F520_Bonds (Part b)
1
Valuation of Debt Contracts
and Their Price Volatility Characteristics
Part b
Please review these notes for next week. I will
answer questions and go over highlights of these
notes briefly next week.
This set of the chapter 18 notes has two purposes.
1. Apply the concepts of chapter 18 to a note that is
between coupon dates.
2. Gain a functional knowledge of your calculators
and Excel.
The following 3 pages provide information about a
Treasury note that was issued on December 31, 2006
and matured on December 31, 2010. The Wall Street
Journal quotes are for March 11, 2007. Our goal is to
provide a set of detailed calculations for this note
using all the concepts in Chapter 18.
The difference between this note and the material
typically used in textbooks is that we are not
evaluating bonds only on their last coupon date. Our
evaluation period is in-between coupon dates. This
will mean that each of our calculations will need
some adjustment.
F520_Bonds (Part b)
2
Outline of the questions we will answer.
1. Determine the accrued interest for this note.
2. If you bought this note today, what would you pay for this
note? Include the quoted price and the accrued interest.
3. If you sold this note today, what would you pay for this
note?
4. Using your time value of money (TVM) keys can you show
how this note’s price was obtained?
5. Using the note function keys on your financial calculator
can you calculate this note’s price, accrued interest, and
yield to maturity (YTM)?
6. Using Excel can you calculate this note’s price, accrued
interest, and yield to maturity (YTM)?
7. What is this note’s nominal rate and its effective annual
rate.
8. Calculate this note’s modified duration using the
approximation method.
9. Calculate this note’s Macaulay duration and modified
duration using the precise method.
10. Using the information just calculated in question 9,
calculate the percentage change in price and the dollar
change in price for this note. Assume interest rates
increase by 30 basis points.
11. Compare your calculations of price changes in question 10
with the price that you obtain from a financial calculator
using a yield-to-maturity that is 30 basis points higher.
12. Calculate the percentage change and the dollar value change
using convexity. Assume a YTM that is 30 basis points higher.
F520_Bonds (Part b)
3
Rewritten from the Wall Street Journal (3/11/07)
Rate
Mo/Yr
Bid
Asked
Bond 1
Bond 2
1.
6 1/8 Dec 31,
2010n
9 1/4 Aug 15,
2007n
101.6875% 101.75%
101.5625% 101.625%
Chg
Ask
Yld
.09375 5.60
5.36
Determine the accrued interest for this note.
The accrued interest is equal to:
AI  PAR *
CPN A
*
2
E
PAR = Par value of bond
CPN = Annual coupon payment
A = Number of days from the beginning of the coupon
period to the settlement date (today)
E = Number of days in coupon period in which the
settlement date falls
Accrued interest in dollars as a percent of par:
Bond 1
.06125 70
 100 *
*
Accrued
2
181
Interest (%) =1.18439%
Bond 2
 100 *
.0925 24
*
2
181
0.61326%
A=70 days from 12/31/06 to 3/11/07
E=181 days from 12/31/06 to 6/30/07
A=24 days from 2/15/07 to 3/11/07
E= 181 days from 2/15/07 to 8/15/07
