Transcript Chapter 1

Chapter 10
Bond Prices and Yields
Straight Bond
• Obligates the issuer of the bond to pay the
holder of the bond:
• A fixed sum of money (principal, par value, or face
value) at the bond’s maturity
• Constant, periodic interest payments (coupons)
during the life of the bond (Sometimes)
• Special features may be attached
• Convertible bonds
• Callable bonds
• Putable bonds
10-2
Straight Bond Basics
• $1,000 face value
• Semiannual coupon payments
AnnualCoupon
(10.1) CouponRate 
Par Value
AnnualCoupon
(10.2) Curre ntYie ld 
Bond Price
10-3
Straight Bonds
• Suppose a straight bond pays a semiannual
coupon of $45 and is currently priced at
$960.
• What is the coupon rate?
• What is the current yield?
$45  2
(10.1) CouponRate 
 9.00%
$1,000
$45  2
(10.2) Curre ntYie ld 
 9.375%
$960
10-4
Straight Bond Prices & Yield to
Maturity
• Bond Price:
• Present value of the bond’s coupon
payments
• + Present value of the bond’s face value
• Yield to maturity (YTM):
• The discount rate that equates today’s
bond price with the present value of the
future cash flows of the bond
10-5
Bond Pricing Formula


C 
1
FV

(10.3) Bond Price 
1

2
M
2M

YTM 
YTM
YTM
1

2  1 
2


PV of coupons


PV of FV
Where:
C = Annual coupon payment
FV = Face value
M = Maturity in years
YTM = Yield to maturity
10-6
Straight Bond Prices
Calculator Solution

C 
1
(10.3) Bond Price 
1

YTM
1  YTM 2


Where:
C = Annual coupon payment
FV = Face value

2M

FV


YTM
 1 
2


2M
N = 2M
I/Y = YTM/2
PMT = C/2
M = Maturity in years
FV = 1000
YTM = Yield to maturity
CPT PV
10-7
Straight Bond Prices


C 
1
FV

(10.3) Bond Price 
1
2M 
2M
YTM 
YTM
YTM
1

2  1 
2


PV of coupons


PV of FV
For a straight bond with 12 years to maturity, a coupon
rate of 6% and a YTM of 8%, what is the current price?


60 
1
  $457.41
PV of Coupons
1
24 
.08 
.
08
1

2 
1000
PV of FV 
 $390.12
24
1  .08 2




Price = $457.41 + $390.12 = $847.53
10-8
Calculating a Straight Bond Price Using Excel
• Excel function to price straight bonds:
=PRICE(“Today”,“Maturity”,Coupon
Rate,YTM,100,2,3)
• Enter “Today” and “Maturity” in quotes, using mm/dd/yyyy
format.
• Enter the Coupon Rate and the YTM as a decimal.
• The "100" tells Excel to us $100 as the par value.
• The "2" tells Excel to use semi-annual coupons.
• The "3" tells Excel to use an actual day count with 365 days per
year.
Note: Excel returns a price per $100 face.
10-9
Spreadsheet Analysis
10-10
Par, Premium and Discount Bonds
Par bonds:
Price = par value
YTM = coupon rate
Premium bonds: Price > par value
YTM < coupon rate
The longer the term to maturity,
the greater the premium over par
Discount bonds: Price < par value
YTM > coupon rate
The longer the term to maturity,
the greater the discount from par
10-11
Premium Bond Price
12 years to maturity
8% coupon rate, paid semiannually
YTM = 6%

80 
1
PV of Coupons
1

.06
.06
1


2


1000
PV of FV 
1  .03 2


  $677.42
24 


24
 $491.93
Price = $457.41 + $390.12 = $1,169.36
10-12
Discount Bonds
Consider two straight bonds with a coupon rate of 6%
and a YTM of 8%.
If one bond matures in 6 years and one in 12, what
are their current prices?

60 
1
1000

Bond Price (6 yr) 
1
 $906.15
12
12
.08 
1  .08 2  1  .08


2



60 
1
Bond Price (12 yr)
1
.08 
1  .08

2




1000

24
.08
 1 
2



24
 $847.53
10-13
Premium Bonds
Consider two straight bonds with a coupon rate of 8% and
a YTM of 6%.
If one bond matures in 6 years and one in 12, what are
their current prices?


