1
F520_Bonds (Part b)
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.
- Apply the concepts of chapter 18 to a note that is between coupon dates.
- 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.
Outline of the questions we will answer.
- Determine the accrued interest for this note.
- If you bought this note today, what would you pay for this note? Include the quoted price and the accrued interest.
- If you sold this note today, what would you pay for this note?
- Using your time value of money (TVM) keys can you show how this note’s price was obtained?
- Using the note function keys on your financial calculator can you calculate this note’s price, accrued interest, and yield to maturity (YTM)?
- Using Excel can you calculate this note’s price, accrued interest, and yield to maturity (YTM)?
- What is this note’s nominal rate and its effective annual rate.
- Calculate this note’s modified duration using the approximation method.
- Calculate this note’s Macaulay duration and modified duration using the precise method.
- 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.
- 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.
- Calculate the percentage change and the dollar value change using convexity. Assume a YTM that is 30 basis points higher.
Rewritten from the Wall Street Journal (3/11/07)
Rate / Mo/Yr / Bid / Asked / Chg / Ask YldBond 1 / 6 1/8 / Dec 31, 2010n / 101.6875% / 101.75% / .09375 / 5.60
Bond 2 / 9 1/4 / Aug 15, 2007n / 101.5625% / 101.625% / 5.36
1.Determine the accrued interest for this note.
The accrued interest is equal to:
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 / Bond 2
Accrued Interest (%) /
=1.18439% /
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 ($) /
=$11.8439 /
=$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
2.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 / Bond 2Quoted Price (%) / 101.75% / 101.625%
Accrued Interest (%) /
=1.18439% /
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 (% of par) / 102.9344% / 102.2383%
Converting percents into price (par value is $1000)
Price per note / $1,029.34 / $1,022.38
3.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 / Bond 2Quoted Price (%) / 101.6875% / 101.5625%
Accrued Interest (%) /
=1.18439% /
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 (% of par) / 102.8719% / 102.1758%
Converting percents into price (par value is $1000)
Price per note / $1,028.72 / $1,021.76
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.
4.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 6/30/0712/31/07 6/30/0812/31/08 6/30/0912/31/09 6/30/1012/31/10
|||||||||
012345678
3.06253.06253.06253.06253.06253.06253.06253.0625Coupon payments (%)
100Maturity Value (%)
Step 1: Find the value on the date of the last coupon payment (12/31/06) just after the coupon was paid.
N8 coupon semi-annual pmts till maturity / I
5.60/2 = 2.80 semi-annual rate / PV
Cpt / PMT
6.125/2 =
3.0625 semi-annual cpn pmt (as %) / FV
100 the residual value (as %)
-101.8583% (value on 12/31/2006)
Step 2: Find the value today, 3/11/07.
N70/181=
0.38674 fraction of cpn period from last coupon (12/31/06)to today (3/11/07) / I
5.60/2 = 2.80 semi-annual rate / PV
-101.8583 / PMT
0 / FV
cpt
(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
Bond 2
This bond has no intermediate bond payments, so it can be valued as a lump sum, reducing one step.