Accrued interest in dollars:
Accrued
Interest ($)
 1000 *
.06125 70
*
2
181
=$11.8439
 1000 *
.0925 24
*
2
181
=$6.1326
A=70 days from 12/31/06 to 3/11/07
E=181 days from 12/31/06 to 6/30/07
A=24 days from 2/15/07 to 3/11/07
E= 181 days from 2/15/07 to 8/15/07
F520_Bonds (Part b)
2.
4
If you bought this note today, what would you pay
for this note? Include the quoted price and the
accrued interest.
Remember investors buy at the ask and sell at the bid
price. Also when buying a bond you pay the quoted
price plus accrued interest.
Bond 1
101.75%
Bond 2
101.625%
Quoted
Price (%)
.0925 24
.06125 70
A=70 days from 12/31/06 to 3/11/07
 100 *
*
 100 *
*
Accrued
E=181 days from 12/31/06 to 6/30/07
2
181
2
181
Interest
=1.18439% 0.61326%
A=24 days from 2/15/07 to 3/11/07
E= 181 days from 2/15/07 to 8/15/07
(%)
Price Paid 102.9344% 102.2383%
(% of par)
Converting percents into price (par value is $1000)
Price per
$1,029.34
$1,022.38
note
F520_Bonds (Part b)
3.
5
If you sold this note today, what would you receive
for this note?
Remember investors buy at the ask and sell at the bid
price. Also when selling a bond you receive the
quoted price plus accrued interest.
Bond 1
101.6875%
Quoted
Price (%)
.06125 70
 100 *
*
Accrued
2
181
Interest (%) =1.18439%
Bond 2
101.5625%
 100 *
.0925 24
*
2
181
0.61326%
A=70 days from 12/31/06 to 3/11/07
E=181 days from 12/31/06 to 6/30/07
A=24 days from 2/15/07 to 3/11/07
E= 181 days from 2/15/07 to 8/15/07
Price Paid 102.8719% 102.1758%
(% of par)
Converting percents into price (par value is $1000)
Price per
$1,028.72
$1,021.76
note
The 0.0625% difference between the bid and the ask is
the dealers revenue. This is separate from a brokerage
commission that may be paid to a broker to contact the
dealer.
F520_Bonds (Part b)
6
4.
Today is
3/11/07
Using your time value of money (TVM) keys can
you show how this note’s price was obtained?
Finding the value of this bond, if the current interest rate for this
risk class is known.
Bond 1:
12/31/06 6/30/07 12/31/07 6/30/08 12/31/08 6/30/09 12/31/09 6/30/10 12/31/10
|
|
|
|
|
|
|
|
|
0
1
2
3
4
5
6
7
8
3.0625 3.0625 3.0625 3.0625 3.0625 3.0625 3.0625 3.0625 Coupon payments (%)
100
Maturity Value (%)
Step 1: Find the value on the date of the last coupon payment
(12/31/06) just after the coupon was paid.
N
I
PV
PMT
FV
8 coupon
5.60/2 =
Cpt
6.125/2 =
100 the
semi-annual
residual value
2.80
semi3.0625
semipmts till
(as %)
maturity
annual rate
annual cpn pmt
(as %)
-101.8583% (value on 12/31/2006)
Step 2: Find the value today, 3/11/07.
N
I
PV
PMT
70/181=
5.60/2 =
-101.8583 0
0.38674
2.80 semi-
FV
cpt
fraction of cpn
annual rate
period from last
coupon
(12/31/06)to
today (3/11/07)
(value 3/11/2007) 102.9529%
Step 3: Find the quoted value by subtracting the accrued interest
from the value in Step 2.
102.9529 – 1.18439 = 101.76851% = $1,017.69
F520_Bonds (Part b)
7
Bond 2
This bond has no intermediate bond payments, so it can be
valued as a lump sum, reducing one step.
N
I
(181-24) / 5.36/2 =
181=
2.68 semiannual rate
0.8674
PV
Cpt
PMT
0
FV
1000 +
1000*.0925/2 =
104.625
(% of par)
fraction period
left till maturity
(8/15/07) to
