1,200
$1,300
0% 2% 4% 6% 8% 10% 12% 14% 16%
Yield
Price
F=$1000
c=7%
semiannual
T=4.5
yrs
Bond
“price
–
yield”
or
P-­‐y
curve
Illustrates
how
price
changes
as
yield-­‐to-­‐maturity
changes
for
a
par-cular
bond
(
c,
m,
M,
and
F
are
constant)
Each
point
represents
a
DCF
calcula-on
∑= +
=
M
1i
t
i
i
)y(1
CF
P
28

29. Determine
the
fair
price
of
a
bond
¨ In
this
case
c,
m,
T,
and
the
relevant
zero
coupon
yield
curve
are
known
¨ Compute
the
fair
value,
P
∑= +
=
M
1i
t
i
i
i
)z(1
CF
P
zt
0
T
for
zero
coupon
bonds
ti
for
bond
cash
flows
CFt
Cash
flow
diagram
Zero
coupon
bond
yield
curve
29

30. Example
of
pricing
a
bond
Price of the bond
F=$1000
c=7%
semiannual
T=4.5
yrs
With
the
following
zero
coupon
yield
curve
$1,000 F t CF z DF DCF
7.00% c
nominal 0 $0 4.00% $0.00
0.5 $35 4.85% 0.977 $34.18
1 $35 5.20% 0.951 $33.27
1.5 $35 5.47% 0.923 $32.31
2 $35 5.70% 0.895 $31.33
2.5 $35 5.90% 0.867 $30.33
3 $35 6.08% 0.838 $29.32
3.5 $35 6.24% 0.809 $28.31
4 $35 6.40% 0.780 $27.31
4.5 $1,035 6.55% 0.752 $778.09
Sum $1,315 P $1,024.46
0%
1%
2%
3%
4%
5%
6%
7%
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
t
z
30

31. Home
Mortgage
Calcula-on
¨ Given
the
nominal
interest
rate,
m=12,
P,
and
N,
what
is
the
monthly
payment,
C?
¨ C
:
monthly
payment
¤ Includes
principal
repayment
and
interest
–
there
is
no
return
of
principal
“F”
¨ N
:
number
of
years
¨ m
:
number
of
compounding
periods
per
year
(12
for
home
loans)
¨ y
:
nominal
fixed
interest
rate
for
the
loan
¨ P
:
loan
principal
(the
mortgage
amount)
¨ Solve
for
C
using
Excel
Goal
Seek
¤ Find
the
value
of
C
that
equates
the
lek
and
right
hand
sides
∑=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
M
1i
i
i
m
y
1
C
P
31

32. Mortgage
Example
¨ You
wish
to
borrow
$300,000
at
6.5%
fixed
for
30
years.
¨ The
following
excel
table
shows
the
calcula-ons
for
the
first
12
months
and
the
last
5
months.
¨ The
monthly
payment
of
$1896
is
determined
using
goal
seek
to
force
the
sum
of
the
last
column
to
$300,000.
¨ Note
that
you
will
pay
out
$682,633
in
principal
and
interest
¤ $300,000
in
principal
¤ $382,633
in
interest
32

33. Mortgage
Example
t i
CF DF DCF
0.000 0 -­‐$
-­‐$
0.083 1 1,896$
0.995 1,886$
0.167 2 1,896$
0.989 1,876$
0.250 3 1,896$
0.984 1,866$
0.333 4 1,896$
0.979 1,856$
0.417 5 1,896$
0.973 1,846$
0.500 6 1,896$
0.968 1,836$
0.583 7 1,896$
0.963 1,826$
0.667 8 1,896$
0.958 1,816$
0.750 9 1,896$
0.953 1,806$
0.833 10 1,896$
0.947 1,796$
0.917 11 1,896$
0.942 1,787$
1.000 12 1,896$
0.937 1,777$
29.667 356 1,896$
0.146 277$
29.750 357 1,896$
0.145 276$
29.833 358 1,896$
0.145 274$
29.917 359 1,896$
0.144 273$
30.000 360 1,896$
0.143 271$
Sum 682,633$
P 300,000$
∑=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
M
1i
i
i
m
y
1
C
P
$300,000 P
6.500% y
nominal
12 m
6.697% y
annual
effective
0.542% y
monthly
effective
33