80 
1
1000

Bond Price (6 yr) 
1
 $1,099.54
12
12

.06 
.06
1

1  .06


2 
2



80 
1
Bond Price (12 yr) 
1

.06
.06
1


2




1000

24 
.06
1


2



24
 $1,169.36
10-14
Bond Value ($) vs Years to Maturity
Premium
CR>YTM
8%>6%
YTM = CR
M
1,000
CR<YTM
6%<8%
Discount
12
6
0
10-15
Premium and Discount Bonds
• In general, when the coupon rate and YTM
are held constant:
For premium bonds: the longer the term to
maturity, the greater the premium over par
value.
For discount bonds: the longer the term to
maturity, the greater the discount from par
value.
10-16
Relationships among Yield
Measures
For premium bonds:
coupon rate > current yield > YTM
For discount bonds:
coupon rate < current yield < YTM
For par value bonds:
coupon rate = current yield = YTM
10-17
A Note on Bond Quotations
• If you buy a bond between coupon
dates:
• You will receive the next coupon payment
• You might have to pay taxes on it
• You must compensate the seller for any
accrued interest.
10-18
A Note on Bond Quotations
• Clean Price = Flat Price
• Bond quoting convention ignores accrued interest.
• Clean price = a quoted price net of accrued
interest
• Dirty Price = Full Price = Invoice Price
• The price the buyer actually pays
• Includes accrued interest added to the clean
price.
10-19
Clean vs. Dirty Prices
Example
• Today is April 1. Suppose you want to buy a
bond with a 8% annual coupon payable on
January 1 and July 1.
• The bond is currently quoted at $1,020
• The Clean price = the quoted price = $1,020
• The Dirty or Invoice price = $1,020 plus
(3mo/6mo)*$40 = $1,040
10-20
Calculating Yields


C 
1
FV

Bond Price 
1

2M
2M

YTM 
YTM
YTM
1
1
2 
2





• Trial and error
• Calculator
• Spreadsheet
10-21
Calculating Yields
Trial & Error
A 5% bond with 12 years to maturity is priced at
90% of par ($900).
Selling at a discount YTM > 5%
Try 6% --- price = $915.32 too high
Try 6.5% --- price = $876.34 too low
Try 6.25% --- price = $895.56 a little low
Actual = 6.1933%
10-22
Calculating Yields
Calculator
A 5% bond with 12 years to maturity is priced at
90% of par ($900).
N = 24
PV = -900
PMT = 25
FV = 1000
CPT I/Y = 3.0966 x 2 = 6.1933%
10-23
Calculating Yields
Spreadsheet
5% bond with 12 years to maturity,priced at 90% of par
=YIELD(“Now”,”Maturity”,Coupon, Price,100,2,3)
“Now” = “06/01/2008”
“Maturity” = “06/01/2020”
Coupon = .05
Price = 90 (entered as a % of par)
100 redemption value as a % of face value
“2” semiannual coupon payments
“3” actual day count (365)
=YIELD(“06/01/2008”,”06/01/2020”,0.05,90,100,2,3) = 0.06193276
10-24
Spreadsheet Analysis
10-25
Callable Bonds
• Gives the issuer the option to:
• Buy back the bond
• At a specified call price
• Anytime after an initial call protection
period.
• Most bonds are callable Yield-to-call may
be more relevant
10-26
Yield to Call


C 
1
CP

Callable Bond Price 
1

2T
2T

YTC 
YTC
YTC
1

2  1 
2




Where:
C = constant annual coupon
CP = Call price of bond
T = Time in years to earliest call date
YTC = Yield to call
10-27
Yield to Call
• Suppose a 5% bond, priced at 104% of par with 12
years to maturity is callable in 2 years with a $20
call premium. What is its yield to call?
N=4
# periods to first call date
PV = -1040
PMT = 25
FV = 1,020
Face value + call premium
CPT I/Y = 1.9368 x 2 = 3.874%
Remember: resulting rate = 6 month rate
10-28
Interest Rate Risk
• Interest Rate Risk = possibility that changes
in interest rates will result in losses in the
bond’s value
• Realized Yield = yield actually earned or
“realized” on a bond
• Realized yield is almost never exactly equal
to the yield to maturity, or promised yield
10-29
Interest Rate Risk and Maturity
10-30
Malkiel’s Theorems
1. Bond prices and bond yields move in
opposite directions.
2. For a given change in a bond’s YTM, the
longer the term to maturity, the greater the
magnitude of the change in the bond’s
price.
3. For a given change in a bond’s YTM, the size
of the change in the bond’s price increases at
a diminishing rate as the bond’s term ot
maturity lengthens.
10-31
Malkiel’s Theorems
4. For a given change in a bond’s YTM, the
absolute magnitude of the resulting
change in the bond’s price is inversely
related to the bond’s coupon rate.
5. For a given absolute change in a bond’s
YTM, the magnitude of the price increase
caused by a decrease in yield is greater than
the price decrease caused by an increase in
yield.
10-32
Bond Prices and Yields
10-33
Duration
• Duration measure the sensitivity of a bond
price to changes in bond yields.
Changein YTM
% Bond Price  Duration
1  YTM
2