today (3/11/07)
102.252 (ask + accrued interest)
Step 2: Find the quoted value by subtracting the accrued interest
from the value in the previous step.
102.252 – 0.61326 = 101.639% = $1,016.39
F520_Bonds (Part b)
8
5.
Using the bond function keys on your financial
calculator can you calculate this note’s price,
accrued interest, and yield to maturity (YTM)?
BAII Plus
BAII Plus
Excel
2nd Bond
2nd Bond
STD
3.1107
3.1107
3/11/2007
3-11-2007
ENTER 
ENTER 
Settlement date
CPN
6.125
9.25
.0925
Coupon
ENTER 
ENTER 
RDT
12.3110
8.1507
8/15/07
Redemption date
ENTER 
ENTER 
or maturity date
For date 12-31-2010
For date 8-15-2007
100
ENTER 
100
ENTER 
100
360
2nd SET 
2nd SET 
Accounting year
will show ACT
will show ACT
2/Y
2nd SET 
2nd SET 
payments per yr
Will show 2/Y
Will show 2/Y
YLD
5.60
ENTER 
CPT
=101.7676
CPT
=1.1844
5.36
ENTER 
CPT
=101.6348
CPT
=0.6133
1=
Actual/Actual
2=
Semi-annual
.0536
RV
Residual value
I used as a
percent.
Yield (calc)
PRI
Quote Price
(calc)
AI
Accrued interest
(calc)
YIELD
PRICE
=101.6348
ACCRINT
=0.6133
F520_Bonds (Part b)
9
Calculations using an HP12c
HP12c
HP12c
Calculate Price
Calculate Price
f CLEAR FIN
f CLEAR FIN
5.60 i
5.36 i
6.125 PMT
9.25 PMT
g M.DY
g M.DY
3.112007 ENTER 3.112007 ENTER
12.312010
8.152007
f PRICE
f PRICE
HP12c
Calculate Yield
f CLEAR FIN
101.75 PV
6.125 PMT
g M.DY
3.112007 ENTER
12.152010
f YTM
computes price of
computes price of
computes price of
101.7676
+
101.6348
+
5.5906
computes total price,
including accrued interest
computes total price,
including accrued interest
102.952
Accrued interest
102.9520-101.7676
=1.1844
102.2481
Accrued interest
102.2481-101.6348
=0.6133
HP10B’s do not allow for bond calculations.
F520_Bonds (Part b)
10
Calculations using an HP17b
HP17b
HP17b
Calculate Price
Calculate Price
FIN Bond
FIN Bond
_ Clear Data
_ Clear Data
Type ACT
Type ACT
Semi Exit
Semi Exit
03.112007 Sett
03.112007 Sett
12.312010 Mat
08.152007 Mat
6.125 cpn%
9.25 cpn%
more
more
5.6 yld%
5.36 yld%
PRICE
PRICE
HP17b
Calculate Yield
FIN Bond
_ Clear Data
Type ACT
Semi Exit
03.112007 Sett
12.312010 Mat
6.125 cpn%
more
101.75 price
YTM
computes price of
computes price of
computes YTM of
101.7676
+
101.6348
+
5.5906
computes total price,
including accrued interest
computes total price,
including accrued interest
102.952
Accrued interest
102.9520-101.7676
=1.1844
102.2481
Accrued interest
102.2481-101.6348
=0.6133
F520_Bonds (Part b)
6.
11
Using Excel can you calculate this note’s price,
accrued interest, and yield to maturity (YTM)?
Calculations using Excel (Excel insert: Please double click
on this page to see the Excel sheet and use the function
wizard to see the cell formulas.)
Today
Rate
6.125%
9.250%
PRICE=
PRICE=
ACCRINT=
ACCRINT=
3/11/2007
Maturity
12/31/2010
8/15/2007
Calculation
101.7676
101.6348
1.1844
0.6133
Bid
101.6875
101.5625
Ask
101.7500
101.6250
Based on Quotes
Bid %
Ask %
101.6875
101.7500
101.5625
101.6250
Chg.
0.0938
YIELD=
YIELD=
Ask
Yield
5.60%
5.36%
Calculation
5.60%
5.36%
100
100
F520_Bonds (Part b)
7.
12
What is this note’s nominal rate and its effective
annual rate.
Bond 1
Nominal rate is 5.60 percent
Effective annual rate is
 0.056 
1