34. Perpetuity
34
Now
in
the
case
that
M=∞
C
is
constant
and
of
course
y
<
1
This
is
a
perpetuity
⎟
⎠
⎞
⎜
⎝
⎛
=
m
y
C
P
P=
Ci
(1+y)i
i=1
M
∑
P=C⋅
1
(1+y)i
i=1
∞
∑
P=
C
y
P=
C
y
P=
Ci
(1+y)i
i=1
M
∑
If
a
nominal
annual
rate,
y,
is
used
then
P
C
i
Example:
How
much
money
do
you
need
to
invest,
P,
to
pay
out
$1
per
year
forever
if
the
pay
out
rate
is
10%
(effec-ve)
per
year?

35. Annuity
35
Now
how
much
money
do
you
need
to
invest
at
10%
to
receive
a
$1
/
year
payout
for
M
years
?
That’s
an
annuity
(a
perpetuity
would
pay
out
forever)
P
C
i
M
M+1
PM+1
=
C
y
P0
M+1
=
C
y
⋅
1
1+y( )
M
=
C
y
⋅ 1+y( )
−M
P = P0
=
C
y
−
C
y
⋅ 1+y( )
−M
=
C
y
⋅ 1− 1+y( )
−M#
$
%
&
'
(
P=
C
y
⋅ 1& 1+y( )
&M"
#
$
%
&
'
C =
P⋅ y
1& 1+y( )
&M
=
P⋅ y⋅ 1+y( )
M
1+y( )
M
−1
P=
C⋅m
y
⋅ 1' 1+
y
m
"
#
$
%
&
'
'M(
)
*
*
+
,
-
-
C=
P⋅
y
m
"
#
$
%
&
'⋅ 1+
y
m
"
#
$
%
&
'
M
1+
y
m
"
#
$
%
&
'
M
'1
M=20
years
C=$1
Y=10%
P=$8.51

36. Annui-es
36

37. Closed
Form
Formulas
¨ Annuity
¤ Home
mortgage
annuity
formula
example
¨ Bonds
¤ Annuity
for
coupon
payment
plus
the
discounted
face
value
20.1896$
1)%542.0(1
0.542%)(1
0.542%$300,000
C 360
360
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+
+⋅⋅
=
MM
m
y
1
F
m
y
1
m
y
1
m
y
1
CP
⎟
⎠
⎞
⎜
⎝
⎛
+
+
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
⋅=
37

38. Closed
Form
Formulas
¨ Bonds
¤ Example
of
bond
w/
F=$1000,
c=7%
semi-­‐annual,
T=4.5yrs,
y
annual
nominal
=
13.011%
¤ Bond
with
frac-onal
ini-al
period
00.800$
2
y
1
$1000
2
13.011%
1
2
13.011%
1
2
13.011%
1
35$P 99
=
⎟
⎠
⎞
⎜
⎝
⎛
+
+
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
⋅=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
+
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⋅⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
+⋅=
d
eMM
m
y
1
1
m
y
1
F
m
y
1
m
y
1
m
y
1
1CP
38

39. Closed
Form
Formulas
.825
.175
last
coupon
next
coupon
e=64
days
d
=
365
days
e/d=.175
8/15/08
8/15/09
8/15/10
8/15/11
8/15/12
8/15/13
8/15/14
6/12/09
F=$100
y=4%
annual
c=5%
annual
y
&
c
are
effec-ve
&
nominal
Clean and Dirty Price example (p. 7.10) using closed form
$100
$101
$102
$103
$104
$105
$106
$107
$108
$109
$110
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Time
Price
39
70.108$
)%4(1
1
)%4(1
$100
)%44%(1
1
4%
1
15$P
365
6455
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
+⎟
⎠
⎞
⎜
⎝
⎛
+
−+⋅=