N(181-24) / 181=
0.8674 fraction period left till maturity (8/15/07) to today (3/11/07) / I
5.36/2 = 2.68 semi-annual rate / PV
Cpt / PMT
0 / FV
1000 +
1000*.0925/2 =
104.625
(% of par)
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
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 / Excel2nd Bond / 2nd Bond
STD
3-11-2007
Settlement date / 3.1107
ENTER / 3.1107
ENTER / 3/11/2007
CPN
Coupon / 6.125
ENTER / 9.25
ENTER / .0925
RDT
Redemption date
or maturity date / 12.3110
ENTER
For date 12-31-2010 / 8.1507
ENTER
For date 8-15-2007 / 8/15/07
RV
Residual value
I used as a percent. / 100
ENTER / 100
ENTER / 100
360
Accounting year / 2ndSET
will show ACT / 2ndSET
will show ACT / 1=
Actual/Actual
2/Y
payments per yr / 2ndSET
Will show 2/Y
/ 2ndSETWill show 2/Y
/ 2=Semi-annual
YLD
Yield (calc) / 5.60ENTER / 5.36
ENTER / .0536
YIELD
PRIQuote Price (calc) / CPT
=101.7676 / CPT
=101.6348 / PRICE
=101.6348
AI
Accrued interest (calc) / CPT
=1.1844 / CPT
=0.6133 / ACCRINT
=0.6133
Calculations using an HP12c
HP12cCalculate Price / HP12c
Calculate Price / HP12c
Calculate Yield
f CLEAR FIN
5.60 i / f CLEAR FIN
5.36 i / f CLEAR FIN
101.75 PV
6.125 PMT / 9.25 PMT / 6.125 PMT
g M.DY
3.112007ENTER / g M.DY
3.112007ENTER / g M.DY
3.112007ENTER
12.312010 / 8.152007 / 12.152010
f PRICE
computes price of
101.7676 / f PRICE
computes price of
101.6348 / f YTM
computes price of
5.5906
+
computes total price, including accrued interest
102.952 / +
computes total price, including accrued interest
102.2481
Accrued interest
102.9520-101.7676
=1.1844 / Accrued interest
102.2481-101.6348
=0.6133
HP10B’s do not allow for bond calculations.
Calculations using an HP17b
HP17bCalculate Price / HP17b
Calculate Price / HP17b
Calculate Yield
FIN Bond
_Clear Data / FIN Bond
_Clear Data / FIN Bond
_Clear Data
Type ACT / Type ACT / Type ACT
Semi Exit / Semi Exit / Semi Exit
03.112007Sett / 03.112007Sett / 03.112007Sett
12.312010Mat / 08.152007Mat / 12.312010Mat
6.125 cpn% / 9.25 cpn% / 6.125 cpn%
more / more / more
5.6 yld% / 5.36 yld% / 101.75 price
PRICE
computes price of
101.7676 / PRICE
computes price of
101.6348 / YTM
computes YTM of
5.5906
+
computes total price, including accrued interest
102.952 / +
computes total price, including accrued interest
102.2481
Accrued interest
102.9520-101.7676
=1.1844 / Accrued interest
102.2481-101.6348
=0.6133
6.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.)
7.What is this note’s nominal rate and its effective annual rate.
Bond 1
Nominal rate is 5.60 percent
Effective annual rate is
Bond 2
Nominal rate is 5.36 percent
Effective annual rate is
8.Calculate this note’s modified duration using the approximation method.
Using a Financial Calculator and our estimation formula:
BAII Plus / BAII Plus / BAII PlusP0 / P+ / P-
STD / 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
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.
9.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)
Note: Since t is measured in 6-month periods, we use the semi-annual rate (equal to the YTM / 2).
Method 2 for calculating exact Duration (alternative):
Note: Since t is measured in 6-month periods, we use the semi-annual rate (equal to the YTM / 2).
Method 3 for calculating exact Duration (alternative):
Note: Since t is measured in years (instead of semi-annual periods), we must use the Effective Annual Rate in discounting.
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:
Price = -0.009980545 * 102.9520% = -1.0275%
= -$10.28
Using the duration calculated in approximate method:
Price = -0.01008 * 102.9520% = -1.0378%
= -$10.38
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.
BAII Plus / BAII PlusSTD / 3-11-07 / 3-11-07
CPN / 6.125 / 6.125
RDT / 12-31-2010 / 12-31-2010
RV / 100 / 100
360 / 2nd set ACT / 2nd set ACT
2/Y / 2/Y / 2/Y
YLD / 5.60 / 5.90
PRI / CPT=101.7676 / CPT=100.7467
101.7676% – 100.7467% = 1.0209% = $10.21 reduction
12.Calculate the percentage change and the dollar value change using convexity.
Duration:
Total Convexity:
Expected Price Change assuming +30 basis points :
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.