2 

2
 1  0.0568  5.68%
Bond 2
Nominal rate is 5.36 percent
Effective annual rate is
2
 0.0536 
1

2 

 1  0.0543  5.43%
F520_Bonds (Part b)
8.
13
Calculate this note’s modified duration using the
approximation method.
Using a Financial Calculator and our estimation formula:
BAII Plus
BAII Plus
BAII Plus
P0
P+
PSTD
3-11-07
3-11-07
3-11-07
CPN
6.125
6.125
6.125
RDT
12-31-2010
12-31-2010
12-31-2010
RV
100
100
100
360
2nd set ACT
2nd set ACT
2nd set ACT
2/Y
2/Y
2/Y
2/Y
YLD
5.60
5.70
5.50
PRI
CPT=101.7676 CPT=101.4259 CPT=102.1107
Macaulay Duration

102.1107  101.4259
D P P 
 3.3645
2 * P * y
2 *101.7676 * .001


0
Using Excel:
Bond 2 Duration equals time to maturity. All cash flows occur
at maturity. 157 days (0.4301 years) from 3/11/07 to 8/15/07.
F520_Bonds (Part b)
9.
14
Calculate this note’s Macaulay duration and
modified duration using the precise method.
Double Checking our Duration Calculation:
Method 1 for Calculating Exact Duration (recommended)
Wall Street Journal Quote 3/11/2007: (Understanding Duration)
6.125% Dec 10n
101.6875
101.7500
0.09375
Maurity
12/31/2010
Today
3/11/2007
Last Coupon Date
12/31/2006
Calculate Exact Duration:
t
6/30/2007
1
12/31/2007
2
6/30/2008
3
12/31/2008
4
6/30/2009
5
12/31/2009
6
6/30/2010
7
12/31/2010
8
t
#
CF
30.625
30.625
30.625
30.625
30.625
30.625
30.625
1030.625
PV(CF)#
$29.79
$28.98
$28.19
$27.42
$26.68
$25.95
$25.24
$826.33
$1,018.58
PV*t
29.79
57.96
84.57
109.69
133.38
155.69
176.69
6,610.67
7,358.45
5.60%
Col. for
convexity only
PV*t*(t+1)
59.58
173.88
338.28
548.45
800.26
1,089.85
1,413.55
59,496.07
63,919.92
is semi-annual periods
Used the 6 month rate (YTM/2)
Duration as of last coupon date is
Adjusting Duration for today.
Exact Duration (as of Trading Date)is
Excel calculation
Total Convexity Calculation
59.38 semi-annual periods
7.22 Semi-annual periods
3.61 Years
Subtract last coupon from today
0.19 =70/365=(3/11/98 - 12/31/97)/365
3.42 Years (Macaulay Duration)
3.42 Years (Macaulay Duration)
14.85 yearly convexity
Note: Since t is measured in 6-month periods, we use the semiannual rate (equal to the YTM / 2).
F520_Bonds (Part b)
15
Method 2 for calculating exact Duration (alternative):
Wall Street Journal Quote 3/11/2007: (Understanding Duration)
6.125% Dec 10n
101.6875
101.7500
0.09375
Maurity
12/31/2010
Today
3/11/2007
Last Coupon Date
12/31/2006
Calculate Exact Duration
t
6/30/2007 0.608219
12/31/2007 1.616438
6/30/2008 2.613699
12/31/2008 3.621918
6/30/2009 4.613699
12/31/2009 5.621918
6/30/2010 6.613699
12/31/2010 7.621918
t
#
CF
30.625
30.625
30.625
30.625
30.625
30.625
30.625
1030.625
PV(CF)#
$30.11
$29.29
$28.49
$27.71
$26.96
$26.22
$25.51
$835.01
$1,029.31
PV*t
18.32
47.34
74.47
100.36
124.39
147.41
168.73
6,364.36
7,045.39
5.60%
Col. for
convexity only
PV*t*(t+1)
29.46
123.87
269.11
463.87
698.30
976.16
1,284.69
54,872.95
58,718.40
is semi-annual periods
Used the 6 month rate (YTM/2)
Caculation of Exact Duration
Macaulay Duration =
Excel calculation
Total Convexity Calculation
53.98 semi-annual periods
6.84 Semi-annual periods
3.42 Years
3.42 Years (Macaulay Duration)
13.50 yearly convexity
Note: Since t is measured in 6-month periods, we use the semiannual rate (equal to the YTM / 2).
F520_Bonds (Part b)
16
Method 3 for calculating exact Duration (alternative):
Wall Street Journal Quote 3/11/2007: (Understanding Duration)
6.125% Dec 10n
101.6875
101.7500
0.09375 5.60%
Maurity
12/31/10
EFF
5.68%
Today
3/11/07
Calculate Exact Duration:
PV(CF)#
t
CF
PV*t
6/30/07 0.30411
30.625
$30.11
9.16
12/31/07 0.808219
30.625
$29.29
23.67
6/30/08 1.306849
30.625
$28.49
37.24
12/31/08 1.810959
30.625
$27.71
50.18
6/30/09 2.306849
30.625
$26.96
62.20
12/31/09 2.810959
30.625
$26.22
73.71
6/30/10 3.306849
30.625
$25.51
84.37
12/31/10 3.810959
1030.625
$835.01
3,182.18
$1,029.31
3,522.69
t
is based on a 365 day year
# Used the effective rate to be correct
Macaulay Duration =
3.42 Days
Excel
3.42 Years
Bond Price plus Accrued Interest
$1,029.31
Accrued Interest
Last Coupon Date
12/31/2006
Next Coupon Date
6/30/2007
Price from Duration Table
$
1,029.31
Less Accrued Interest
11.84
Quoted Price
$
1,017.46
Price Calculated using Excel
$
1,017.68
which matches the WSJ quote price
Note: Since t is measured in years (instead of semi-annual
periods), we must use the Effective Annual Rate in discounting.
F520_Bonds (Part b)
17
10. Using the information just calculated in question 9,
calculate the percentage change in price and the dollar
change in price for this note. Assume interest rates
increase by 30 basis points.
Using the duration calculated in precise method:
dP
  MD * R
P
dP
3.42