Two bonds with the same duration, but not
necessarily the same maturity, will have
approximately the same price sensitivity
to a (small) change in bond yields.
10-34
Macaulay Duration
(10.5)
ΔYTM
% Bond Price ≈ -Duration x 1  YTM 2
A bond has a Macaulay Duration = 10 years,
its yield increases from 7% to 7.5%.
How much will its price change?
Duration = 10
Change in YTM = .075-.070 = .005
YTM/2 = .035
%Price ≈ -10 x (.005/1.035) = -4.83%
10-35
Modified Duration
• Some analysts prefer a variation of Macaulay’s
Duration, known as Modified Duration.
M acaulayDuration
M odifie dDuration 
YTM 

1



2 

• The relationship between percentage changes in
bond prices and changes in bond yields is
approximately:
Pct. Changein Bond Price  - ModifiedDuration Changein YTM
10-36
Modified Duration
(10.6)
MacaulayDuration
Modified Duration
1  YTM 2
(10.7) %Bond Price ≈ -Modified Duration x YTM
A bond has a Macaulay duration of 9.2
years and a YTM of 7%. What is it’s
modified duration?
Modified duration = 9.2/1.035 = 8.89 years
10-37
Modified Duration
(10.6)
MacaulayDuration
Modified Duration
1  YTM 2
(10.7) %Bond Price ≈ -Modified Duration x YTM
A bond has a Modified duration of 7.2 years
and a YTM of 7%. If the yield increases to
7.5%, what happens to the price?
%Price ≈ -7.2 x .005 = -3.6%
10-38
Calculating Macaulay’s Duration
• Macaulay’s duration values stated in years
• Often called a bond’s effective maturity
• For a zero-coupon bond:
Duration = maturity
• For a coupon bond:
Duration = a weighted average of individual
maturities of all the bond’s separate cash flows,
where the weights are proportionate to the
present values of each cash flow.
10-39
Calculating Macaulay Duration

1
(10.8) Par Value bond duration  1  YTM 2  1 

2M 
YTM


1

YTM
2


Suppose a par value bond has 12 years to
maturity and an 8% coupon. What is its
duration?
Par Value bond duration 
1  .08 2 
.08
1
1


24


1

.08
2


  10.47 years

10-40
General Macaulay Duration Formula
(10.9)
1  YTM 2 1  YTM 2  M(CPR YTM)
Duration 

2M
YTM
YTM  CPR 1  YTM 2  1


Where:
CPR = Constant annual coupon rate
M = Bond maturity in years
YTM = Yield to maturity assuming semiannual coupons
10-41
Calculating Macaulay’s Duration
• In general, for a bond paying constant semiannual
coupons, the formula for Macaulay’s Duration is:
Duration 
1  YTM2
YTM

1  YTM2  MC  YTM


2M


YTM
YTM  C 1 

1
2


• In the formula, C is the annual coupon rate, M is the
bond maturity (in years), and YTM is the yield to
maturity, assuming semiannual coupons.
10-42
Using the General Macaulay Duration
Formula
•
What is the modified duration for a bond that
matures in 15 years, has a coupon rate of
5% and a yield to maturity of 6.5%?
• Steps:
1. Calculate Macaulay duration using 10.9
2. Convert to Modified duration using 10.6
10-43
Using the General Macaulay Duration
Formula
Bond matures in 15 years
Coupon rate = 5%
Yield to maturity = 6.5%
1. Calculate Macaulay duration using 10.9
1  YTM 2 1  YTM 2  M(CPR YTM)
Duration 

2M
YTM
YTM  CPR 1  YTM 2  1


1  .065 2 1  .065 2  15(.05  .065)
Duration 

30
.065
.065  .05 1  .065 2  1
Duration 10.34 years


10-44
Using the General Macaulay Duration
Formula
Bond matures in 15 years
Coupon rate = 5%
Yield to maturity = 6.5%
Macaulay Duration = 10.34 years
2. Convert to Modified duration using 10.6
Modified Duration
MacaulayDuration
1  YTM 2
Modified Duration
10.34
 10.01 years
1  .065 2
10-45
Calculating Duration Using Excel
• Macaulay Duration -- DURATION function
• Modified Duration -- MDURATION function
=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)
10-46
Calculating Macaulay and
Modified Duration
10-47
Duration Properties: All Else Equal
The longer a bond’s maturity, the longer its
duration.
2. A bond’s duration increases at a decreasing
rate as maturity lengthens.
3. The higher a bond’s coupon, the shorter is
its duration.
4. A higher yield to maturity implies a shorter
duration, and a lower yield to maturity
implies a longer duration.
1.
10-48
Properties of Duration
10-49
Bond Risk Measures based on
Duration
• Dollar Value of an 01:
(10.10)
≈ Modified Duration x Bond Price x 0.0001
= Value of a basis point change
• Yield Value of a 32nd:
≈
(10.11)
1
32  Dollar Value of an 01
In both cases, the bond price is per $100 face value.
10-50
Calculating Bond Risk Mesures
Bond matures in 15 years
M = 15
Coupon rate = 5%
C = $50
Yield to maturity = 6.5%
YTM = .065
Macaulay Duration = 10.34 years
Modified Duration = 10.01 years
• First we have to find the price of the bond:

C 
1
(10.3) Bond Price 
1
YTM 
1  YTM 2



50 
1
Bond Price 
1
.065 
1  .065 2



2M

FV


1  YTM 2



1000

30

1  .065 2





30
2M
 $857.64
10-51
Bond Risk Measures based on Duration
Bond matures in 15 years
M = 15
Coupon rate = 5%
C = $50
Yield to maturity = 6.5%
YTM = .065
Macaulay Duration = 10.34 years
Modified Duration = 10.01 years
Price = 85.764
• Dollar Value of an 01:
≈ Modified Duration x Bond Price x 0.0001
≈ 10.01 x 85.764 x 0.0001 = 0.0859
(10.10)
10-52
Bond Risk Measures based on Duration
Bond matures in 15 years
M = 15
Coupon rate = 5%
C = $50
Yield to maturity = 6.5%
YTM = .065
Macaulay Duration = 10.34 years
Modified Duration = 10.01 years
Price = 85.764
• Yield Value of a 32nd:
(10.11)
1
1
≈ 32  Dollar Value of an 01  32  0.0859  0.364
10-53
Dedicated Portfolios
• Bond portfolio created to prepare for a
future cash payment, e.g. pension funds
• Target Date = date the payment is due
10-54
Reinvestment Risk & Price Risk
• Reinvestment Rate Risk:
• Uncertainty about the value of the portfolio on the
target date
• Stems from the need to reinvest bond coupons at
yields not known in advance
• Price Risk:
• Risk that bond prices will decrease
• Arises in dedicated portfolios when the target date
value of a bond is not known with certainty
10-55
Price Risk vs. Reinvestment Rate
Risk For a Dedicated Portfolio
• Interest rate increases have two effects:
 in interest rates decrease bond prices, but
 in interest rates increase the future value of
reinvested coupons
• Interest rate decreases have two effects:
 in interest rates increase bond prices, but
Decreases in interest rates decrease the future
value of reinvested coupons
10-56
Immunization
• Immunization = constructing a dedicated
portfolio that minimizes uncertainty
surrounding the target date value
• Engineer a portfolio so that price risk and
reinvestment rate risk offset each other
(just about entirely).
• Duration matching = matching the
duration of the portfolio to its target date
10-57
Immunization by Duration Matching
10-58
Dynamic Immunization
• Periodic rebalancing of a dedicated
bond portfolio for the purpose of
maintaining a duration that matches the
target maturity date
• Advantage = reinvestment risk greatly
reduced
• Drawback = each rebalancing incurs
management and transaction costs
10-59
Constructing a Dedicated Portfolio
• Suppose a Pension Fund estimates it will
need to pay out about $50 million in 4 years.
The fund decides to buy coupon bonds
paying 6%, maturing in 4 years and selling at
par.
• Assuming interest rates do not change over
the next four years, how much should the
fund invest to have $50 million in 4 years?
10-60
Constructing a Dedicated Portfolio
10.12
P(1  YTM 2) 2M 
10.13
Face Value 


C
1  YTM 22M  1  Face value
YTM
Future Value Re quire d
(1 YTM 2) 2M
Future Value required = $50 million
Time = 4 years (M=8)
Par value bonds paying 6%
Face Value 
$50,000,000
 $39,470,462
8
(1.03)
10-61
Constructing a Dedicated Portfolio
Year
1
2
3
4
6-month
Period
1
2
3
4
5
6
7
8
Payment
$1,184,113.86
$1,184,113.86
$1,184,113.86
$1,184,113.86
$1,184,113.86
$1,184,113.86
$1,184,113.86
$1,184,113.86
Future Value of Coupons
Face Value
Value at End
of Year 5
$ 1,456,310.69
$ 1,413,893.87
$ 1,372,712.50
$ 1,332,730.58
$ 1,293,913.19
$ 1,256,226.39
$ 1,219,637.28
$ 1,184,113.86
$ 10,529,538.36
$ 39,470,462.00
$ 50,000,000.36
10-62
Useful Internet Sites
•
•
•
•
•
www.sifma.org (check out the bonds section)
www.jamesbaker.com (a practical view of bond
portfolio management)
www.bondsonline.com (bond basics and current
market data)
www.investinginbonds.com (bond basics and
current market data)
www.bloomberg.com (for information on
government bonds)
10-63