* .0030  .009980545  0.9981%
.056
P
1
2
Price = -0.009070545 * 102.9520% = -1.0275%
= -$10.28
Using the duration calculated in approximate method:
dP
  MD * R
P
dP
 3.36 * .0030  .01008  1.008%
P
Price = -0.01008 * 102.9520% = -1.0378%
= -$10.38
F520_Bonds (Part b)
18
11. Compare your calculations of price changes in
question 10 with the price that you obtain from a
financial calculator using a yield-to-maturity that is
30 basis points higher.
STD
CPN
RDT
RV
360
2/Y
YLD
PRI
BAII Plus
3-11-07
6.125
12-31-2010
100
2nd set ACT
2/Y
5.60
CPT=101.7676
BAII Plus
3-11-07
6.125
12-31-2010
100
2nd set ACT
2/Y
5.90
CPT=100.7467
101.7676% – 100.7467% = 1.0209% = $10.21 reduction
F520_Bonds (Part b)
19
12. Calculate the percentage change and the dollar value
change using convexity.
Duration:
D
P  P  102.1107  101.4259  3.3645
2 * P * y
2 *101.7676 * .001


0
Total Convexity:
CX 
P  P  2* P
P * y 


2
0
0

102.1107  101.4259  2 *101.7676
101.7676 *
0.001
2

0.0014
 13.7568
0.000101768
Expected Price Change assuming +30 basis points :
2
dP
1
  MD * R  CX (R)
P
2
2
dP
1
 3.3645 * .0030  13.7568 (.0030)
P
2
dP
 .0100935  .000061907  .010031594  1.00%
P
Price = -0.01 * 102.9520% = -1.02952%
= -$10.30 reduction in price
The convexity measurement is slightly better than the
approximate of –$10.38 found in part 10.